i
Moduli stacks of étale (ϕ, Γ)-modules and the existence of crystalline lifts

iiModuli stacks of étale (ϕ, Γ)-modules and theexistence of crystalline lifts
Matthew Emertoniii

Toby Gee

iv

Dedication

In memory of Jean-Marc Fontaine.

Contents
Dedicationv

1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Our main theorems . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 (ϕ, Γ)-modules with coefficients . . . . . . . . . . . . . . . . . . .
1.4 Families of extensions . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Crystalline lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Crystalline and semistable moduli stacks . . . . . . . . . . . . . .
1.7 The geometric Breuil–Mézard conjecture and the weight part of
Serre’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Further questions . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 An outline of the book . . . . . . . . . . . . . . . . . . . . . . . .
1.11 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . .
1.12 Notation and conventions . . . . . . . . . . . . . . . . . . . . . .

1
1
2
4
5
6
8
10
12
13
14
15
15

2 Rings and coefficients19
2.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Almost Galois descent for profinite group actions . . . . . . . . . 34
2.4 An application of almost Galois descent . . . . . . . . . . . . . . 43
2.5 Étale ϕ-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6 Frobenius descent . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.7 (ϕ, Γ)–modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Moduli stacks of ϕ-modules and (ϕ, Γ)-modules
55
3.1 Moduli stacks of ϕ-modules . . . . . . . . . . . . . . . . . . . . . 55
3.2 Moduli stacks of (ϕ, Γ)-modules . . . . . . . . . . . . . . . . . . . 57
3.3 Weak Wach modules . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 Xd is an Ind-algebraic stack . . . . . . . . . . . . . . . . . . . . . 67
3.5 Canonical actions and weak Wach modules . . . . . . . . . . . . 72
3.6 The connection with Galois representations . . . . . . . . . . . . 74
3.7 (ϕ, GK )-modules and restriction . . . . . . . . . . . . . . . . . . 78
vii

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CONTENTS
3.8

Tensor products and duality . . . . . . . . . . . . . . . . . . . . .

81

4 Crystalline and semistable moduli stacks
83
4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Breuil–Kisin modules and Breuil–Kisin–Fargues modules . . . . . 84
4.3 Canonical extensions of GK∞ -actions . . . . . . . . . . . . . . . . 86
4.4 Breuil–Kisin–Fargues GK -modules and canonical actions . . . . . 90
4.5 Stacks of semistable and crystalline Breuil–Kisin—Fargues modules 90
4.6 Inertial types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.7 Hodge–Tate weights . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.8 Moduli stacks of potentially semistable representations . . . . . . 111
5 Families of extensions
121
5.1 The Herr complex . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Residual gerbes and isotrivial families . . . . . . . . . . . . . . . 135
5.3 Twisting families . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.4 Dimensions of families of extensions . . . . . . . . . . . . . . . . 142
5.5 Xd is a formal algebraic stack . . . . . . . . . . . . . . . . . . . . 150
6 Crystalline lifts and the finer structure of Xd,red
159
6.1 The fibre dimension of H 2 on crystalline deformation rings . . . 159
6.2 Two geometric lemmas . . . . . . . . . . . . . . . . . . . . . . . . 161
6.3 Crystalline lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.4 Potentially diagonalizable crystalline lifts . . . . . . . . . . . . . 167
6.5 The irreducible components of Xd,red . . . . . . . . . . . . . . . . 170
6.6 Closed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.7 The substack of GK -representations . . . . . . . . . . . . . . . . 174
7 The rank one case177
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.2 Moduli stacks in the rank one case . . . . . . . . . . . . . . . . . 182
7.3 A ramification bound . . . . . . . . . . . . . . . . . . . . . . . . . 189
8 A geometric Breuil–Mézard conjecture201
8.1 The qualitative geometric Breuil–Mézard conjecture . . . . . . . 202
8.2 Semistable and crystalline inertial types . . . . . . . . . . . . . . 204
8.3 The relationship between the numerical, refined and geometric
Breuil–Mézard conjectures . . . . . . . . . . . . . . . . . . . . . . 205
8.4 The weight part of Serre’s conjecture . . . . . . . . . . . . . . . . 207
8.5 The case of GL2 (Qp ) . . . . . . . . . . . . . . . . . . . . . . . . . 208
8.6 GL2 (K): potentially Barsotti–Tate types . . . . . . . . . . . . . . 208
8.7 Brief remarks on GLd , d > 2 . . . . . . . . . . . . . . . . . . . . . 213
A Formal algebraic stacks

215

CONTENTS

ix

B Graded modules and rigid analysis
233
B.1 Associated graded algebras and modules . . . . . . . . . . . . . . 233
B.11 A general setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
B.24 Loci of vanishing and of equivariance . . . . . . . . . . . . . . . . 239
B.30 Extending actions . . . . . . . . . . . . . . . . . . . . . . . . . . 242
B.35 A rigid analytic perspective . . . . . . . . . . . . . . . . . . . . . 243
b R C-modules . . . . . . . . . . . . . . . . . . . . . 245
B.37 Descent for A ⊗C Topological groups and modules

247

D Tate modules and continuity
251
D.1 Topologies and lattices . . . . . . . . . . . . . . . . . . . . . . . . 251
D.12 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
D.16 T -quasi-linear endomorphisms . . . . . . . . . . . . . . . . . . . . 257
E Points, residual gerbes, and isotrivial families

265

F Breuil–Kisin–Fargues modules and potentially semistable representations (by Toby Gee and Tong Liu)
269
F.21 Potentially semistable representations . . . . . . . . . . . . . . . 276
F.24 Hodge and inertial types . . . . . . . . . . . . . . . . . . . . . . . 276

x

CONTENTS

Chapter 1

Introduction
In this book we construct moduli stacks of étale (ϕ, Γ)-modules (projective, ofsome fixed rank, and with coefficients in p-adically complete rings), and establish
some of their basic properties. We also present some first applications of thisconstruction to the theory of Galois representations.

1.1

Motivation

Mazur’s theory of deformations of Galois representations [Maz89] is modelled on
the geometric study of infinitesimal neighbourhoods of points in moduli spaces
via formal deformation theory. In the mid-2000s, Kisin suggested that somekind of moduli spaces of local Galois representations should exist; that is, there
should be formal algebraic stacks over Zp whose closed points correspond torepresentations ρ : GK → GLd (Fp ), and whose versal rings at such points
should recover appropriate Galois deformation rings. This expectation is borneout by the results of this book. (In fact, Kisin was motivated by calculations ofcrystalline deformation rings for GL2 (Qp ) that had been carried out by Berger–
Breuil using the p-adic local Langlands correspondence, and suggested that theversal rings should give crystalline deformation rings. Thus his suggestion iscrys,λrealized by the stacks Xdof Theorem 1.2.4 below.)A natural way to construct such a stack would be to consider a literal moduli
stack of continuous representations ρ : GK → GLd (A), for K a p-adic field andA a p-adically complete Zp -algebra; indeed such stacks were constructed by Carl
Wang-Erickson [WE18]. However, the stacks constructed in this way are less“global” than one would wish, and in particular the corresponding families ofmod p representations ρ : GK → GLd (Fp ) have constant semisimplification.
In this book, we instead consider moduli stacks of étale (ϕ, Γ)-modules.
These contain Wang-Erickson’s stacks as substacks, and coincide with them on
the level of Fp -points, but their geometry is quite different; in particular, we
see much larger families, exhibiting some unexpected features (for example, irreducible representations arising as limits of reducible representations). The
1

2

CHAPTER 1. INTRODUCTION

relationship between the theory and constructions that we develop here, andthe usual formal deformation theory of Galois representations, is the same asthat between the theory of moduli spaces of algebraic varieties, and the formal
deformation theory of algebraic varieties: the latter gives valuable local information about the former, but moduli spaces, when they can be constructed,
capture global aspects of the situation inaccessible to the purely infinitesimaltools of formal deformation theory.

1.2

Our main theorems

Our goal in this book is to construct, and establish the basic properties of, moduli stacks of étale (ϕ, Γ)-modules. More precisely, if we fix a finite extension Kof Qp , and a non-negative integer d (the rank), then we let Xd denote the category fibred in groupoids over Spf Zp whose groupoid of A-valued points, for any
p-adically complete Zp -algebra A, is equal to the groupoid of rank d projectiveétale (ϕ, Γ)-modules with A-coefficients. (See Section 1.3 below for a definitionof these.) Our first main theorem is the following. (See Corollary 5.5.18 andTheorem 6.5.1.)
1.2.1 Theorem. The category fibred in groupoids Xd is a Noetherian formal algebraic stack. Its underlying reduced substack Xd,red (which is an algebraic stack)is of finite type over Fp , and is equidimensional of dimension [K : Qp ]d(d−1)/2.
The irreducible components of Xd,red admit a natural labelling by Serre weights.
We will elaborate on the labelling of components by Serre weights further below. For now, we mention that, under the usual correspondence between étale(ϕ, Γ)-modules and Galois representations, the groupoid of Fp -points of Xd ,which coincides with the groupoid of Fp -points of the underlying reduced substack Xd,red , is naturally equivalent to the groupoid of continuous representations ρ : GK → GLd (Fp ). (More generally, if A is any finite Zp -algebra, thenthe groupoid Xd (A) is canonically equivalent to the groupoid of continuous representations GK → GLd (A).) It is expected that our labelling of the irreducible
components can be refined (by adding further labels to some of the components)
to give a geometric description of the weight part of Serre’s conjecture, so that
ρ corresponds to a point in a component of Xd,red which is labeled by the Serreweight k if and only if ρ admits k as a Serre weight; we discuss this expectation, and what is known about it, in Section 1.7 below (and in more detail in
Chapter 8).
Again using the correspondence between étale (ϕ, Γ)-modules and Galois
representations, we see that the universal lifting ring of a representation ρ asabove will provide a versal ring to Xd at the corresponding Fp -valued point.
Accordingly we expect that the stacks Xd will have applications to the studyof Galois representations and their deformations. As a first example of this,
we prove the following result on the existence of crystalline lifts; although thestatement of this theorem involves a fixed ρ, we do not know how to prove it
without using the stacks Xd , over which ρ varies.

1.2. OUR MAIN THEOREMS

3

1.2.2 Theorem (Theorem 6.4.4). If ρ : GK → GLd (Fp ) is a continuous representation, then ρ has a lift ρ◦ : GK → GLd (Zp ) for which the associated p-adic
representation ρ : GK → GLd (Qp ) is crystalline of regular Hodge–Tate weights.
We can furthermore ensure that ρ◦ is potentially diagonalizable.
(The notion of a potentially diagonalizable representation was introduced
in [BLGGT14], and is recalled as Definition 6.4.2 below.) In combination withpotential automorphy theorems, this has the following application to the globalization of local Galois representations.
1.2.3 Theorem (Corollary 6.4.7). Suppose that p - 2d, and fix ρ : GK →GLd (Fp ). Then there is an imaginary CM field F and an irreducible conjugate
self dual automorphic Galois representation r : GF → GLd (Fp ) such that forevery v|p, we have Fv ∼= K and either r|GFv ∼= ρ or r|GFvc ∼= ρ.
Another key result of the book is the following theorem, describing moduli
stacks of étale (ϕ, Γ)-modules corresponding to crystalline and semistable Galois
representations.
1.2.4 Theorem (Theorem 4.8.12). If λ is a collection of labeled Hodge–Tateweights, and if O denotes the ring of integers in a finite extension E of Qp
containing the Galois closure of K (which will serve as the ring of coefficients),
crys,λof (Xd )O which is a p-adic formal algethen there is a closed substack Xdbraic stack and is flat over O, and which is characterized as being the uniqueclosed substack of (Xd )O which is flat over O and whose groupoid of A-valued
points, for any finite flat O-algebra A, is equivalent (under the equivalence between étale (ϕ, Γ)-modules and continuous GK -representations) to the groupoidof continuous representations GK → GLd (A) which become crystalline after extension of scalars to A ⊗O E, and whose labeled Hodge–Tate weights are equal
to λ.
ss,λ
Similarly, there is a closed substack Xd of (Xd )O which is a p-adic formal
algebraic stack and is flat over O, and which is characterized as being the uniqueclosed substack of (Xd )O which is flat over O and whose groupoid of A-valued
points, for any finite flat O-algebra A, is equivalent to the groupoid of continuous representations GK → GLd (A) which become semistable after extension ofscalars to A ⊗O E, and whose labeled Hodge–Tate weights are equal to λ.
1.2.5 Remark. In fact, Theorem 4.8.12 also proves the analogous result for potentially crystalline and potentially semistable representations of arbitrary inertial type, but for simplicity of exposition we restrict ourselves to the crystalline
and semistable cases in this introduction.
crys,λA crucial distinction between the stacks Xd and their closed substacks Xd
ss,λand Xdis that while Xd is a formal algebraic stack lying over Spf Zp , it is
not actually a p-adic formal algebraic stack (in the sense of Definition A.7);
crys,λss,λsee Proposition 6.5.2. On the other hand, the stacks Xd
and Xdare padic formal algebraic stacks, which implies that their mod pa reductions are infact algebraic stacks. This gives in particular a strong interplay between the

4

CHAPTER 1. INTRODUCTION

structure of the mod p fibres of crystalline and semistable lifting rings and thegeometry of the underlying reduced substack Xd,red . This plays an importantrole in determining the structure of this reduced substack, and also in the proofof Theorem 1.2.2. As we explain in more detail in Section 1.7 below, it also
allows us to reinterpret the Breuil–Mézard conjecture in terms of the interactionss,λcrys,λand Xd , andbetween the structure of the mod p fibres of the stacks Xd
the geometry of Xd,red .

1.3(ϕ, Γ)-modules with coefficients

There is quite a lot of evidence, for example from Colmez’s work on the p-adic
local Langlands correspondence [Col10], and work of Kedlaya–Liu [KL15], thatrather than considering families of representations of GK , it is more natural toconsider families of étale (ϕ, Γ)-modules.
The theory of étale (ϕ, Γ)-modules for Zp -representations was introduced by
Fontaine in [Fon90]. There are various possible definitions that can be made,with perfect, imperfect or overconvergent coefficient rings, and different choicesof Γ; we discuss the various variants that we use, and the relationships between
them, at some length in the body of the book. For the purpose of this introduction we simply let AK = W (k)((T ))∧ , where k is a finite extension of Fp(depending on K), and the hat denotes the p-adic completion. This ring isendowed with a Frobenius ϕ and an action of a profinite group Γ (an open subgroup of Z×p ) that commutes with ϕ; the formulae for ϕ and for this action canbe rather complicated for general K, although they admit a simple description
if K/Qp is abelian. (See Definition 2.1.12 and the surrounding material.)
An étale (ϕ, Γ)-module is then, by definition, a finite AK -module endowedwith commuting semilinear actions of ϕ and Γ, with the property that thelinearized action of ϕ is an isomorphism. There is a natural equivalence of
categories between the category of étale (ϕ, Γ)-modules and the category ofcontinuous representations of GK on finite Zp -modules.
b Zp AKLet A be a p-adically complete Zp -algebra. We let AK,A := A ⊗(where the completed tensor product is taken with respect to the p-adic topology
on A and the so-called weak topology on AK ), and define an étale (ϕ, Γ)-modulewith A-coefficients just as in the case A = Zp described above, but now using
the ring AK,A . In the case that A is finite as a Zp -module there is again
an equivalence of categories with the category of continuous representationsof GK on finite A-modules, but for more general A no such equivalence exists.
Our moduli stack Xd is defined to be the stack over Spf Zp with the property
that Xd (A) is the groupoid of projective étale (ϕ, Γ)-modules of rank d with
A-coefficients. (That this is indeed an étale stack, indeed even an fpqc stack,follows from results of Drinfeld.) Using the machinery of our paper [EG21] weare able to show that Xd is an Ind-algebraic stack, but to prove Theorem 1.2.1
we need to go further and make a detailed study of its special fibre and of theunderlying reduced substack. This study is guided by ideas coming from Galois
deformation theory and the weight part of Serre’s conjecture, in a manner that

1.4. FAMILIES OF EXTENSIONS

5

we now describe.

1.4

Families of extensions

As we have already explained, over a general base A there is no longer an equivalence between (ϕ, Γ)-modules and representations of GK . Perhaps surprisingly,
from the point of view of applications of our stacks to the study of p-adic Galois
representations, this is a feature rather than a bug. For example, an examination of the known results on the reductions modulo p of 2-dimensional crystalline
representations of GQp (see for example [Ber11, Thm. 5.2.1]) suggests that any
moduli space of mod p representations of GK should have the feature that the
representations are generically reducible, but can specialise to irreducible representations. A literal moduli space of representations of a group cannot behavein this way (essentially because Grassmannians are proper), but it turns outthat the underlying reduced substack Xd,red of Xd does have this property. (Seealso Section 6.7 and Remark 7.2.19 for further discussions of the relationshipbetween our stacks of (ϕ, Γ)-modules and stacks of representations of a Galois
or Weil–Deligne group.)
More precisely, the results of [Ber11, Thm. 5.2.1], together with the weightpart of Serre’s conjecture, suggest that each irreducible component of Xd,red
should contain a dense set of Fp -points which are successive extensions of characters GK → Fp , with the extensions being as non-split as possible. This turns
out to be the case. The restrictions of these characters to the inertia subgroup IK are constant on the irreducible components, and the discrete dataof these characters, together with some further information about peu- andtrès ramifiée extensions, determines the components. This discrete data can beconveniently and naturally organised in terms of “Serre weights” k, which aretuples of integers which biject with the isomorphism classes of the irreducible
Fp -representations of GLd (OK ). The relationship between Serre weights andGalois representations is important in the p-adic Langlands program, and inproving automorphy lifting theorems, and we discuss it further in Section 1.7.
Having guessed that the Fp -points of Xd should be arranged in irreducible
components in this way, an inductive strategy to prove this suggests itself. It
is easy to see that irreducible representations of GK are “rigid”, in that there
are up to twist by unramified characters only finitely many in each dimension;
furthermore, it is at least intuitively clear that each such family of unramified twists of a d-dimensional irreducible representation should give rise to azero-dimensional substack of Xd (there is a Gm of twists, but also a Gm of×
automorphisms). On the other hand, given characters χ1 , . . . , χd : GK → Fp , a
Galois cohomology calculation suggests that there should be a substack of Xd,redof dimension [K : Qp ]d(d − 1)/2 given by successive extensions of unramified
twists of the χi . Accordingly, one could hope to construct the stacks corresponding to the Serre weights k by inductively constructing families of extensions ofrepresentations.
To confirm this expectation, we use the machinery originally developed by

6

CHAPTER 1. INTRODUCTION

Herr [Her98, Her01], who gave an explicit complex which is defined in termsof (ϕ, Γ)-modules and computes Galois cohomology. This definition goes overunchanged to the case with coefficients, and with some effort we are able toadapt Herr’s arguments to our setting, and to prove finiteness and base change
properties (following Pottharst [Pot13], we in fact find it helpful to think of
the Herr complex of a (ϕ, Γ)-module with A-coefficients as a perfect complex of
A-modules). Using the Herr complex, we can inductively construct irreduciblekclosed substacks Xd,red of Xd,red of dimension [K : Qp ]d(d − 1)/2 whose genericFp -points correspond to successive extensions of characters as described above(the restrictions of these characters to IK being determined by k). Furthermore,kby a rather involved induction, we can show that the union of the Xd,red , together
possibly with a closed substack of Xd,red of dimension strictly less than [K :k
Qp ]d(d − 1)/2, exhausts Xd,red . In particular, each Xd,red is an irreducible
component of Xd,red , and any irreducible component that is not one of thek
Xd,red is of strictly smaller dimension than these components.
k

One way to show that the Xd,red exhaust the irreducible components of Xd,red
would be to show that every representation GK → GLd (Fp ) occurs as an Fp kvalued point of some Xd,red . We expect this to be difficult to show directly;
indeed, already for d = 2 the paper [CEGS19] shows that the closed points
kof Xd,red are governed by the weight part of Serre’s conjecture, and the explicit
description of this conjecture is complicated (see e.g. [BDJ10, DDR16]). Furthermore it seems hard to explicitly understand the way in which families of
reducible (ϕ, Γ)-modules degenerate to irreducible ones, or to reducible representations with different restrictions to IK (phenomena which are implied by
the weight part of Serre’s conjecture).
Instead, our approach is to show by a consideration of versal rings that Xd,red
is equidimensional of dimension [K : Qp ]d(d−1)/2; this suffices, since our inductive construction showed that any other irreducible component would necessarily
have dimension strictly less than [K : Qp ]d(d − 1)/2. Our proof of this equidimensionality relies on Theorems 1.2.2 and 1.2.4, as we explain in Remark 1.5.4
below.

1.5

Crystalline lifts

Theorem 1.2.2 solves a problem that has been considered by various authors,in particular [Mul13, GHLS17]. It admits a well-known inductive approach(which is taken in [Mul13, GHLS17]): one writes ρ as a successive extensionof irreducible representations, lifts each of these irreducible representations toa crystalline representation, and then attempts to lift the various extension
classes. The difficulty that arises in this approach (which has proved an obstacle
to obtaining general statements along the lines of Theorem 1.2.2 until now) is
showing that the mod p extension classes that appear in this description of ρ canactually be lifted to crystalline extension classes in characteristic zero. The basic

1.5. CRYSTALLINE LIFTS

7

source of the difficulty is that the local Galois H 2 can be non-zero, and non-zeroclasses in H 2 obstruct the lifting of extension classes (which can be interpreted asclasses lying in H 1 ). In fact, the difficulty is not so much in obtaining crystalline
extension classes, as in lifting to any classes in characteristic zero; indeed, it was
not previously known that an arbitrary ρ had any lift to characteristic zero atall. (Subsequently a different proof of the existence of such a lift has been foundby Böckle–Iyengar–Paškūnas [BIP21].)
Our proof of Theorem 1.2.2 relies on the inductive strategy described in thepreceding paragraph, but we are able to prove the following key result, which
controls the obstructions that can be presented by H 2 , and is a consequence ofTheorems 5.5.12 and 6.5.1 (see also Remark 1.5.4).
1.5.1 Proposition. The locus of points ρ ∈ Xd,red (Fp ) at whichdim H 2 (GK , ρ) ≥ r
is Zariski closed in Xd,red (Fp ), and is of codimension ≥ r.
Let Rρ denote the universal lifting ring of ρ, with universal lifting ρuniv . For
crys,λeach regular tuple of labeled Hodge–Tate weights λ, we let Rρ
denote thequotient of Rρ corresponding to crystalline lifts of ρ with Hodge–Tate weights λ(of course, this quotient is zero unless ρ admits such a crystalline lift). Then
H 2 (GK , ρuniv ) is an Rρ -module, and Proposition 1.5.1 implies the following
corollary.
1.5.2 Corollary. For any regular tuple of labeled Hodge–Tate weights λ thecrys,λlocus of points x ∈ Spec Rρ/p for whichdimκ(x) H 2 (GK , ρuniv ) ⊗R κ(x) ≥ rρ

has codimension ≥ r.
1.5.3 Remark. Tate local duality, together with the compatibility of H 2 withbase-change, shows thatdimκ(x) H 2 (GK , ρuniv ) ⊗R κ(x) = dimκ(x) H 2 GK , ρuniv ⊗R κ(x)ρρ= dimκ(x) HomGK (ρuniv )∨ ⊗R κ(x), ρ(where  denotes the mod p cyclotomic character, thought of as taking valuesin κ(x)× ). Thus the statement of Corollary 1.5.2 is related to the way in whichcrys,λSpec Rρ/p intersects the reducibility locus in Spec Rρ .
Given Corollary 1.5.2, we prove Theorem 1.2.2 by working purely within thecontext of formal lifting rings. However we don’t know how to prove the corollary while staying within that context. Indeed, as Remark 1.5.3 indicates, thiscorollary is related to the way in which the special fibre of a potentially crystalline deformation ring intersects another natural locus in Spec Rρ (namely,

8

CHAPTER 1. INTRODUCTION

the reducibility locus). Since the special fibre of a potentially crystalline lifting ring is not directly defined in deformation-theoretic terms, such questions
are notoriously difficult to study directly. Our proof of the corollary proceeds
differently, by replacing a computation on the special fibre of the potentiallycrystalline deformation ring by a computation on Xd,red ; this latter space has
a concrete description in terms of families of varying ρ, whose H 2 we are able
to compute, as a result of the inductive construction of families of extensions
described in Section 1.4.
In order to deduce Corollary 1.5.2 from Proposition 1.5.1, it is crucial that
we know that the natural morphism Spf Rcrys,λ /p → Xd is effective, in thesense that it arises from a morphism Spec Rcrys,λ /p → Xd . More concretely, theuniversal representation ρuniv gives an étale (ϕ, Γ)-module over each Artinian
quotient of Rρ . By passing to the limit over these quotients, we obtain a“universal formal étale (ϕ, Γ)-module” over the completion of (k⊗Zp Rρ /p)((T ))with respect to the maximal ideal m of Rρ . Since the special fibre of Xd is formal
algebraic but not algebraic (see Section 1.8 below), there is no corresponding(ϕ, Γ)-module with Rρ /p-coefficients; the ϕ and Γ actions on the universal
formal étale (ϕ, Γ)-module involve Laurent tails of unbounded degree (with thecoefficients of T −n tending to zero m-adically as n → ∞).
The assertion that Spf Rcrys,λ /p → Xd is effective is equivalent to showingthat the base change of the universal formal étale (ϕ, Γ)-module to Rcrys,λ /parises from a genuine (ϕ, Γ)-module, i.e. from one that involves only Laurent
tails of bounded degree. We deduce this from Theorem 1.2.4. Indeed, thering Rcrys,λ /p is a versal ring for the special fibre of the p-adic formal algebraic
crys,λstack Xd
, and (by the very definition of a p-adic formal algebraic stack) thisspecial fibre is an algebraic stack; and the versal rings for algebraic stacks arealways effective.
1.5.4 Remark. As our citation of both Theorems 5.5.12 and 6.5.1 for the proofof Proposition 1.5.1 may indicate, our proof of Proposition 1.5.1 is somewhat
intricate. Indeed, in Theorem 5.5.12, we show that Xd,red has dimension at
most [K : Qp ]d(d − 1)/2, and that the locus considered in Proposition 1.5.1 hasdimension at most [K : Qp ]d(d − 1)/2 − r. This is in fact enough to deduce
Corollary 1.5.2, as Spec Rcrys,λ /p is known to be equidimensional.
Given Corollary 1.5.2, we prove Theorem 1.2.2. In combination with the effective versality of the crystalline deformation rings discussed above we are thenable to deduce the equidimensionality of Xd,red , and then also prove Proposition 1.5.1 as stated.

1.6

Crystalline and semistable moduli stacks

We now explain the proof of Theorem 1.2.4; the proof is essentially identical in
the crystalline and semistable cases, so we concentrate on the crystalline case.
To prove the theorem, it is necessary to have a criterion for a (ϕ, Γ)-module tocome from a crystalline Galois representation. In the case that K/Qp is unrami-1.6. CRYSTALLINE AND SEMISTABLE MODULI STACKS

9fied, it is possible to give an explicit criterion in terms of Wach modules [Wac96],
but no such direct description is known for general K. Instead, following Kisin’s
construction of the crystalline deformation rings Rρcrys,λ in [Kis08], we use thetheory of Breuil–Kisin modules. More precisely, Kisin shows that crystalline
representations of GK have finite height over the (non-Galois) Kummer extension K∞ /K obtained by adjoining a compatible system of p-power roots of a
uniformizer of K; here being of finite height means that the corresponding étale
ϕ-modules admit certain ϕ-stable lattices, called Breuil–Kisin modules.
While not every representation of finite height over K∞ comes from a crystalline representation, we are able to show in Appendix F (jointly written by
T.G. and Tong Liu) that a representation GK → GLd (Zp ) is crystalline if andonly if it is of finite height for every choice of K∞ , and if the corresponding
Breuil–Kisin modules satisfy certain natural compatibilities. (These compatibilities are best expressed in terms of Bhatt–Scholze’s prismatic site, as in [BS21],
but we do not make use of that perspective in this book. Instead, we write
down explicit conditions on the corresponding Breuil–Kisin–Fargues modules;
recall that Breuil–Kisin–Fargues modules are a variant of Breuil–Kisin modules
introduced by Fargues, see e.g. [BMS18, §4].)
We use this description of the crystalline representations to prove the exiscrys,λcrys,λtence of the stacks Xd. The proof that Xd
is a p-adic formal algebraic
stack relies on an analogue of results of Caruso–Liu [CL11] on extensions of
the Galois action on Breuil–Kisin modules, which roughly speaking says thatthe action of GK∞ determines the action of GK up to a finite amount of ambiguity. More precisely, given a Breuil–Kisin module over a Z/pa -algebra forsome a ≥ 1, there is a finite subextension Ks /K of K∞ /K depending only on a,
K and the height of the Breuil–Kisin module, such that there is a canonicalaction of GKs on the corresponding Breuil–Kisin–Fargues module. This canonical action is constructed by Frobenius amplification, and in the case that theBreuil–Kisin module arises from the reduction modulo pa of a crystalline representation of GK , the canonical action coincides with the restriction to GKsof the GK -action on the Breuil–Kisin–Fargues module. (In [CL11] a versionof this canonical action is used to prove ramification bounds on the reductions
modulo pa of crystalline representations; in Chapter 7, we use analogous arguments in the setting of (ϕ, Γ)-modules to relate our stacks to stacks of Weilgroup representations in the rank one case.)
There is one significant technical difficulty, which is that we need to definemorphisms of stacks that correspond to the restriction of Galois representations
from K to K∞ . In order to do this we have to compare (ϕ, Γ)-modules with
A-coefficients (which are defined via the cyclotomic extension K(ζp∞ )/K) to ϕmodules with A-coefficients defined via the extension K∞ /K. We do not knowof a direct way to do this; we proceed by proving a correspondence between ϕmodules over Laurent series rings with ϕ-modules over the perfections of these
Laurent series rings, and proving the following descent result which may be of
independent interest; in the statement, C denotes the completion of an algebraic
closure of Qp .

10

CHAPTER 1. INTRODUCTION

1.6.1 Theorem (Theorem 2.4.1). Let A be a finite type Z/pa -algebra, for somea ≥ 1. Let F be a closed perfectoid subfield of C, with tilt F [ , a closed perfectoid
subfield of C[ . Write W (F [ )A := W (F [ ) ⊗Zp A.
Then the inclusion W (F [ )A → W (C[ )A is a faithfully flat morphism of
Noetherian rings, and the functor M 7→ W (C[ )A ⊗W (F [ )A M induces an equivalence between the category of finitely generated projective W (F [ )A -modules andthe category of finitely generated projective W (C[ )A -modules endowed with acontinuous semi-linear GF -action.
The existence of the required morphism of stacks follows easily from two
applications of Theorem 1.6.1, applied with F equal to respectively the completion of K∞ and the completion of K(ζp∞ ). Furthermore, this construction gives
an alternative description of our stacks, as moduli spaces of W (C[ )A -modules
endowed with commuting semi-linear actions of GK and ϕ. It seems plausiblethat this description will be useful in future work, as it connects naturally tothe theory of Breuil–Kisin–Fargues modules (and indeed we use this connectionin our construction of the potentially semistable moduli stacks). Note thoughthat the description in terms of (ϕ, Γ)-modules is important (at least in our
approach) for establishing the basic finiteness properties of our stacks.

1.7

The geometric Breuil–Mézard conjecture andthe weight part of Serre’s conjecture

We will now briefly explain our results and conjectures relating our stacks to
the Breuil–Mézard conjecture and the weight part of Serre’s conjecture. Further
explanation and motivation can be found throughout Chapter 8. Some of theseresults were previewed in [GHS18, §6], and the earlier sections of that paper (inparticular the introduction) provide an overview of the weight part of Serre’s
conjecture and its connections to the Breuil–Mézard conjecture that may be
helpful to the reader who is not already familiar with them. As in the restof this introduction, we ignore the possibility of inertial types, and we also
restrict to crystalline representations for the purpose of exposition. Everything
in this section extends to the more general setting of potentially semistable
representations, and indeed as we explain in Section 8.6 when discussing thepapers [CEGS19] and [GK14], the additional information provided by non-trivial
inertial types is very important.
Let ρ : GK → GLd (F) be a continuous representation (for some finite extension F of Fp ), and let Rρ be the corresponding universal lifting ring. The corresponding formal scheme Spf Rρ doesn’t carry a lot of evident geometry in and of
itself; for example, its underlying reduced subscheme is simply the closed point
Spec F, corresponding to ρ itself. On the other hand, Xd has a quite non-trivial
underlying reduced substack Xd,red , which parameterizes all the d-dimensional
residual representations of GK . It is natural to ask whether this underlying
reduced substack has any significance in formal deformation theory. More precisely, we could ask for the meaning of the fibre product Spf Rρ ×Xd Xd,red .

1.7. THE GEOMETRIC BREUIL–MÉZARD CONJECTURE

11This fibre product is a reduced closed formal subscheme of Spf Rρ of dimension d2 + [K : Qp ]d(d − 1)/2. It arises (via completion at the closed point) froma closed subscheme of Spec Rρ (as does any closed formal subscheme of theSpf of a complete Noetherian local ring), whose irreducible components, whenthought of as cycles on Spec Rρ , are precisely the cycles that (conjecturally)
appear in the geometric Breuil–Mézard conjecture of [EG14]. More precisely,
we obtain the following qualitative version of the geometric Breuil–Mézard conjecture [EG14, Conj. 4.2.1].
1.7.1 Theorem (Theorem 8.1.4). If ρ : GK → GLd (F) is a continuous representation, then there are finitely many cycles of dimension d2 + [K : Qp ]d(d −1)/2 in Spec Rρ /p, such that for any regular tuple of labeled Hodge–Tate weights
crys,λλ, the special fibre Spec Rρ
/p is set-theoretically supported on the union ofsome number of these cycles.
The cycles in the statement of the theorem are precisely those arising fromk
the fibre products Spf Rρ ×Xd Xd,red , where k runs over the Serre weights.
While Theorem 1.7.1 is a purely local statement, we do not know how to prove
it without using the stacks Xd .
The full geometric Breuil–Mézard conjecture of [EG14] makes precise prediccrys,λtions about the multiplicities of the cycles of the special fibres of Spec Rρ/p;passing from cycles to Hilbert–Samuel multiplicities then recovers the original Breuil–Mézard conjecture [BM02] (or rather a natural generalisation of it
to GLd ), which we refer to as the “numerical Breuil–Mézard conjecture”. In
particular, the multiplicities are expected to be computed in terms of quantities ncrys
k (λ) that are defined as follows: one associates an irreducible algebraic
representation σ crys (λ) of GLd /K to λ, defined to have highest weight (a certain
shift of) λ. The semisimplification of the reduction mod p of σ crys (λ) can be
written as a direct sum of irreducible representations of GLd (k), and ncrys
k (λ) isdefined to be the multiplicity with which the Serre weight k occurs.
In Chapter 8 we explain that as we run over all ρ, the geometric Breuil–Mézard conjecture is equivalent to the following analogous conjecture for the
special fibres of our crystalline and semistable stacks. Here by a “cycle” in Xd,redk
we mean a formal Z-linear combination of its irreducible components Xd .
1.7.2 Conjecture (Conjecture 8.2.2). There are cycles Zk in Xd,red with theproperty that for each regular tuple of labeled Hodge–Tate weights λ, the underP
crys,λlying cycle of the special fibre of Xd
is k ncrys
k (λ) · Zk .
In fact we expect that the cycles Zk are effective, i.e. that they are a linear
k0

combination of the irreducible components Xd with non-negative integer coefficients. Since there are infinitely many possible λ, the cycles Zk , if they exist,
are hugely overdetermined by Conjecture 1.7.2.
As first explained in [Kis09a], the (numerical) Breuil–Mézard conjecture has
important consequences for automorphy lifting theorems; indeed proving theconjecture is closely related to proving automorphy lifting theorems in situations

12

CHAPTER 1. INTRODUCTION

with arbitrarily high weight or ramification at the places dividing p. Conversely,
following [Kis10], one can use automorphy lifting theorems to deduce cases of
the Breuil–Mézard conjecture. Automorphy lifting theorems involve a fixed ρ,and in fact we can deduce Conjecture 1.7.2 from the Breuil–Mézard conjecture
for a finite set of suitably generic ρ.
In particular, we are able to combine results in the literature to show thatfor GL2 the cycles Zk in Conjecture 1.7.2 must have a particularly simple form:kwe necessarily have Zk = Xd,red unless k is a so-called “Steinberg” weight, inkwhich case Zk is the sum of Xd,red and one other irreducible component. (Moreprecisely, what we show, following [CEGS19, GK14], is that with these cycles Zk ,Conjecture 1.7.2 holds for all “potentially Barsotti–Tate” representations.)
The weight part of Serre’s conjecture predicts the weights in which particular
Galois representations contribute to the mod p cohomology of locally symmetricspaces. Following [GK14], this conjecture is closely related to the Breuil–Mézard
conjecture; indeed, if Conjecture 1.7.2 holds, then the set of Serre weights associated to a representation ρ : GK → GLd (Fp ) should be precisely the weights kfor which Zk is supported at ρ. In other words, if we refine our labelling of theirreducible components of Xd,red by labelling each component by the union of
the weights k for which that component contributes to Zk , then we expect theset of Serre weights for ρ to be the union of the sets of weights labelling theirreducible components containing ρ. This expectation holds for GL2 by themain results of [CEGS19].

1.8

Further questions

There are many other questions one could ask about the stacks Xd , which wehope to return to in future papers. For example, we show in Proposition 6.5.2
that Xd is not a p-adic formal algebraic stack. Indeed, if it were p-adic formal
algebraic, then its special fibre would be an algebraic stack, whose dimension
would be equal to the dimension of its underlying reduced substack. In turn,this would imply that the versal rings Rρ would have dimension equal to thecrys,λdimensions of the crystalline deformation rings Rρ
, and this is known not tobe true. In fact, it is a folklore conjecture, recently proved by Böckle–Iyengar–
Paškūnas [BIP21], that the lifting rings Rρ are Zp -flat local complete intersections of dimension 1 + d2 + [K : Qp ]d2 , which should imply that the stacks Xd
are Zp -flat local complete intersections of dimension 1 + [K : Qp ]d2 (a notionthat we do not attempt to make precise for formal algebraic stacks).
It is natural to ask about the rigid analytic generic fibre of Xd ; this should
exist as a rigid analytic stack in an appropriate sense. The generic fibres ofkthe substacks Xd should admit morphisms to the stacks of Hartl and Hellmann [HH20] (although these morphisms won’t be isomorphisms, since for anykfinite extension E of Qp , the OE -points of Xd , which would coincide with the
E-points of its generic fibre, correspond to lattices in crystalline representations,whereas the stacks of [HH20] parameterize crystalline or semistable representa-

1.9. PREVIOUS WORK

13

tions themselves).
We expect that the Xd will have a role to play in generalisations of the p-adic
local Langlands correspondence. For example, we expect that when K = Qp
the p-adic local Langlands correspondence for GL2 (Qp ) can be extended to giverise to sheaves of GL2 (Qp )-representations on X2 . More generally, we expectthat there will be a p-adic analogue of the work of Fargues–Scholze on the local
Langlands correspondence [FS21] involving the stacks Xd .

1.9

Previous work

The description of local Galois representations in terms of étale (ϕ, Γ)-modules isdue to Fontaine [Fon90]. The importance of “height” as an aspect of the theory
was already emphasized in [Fon90], and was further developed by Wach [Wac96],
who explored the relationship between the finite height condition and crystallinity of Galois representations in the absolutely unramified context.
The use of what are now called Breuil–Kisin modules as a tool for studying
crystalline and semistable representations for general (i.e. not necessarily absolutely unramified) p-adic fields (a study which, apart from its intrinsic importance, is crucial for treating potentially crystalline or semistable representations,even in the absolutely unramified context) was due originally to Breuil [Bre98]
and was extensively developed by Kisin [Kis09b, Kis08], who used them to study
Galois deformation rings.
The algebro-geometric and moduli-theoretic perspectives that already played
key roles in Kisin’s work were further developed by Pappas and Rapoport[PR09], who introduced moduli stacks of Breuil–Kisin modules and of étale
ϕ-modules; it this work of Pappas and Rapoport, which can be very roughly
thought of as constructing moduli stacks of representations of the absolute Galois groups of certain perfectoid fields, which is the immediate launching point
for our work in this book, as well as for our paper [EG21]. (We should alsomention Drinfeld’s work [Dri06], which underpins the verification of the stackproperty for the constructions of [PR09], as well as for those of the present
book.) Our use of moduli stacks of Breuil–Kisin–Fargues modules (in the construction of the potentially crystalline and semistable substacks) was in part
inspired by the work of Fargues and Bhatt–Morrow–Scholze (see in particular[BMS18, §4]), which taught us not to be afraid of Ainf .
Moduli stacks parameterizing crystalline and semistable representations havealready been constructed by Hartl and Hellmann [HH20]; as remarked uponk
above, these stacks should have a relationship to the stacks Xd that we construct. See also the related papers of Hellmann [Hel16, Hel13].
As far as we are aware, the first construction of moduli stacks of representations of GK in which the residual representation ρ can vary is the work of Carl
Wang-Erickson [WE18] mentioned above, which constructs and studies suchstacks in the case that ρ has fixed semisimplification. These are literally moduli
stacks of representations of GK ; they are isomorphic to certain substacks of our
stacks Xd , as we explain in Section 6.7.

14

1.10

CHAPTER 1. INTRODUCTION

An outline of the bookWe finish this introduction with a brief overview of the contents of this book.
The reader may also wish to refer to the introductions to each chapter, as wellas to the overview of this book provided by the notes [EG20].
In Chapter 2 we recall various of the coefficient rings used in the theories of(ϕ, Γ)-modules and Breuil–Kisin modules, and introduce versions of these rings
with coefficients in a p-adically complete Zp -algebra. We also prove almost
Galois descent results for projective modules, and deduce Theorem 1.6.1.
In Chapter 3 we recall the results of [EG21] on moduli stacks of ϕ-modules,
and use them to define our stacks Xd of étale (ϕ, Γ)-modules. With some effort,
we prove that Xd is an Ind-algebraic stack. Chapter 4 defines various moduli
stacks of Breuil–Kisin and Breuil–Kisin–Fargues modules, and uses them toconstruct our moduli stacks of potentially semistable and potentially crystalline
representations, and in particular to prove Theorem 1.2.4.
Chapter 5 develops the theory of the Herr complex, proving in particular that
it is a perfect complex and is compatible with base change. We show how to
use the Herr complex to construct families of extensions of (ϕ, Γ)-modules, andkwe use these families to define the irreducible substack Xd,red corresponding
to a Serre weight k. By induction on d we prove that Xd is a Noetherian
formal algebraic stack, and establish a version of Proposition 1.5.1 (although as
discussed in Remark 1.5.4, we do not prove Proposition 1.5.1 as stated at thispoint in the argument).
It may help the reader for us to point out that Chapters 4 and 5 are essentially independent of one another, and are of rather different flavours. Chapter 4 involves an interleaving of stack-theoretic arguments with ideas from p-adic
Hodge theory and the theory of Breuil–Kisin modules, while in Chapter 5, once
we complete our analysis of the Herr complex, our perspective begins to shift:although at a technical level we of course continue to work with (ϕ, Γ)-modules,
we begin to think in terms of Galois representations and Galois cohomology,
and the more foundational arguments of the preceding chapters recede somewhat into the background.
In Chapter 6 we combine the results of Chapters 4 and 5 with a geometricargument on the local deformation ring to prove Theorem 1.2.2. Having done
this, we are then able to improve on the results on Xd established in the earlier
chapters by proving Theorem 1.2.1. We also deduce Theorem 1.2.3, as well
as determining the closed points of Xd , and describing the relationship of our
stacks with Wang–Erickson’s stacks of Galois representations.
Chapter 7 gives explicit descriptions of various of our moduli stacks in thecase d = 1, relating them to moduli stacks of Weil group representations.
Chapter 8 explains our geometric version of the Breuil–Mézard conjecture, andproves some results towards it, particularly in the case d = 2.
Finally the appendices for the most part establish various technical results
used in the body of the book. We highlight in particular Appendix A, whichsummarises the theory of formal algebraic stacks developed in [Eme], and Appendix F, which combines the theory of Breuil–Kisin–Fargues modules with

1.11. ACKNOWLEDGEMENTS

15bTong Liu’s theory of (ϕ, G)-modules
to give a new characterisation of integral
lattices in potentially semistable representations, of which we make crucial use
in Chapter 4.

1.11

Acknowledgements

We would like to thank Robin Bartlett, Laurent Berger, Bhargav Bhatt, Xavier
Caruso, Pierre Colmez, Yiwen Ding, Florian Herzig, Ashwin Iyengar, Mark
Kisin, Tong Liu, George Pappas, Dat Pham, Léo Poyeton, Michael Rapoport,
and Peter Scholze for helpful correspondence and conversations.We would particularly like to thank Tong Liu for his contributions to Appendix F. We would
also like to thank the organisers (Johannes Anschütz, Arthur-César Le Bras, andAndreas Mihatsch) of the 2019 Hausdorff School on “the Emerton–Gee stackand related topics”, as well as all the participants, both for the encouragement
to finish this book and for the many helpful questions and corrections resulting
from our lectures.
Our mathematical debt to the late Jean-Marc Fontaine will be obvious to
the reader. This book benefited from several conversations with him over theyears 2011-2018, and from the interest he showed in our results; in particular, his
explanations to us of the relationship between framing Galois representations
and Fontaine–Laffaille modules during his visit to Northwestern University inthe spring of 2011 provided an important clue as to the correct definitions of
our stacks.
We owe special thanks to Colette and Therese for all of their patience andsupport during the writing and revision of this book.
The first author was supported in part by the NSF grants DMS-1303450,
DMS-1601871, and DMS-1902307. The second author was supported in partby a Leverhulme Prize, EPSRC grant EP/L025485/1, Marie Curie Career Integration Grant 303605, ERC Starting Grant 306326, ERC Advanced Grant
884596, and a Royal Society Wolfson Research Merit Award. This project hasreceived funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 884596).

1.12

Notation and conventions

p-adic Hodge theory
Let K/Qp be a finite extension. If ρ is a de Rham representation of GK on
a Qp -vector space W , then we will write WD(ρ) for the corresponding Weil–
Deligne representation of WK (see e.g. [CDT99, App. B]), and if σ : K ,→ Qp
is a continuous embedding of fields then we will write HTσ (ρ) for the multisetof Hodge–Tate numbers of ρ with respect to σ, which by definition contains ibGKwith multiplicity dimQp (W ⊗σ,K K(i)). Thus for example if  denotes the16

CHAPTER 1. INTRODUCTION

p-adic cyclotomic character, then HTσ () = {−1}.
By a d-tuple of labeled Hodge–Tate weights λ, we mean a tuple of integers {λσ,i }σ:K,→Qp ,1≤i≤d with λσ,i ≥ λσ,i+1 for all σ and all 1 ≤ i ≤ d − 1. We
will also refer to λ as a Hodge type. By an inertial type τ we mean a representation τ : IK → GLd (Qp ) which extends to a representation of WK with open
kernel (so in particular, τ has finite image).
Then we say that ρ has Hodge type λ (or labeled Hodge–Tate weights λ) if
for each σ : K ,→ Qp we have HTσ (ρ) = {λσ,i }1≤i≤d , and we say that ρ hasinertial type τ if WD(ρ)|IK ∼= τ.
We often somewhat abusively write that a representation ρ : GK → GLd (Zp )is crystalline (or potentially crystalline, or semistable, or. . . ) if the corresponding representation ρ : GK → GLd (Qp ) is crystalline (or potentially crystalline,or. . . ).

Serre weights and Hodge–Tate weights
By a Serre weight k we mean a tuple of integers {kσ,i }σ:k,→Fp ,1≤i≤d with theproperties that• p − 1 ≥ kσ,i − kσ,i+1 ≥ 0 for each 1 ≤ i ≤ d − 1, and• p − 1 ≥ kσ,d ≥ 0, and not every kσ,d is equal to p − 1.
The set of Serre weights is in bijection with the set of irreducible Fp -representationsof GLd (k), via passage to highest weight vectors (see for example the appendix
to [Her09]).
Each embedding σ : K ,→ Qp induces an embedding σ : k ,→ Fp ; if K/Qp
is ramified, then each σ corresponds to multiple embeddings σ. We say that λis a lift of k if for each embedding σ : k ,→ Fp , we can choose an embedding σ :K ,→ Qp lifting σ, with the properties that:• λσ,i = kσ,i + d − i, and• if σ 0 : K ,→ Qp is any other lift of σ, then kσ0 ,i = d − i.

Lifting rings
Let K/Qp be a finite extension, and let ρ : GK → GLd (Fp ) be a continuous
representation. Then the image of ρ is contained in GLd (F) for any sufficiently
large finite extension F/Fp . Let O be the ring of integers in some finite extension E/Qp , and suppose that the residue field of E is F. Let Rρ,O be theuniversal lifting O-algebra of ρ; by definition, this (pro-)represents the functor given by lifts of ρ to representations ρ : GK → GLd (A), for A an Artin
local O-algebra with residue field F. The precise choice of E is unimportant,in the sense that if O0 is the ring of integers in a finite extension E 0 /E, then0by [BLGGT14, Lem. 1.2.1] we have Rρ,O = Rρ,O ⊗O O0 .
Fix some Hodge type λ and inertial type τ . If O is chosen large enoughthat the inertial type τ is defined over E = O[1/p], and large enough that E1.12. NOTATION AND CONVENTIONS

17

contains the images of all embeddings σ : K ,→ Qp , then we have the usual
crys,λ,τ,Oss,λ,τ,O
lifting O-algebras Rρ
and Rρ
. By definition, these are the uniqueO-flat quotients of Rρ,O with the property that if B is a finite flat E-algebra,
crys,λ,τ,Othen an O-algebra homomorphism Rρ,O → B factors through Rρ(resp.

ss,λ,τ,Othrough Rρ
) if and only if the corresponding representation of GK is po-

tentially crystalline (resp. potentially semistable) of Hodge type λ and inertial
type τ . If τ is trivial, we will sometimes omit it from the notation. By the main
theorems of [Kis08], these rings are (when they are nonzero) equidimensional ofdimensionX
1 + d2 +#{1 ≤ i < j ≤ d|λσ,i > λσ,j }.
σNote that this quantity is at most 1+d2 +[K : Qp ]d(d−1)/2, with equality if andonly if λ is regular, in the sense that λσ,i > λσ,i+1 for all σ and all 1 ≤ i ≤ d − 1.
crys,λ,τ,O 0
crys,λ,τ,Oss,λ,τ,O 0
As above, we have Rρ= Rρ
⊗O O0 , and similarly for Rρ
.
crys,λ,τ,O

By [Kis08, Thm. 3.3.8] the localized rings Rρ

[1/p] are regular, and thus

crys,λ,τ,Othe rings Rρ
(which embed into their localizations away from p, since

they are O-flat) are reduced.

AlgebraOur conventions typically follow [Sta]. In particular, if M is an abelian topological group with a linear topology, then as in [Sta, Tag 07E7] we say that M is complete if the natural morphism M → limi M/Ui is an isomorphism, where {Ui }i∈I←−
is some (equivalently any) fundamental system of neighbourhoods of 0 consisting of subgroups. Note that in some other references this would be referred toas being complete and separated.
If R is a ring, we write D(R) for the (unbounded) derived category of Rmodules. We say that a complex P • is good if it is a bounded complex of finite
projective R-modules; then an object C • of D(R) is called a perfect complex
if there is a quasi-isomorphism P • → C • where P • is good. In fact, C • isperfect if and only if it is isomorphic in D(R) to a good complex P • : if we haveanother complex D• and quasi-isomorphisms P • → D• , C • → D• , then there
is a quasi-isomorphism P • → C • ([Sta, Tag 064E]).

StacksOur conventions on algebraic stacks and formal algebraic stacks are those of [Sta]and [Eme]. We recall some terminology and results in Appendix A. Throughout
the book, if A is a topological ring and C is a stack we write C(A) for C(Spf A);
if A has the discrete topology, this is equal to C(Spec A).

18

CHAPTER 1. INTRODUCTION

Chapter 2

Rings and coefficients
In this chapter we study various rings which will be the coefficients of the ϕmodules and (ϕ, Γ)-modules that we study in the subsequent chapters. Throughout the chapter, we fix a finite extension K of Qp , which we regard as a subfieldof some fixed algebraic closure Qp of Qp .
We begin by recalling some rings used in integral p-adic Hodge theory (in particular, in defining (ϕ, Γ)-modules and Breuil–Kisin modules), before introducing versions of them with coefficients in certain p-adically complete Zp -algebras.
We then prove (almost) descent results that allow us to relate ϕ-modules overvarious different rings, before introducing the (ϕ, Γ)-modules which we workwith throughout the book.

2.1

Rings

2.1.1

Perfectoid fields and their tilts

As usual, we let C denote the completion of the algebraic closure Qp of Qp . It
is a perfectoid field, whose tilt C[ is a complete non-archimedean valued perfectfield of characteristic p. If F is a perfectoid closed subfield of C, then its tilt F [is a closed, and perfect, subfield of C[ .
[We let OC denote the ring of integers in C. Its tilt OC
is then the ring of[integers in C . Similarly, if OF denotes the ring of integers in F , then OF[ isthe ring of integers in F [ .
We may form the rings of Witt vectors W (C[ ) and W (F [ ), and the ringsof Witt vectors W (OC[ ) and W (OF[ ); following the standard convention, we
typically denote W (OC[ ) by Ainf .
Each of these rings of Witt vectors is a p-adically complete ring, but wealways consider them as topological rings by endowing them with a finer topology, the so-called weak topology, which admits the following description: If R is[
any of C[ , F [ , OC
, or OF[ , endowed with its natural (valuation) topology, then19

20

CHAPTER 2. RINGS AND COEFFICIENTS

there is a canonical identification (of sets):∼

Wa (R) −→ R × · · · × R (a factors),
and we endow Wa (R) with the product topology. We then endow W (R) :=lima Wa (R) with the inverse limit topology.
←−
This topology admits the following more concrete description (in the general case of a perfectoid F ; setting F = C recovers that particular case): If x
is any element of OF[ of positive valuation, and if [x] denotes the Teichmüller
lift of x, then we endow Wa (OF[ ) with the [x]-adic topology, so that W (OF[ )is then endowed with the (p, [x])-adic topology. The topology on Wa (F [ ) isthen characterized by the fact that Wa (OF[ ) is an open subring — concretely
1] = limi Wa (OF[ ) (the transition maps being given by
Wa (F [ ) = Wa (OF[ )[ [x]−→multiplication by [x], and each transition map being an open and closed embedding), while the topology on W (F [ ) is the inverse limit topology — concretely,\[ )[ 1 ].
W (F [ ) is the p-adic completion W (OF

[x]Apart from the case of C itself, there are two main examples of perfectoid
F that will be of importance to us; see Example 2.1.7 for a justification of theclaim that these fields are indeed perfectoid.
2.1.2 Example (The cyclotomic case). We write K(ζp∞ ) to denote the extension of K obtained by adjoining all p-power roots of unity. It is an infinite
degree Galois extension of K, whose Galois group is naturally identified with anopen subgroup of Z×
p . We let Kcyc denote the unique subextension of K(ζp∞ )whose Galois group over K is isomorphic to Zp (so Kcyc is the “cyclotomic Zp b cyc denote the closure of Kcyc in C, then Kb cyc isextension” of K). If we let K
a perfectoid subfield of C.
2.1.3 Example (The Kummer case). If we choose a uniformizer π of K, asnwell as a compatible system of p-power roots π 1/p of π (here, “compatible”
n+1
n
has the obvious meaning, namely that (π 1/p )p = π 1/p ), then we defineS∞nb ∞ denote the closure of K∞ in C,
K∞ = K(π 1/p ) := n K(π 1/p ). If we let K
b ∞ is again a perfectoid subfield of C.
then Kb be the closure
2.1.4 Remark. Let L be an algebraic extension of Qp , and let L
of L in C. By Krasner’s lemma (see [GR04, Prop. 9.1.16] for full details), theb ⊗L Qp is an algebraic closure of L,b so that the absolute Galois groups
field Lb are canonically identified. The action of GL on Qp extends to anof L and L
b We
action on C, and by a theorem of Ax–Tate–Sen [Ax70], we have CGL = L.
will in particular make use of these facts in the cases L = Kcyc and L = K∞ .

2.1.5

Fields of norms

If L is an infinite strictly arithmetically profinite (strictly APF) extension of K
in Qp , then we may form the field of norms XK (L), as in [FW79] and [Win83,§2]. This is a complete discretely valued field of characteristic p, whose residue
field is canonically identified with that of L. (We don’t define the notion of2.1. RINGS

21

strictly APF here, but simply refer to [Win83, Def. 1.2.1] for the definition. We
will only apply these notions to Examples 2.1.2 and 2.1.3, in which case the end
result of the field of norms construction can be spelt out quite explicitly.) The
following theorem is well known.
2.1.6 Theorem. Let L be an infinite strictly APF extension of K in Qp .
b of L in C is perfectoid.
1. The closure Lb [ , which identifies the target
2. There is a canonical embedding XK (L) ,→ (L)with the completion of the perfect closure of the source.
b is perfectoid, it suffices to show that it is not discretely
Proof. To show that L
valued, and that the absolute Frobenius on OL /pOL is surjective. This surjectivity follows from [Win83, Cor. 4.3.4]. It follows immediately from the description in [Win83, §1.4] of APF extensions as towers of elementary extensions
b is not discretely valued. The canonical embedding of (2)that L, and thus L,
is constructed in [Win83, §4.2], and its claimed property is proved in [Win83,
Cor. 4.3.4].
2.1.7 Example. A theorem of Sen [Sen72] shows that if the Galois group of the
Galois closure of L over K is a p-adic Lie group, and the induced extension ofresidue fields is finite, then L is a strictly APF extension of K. (Sen’s theorem
shows that the Galois closure of L is strictly APF over K; see [Win83, Ex. 1.2.2].
It then follows from [Win83, Prop. 1.2.3 (iii)] that L itself is strictly APF overK.) As a consequence, we see that the theory of the field of norms, and inparticular Theorem 2.1.6, applies in the cases of Examples 2.1.2 and 2.1.3.

2.1.8

Thickening fields of normsb [ to the flat
In the context of Theorem 2.1.6, given that one may thicken (L)a[bZ/p -algebra Wa (L) , it is natural to ask whether XK (L) admits a similarb [ lifts to an embedding
such thickening, such that the embedding XK (L) ,→ (L)of the corresponding thickenings. One would furthermore like such a lifted
embedding to be compatible
 with various auxiliary structures, such as the actionb [ , or the action of the Galois group Gal(L/K), in theof Frobenius on Wa (L)case when L is Galois over K (i.e. one would like the image of this embedding
to be stable under these actions).
Since XK (L) is imperfect, there is no canonical thickening of XK (L) overZ/pa , and as far as we know, there is no simple or general answer to the question of whether such thickenings and embeddings exist with desirable extra
properties, such as being Frobenius or Galois stable. However, in the cases of
Examples 2.1.2 and 2.1.3, such thickenings and thickened embeddings can beconstructed directly, as we now recall.

22

2.1.9

CHAPTER 2. RINGS AND COEFFICIENTSThe cyclotomic case

The extension K(ζp∞ ) of K is infinite and strictly APF, as well as being Galoise K := Gal(K(ζp∞ )/K), then the cyclotomic character
over K. If we write Γe K ,→ Z×
induces an embedding χ : Γ
p . Consequently, there is an isomorphism∼∼eΓK = ΓK × ∆, where ΓK = Zp and ∆ is finite. We have Kcyc = (K(ζp∞ ))∆ .
Later in the book, K will typically be fixed, and we will write Γ for ΓK .
Suppose for a moment that K = Qp . If we choose a compatible systemof pn th roots of 1, then these give rise in the usual way to an element ε ∈[(Q\p (ζp∞ )) . If we identify the field of norms XQp (Qp (ζp∞ )) with a subfield of
[
\ [
(Q\
p (ζp∞ )) via the embedding XQp (Qp (ζp∞ )) ,→ (Qp (ζp∞ )) , then ε − 1 is a
uniformizer of XQp (Qp (ζp∞ )). As usual, let [ε] denote the Teichmüller lift of εto an element of W O[ \ ). There is then a continuous embedding
Qp (ζp∞ )Zp [[T ]] ,→ W (O[ \∞ )Qp (ζp)(the source being endowed with its (p, T )-adic topology, and the target with its
weak topology), defined via T 7→ [ε]−1. We denote the image of this embeddingby (A0Qp )+ . This embedding extends to an embedding\ [Z\p ((T )) ,→ W (Qp (ζp∞ ))(here the source is the p-adic completion of the Laurent series ring Zp ((T ))),
whose image we denote by A0Qp .
We now return to the case of general K. We will compare this case with thecase of Qp . Since KQp (ζp∞ ) = K(ζp∞ ), the theory of the field of norms gives
an identification XK (K(ζp∞ )) = XQp (K(ζp∞ )) [Win83, Rem. 2.1.4], and showsthat this field is a separable extension of XQp (Qp (ζp∞ )) [Win83, Thm. 3.1.2],
if we regard both as embedded in C[ via the embeddings of Theorem 2.1.6. To
ease notation, from now on we write E0K = XK (K(ζp∞ )), and EK := (E0K )∆ , sothat E0K /EK is a separable extension. We write (E0K )+ , E+
K for the respective0, EK , (E0K )+rings of integers. We write ϕ for the (p-power) Frobenius on EK+
and EK .
0
We note now that BQ:= A0Qp [1/p] is a discretely valued field admitting pp
as a uniformizer, with residue field E0Qp . There is then a unique finite unramified
extension of B0Qp contained in the field W (C[ )[1/p] with residue field E0K . We
denote this extension by B0K ; it is again a discretely valued field, admitting p asa uniformizer, and we let A0K denote its ring of integers; equivalently, we have
A0K = B0K ∩ W (C[ ). We see that A0K is a discrete valuation ring, admitting
p as a uniformizer, and that A0K /pA0K = E0K . There is a natural lift of the
Frobenius ϕ from E0K to A0K .
e K on E0 = XK (K(ζp∞ )) induces an action of Γe K on A0 ,The action of ΓKK
0 ∆and we write AK := (AK ) . This is the ring of integers in the discretely valued
field BK := AK [1/p] = (B0K )∆ , and has residue field equal to EK . The actionse K on A0 and of ΓK on AK commute with ϕ.
of ΓK

2.1. RINGS

23

0
0
If we let TK
denote a lift of a uniformizer of E0K , and let k∞denote the∼00\residue field of K(ζp∞ ), then there is an isomorphism W (k∞ )((TK )) −→ A0K .
(Here the source denotes the p-adic completion.) Similarly if k∞ denotes theresidue field of Kcyc , and TK is a lift of a uniformizer of EK , then there is an∼
isomorphism W (k\∞ )((TK )) −→ AK .
For a general extension K/Qp it is hard to give an explicit formula for the00e K on W (k∞actions of ϕ and Γ)((TK))∧ , but in some of our arguments it isuseful to reduce to a special case where we can use explicit formulae. If K = K0
(that is, if K/Qp is unramified) then we have such a description as follows: we0have E0K0 = k((ε − 1)), and A0K0 = W (k((ε − 1))), and we set TK= [ε] − 1,0with the square brackets denoting the Teichmüller lift.
e K on T 0 ∈ A0 are given by the explicit
The actions of ϕ and γ ∈ Γ0K0K0formulae00ϕ(TK) = (1 + TK
)p − 1,(2.1.10)0000γ(1 + TK
) = (1 + TK
)χ(γ) ,
00(2.1.11)0+e K → Z×=
where χ : Γ
p denotes the cyclotomic character. We set (AK0 )00e K )-stable. We set TK = tr A0 /A (T 0 )W (k)[[TK0 ]], which is visibly (ϕ, Γ
0
0
K0
K0
K
0

+
∧
and A+
K0 = W (k)[[TK0 ]]; then we have AK0 = W (k)((TK0 )) , and AK0 is(ϕ, ΓK0 )-stable (if p > 2 this is [Fon90, Prop. A.3.2.3], and if p = 2 the samestatements hold, as explained in [Her98, §1.1.2.1]). After possibly replacing TK0
by (TK0 − λ) for some λ ∈ pW (k), by [Ber14, Prop. 4.2, 4.3] we can and do+
assume that ϕ(TK0 ) ∈ TK0 A+
K0 , and that g(TK0 ) ∈ TK0 AK0 for all g ∈ ΓK0 .

2.1.12 Definition. We say that K is basic if it is contained in K0 (ζp∞ ). (Although it won’t play a role in what follows, we remark that this implies inparticular that K is abelian over Qp .)If K is basic then we have K(ζp∞ ) = K0 (ζp∞ ), so that E0K = E0K0 , and we00, (A0K )+ = (A0K0 )+ , and similarly for AK , with the= TKtake A0K = A0K0 , TK0e K and ΓK being the restrictions of the actions of Γe K and ΓK .
actions of Γ000
If K is not basic then as explained above, we may still choose some TK
0
0
0
∧
∧
so that AK = W (k∞ )((TK )) , and some TK so that AK = W (k∞ )((TK )) .
00Having done so, we set (A0K )+ = W (k∞)[[TK]], A+
K = W (k∞ )[[TK ]], where thetopology on W (k∞ )[[T ]] is as usual the (p, T )-adic topology.
2.1.13 Remark. For an arbitrary K, it is not necessarily possible to choose TK
0 +so that A+
K is ϕ-stable and ΓK -stable (and similarly for (AK ) ). In fact, it is
not even possible to choose TK so that A+is
ϕ-stable.
Indeed,by [Ber14, Prop.
K
4.2, 4.3], if A+
isϕ-stable
then
it
is
also
Γ
-stable.
As
explained
in [Her98,
KK§1.1.2.2], it is possible to choose TK so that A+
is
ϕ
and
Γ
-stablepreciselyK
K
when K is contained in an unramified extension of K0 (ζp∞ ), i.e. when K isabelian over Qp .

24

CHAPTER 2. RINGS AND COEFFICIENTS

2.1.14

The Kummer case

Set S = W (k)[[u]], with a ϕ-semi-linear endomorphism ϕ determined by ϕ(u) =up , and let OE be the p-adic completion of S[1/u]. The extension K∞ =∞K(π 1/p ) of K is infinite and strictly APF (but not Galois). The choice of∞[compatible system of p-power roots of π gives an element π 1/p ∈ OK
, and
d
∞
there is a continuous ϕ-equivariant embedding[
S ,→ W (OKd)∞

b ∞ )[ , defined by sending u 7→ [π 1/p∞ ].
which lifts the morphism XK (K∞ ) ,→ (KThis embedding extends to a continuous ϕ-equivariant embedding[dOE ,→ W ((K∞ ) ).

2.1.15

Graded and rigid analytic techniques

In Appendix B we prove a number of results using rigid analysis and gradedtechniques, which we will apply in the next section in order to prove some basic
facts about our coefficient rings. In order to apply these results to the various
rings introduced above, we need to verify the hypotheses introduced in B.11,
which are as follows. There is an Artinian local ring R, which in our present
context we take to be the ring Z/pa . Then we work with a Z/pa -algebra C + ,and an element u ∈ C + , satisfying the following properties:(A) u is a regular element (i.e. a non-zero divisor) of C + .
(B) C + /u is a flat (equivalently, free) Z/pa -algebra.
(C) C + /p is a rank one complete valuation ring, and the image of u in C + /p
(which is necessarily non-zero, by ((A))) is of positive valuation (i.e. liesin the maximal ideal of C + /p).
The following lemma gives the examples of this construction that we will use in
the next section.
2.1.16 Lemma. The following pairs (C + , u) satisfy axioms (A)–(C) above.
1. C + = Wa (OF[ ), where F is any perfectoid field, and u is any elementof Wa (OF[ ) whose image in OF[ is of positive valuation.
a2. C + = A+
K /p , u = TK .

3. C + = Wa (k)[[u]].
Proof. Axioms (A) and (C) are clear from the definitions in each case, so weneed only verify that C + /u is flat over Z/pa in each case. This is immediate incases (2) and (3), so we focus on case (1). Rather than checking (B) directlyin this case, we instead check that W (OF[ )/u is flat over Zp ; base-changing to2.2. COEFFICIENTS

25

Z/pa then gives us (B). For this, it suffices to observe that multiplication by u
is injective on each term of the short exact sequencep·

0 → W (OF[ ) −→ W (OF[ ) −→ OF[ → 0 ;the snake lemma then shows that multiplication by p is injective on W (OF[ )/u,as required.
We also note the following lemma.
2.1.17 Lemma. The actions of GK on Wa (C[ ), and of ΓK on AK /pa ande K /pa , are continuous and bounded in the sense of Definition B.32.
A
Proof. It suffices to show that the action of GK on Wa (C[ ) is continuous andbounded, as the other cases follow by restricting this action to the corresponding
subgroup and subring. By Remark B.33, we may assume that u = [v] for[[some v ∈ OC
with positive valuation; since the action of GK on OC
preservesM[M[
the valuation, we then have GK · u Wa (C ) = u Wa (C ) for all M ∈ Z, so
the action is continuous and bounded.

2.2

Coefficients

Our main concern throughout this book will be families of étale (ϕ, Γ)-modules;
an auxiliary role will also be played by families of (various flavours of) étale
ϕ-modules. Such a family will be parameterized by an algebra of coefficients,
typically denoted by A. Since we will work throughout in the context of formal
algebraic stacks over Spf Zp (or closely related contexts), we will always assume
that our coefficient ring A is a p-adically complete Zp -algebra. Frequently, we
will work modulo some a fixed power pa of p, and thus assume that A is actuallya Z/pa -algebra (and sometimes we impose further conditions on A, such as thatof being Noetherian, or even of finite type over Z/pa ).
It is often convenient to introduce an auxiliary base ring for our coefficients,
which we will take to be the ring of integers O in a finite extension E of Qp ;in this case, we will let $ denote a uniformizer of O, and A will be taken to bea p-adically complete (or, equivalently, a $-adically complete) O-algebra, or,quite frequently, an O/$a -algebra (perhaps Noetherian, or even of finite type),for some power $a of $.
For the moment, we put ourselves in the most general case; that is, we
assume that A is a p-adically complete Zp -algebra, and define versions of thevarious rings considered in Section 2.1 “relative to A”.
Let F be a perfectoid closed subfield of C, and if a ≥ 1, let v denote anelement of the maximal ideal of Wa (OF[ ) whose image in OF[ is non-zero. We
then setb Zp A := lim Wa (OF[ ) ⊗Zp A /v i
Wa (OF[ )A = Wa (OF[ ) ⊗
←−
i

26

CHAPTER 2. RINGS AND COEFFICIENTS(so that the indicated completion is the v-adic completion). Note that any twob Zp Achoices of v induce the same topology on Wa (OF[ )⊗Zp A, so that Wa (OF[ ) ⊗is well-defined independent of the choice of v. We then defineb Zp A := Wa (OF[ )A [1/v];
Wa (F [ )A = Wa (F [ ) ⊗this ring is again well-defined independently of the choice of v.
There are natural reduction mapsb Zp A → Wb (OF[ ) ⊗b Zp AWa (OF[ ) ⊗andb Zp A → Wb (F [ ) ⊗b Zp A,Wa (F [ ) ⊗if a ≥ b, so that we may defineb Zp A := lim Wa (OF[ )A ,W (OF[ )A = W (OF[ ) ⊗←−aand similarlyb Zp A := lim Wa (F [ )A .
W (F [ )A = W (F [ ) ⊗←−aFor certain choices of F , we introduce alternative notation for the preceding
constructions. In the case when F is equal to C itself, we write[
Ainf,A := W (OC)A .

b cyc , we write
In the case when F is equal to K
e K,A := W (Kb cyc )[ .
AAb ∞ , we write
In the case when F is equal to K
eE,A := W (Kb ∞ )[ .
OAWe next introduce various imperfect coefficient rings with A-coefficients: If
TK denotes a lift to AK of a uniformizer of EK , defining a subring A+
K of AKas in Section 2.1, then we write+ bnA+lim A+/(p, TK )n ⊗Zp A = lim(lim A+/(pm , TK) ⊗Zp A),
K,A = AK ⊗Zp A := ←
− K
←
−←
− K
n
m
n

and

m
n
b Zp A := lim lim(A+AK,A = AK ⊗K /(p , TK ) ⊗Zp A)[1/TK ] .
←
−
←
−
m
n
Similarly, if u denotes the usual element of S, whose reduction modulo p is a
uniformizer of XK (K∞ ), then we writeb Zp A := lim S/(p, u)n ⊗Zp A = lim(lim S/(pm , un ) ⊗Zp A),
SA = S ⊗
←−
←− ←−
n

m

n

and

b Zp A := lim lim(S/(pm , un ) ⊗Zp A)[1/u] .
OE,A = OE ⊗
←− ←−
m

n

Note that by definition the ring SA is a power series ring over W (k) ⊗A A,
and OE,A is the p-adic completion of the corresponding Laurent series ring; andsimilarly for the rings A+
K,A and AK,A .

2.2. COEFFICIENTS

2.2.1

27

A digression on flatness and completion

We record some results giving sufficient conditions for flatness to be preserved
after passage to certain inverse limits, beginning with the following resultsfrom [FGK11].
2.2.2 Proposition. Let R be a ring which is x-adically complete for somex ∈ R, and let S be an R-algebra which is also x-adically complete, and forwhich R[1/x] and S[1/x] are both Noetherian. If for each k ≥ 1, the induced
morphism R/xk → S/xk S is (faithfully) flat, then the morphism R → S is
(faithfully) flat.
Proof. The flatness claim is a special case of the discussion at the beginningof [FGK11, §5.2]. The faithful flatness claim is a special case of [FGK11, Prop.
5.2.1 (2)]. It also follows from the flatness claim together with Lemma 2.2.9
below.
2.2.3 Remark. Under the faithful flatness hypothesis of the preceding proposition, it is proved in [FGK11, Prop. 5.2.1 (1)] that Noetherianness of R[1/x] isa consequence of Noetherianness of S[1/x]. This is a kind of fpqc descent result
for this property, which however we won’t need in the present book.
We next present the following variation on a result of Bhatt–Morrow–Scholze.
2.2.4 Proposition. Let R be a Noetherian ring which is x-adically complete,
for some element x ∈ R, let S be an x-adically complete R-algebra, and supposethat S/xn is (faithfully) flat over R/xn for every n ≥ 1. Then S is (faithfully)
flat over R.
2.2.5 Remark. In [BMS18, Rem. 4.31], the authors prove the above propositionin the particular case when R and S are flat Zp -algebras, and x equals p. In thiscase they only need to assume that S/p is flat over R/p. (Since R and S are bothflat over Zp , it follows automatically that each S/pn is flat over R/pn .) Their
proof makes use of the following key ingredients: that for a p-adically complete
ring, any pseudo-coherent complex is derived complete; that for a complex overa p-torsion free ring, the derived p-adic completion may be computed “naively”;
and the Artin–Rees lemma for the p-adically complete and Noetherian ring R.
Our argument is identical to theirs, once we confirm that these ingredientsremain available (with p-adic completions being replaced by x-adic completions).
The first statement holds quite generally [Sta, Tag 0A05], and of course Artin–
Rees holds for any Noetherian ring R. Thus the main point is to verify that
we may compute derived x-adic completions for the (generally non-Noetherianring) S “naively”. By [Sta, Tag 0923], this is possible provided that the x-power
torsion in S is bounded. So our task is to verify this boundedness (under our
given hypotheses).
We begin with the following general criterion for a ring to have bounded
torsion.
2.2.6 Lemma. If x is an element of the ring R, then the following are equivalent:

28

CHAPTER 2. RINGS AND COEFFICIENTS
1. R[xm ] = R[x∞ ]2. The morphism R[xi ] → R/xm is injective for some i ≥ 1.
3. The morphism R[xi ] → R/xm is injective for every i ≥ 1.

Proof. This is straightforward. Indeed, if the morphism R[xi ] → R/xm is injective for some i ≥ 1, then it is in particular injective if i = 1. Suppose then that
injectivity holds for i = 1. If y ∈ R is such that xm+n y = 0 for some n ≥ 1,then xm+n−1 y is in the kernel of the morphism R[x] → R/xm , so xm+n−1 y = 0,
and by an easy induction we have xm y = 0, as required.
Conversely, if t is in the kernel of the morphism R[xi ] → R/xm for some i,then we can write t = xm y, and we have xi t = 0; so xm+i y = 0. If R[xm ] =R[x∞ ], then we have xm y = 0, so t = 0, as required.
2.2.7 Lemma. If x is any element of the ring R, then for any i, n ≥ 1 we havethe exact sequencexi

R[xi ] → R/xn −→ R/xn+i → R/xi → 0.
Proof. This is immediately verified.
2.2.8 Lemma. If x is an element of the ring R, and if R[xm ] = R[x∞ ], thenxi

0 → R[xi ] → R/xn −→ R/xn+i → R/xi → 0
is an exact sequence, for any n ≥ m and any i ≥ 1.
Proof. This follows from Lemmas 2.2.6 and 2.2.7.
Proof of Proposition 2.2.4. Since R is Noetherian, we see that R[xm ] = R[x∞ ]for some m ≥ 0. Thus, by Lemma 2.2.8, for each n ≥ m and each i ≥ 1, we
obtain an exact sequencexi

0 → R[xi ] → R/xn −→ R/xn+i → R/xi → 0
Tensoring with the flat R/xn+i -algebra S/xn+i , we obtain an exact sequencexi

0 → S ⊗R R[xi ] → S/xn −→ S/xn+i → S/xi → 0
Passing to the inverse limit over n (and taking into account that all the transition
morphisms are surjective), we obtain an exact sequence of R/xn+i -modules
xi

0 → S ⊗R R[xi ] → S −→ S → S/xi → 0∼

In particular, we find that S ⊗R R[xi ] −→ S[xi ], for every i, and so in particular
S[xm ] = S[x∞ ], so that S has bounded x-power torsion.
We now follow the proof of [BMS18, Rem. 4.31]. By (for example) [Sta,Tag 00M5], it is enough to show that if M is a finitely generated R-module,

2.2. COEFFICIENTS

29

then M ⊗L
R S has cohomology concentrated in degree 0. Since R is Noetherian,[Sta, Tag 066E] shows that M is pseudo-coherent when regarded as a complexof R-modules in a single degree, and so M ⊗L
R S is a pseudo-coherent complex
of S-modules, by [Sta, Tag 0650]. Also, since R and S are x-adically complete,it then follows from [Sta, Tag 0A05] that both M and M ⊗L
R S are derived
x-adically complete.
Since S has bounded x-power torsion, it follows from [Sta, Tag 0923] that
we may compute derived x-adic completions naively, i.e. via R lim(– ⊗S S/xn ).
Thus

∼
∼
L
n
L
n
L
n
M ⊗L
R S −→ R lim(M ⊗R S/x ) −→ R lim (M ⊗R R/x ) ⊗R/xn S/x .
nArtin–Rees allows us to replace the pro-system {M ⊗L
R R/x } with the pronsystem {M/x }, and so we find that in fact
∼

n L
n
M ⊗L
R S −→ R lim(M/x ⊗R/xn S/x )∼∼−→ R lim(M ⊗R S/xn ) −→ lim M ⊗R S/xn ,the penultimate isomorphism holding since S/xn is flat over R/xn , and thefinal isomorphism following from the fact that the transition morphisms in the
pro-system {M ⊗R S/xn } are surjective. Thus indeed M ⊗L
R S has cohomology
supported in a single degree, as required.
The claim about faithful flatness follows from the immediately following
Lemma 2.2.9.
2.2.9 Lemma. If R → S is a flat morphism of rings, if I is an ideal in R for
which R is I-adically complete, and if the morphism R/I → S/I is faithfully
flat, then the morphism R → S is faithfully flat.
Proof. Since R is I-adically complete, we see that I is contained in the Jacobsonradical of R. Thus Max Spec R/I = Max Spec R, and so our assumption that
R/I → S/I is faithfully flat shows that Max Spec R is contained in the image ofSpec S/I ⊆ Spec S in Spec R. Since flat morphisms satisfy going down, we findthat in fact Spec S → Spec R is surjective, as claimed.
We also note the following lemma.
2.2.10 Lemma. If F ,→ F 0 is an inclusion of perfectoid fields in characteristic p, then the induced morphism Wa (OF ) → Wa (OF 0 ) is faithfully flat, for
any a ≥ 1.
Proof. Since OF is a Bézout ring, being flat over OF is the same as being torsion
free, and so the inclusion OF ,→ OF 0 is a flat morphism, and thus even faithfully flat, being a local morphism of local rings. A standard grading argument,
applying Lemma B.8 to the p-adic filtrations on source and target, then showsthat the inclusion Wa (OF ) ,→ Wa (OF 0 ), is faithfully flat, for each a ≥ 1.

30

CHAPTER 2. RINGS AND COEFFICIENTS

2.2.11

Back to coefficient rings

We now return to our discussion of coefficient rings, and record that, at least
if A is a finite type Z/pa Z-algebra, the various natural maps between theserings are in fact (faithfully) flat injections. We also show that the various mapsof coefficient rings induced by a (faithfully) flat morphism of finite type Z/pa algebras are again (faithfully) flat.
2.2.12 Proposition. Suppose that A → B is a flat homomorphism of finite type
Z/pa Z-algebras for some a ≥ 1. Then all the maps in the following diagram
are flat. Furthermore the vertical arrows are all injections, while the horizontal
arrows are all faithfully flat (and so in particular also injections). If A → B isfurthermore faithfully flat, then the same is true of the diagonal arrows.

</ Ainf,B o
9

A+
K,B

/ Ainf,A o

A+
K,A


AK,A

< SB

SA


AK,B
<

e
/A
: K,B


/ W (C[ )B o
9

e K,A
/A


/ W (C[ )A oeE,A oO:

eE,B o
O


OE,B
<


OE,A

Proof. From left to right, the vertical maps are given by inverting the elements
T , v, and u respectively (where as above, v is any element of the maximal ideal[of Wa (OC), whose image in O[ is nonzero; in particular, we can take v equal
to either T or u). Since localizations are flat, to prove the proposition for thesemaps it is enough to note that the sources of the vertical maps are respectively
T -, v-, and u-torsion free, by Lemma B.19.
We now turn to the horizontal maps, where we will make repeated use of
Proposition 2.2.2. It evidently suffices to treat the maps with A-coefficients.
e K,A , W (C[ ))A , O
eE,A , and OE,A is Noetherian by PropoSince each of AK,A , A
sition B.36 (1), it follows from Proposition 2.2.2 (and the flatness of the vertical[imaps) that we need only show that the maps SA /ui → (Wa (OKd )⊗Zp A)/u →
∞

i
[
i
i
Ainf,A /ui and A+
K,A /T → (Wa (O [ ) ⊗Zp A)/T → Ainf,A /T are faithfully
Kcyc

flat for each i ≥ 1.
[i
For the faithful flatness of SA /ui → (Wa (OKd ) ⊗Zp A)/u , it is enough to∞[show that S/pa → Wa (OKd ) is faithfully flat. By a standard grading argument∞[(see Lemma B.8), it suffices in turn to prove that k[[u]] → OK
d is faithfully flat,∞[so in turn it is enough to check that OK
d is u-torsion free, which is clear. To∞[ii
see that (Wa (OKd ) ⊗Zp A)/u → Ainf,A /u is faithfully flat, it suffices to note∞

2.2. COEFFICIENTS

31[[
that Wa (OK) → Wa (OC) is faithfully flat, by Lemma 2.2.10. The faithfuld∞flatness of the remaining horizontal maps is proved in exactly the same way.
Finally, the (faithful) flatness of the diagonal maps is now immediate fromanother application of Proposition 2.2.2, together with the (faithful) flatness of
A → B.

Recall that if A◦ is a p-adically complete O-algebra, then A◦ is said to
be topologically of finite type over O if it can be written as a quotient of a
restricted formal power series ring in finitely many variables OhhX1 , . . . , Xn ii;
equivalently, if and only if A◦ ⊗O k is a finite type k-algebra ([FK18, §0, Prop.
8.4.2]). In particular, if A◦ is a O/$a -algebra for some a ≥ 1, then A◦ is
topologically of finite type over O if and only if it is of finite type over O/$a .
Since O is finite over Zp , it is equivalent for A to be topologically of finite typeover O or over Zp , and so for the moment we consider the maximally generalcase of a topologically finite type Zp -algebra.
2.2.13 Remark. We don’t know if the analogue of Proposition 2.2.12 holds when
A and B are taken to be merely p-adically complete and topologically of finite
type over Zp , rather than of finite type over Z/pa for some a ≥ 1. Each of thevarious coefficient rings over A and B is (by definition) formed by first forming
the corresponding coefficient ring over each A/pa or B/pa , and then taking an
inverse limit. Since the formation of inverse limits is left exact, we see that thehorizontal and vertical arrows in the diagram are injective, but we don’t know
in general that the various arrows are flat (although we have no reason to doubtit). One can however establish some partial results, using Proposition 2.2.4. We
record here one such result, which we will need later on.
2.2.14 Proposition. If A is a p-adically complete Zp -algebra which is topologically of finite type, then the natural morphism SA → Ainf,A is faithfully
flat.
Proof. This follows from Propositions 2.2.4 and 2.2.12, once we note that SA
is Noetherian when A is p-adically complete and topologically of finite type;indeed since A is in particular Noetherian, so is the power series ring W (k) ⊗Zp
A[[u]].
We conclude this initial discussion of coefficient rings by explaining how theaction of ϕ on the various rings S, Ainf and so on extends to a continuous action
on the corresponding rings SA , Ainf,A , etc., and similarly for the various Galois
actions. For this, it is convenient for us to briefly digress, and to introduce thefollowing situation, which will also be useful for us in Chapter 3.
2.2.15 Situation. Fix a finite extension k/Fp and write A+ := W (k)[[T ]].
Write A for the p-adic completion of A+ [1/T ].
If A is a p-adically complete Zp -algebra, we write A+
A := (W (k) ⊗Zp A)[[T ]];we equip A+
with
its
(p,
T
)-adictopology,
so
that
it
is
a topological A-algebraA(where A has the p-adic topology). Let AA be the p-adic completion of A+
A [1/T ],which we regard as a topological A-algebra by declaring A+
to
be
an
open
A

32

CHAPTER 2. RINGS AND COEFFICIENTS

subalgebra. Note that the formation of SA , OE,A , A+
K,A and AK,A above areparticular instances of this construction.
Let ϕ be a ring endomorphism of A which is congruent to the (p-power)
Frobenius endomorphism modulo p. We say that A+ is ϕ-stable if ϕ(A+ ) ⊆ A+ .
By [EG21, Lem. 5.2.2 and 5.2.5], if A+ is ϕ-stable, then ϕ is faithfully flat,and induces the usual Frobenius on W (k); the same arguments show that thisis true for ϕ on A, even if A+ is not ϕ-stable.
We have the following variant of [EG21, Lem. 5.2.3].
2.2.16 Lemma. Suppose that we are in Situation 2.2.15. Then for each integera ≥ 1 there is an integer C ≥ 0 such that for all n ≥ 1, we have ϕ(T n ) ∈T pn−C A+ + pa A. In particular, the action of ϕ on A is continuous.
Proof. For some h ≥ 0 we have ϕ(T ) ∈ T −h A+ +pa A. Write ϕ(T ) = T p +pY , sothat ϕ(T n ) = (T p + pY )n . If we expand using the binomial theorem, then every
term on the right hand side is either divisible by pa , or is a multiple of T p(n−r) Y r
for some 0 ≤ r ≤ a − 1. It follows that we can take C = (a − 1)(p + h).
2.2.17 Lemma. If A is a p-adically complete Zp -algebra, then in Situation
2.2.15, the endomorphism ϕ of A extends uniquely to an A-linear continuous
endomorphism of AA (which we continue to denote by ϕ). If A+ is ϕ-stable,
then A+A is ϕ-stable for all A.
Proof. Since ϕ is Zp -linear by definition, and since the topologies on A+
A and AAare defined by passage to the limit modulo pa as a → ∞, we are immediately reduced to the case that A is a Z/pa Z-algebra. In this case the result is immediatefrom Lemma 2.2.16 and Lemma B.31.
2.2.18 Lemma. Let A be a p-adically complete Zp -algebra.
1. The endomorphism ϕ of S (resp. OE , AK , W (OF[ ), W (F [ )) extends
uniquely to an A-linear continuous endomorphism of SA (resp. OE,A ,AK,A , W (OF[ ), W (F [ )A ), which we again denote by ϕ. If K is furthermore basic, then the endomorphism ϕ of AK,A preserves A+
K,A .
[2. The continuous GK -action on W (OC) and W (C[ ) (resp. the continue K ) extends to a continuous A-linear actionous ΓK -action on AK and A[
e K,A ).
of GK on W (OC )A and W (C[ )A (resp. of ΓK on AK,A and A

Proof. As in the proof of Lemma 2.2.17, we can immediately reduce to the casethat A is a Z/pa -algebra. Part (1) then follows from Lemma B.31 as in the
proof of Lemma 2.2.17, while part (2) follows from Lemma B.34, bearing in
mind Lemmas 2.1.16 and 2.1.17.
For our final result of this section, we compute the ϕ-invariants in W (C[ )A .

2.2. COEFFICIENTS

33

2.2.19 Lemma. If A is any p-adically complete Zp -algebra, then for each a ≥1, the natural morphism A/pa → Wa (C[ )A induces an identification A/pa =ϕ=1
Wa (C[ )A, and (consequently) the natural morphism A → W (C[ )A induces
ϕ=1
an identification A = W (C[ )A.
Proof. Since W (C[ )A = lima Wa (C[ )A , we see that
←−
ϕ=1
ϕ=1
W (C[ )A
= lim Wa (C[ )A,←−
and thus (as the phrasing of the lemma indicates), the claim for W (C[ )A follows
from the claims for each Wa (C[ )A . Thus for the remainder of the proof, we may
and do assume that A is a Zp /pa -module for some a ≥ 1, and prove that
ϕ=1
A = Wa (C[ )A. In fact, for any Z/pa -module M , we may define[Wa (C[ )M := lim Wa (OC)/T n ⊗Z/pa M [1/T ]←−n(this extends our preceding definition for Z/pa -algebras), and we will proveϕ=1that M = Wa (C[ )M. (In fact this statement for modules, which obviously
implies the statement for algebras, is actually equivalent to it, as one sees byapplying the statement for algebras to Zp /pa -algebras of the form Zp /pa ⊕ M,
where M is an arbitrary Zp /pa -module equipped with the square zero multiplication.)
Suppose first that a = 1, so that M is simply an Fp -vector space.. If we
Q0∼[[choose a basis for M , i.e. an isomorphism F⊕I
p −→ M , then (OC )M =i∈I OC ,where the prime indicates we consider the set of tuples (xi )i∈I for which limi xi =0, where the limit is taken with respect Q
to the filter of cofinite subsets of I. Thus0[
we have, correspondingly, that CM= i∈I C [ (where the prime has the same
meaning). Now (C [ )ϕ=1 = Fp , and so
[ ϕ=1
(CM
)
=

Y

0

(C [ )ϕ=1 = ⊕i∈I Fp = M.

i∈I

This establishes the case a = 1.
Now consider the case of general a. An application of Lemma B.20 (1)(taking R = Z/pa and C + = Wa (OC[ ), and considering the exact sequence of
Z/pa -modules 0 → pM → M → M/pM → 0; see also Lemma 2.1.16) showsthat the sequence[
0 → Wa−1 (C [ )pM = Wa (C [ )pM → Wa (C [ )M → Wa (C [ )M/pM = CM/pM→0

ϕ=1
is short exact. Suppose that x ∈ Wa (C[ )M. The case a = 1 already proved
then shows that we may find m ∈ M so that y := x − m lies in Wa−1 (C [ )pM .
Of course ϕ(y) = y, and so, by induction on a, we find that y ∈ pM . Thusx = m + y ∈ M , as claimed.

34

CHAPTER 2. RINGS AND COEFFICIENTS

We will also use the following variant on Lemma 2.2.19. Suppose that A is a\complete local Noetherian O-algebra with finite residue field, and write W(C[ )A[for the mA -adic completion of W (C )A .
2.2.20 Lemma. Let A be a complete local Noetherian O-algebra with finite\
residue field. Then (W(C[ )A )ϕ=1 = A.
\
Proof. Since W(C[ )A = limn W (C[ )A/mnA , we have
←−
ϕ=1
ϕ=1
\
W
(C[ )A= lim W (C[ )A/mnA,←−nso the result follows from Lemma 2.2.19 (applied to each A/mnA ).

2.3

Almost Galois descent for profinite group
actions

We will be interested in descent results for profinite group actions, and in thissection we establish the key result that we will need. Our setup is slightly
elaborate, but accords with the situations that will arise in practice. We begin
with the case of finite group actions.
Suppose that R is a ring, and that I is an ideal in R such that I 2 = I.
Assume further that I ⊗R I is a flat R-module; by [GR03, Prop. 2.1.7], thisholds if I is a filtered union of principal ideals. In particular, these assumptions
hold if R is a valuation ring with a non-discrete rank one valuation, and I is the
maximal ideal of R.
The full subcategory of the category of R-modules whose objects are themodules annihilated by I is a Serre subcategory, and so we can form the quotientcategory of almost R-modules.
Suppose that S is an R-algebra equipped with an action of a finite group G
by R-algebra automorphisms (i.e. the structural morphism R → S is equivariant
with respect to the given G-action on S, and the trivial G-action on R).
2.3.1 Definition. We say that the morphism R → S makes S an almost Galois
extension of R, with Galois group G, if the natural G-equivariant and S-linear
morphismY
S ⊗R S →Sg∈G(here G acts on the source through its action on the second factor, and on thetarget by permuting the factors, while S acts on the source through its action
on the first factor and on the target through its action on each factor) defined bys1 ⊗ s2 7→ s1 g(s2 ) g∈G is an almost isomorphism (i.e. induces an isomorphismin the category of almost R-modules).

2.3. ALMOST GALOIS DESCENT

35

2.3.2 Remark. Suppose that R → R0 is a morphism, and define I 0 = IR0 , sothat I 02 = I 0 , allowing us to also define the category of almost R0 -modules. If
R → S is almost Galois with Galois group G, then evidently R0 → S 0 := R0 ⊗R S
is almost Galois with Galois group G.
Conversely, if R → R0 is faithfully flat, and R0 → S 0 is almost Galois with
Galois group G, then so is R → S; indeed we may write R0 ⊗R (S ⊗R S) =(R0 ⊗R S) ⊗R0 (R0 ⊗R S) = S 0 ⊗R0 S 0 .
2.3.3 Lemma. Suppose that R → S makes S an almost Galois extension of S
with Galois group G, and furthermore that R → S is faithfully flat. Then themorphism R → S G is an almost isomorphism.
Proof. Since R → S is faithfully flat, it suffices to verify that the induced
morphism
S → (S ⊗R S)G(2.3.4)Q
is an almost isomorphism. By assumption the morphism S ⊗R S → g∈G S
is an almost isomorphism. One easily verifies that the induced morphism on
invariant subrings
Y(S ⊗R S)G → (S)G = Sg∈G

is then also an almost isomorphism. The composite of this map with the morphism (2.3.4) is just the identity, and so we find that (2.3.4) is indeed an almost
isomorphism.
2.3.5 Lemma. Suppose that R → S makes S an almost Galois extension of
S with Galois group G, and furthermore that R → S is faithfully flat. Then if
M is an S-module equipped with a semi-linear G-action, the induced morphism
S ⊗R M G → M is an almost isomorphism of S-modules.
Proof. The semi-linear G-action on M can be reinterpreted as an isomorphism
YY(S) ⊗S M ∼S)
= M ⊗S (g∈Gg∈G(where the tensor product on the left hand side is twisted by the G-action) of
S ⊗R S-modules, and hence as an almost isomorphism
S ⊗R M

almost∼=M ⊗R Sof S ⊗R S-modules. The claim of the lemma now follows from faithfully flat
descent in the almost category (for which see [GR03, §3.4.1]; it is in order tomake this citation that we have assumed that I ⊗R I is R-flat).
The following lemma explains our interest in almost Galois extensions.
2.3.6 Lemma. If F ⊆ F 0 is a finite Galois extension of perfectoid fields, with
Galois group G, then the corresponding inclusion of rings of integers OF ⊆ OF 0
realizes OF 0 as an almost Galois extension of OF , with Galois group G (the“almost” structure being understood with respect to the maximal ideal of OF ).

36

CHAPTER 2. RINGS AND COEFFICIENTSProof. There are various more-or-less concrete ways to see this. For example, [Sch12, Prop. 5.23] shows that the morphism OF → OF 0 is almost étale,which by definition [Sch12, Def. 4.12] means that the surjection of OF 0 -algebras
OF 0 ⊗OF OF 0 → OF 0 may be “almost” split, so that OF 0 is almost a direct
factor of the source. Permuting such a splitting under the action of the
Q Galoisgroup G then allows one to show that the morphism OF 0 ⊗OF OF 0 → g∈G OF 0
is an almost isomorphism.
However, a more direct (but less explicit) way to prove the lemma is to use
the first equivalence of categories in [Sch12, Thm. 5.2], namely the equivalence
between the category of perfectoid F -algebras and the category of perfectoid
almostalgebras over OF . Under this equivalence,Qthe morphism OF 0 ⊗OF OF 0 →
Q
0
0
0
0
O
g∈G F is taken to the morphism F ⊗F F →g∈G F . This latter morphism
0
is an isomorphism, since F is Galois over F with Galois group G, and so weconclude that the former morphism is an almost isomorphism, as required.
In the case of perfectoid fields of characteristic p, we may extend the statement of the preceding lemma to the context of truncated rings of Witt vectors.
2.3.7 Lemma. If F ⊆ F 0 is a finite Galois extension of perfectoid fields incharacteristic p, with Galois group G, then for each a ≥ 1, the inclusion
Wa (OF ) ,→ Wa (OF 0 ) is almost Galois, with Galois group G (the “almost” structure being understood with respect to the maximal ideal of Wa (OF )).
Proof. For simplicity of notation, write R := Wa (OF ) and S := Wa (OF 0 ). Eachof R and S is flat over Z/pa , and S is also flat over R, by Lemma 2.2.10. Thusa
S ⊗R S is flat over S, hence over R, and hence over
Q Z/p as well.
To see that the natural morphism S ⊗R S → g∈G S is an almost isomorphism, it suffices to check the analogous condition after passing to associatedgraded rings for the p-adic filtration. Lemma B.8 allows us to rewrite thisinduced morphism on associated graded rings in the formY
Fp [T ]/(T a ) ⊗Fp (S/pS) ⊗(R/pR) (S/pS) → Fp [T ]/(T a ) ⊗Fp(S/pS)
g∈G(here Fp [T ]/(T a ) appears as the associated graded ring to Z/pa with its p-adic
filtration), which may be identified with the base-change over Fp [T ]/(T a ) of thenatural morphism
Y(S/pS) ⊗(R/pR) (S/pS) →(S/pS).
g∈G

This latter morphism is an indeed an almost isomorphism, by Lemma 2.3.6.
We now pass to the profinite setting. We continue to suppose that we are
given the ring R endowed with an idempotent ideal I. We suppose additionallythat v ∈ I is a regular element of R (i.e. a non-zero divisor) and that R is
v-adically complete.

2.3. ALMOST GALOIS DESCENT

37

We also suppose given a v-adically complete R-algebra S, equipped with anaction of a profinite group G as R-algebra automorphisms. We assume that thisaction is continuous, in the sense that the action map G × S → S is continuous
when S is endowed with its v-adic topology.
S
We further suppose that we may write S = \n Sn as the v-adic completion of
an increasing union of G-invariant v-adically complete R-subalgebras Sn , with
R = S0 ; that the G-action on Sn factors through a finite quotient Gn := G/Hnof G, where Hn is a normal open subgroup of G; and that for each i ≥ 0 andeach m ≤ n, the morphism Sm /v i → Sn /v i is faithfully flat, and realizes Sn /v i
as an almost Galois extension of Sm /v i , having the subgroup Hm /Hn of Gn as
Galois group.
2.3.8 Lemma. For any value of m, the morphism Sm → S Hm is an almost
isomorphism, as is the induced morphism Sm /v i → (S/v i )Hm for any i ≥ 0.
Proof. Our hypotheses, together with Lemma 2.3.3, imply that the morphism
Sm /v i → (Sn /v i )Hm /Hn
is an almost isomorphism for each n ≥ m and each i ≥ 0. Passing to the direct
limit over n gives the second claim, and then additionally passing to the inverse
limit over i gives the first claim.
2.3.9 Corollary. The injection R[1/v] ,→ S[1/v]G is an isomorphism.
Proof. Note that S[1/v]G = (S G )[1/v], since v is G-invariant (being an elementof R) and localization is exact. The corollary thus follows from the m = 0 caseof Lemma 2.3.8, as inverting v converts the almost isomorphism into a genuine
isomorphism.
2.3.10 Definition. We say that an S-module M is iso-projective of finite rank
if it is finitely generated and v-torsion free, and if S[1/v] ⊗S M is a projective
S[1/v]-module.
2.3.11 Lemma. Any iso-projective S-module of finite rank is v-adically complete.
Proof. If M is iso-projective, then we may write M [1/v] = eF where F is a
finite free S[1/v]-module and e ∈ EndS[1/v] F is an idempotent. Write F =F0 ⊗S S[1/v], where F0 is a finite free S-module. Then, since F0 is finitely
generated, we find that eF0 is contained in v −a F0 for some sufficiently large
value of a; the latter S-module is v-adically separated, and thus so is the former.
Since eF0 is furthermore the image of the v-adically complete S-module F0 , itis in fact v-adically complete. Since eF0 [1/v] = M [1/v], and both eF0 and
M are finitely generated S-modules, we find that v b eF0 ⊆ M ⊆ v −b eF0 forsome sufficiently large value of b, and thus M is also v-adically complete, as
claimed.

38

CHAPTER 2. RINGS AND COEFFICIENTS

2.3.12 Theorem. If M is an iso-projective S-module of finite rank, equippedwith a semi-linear G-action that is continuous with respect to the v-adic topologybR MG → M
on M , then the kernel and cokernel of the induced morphism S ⊗(the source being the v-adically completed tensor product) are each annihilated
by a power of v.
Proof. Write P := S[1/v] ⊗S M, so that (by assumption) P is a finitely presented projective module over S[1/v]. Since M is v-adically complete (by∼Lemma 2.3.11), we have an isomorphism M −→ limn M/v n M . This induces←−∼a corresponding isomorphism M G −→ limn (M/v n M )G . (We should also note←−that the topology on M G induced by the v-adic topology on M coincides withthe v-adic topology on M G , as follows immediately from that fact that multiplication by v on M is injective (as M is iso-projective) and commutes with theG-action.) Thus, to prove the theorem, it suffices to show that the kernel andcokernel of each of the morphisms
S ⊗R (M/v n M )G → M/v n M
is annihilated by a power of v that is bounded independently of n.
∼Multiplication by v −n induces an isomorphism M/v n M −→ v −n M/M, andwe will actually prove the equivalent statement that each of the morphisms
S ⊗R (v −n M/M )G → v −n M/M(2.3.13)has kernel and cokernel annihilated by a power of v that is bounded independently of n.
The reason for formulating our argument in terms of the quotients v −n M/M
is that these may be conveniently be regarded as submodules of the quotient
P/M . Our argument will proceed by constructing, for each n, a G-invariant
S-submodule Zn of P/M , such that v −n M/M ⊆ Zn ⊆ v −(n+c) M/M (where cis independent of n), and such that each of the morphisms
S ⊗R ZnG → Zn

(2.3.14)has kernel and cokernel annihilated by a power of v that is bounded independently of n. This implies the corresponding statement for the morphisms (2.3.13),
and thus establishes the theorem. The remainder of the argument is devoted
to constructing the submodules Zn , and proving the requisite properties of themorphisms (2.3.14).
Since P is a projective S[1/v]-module, we may choose a finite rank freemodule F over S[1/v], and an idempotent e ∈ EndS[1/v] (F ), such that P = eF.
∼
We choose a free R-submodule F0 of F such that S[1/v] ⊗R F0 −→ F. (Moreconcretely, F0 is simply the R-span of some chosen S[1/v]-basis of F .) The
endomorphism e may not preserve the S-submodule S ⊗R F0 of F , but if wechoose a sufficiently large, then e0 := v a e will preserve S ⊗R F0 .
Since M is finitely generated over S, we may and do assume that we havechosen F0 in such a manner that M ⊆ v a (S ⊗R F0 ). (Simply replace F0 by2.3. ALMOST GALOIS DESCENT

39v −b F0 for some sufficiently large value of b.) Then in fact M ⊆ e0 (S ⊗R F0 ).
Furthermore, since M and e0 (S ⊗R F0 ) both span P as an S[1/v]-module, wefind that v c e0 (S ⊗R F0 ) ⊆ M for someS sufficiently large value of c.
For any n ≥ 0 we have S/v n = i≥0 Si /v n . Thus e0 mod v n , which is an endomorphism of (S/v n )⊗R F0 , descends to an endomorphism of (Si /v n )⊗R F0 for
all sufficiently large i. Replacing (Sn ) by an appropriately chosen subsequence,
we may and do assume that in fact for each n, e0 descends to an endomorphism
e0n of (Sn /v n ) ⊗R F0 .
Since G acts continuously on P and preserves M , it acts continuously on
P/M . The topology on P/M is discrete, and thus any finite subset of P/M isfixed by some open subgroup of G. In particular, we find thatv −n e0 F0 / v −n e0 F0 ∩ M(which we regard in the natural way as a submodule of P/M , and which wenote is finitely generated over R) is pointwise fixed by some open subgroupof G. Since the Hn form a cofinal sequence of open subgroups of G, if weagain replace Sn and Hn by appropriately chosen
subsequences, we may and doassume that in fact v −n e0 F0 / v −n e0 F0 ∩ M is pointwise fixed by Hn .
c 0
−n 0
−n 0
Since M ⊇ v e (S ⊗R F0 ), we see that v e F0 / v e F0 ∩ M is annihilated
by v n+c , and thus that
G v −n e0 F0 / v −n e0 F0 ∩ M ⊆ v −(n+c) M/M ⊆ v −(n+c) e0 (S ⊗R F0 )/M.

Since G/Hn is finite, and since Hn fixes v −n e0 F0 / v −n e0 F0 ∩ M pointwise, we
see that in fact
G v −n e0 F0 / v −n e0 F0 ∩ M ⊆ v −(n+c) e0 (T ⊗R F0 )/M,
for some finitely generated R-subalgebra T of S/v n+c S. Passing to a subsequence
one more time, we may assume that T is contained in Sn .
We let Yn , respectively Zn , denote the Sn -submodule, respectively the S
submodule, of P/M generated by the G-translates of v −n e0 F0 / v −n e0 F0 ∩ M .
It remains to prove the requisite properties of Zn . We begin by noting theinclusions
v −n e0 (S ⊗R F0 )/M ⊆ Zn ⊆ v −(n+c) M/M ⊆ v −(n+c) e0 (S ⊗R F0 )/M,
which imply thatv c Zn ⊆ v −n e0 (S ⊗R F0 )/M ⊆ Zn ,and thus thatv c ZnHn ⊆ (v −n e0 (S ⊗R F0 )/M )Hn ⊆ ZnHn .

(2.3.15)

Lemma 2.3.8 shows that Sn /v n → (S/v n )Hn is an almost isomorphism.
Thus (v −n Sn /Sn ) ⊗R F0 → (v −n S/S ⊗R F0 )Hn is an almost isomorphism, if wedeclare that Hn acts trivially on F0 , and thus via its action on the first factor

40

CHAPTER 2. RINGS AND COEFFICIENTS

in the target tensor product. Since (e0 )2 = v a e0 , we find that the cokernel of the
inclusion
H n
H n,→ e0 (v −n S ⊗R F0 )/(S ⊗R F0 )e0 (v −n S ⊗R F0 )/(S ⊗R F0 )is annihilated by v a , and thus that the natural morphism Hne0 (v −n Sn ⊗R F0 )/(Sn ⊗R F0 ) → e0 (v −n S ⊗R F0 )/(S ⊗R F0 )has its kernel annihilated by I, and its cokernel annihilated by v a I.
H n
If we let Xn denote the image of e0 (v −n Sn ⊗R F0 ) in e0 (v −n S ⊗R F0 )/M,
then certainly
Xn ⊆ Yn ⊆ ZnHn .
It follows from the conclusion of the preceding paragraph, along with the factthat v c e0 (S0 ⊗R F0 ) ⊆ M, that the cokernel of the inclusion
Xn ,→ e0 (v −n S ⊗R F0 )/MH n

is annihilated by v a+c I, while the chain of inclusions (2.3.15) shows that thecokernel of the inclusion
H ne0 (v −n S ⊗R F0 )/M⊆ ZnHnis annihilated by v c . Thus the cokernel of the inclusion Xn ⊆ ZnHn is annihilated
by v a+2c I, and hence so is the cokernel of the inclusion Yn ⊆ ZnHn . Passing to
Gn -invariants in the inclusion just mentioned, we find that there is an inclusion
YnGn ⊆ ZnG , whose cokernel is annihilated by v a+2c I. Extending scalars to Sn ,we obtain a morphism
Sn ⊗R YnGn → Sn ⊗R ZnG(which is in fact an embedding, since Sn /v n+c is flat over R/v n+c , although we
don’t need this here), whose cokernel is annihilated by v a+2c I. On the other
hand, Lemma 2.3.5 implies that the natural morphism Sn ⊗R YnGn → Yn is analmost isomorphism.
Putting all these results together, we find that the kernel and cokernel of thenatural morphism
Sn ⊗R ZnG → ZnHn
are each annihilated by v a+2c I. Indeed, the composite Sn ⊗R YnGn → Sn ⊗R
ZnG → ZnHn factors through the almost isomorphism Sn ⊗R YnGn → Yn , and wehave shown that the natural morphism Yn → ZnHn is an injection whose cokernel
is killed by v a+2c I, while the cokernel of Sn ⊗R YnGn → Sn ⊗R ZnG is also killedby v a+2c I.
Since Yn ⊆ ZnHn , and since Yn generates Zn over S by their very definitions,
we see that the natural map
S ⊗Sn ZnHn → Zn

(2.3.16)

2.3. ALMOST GALOIS DESCENT

41

is surjective. We bound the exponent of its kernel as follows: The inclusion
Xn ⊆ ZnHn , whose cokernel we have shown above to be annihilated by v a+2c I,
induces an inclusion
S ⊗Sn Xn ⊆ S ⊗Sn ZnHn ,whose cokernel is again annihilated by v a+2c I, while the defining surjectione0 (v −n Sn /v c Sn ) ⊗R F0 → Xn
induces a surjectione0 (v −n S ⊗R F0 )/(v c S ⊗R F0 ) → S ⊗Sn Xn .
Now the natural morphisme0 (v −n S ⊗R F0 )/(v c S ⊗R F0 ) → M [1/v]/M(2.3.17)

has kernel annihilated by v c , and so we find that the kernel of (2.3.16) is annihilated by v a+3c I. (Indeed, if x is an element of the kernel of (2.3.16), then for
any i ∈ I we can lift v a+2c ix to an element of the kernel of (2.3.17).)
Putting together the results of the preceding two paragraphs, we find thatthe cokernel of the natural morphism
S ⊗R ZnG → Zn
is annihilated by v a+2c I, while its kernel is annihilated by v 2a+5c I. Recallingthat v ∈ I, and noting that the powers of v just mentioned are independent ofn, and also that we have the inclusions v −n M/M ⊆ Zn ⊆ v −(n+c) M/M, we seethat the proof of the theorem is completed.
We will now deduce a descent result for projective modules over S[1/v]. For
this, we need to make some additional hypotheses, which we now describe.
2.3.18 Hypothesis.
1. S/vS is countable.
2. S[1/v] is Noetherian.
3. The morphism R[1/v] → S[1/v] is faithfully flat.
We write S[1/v] = limn v −n S. If we identify v −n S with S via multiplica−→
tion by v n , then each of the transition maps becomes identified with the closed
embedding vS ,→ S. Thus if we equip S[1/v] with the inductive limit topology, then S[1/v] becomes a topological ring, which is completely metrizable(since S is v-adically complete) and in fact Polish (since S/vS is countable, sothat S and thus S[1/v] is separable). Note also that vS is an open additivesubgroup of S[1/v] which is closed under multiplication, and consists of topologically nilpotent elements. Proposition C.6 then shows that finitely generated
S[1/v]-modules have a canonical topology, with respect to which all S[1/v]homomorphisms are continuous, with closed image.
We now establish the following descent result in this situation.

42

CHAPTER 2. RINGS AND COEFFICIENTS

2.3.19 Theorem. If Hypothesis 2.3.18 holds, and if M is a finitely generated
projective S[1/v]-module, equipped with a continuous (when endowed with its
canonical topology) semi-linear G-action, then M G is a finitely generated andprojective R[1/v]-module, and the natural morphism
S[1/v] ⊗R[1/v] M G → M(2.3.20)

is an isomorphism.
Proof. Let M 0 denote any finitely generated S-submodule of M that generates
M over S[1/v]; then the S-span M 00 of GM 0 is finitely generated (note thatsince M 0 is finitely generated, there is an open subgroup H of G such that
HM 0 ⊆ M 0 , and H has finite index in the profinite group G), so it is an isoprojective S-submodule of M which is G-invariant. Applying Theorem 2.3.12
to M 00 , and noting thatM G = (M 00 [1/v])G = (M 00 )G [1/v](because localisation is exact), we find that the natural morphism (2.3.20) has
dense image. Since its target is finitely generated over S[1/v], its image isalso finitely generated (because S[1/v] is Noetherian, by assumption), and thus
closed in its target; combined with the density, we find that (2.3.20) is surjective.
Since S[1/v] ⊗R[1/v] M G → M is surjective, while M is finitely generated
over S[1/v], we see that if N is any sufficiently large finitely generated R[1/v]submodule of M G , then S[1/v] ⊗R[1/v] N → M is surjective. Choose a finitely
generated free R[1/v]-module F that surjects onto N , and let E denote thekernel of the induced surjection
S[1/v] ⊗R[1/v] F → S[1/v] ⊗R[1/v] N → M,
so that we have a short exact sequence
0 → E → S[1/v] ⊗R[1/v] F → M → 0.
Passing to G-invariants, and taking into account Corollary 2.3.9, we obtain aleft exact sequence
0 → EG → F → M G,
which (by the choice of F ) induces a short exact sequence
0 → E G → F → N → 0.
Tensoring back up with S[1/v] (which is flat over R[1/v] by assumption), we
obtain a morphism of short exact sequences0/ S[1/v] ⊗R[1/v] E G/ S[1/v] ⊗R[1/v] F/ S[1/v] ⊗R[1/v] N

/0

0


/E


/ S[1/v] ⊗R[1/v] F


/M

/0

2.4. AN APPLICATION OF ALMOST GALOIS DESCENT

43

Evidently the middle vertical arrow is the identity, and thus the natural morphism S[1/v] ⊗R[1/v] E G → E is injective. Since E is finitely generated andprojective (being the kernel of a surjection from a finitely generated free module to a projective module), it follows from what we have already proved that
this morphism is also surjective.
Since the left hand two vertical arrows are isomorphisms, so is the third,
so that S[1/v] ⊗R[1/v] N → M is an isomorphism. This is true for any sufficiently large choice of N , and thus we find (using the faithful flatness of S[1/v]
over R[1/v], which holds by assumption) that all these sufficiently large choices
of N coincide, implying that M G is finitely generated and that (2.3.20) is an
isomorphism. The faithful flatness of S[1/v] over R[1/v] then implies that M G
is projective over R[1/v] [Sta, Tag 058S], as required.
The following corollary provides a convenient reformulation of the preceding
theorem.
2.3.21 Corollary. If Hypothesis 2.3.18 holds, then the functor
M 7→ S[1/v] ⊗R[1/v] M
induces an equivalence between the category of finitely generated projective R[1/v]modules and the category of finitely generated projective S[1/v]-modules endowedwith a continuous semi-linear action of G. A quasi-inverse functor is given by
N 7→ N G .
Proof. Theorem 2.3.19 shows that if N is a finitely generated and projective
S[1/v]-module, endowed with a continuous semi-linear G-action, then the natural map S[1/v] ⊗R[1/v] N G → N is an isomorphism. To complete the proofof the corollary, then, it suffices to show that if M is a finitely generated andprojective R[1/v]-module, then the natural map M → (S[1/v] ⊗R[1/v] M )G is an
isomorphism. Writing M as the direct summand of a finite rank free module,
we reduce to the case when M is free, which (as was already observed in the
proof of Theorem 2.3.19) is established by Corollary 2.3.9.

2.4

An application of almost Galois descent

Let F be a closed perfectoid subfield of C, with tilt F [ , a closed perfectoid
subfield of C[ . Recall that, for any p-adically complete Zp -algebra A, we defined
W (OF[ )A and W (F [ )A in Section 2.2.
2.4.1 Theorem. Let A be a finite type Z/pa -algebra, for some a ≥ 1. The
inclusion W (F [ )A → W (C[ )A is a faithfully flat morphism of Noetherian rings,
and the functor M 7→ W (C[ )A ⊗W (F [ )A M induces an equivalence between the
category of finitely generated projective W (F [ )A -modules and the category offinitely generated projective W (C[ )A -modules endowed with a continuous semilinear GF -action. A quasi-inverse functor is given by N 7→ N GF .

44

CHAPTER 2. RINGS AND COEFFICIENTSThe proof will be an application of the almost Galois descent results ofSection 2.3. Thus we have to place ourselves in the framework of that section.
b
To this end, we note that C may be regarded
S as a completion F of the algebraic
closure F of F . We then write F = n≥1 Fn as the increasing union of asequence of finite Galois extensions of F ; to match the notation of Section 2.3,
we also write F0 := F . We write Gn := Gal(Fn /F ), Hn := Gal(F /Fn ), and∼∼G := H0 = Gal(F /F ), so that G −→ limn Gn and G/Hn −→ Gn . Note that←−each of the finite Galois extensions Fn of F is again perfectoid.
Assume now that A is a finitely generated Z/pa -algebra. Then W (F [ )A =Wa (F [ )A , and similarly W (C[ )A = Wa (C[ )A , so that from here on we may
work with rings of a-truncated Witt vectors, rather than with full rings of Witt
vectors themselves. To accord with the notation of Section 2.3, we also denotethese rings by R and S respectively. Recall from Section 2.2 that v denotes a
non-zero element of the maximal ideal of Wa (OF[ ), and that by definition
R := Wa (OF[ )A = lim Wa (OF[ ) ⊗Z/pa A /v i ,
←−
i

while

[
[
S := Wa (OC)A = lim Wa (OC) ⊗Z/pa A /v i ;←−iso that both R and S are v-adically complete.
We let I denote the ideal in R generated by the maximal ideal of Wa (OF[ ),and let v denote the image in I of the chosen element v of that maximal ideal(no confusion should result from this duplication of notation). We further set
Sn := Wa (OF[ n )A = lim Wa (OF[ n ) ⊗Z/pa A /v i←−ifor each n ≥ 0 (so in particular S0 = R). By construction we have that
S coincides with the v-adic completion of limn Sn . As we will see below, the−→transition morphisms Sm → Sn are in fact faithfully flat, and thus injective, andS
so in fact this direct limit is simply a union; thus S = \n Sn , as required forthe setup of Section 2.3. Furthermore, Lemma 2.2.18 ensures that the G-action
on S is continuous.
2.4.2 Proposition. For each i ≥ 0, and each m ≤ n, the morphism Sm /v i →Sn /v i is faithfully flat, and realizes Sn /v i as an almost Galois extension of
Sm /v i , with Galois group Hm /Hn .
Proof. Lemma 2.2.10 shows that each of the inclusions Wa (OF[ m ) ,→ Wa (OF[ n ) is
faithfully flat, thus so is the morphism Wa (OF[ m ) ⊗Z/pa A → Wa (OF[ n ) ⊗Z/pa A,
and hence so are each of the morphisms
Sm /v i Sm = Wa (OF[ m ) ⊗Z/pa A) /v i Wa (OF[ m ) ⊗Z/pa A)→ Wa (OF[ n ) ⊗Z/pa A) /v i Wa (OF[ n ) ⊗Z/pa A) = Sn /v i Sn .

2.5. ÉTALE ϕ-MODULES

45

An identical argument, taking into account Lemma 2.3.7 and Remark 2.3.2,
shows that Sm /v i Sm → Sn /v i Sn is almost Galois (with respect to the ideal in
Sm generated by the maximal ideal of Wa (OF[ m ), and hence also with respect
to ISm , since the latter ideal is contained in the former), with Galois group
Hm /Hn .
2.4.3 Proposition.
1. Each Sm [1/v], as well as S[1/v], is Noetherian. (Setting m = 0 gives inparticular that R[1/v] is Noetherian.)
2. Each of the morphisms Sm → Sn (for m ≤ n), as well as each morphism Sm → S, is faithfully flat. In particular (setting m = 0 and theninverting v) the morphism R[1/v] → S[1/v] is faithfully flat.
Proof. The Noetherian claims of (1) follow from Proposition B.36 (1).
We already noted in Proposition 2.4.2 that the morphisms Sm /v i → Sn /v iare faithfully flat, and an identical argument shows that each morphism Sm /v i →S/v i is faithfully flat. Taking into account that statement of (1), we find thatthe claims of (2) follow from Proposition 2.2.2.
Proof of Theorem 2.4.1. Proposition 2.4.2 verifies that the running assumptions
imposed at the beginning of Section 2.3 are satisfied. Proposition 2.4.3 verifies
that Hypothesis 2.3.18 is satisfied, and also establishes the Noetherian and faithful flatness claims of Theorem 2.4.1. The remainder of the theorem then follows
from Corollary 2.3.21.

2.5

Étale ϕ-modules

In this section we briefly recall and generalize some definitions and resultsfrom [EG21]. Let R be a Zp -algebra, equipped with a ring endomorphism ϕ,which is congruent to the (p-power) Frobenius modulo p. If M is an R-module,
we writeϕ∗ M := R ⊗R,ϕ M.
2.5.1 Definition. An étale ϕ-module over R is a finite R-module M , equippedwith a ϕ-semi-linear endomorphism ϕM : M → M , which has the property thatthe induced R-linear morphism
1⊗ϕMΦM : ϕ∗ M −→ M
is an isomorphism. A morphism of étale ϕ-modules is a morphism of the underlying R-modules which commutes with the morphisms ΦM . We say that M is
projective (resp. free) if it is projective of constant rank (resp. free of constantrank) as an R-module.

46

CHAPTER 2. RINGS AND COEFFICIENTS

We will typically apply this definition with R taken to be one of the coefficient rings defined in Section 2.2. Of particular interest to us will be the cases
corresponding to imperfect fields of norms, as it is these cases which fit into theframework of [EG21, §5], and we will use the results of that paper to prove thebasic algebraicity properties of our moduli stacks.
2.5.2 Definition. Let S be an R-algebra, and let ϕS be a ring endomorphismof S, which is congruent to Frobenius modulo p, and is compatible with ϕ on R.
Then if M is an étale ϕ-module over R, the extension of scalars S ⊗R M isnaturally an étale ϕ-module over S, with ϕS⊗R M := ϕS ⊗ ϕM .

2.5.3

Multilinear algebra

We briefly recall the multilinear algebra of projective étale ϕ-modules. Firstly,
if P is a projective étale ϕ-module over R, then we give its R-dual P ∨ :=HomR (P, R) the structure of an étale ϕ-module by defining the isomorphismϕ∗ P ∨ → P ∨ to be the inverse of the transpose of ΦP . Secondly, if M and N
are projective étale ϕ-modules, then we endow M ⊗R N with the structure ofan étale ϕ-module by defining ϕM ⊗N := ϕM ⊗ ϕN .
2.5.4 Lemma. If M, N are projective étale ϕ-modules over R then we have anatural identification HomR,ϕ (M, N ) = (M ∨ ⊗R N )ϕ=1 .
Proof. We have HomR (M, N ) = M ∨ ⊗R N . Given f ∈ HomR (M, N ), regarded
as an element of P := M ∨ ⊗R N , we have f ∈ P ϕ=1 if and only if ΦP (1 ⊗f ) = f , and by the definition of the ϕ-structure on P , this is equivalent to fintertwining ΦM and ΦN , as required.

2.6

Frobenius descent

Suppose that A is as in Situation 2.2.15, and that we have a continuous ϕequivariant embedding A ,→ W (C[ ). Let A be a finite type Z/pa -algebra forsome a ≥ 1. Assume that the induced map AA → W (C[ )A is a faithfully flat
injection. Since ϕ is bijective on W (C[ ), we have an increasing unionAA ⊂ ϕ−1 (AA ) ⊂ ϕ−2 (AA ) ⊂ · · · ⊂ W (C[ )A .
e A be the closure of ∪n≥0 ϕ−n (AA ) in W (C[ )A , and we set A
e + :=We let AA
e A ∩Ainf,A . Note that ϕ extends to a bijection on A
e A , which induces a bijection
A
e +.
on A
A

2.6.1 Remark. We will apply the results of this section in the setting introducedin Section 2.1, taking A = AK or A = S. In either case it follows from
Proposition 2.2.12 that AA → W (C[ )A is a faithfully flat injection, and it
eA = Ae K,A ,
follows easily from Theorem 2.1.6 that in the former case we have A
eA = OeE,A .
and in the latter case we have A

2.6. FROBENIUS DESCENT

47

+
e+
2.6.2 Remark. If A+
A is not ϕ-stable, then AA is not a subring of AA , and in+e .
particular it is not equal to AA ∩ A
A
We will make use of the following variant of Lemma 2.2.16; in view of Ree + -modules.
mark 2.6.2 we cannot apply Lemma 2.2.16 in the setting of AA

2.6.3 Lemma. There is an integer r ≥ 0 such that for all s ≥ r we have T s ∈e + , and an integer C ≥ 0 such that for all n ≥ 1, we have ϕ(T n ) ∈ T pn−C A
e +.
A
A
A
Proof. Since T is topologically nilpotent in A+
A it is topologically nilpotent
e A , so in particular T s ∈ A
e + for all s sufficiently large. For the secondin AAstatement, we follow the proof of Lemma 2.2.16, and write ϕ(T ) = T p + pY forsome Y ∈ AA , so that ϕ(T n ) = (T p + pY )n . If we expand using the binomial
theorem, and recall that pa = 0, then we see that ϕ(T n ) − T np is a sum ofterms of the form T p(n−r) Y r for some 0 ≤ r ≤ a − 1. It therefore suffices to
e + for 0 ≤ r ≤ a − 1, which we can do by thechoose C such that T C T −pr Y r ∈ A
Atopological nilpotence of T .
e + need not contain A+ , and2.6.4 Remark. As already noted in Remark 2.6.2, AAAee+so, although AA = A+A [1/T ], it doesn’t make sense to write AA = AA [1/T ].
e + for some r > 0, andOn the other hand, Lemma 2.6.3 does ensure that T r ∈ A
Athen it does make sense to write, and is true (since T is topologically nilpotent
e A , while A
e + is open in Ae A ), that A
eA = Ae + [1/T r ].
in A
A
A
f :=
Given an étale ϕ-module M over AA , we may form its perfection Mee
AA ⊗AA M , which is an étale ϕ-module over AA .

2.6.5

Descending morphisms

2.6.6 Proposition. Let M, N be projective étale ϕ-modules. Then the naturalf e
map HomAA ,ϕ (M, N ) → HomAe A ,ϕ (M , N ) is a bijection.
Proof. Write P := M ∨ ⊗ N , so that P is a projective étale ϕ-module over AA .
Then by Lemma 2.5.4, we are reduced to checking that the natural morphismof A-modules P ϕ=1 → Peϕ=1 is an isomorphism. It is certainly injective, so weneed only show that it is surjective.
Since the formation of ϕ-invariants is compatible with direct sums, it follows
from [EG21, Lem. 5.2.14] that it is enough to consider the case that P is infact free. By Lemma 2.6.3 (cf. Remark 2.6.4) we can choose a ϕ-stable free
e of Pe, which generates Pe over AA (indeed, choose any basis
e + -submodule PAA
e be the submoduleof Pe, multiply by a sufficiently large power of T , and let P
generated by this scaled basis).
Consider an element x ∈ Peϕ=1 . Choose an integer r ≥ 1 such that if s ≥ re + ) ⊆ T s+1 A
e + (such an r exists by Lemma 2.6.3). We may write
then ϕ(T s AAAe and x1 ∈ ϕ−n (AA )⊗A P for some n ≥ 0. Choose nx = x1 +x2 where x2 ∈ T r P
A
to be minimal with this property. If n > 0, then since x = ϕ(x) = ϕ(x1 )+ϕ(x2 ),e we see that we may replace x1 with ϕ(x1 ) ∈ ϕ1−n (AA )⊗A P ,and ϕ(x2 ) ∈ T r P,A
a contradiction.

48

CHAPTER 2. RINGS AND COEFFICIENTSe ∩ P . The sum
Thus x1 ∈ P , and we have x1 − ϕ(x1 ) = ϕ(x2 ) − x2 ∈ T r P
P∞ i
rex := x1 + i=0 ϕ (ϕ(x1 ) − x1 ) converges to an element of T P, and the sum
therefore converges in P . By definition we have ϕ(x0 ) = x0 . We have x − x1 ,e so x − x0 ∈ T r P;
e then x − x0 = ϕ(x − x0 ) ∈ T r+1 P,
e and iteratingx0 − x1 ∈ T r P,
0
gives x = x ∈ P , as required.
0

2.6.7

Descending objects

e A arises as the perWe will show that every projective étale ϕ-module over A
fection of an étale ϕ-module over AA . We first note the following analoguee A.
of [EG21, Lem. 5.2.14] for étale ϕ-modules over A
e A , then M is a direct
2.6.8 Lemma. If M is a projective étale ϕ-module over A
e A.
summand of a free étale ϕ-module over A
Proof. This can be proved in an identical fashion to [EG21, Lem. 5.2.14].
e A . Then there
2.6.9 Proposition. Let M be a projective étale ϕ-module over A∼ f
is a projective étale ϕ-module M0 over AA and an isomorphism M −→ M
0.
e A -module. Let X
Proof. Suppose firstly that M is free of some rank d as an A
denote the matrix of ϕ with respect to a choice of basis e1 , . . . , ed . As in the
proof of Proposition 2.6.6, after possibly scaling the ei by powers of T , we may
e + . Since ΦM is an isomorphism, it follows fromassume that X has entries in AAe + ).
Remark 2.6.4) that there is an integer h ≥ 0 such that X −1 ∈ T −h Md (AA+e ) ∩ GLd (Ae A ), the new matrix
If we change basis via a matrix Y ∈ Md (AA
for ϕ is ϕ(Y )XY −1 . It suffices to show that we can choose Y so that this matrix
has entries in AA (as we can then let M0 be the AA -span of this new basis).
Fix some H > d(C + h + 1)/(p − 1)e, where C is as in Lemma 2.6.3, ande + ) and X 00 ∈ Md (ϕ−n (AA ))write X = X 0 + X 00 where X 0 ∈ T H+h Md (AA
for some sufficiently large n. (The reason for this choice of H is to be able
to apply Lemma 2.6.10 below.) Taking Y = X, we may replace X, X 0 , X 00
by ϕ(X), ϕ(X 0 ), ϕ(X 00 ) respectively (note that by Lemma 2.6.3, we have ϕ(X 0 ) ∈e + ), because p(H + h) − C ≥ H + h by our choice of C), which has
T H+h Md (AA
the effect of replacing n by (n − 1). Iterating this procedure, we can assume
that n = 0, so X 00 ∈ Md (AA ). By Lemma 2.6.10 below, we can find Y suchthat ϕ(Y )XY −1 = X 00 , completing the proof in the case that M is free.
We now return to the general case in which M is only assumed finitely
generated projective, rather than free. By Lemma 2.6.8, we may write M ase A . By the case alreadya direct summand of a free étale ϕ-module F over A∼ eproved, we may write F −→ F0 for some free étale ϕ-module F0 . By Proposition 2.6.6, the idempotent in End(F ) corresponding to M comes from an idempotent in End(F0 ), and we may take M0 to be the étale ϕ-module corresponding
to this idempotent.
The following lemma and its proof are based on [PR09, Prop. 2.2].

2.7. (ϕ, Γ)–MODULES

49e + ) ∩ GLd (Ae A ), and that X −1 ∈2.6.10 Lemma. Suppose that X ∈ Md (A
A
+
−h
00
e ) for some h ≥ 0. Suppose that X ∈ Md (Ae + ) ∩ GLd (Ae A ) is such
T Md ( A
A
A
+
−1 00
d(C+h+1)/(p−1)eethat X X ∈ 1 + T
Md (AA ), where C is as in Lemma 2.6.3.
e + ) ∩ GLd (Ae A ) with X 00 = ϕ(Y )XY −1 .
Then there exists Y ∈ Md (AA
Proof. Define sequences Xi , hi by X0 = X, h0 = (X 00 )−1 X, and for each i ≥ 1,
00 −1
Xi = ϕ(hi−1 )Xi−1 h−1
Xi . Then if we set yi = hi hi−1 · · · h0 , we
i−1 , hi = (X )−1−1have X 00 = Xi hi = ϕ(yi−1 )Xyi for each i. We claim that the yi tend to alimit Y as i → ∞; then we have X 00 = ϕ(Y )XY −1 , as required.
To see that the yi tend to a limit, it is enough to show that hi → 1 as i → ∞.
e + ) for some s ≥ (C + h + 1)/(p − 1).
To see this, suppose that hi ∈ 1 + T s Md (AA
We have
X 00 hi+1 = Xi+1 = ϕ(hi )Xi h−1= ϕ(hi )X 00 ,iso that hi+1 = (X 00 )−1 ϕ(hi )X 00 . Using Lemma 2.6.3 and the assumption thate + ), we see that hi+1 ∈ 1 + T ps−C−h Md (Ae + ). Since ps −(X 00 )−1 ∈ T −h Md (AAA
C − h ≥ s + 1, we are done.

2.6.11

An equivalence of categories

We summarise the results of this section in the following proposition.
2.6.12 Proposition. Let A be a finite type Z/pa -algebra for some a ≥ 1.
f is an equivalence of categories from the category ofThen the functor M 7→ M
projective étale ϕ-modules over AA to the category of projective étale ϕ-modules
e A.
over A
Proof. The functor is essentially surjective by Proposition 2.6.9, and fully faithful by Proposition 2.6.6.

2.7(ϕ, Γ)–modules

By definition, an étale (ϕ, ΓK )-module is a finitely generated AK -module M ,equipped with• a ϕ-linear morphism ϕ : M → M with the property that the corresponding morphism ΦM : ϕ∗ M → M is an isomorphism (i.e. M is given the
structure of an étale ϕ-module over AK ), and• a continuous semi-linear action of ΓK that commutes with ϕ.

2.7.1

The relationship with Galois representations

There is an equivalence of categories between the category of continuous representations of GK on finite Zp -modules, and the category of étale (ϕ, ΓK )b ur
modules, which is given by functors D and T that are defined as follows. Let A
K
denote the p-adic completion of the ring of integers of the maximal unramified

50

CHAPTER 2. RINGS AND COEFFICIENTS

extension of AK [1/p] in W (C[ )[1/p]; this is preserved by the natural actionsof ϕ and GK on W (C[ )[1/p]. Then for a GK -representation T , we defineb ur ⊗Z T )GKcyc ,D(T ) := (AKpwhile for an étale (ϕ, ΓK )-module M we defineb ur ⊗A M )ϕ=1 .
T (M ) := (AK
K
The action of GKcyc (resp. ϕ) in the definition of D (resp. of T ) is the diagonal one (with the GK -action on M being that inflated from the actione K )-modules, and takof ΓK ). There is a completely analogous theory of (ϕ, Γe K )-modules
ing ∆K -invariants gives an equivalence of categories between (ϕ, Γand (ϕ, ΓK )-modules.
The functor T (M ) is also defined for finitely generated étale ϕ-modules
over AK (i.e. in the absence of a Γ-action on M ), and in this context yields anequivalence of categories between finitely generated étale ϕ-modules over AK
and continuous GKcyc -representations on finite Zp -modules.
Suppose now that L/K is a finite extension, and suppose further that K =Kcyc ∩ L (or, equivalently, that the natural embedding ΓL ,→ ΓK induced by
the inclusion Kcyc ⊆ Lcyc is an isomorphism). We will recall the description ofK
the functor IndG
GL in terms of (ϕ, Γ)-modules. (We will use this description in
Section 7.3.)
b ur , which are unramified
By construction, there are inclusions AK ⊆ AL ⊆ A
Kembeddings of DVRs. The composite of these embeddings is GK -equivariant(where GK -acts on AK through its quotient ΓK ), while the second is GL equivariant (where GL -acts on AL through its quotient ΓL ). Regarding thesecond of these embeddings as an AK -linear morphism of AK -modules, it inb ur -linear surjection Ab ur ⊗A AL → A
b ur . The source of this surjecduces an AKKKKtion has a diagonal action of GK (the GK -action on AL being via its quotient∼ΓK −→ ΓL ), while the surjection itself is GL -equivariant. Thus it induces a
GK -equivariant morphismb ur ⊗A AL → IndGK A
b ur ,A
K
K
K
GL
which is easily seen to be an isomorphism. Using the description of the induction
as a tensor product, we can express this as an isomorphism∼b ur ⊗A AL −→b ur .
A
Zp [GK ] ⊗Zp [GL ] A
K
KK(2.7.2)

If M is an étale (ϕ, Γ)-module over AL , then we let M 0 denote M regarded
as an étale (ϕ, Γ)-module over the subring AK of AL . (Recall again that we are∼assuming ΓL −→ ΓK .) The isomorphism (2.7.2) then yields an isomorphismb ur ⊗A M 0 )ϕ=1 = (Ab ur ⊗A AL ) ⊗A M ϕ=1
T (M 0 ) := (A
K
K
K
K
L


b ur ⊗A M ϕ=1 = Zp [GK ] ⊗Z [G ] (Ab ur ⊗A M ϕ=1= (Zp [GK ] ⊗Zp [GL ] A
K
K
L
L
p
L
K
= IndGGL T (M ). (2.7.3)

2.7. (ϕ, Γ)–MODULES

51

In short, induction of Galois representations corresponds to the restriction ofcoefficients on the (ϕ, Γ)-module side.
The preceding discussion also applies in the context of étale ϕ-modules (in∼the sense of Definition 2.5.1). Our assumption that ΓL −→ ΓK implies that
GKcyc /GLcyc → GK /GL is a bijection, so that (2.7.2) may also be interpreted
as an isomorphism∼b ur ⊗A AL −→b ur .
A
Zp [GKcyc ] ⊗Zp [GLcyc ] A
K
KK

Then, if M is a finitely generated étale ϕ-module over AL , and if M 0 denotes its
restriction to an étale ϕ-module over AK , the preceding isomorphism induces
an isomorphism
GKcyc
T (M ).
(2.7.4)
T (M 0 ) = IndGLcyc
We also need a variant of the preceding theory which again follows from theresults of [Fon90], using the Kummer extension K∞ /K introduced in Example 2.1.3 and Section 2.1.14; the integral version of this theory was first studied
by Breuil and Kisin, see [Kis09b, §1]. Namely, there is an equivalence of categories between the category of continuous representations of GK∞ on finite Zp algebras, and the category of étale ϕ-modules over OE , which is given by functors
D∞ , T∞ that are defined as follows: Let OEd
ur denote the p-adic completion of thering of integers in the maximal unramified extension of Frac(OE ) in W (C[ )[1/p];
this is preserved by the natural actions of ϕ and GK∞ on W (C[ )[1/p]. We defineGK∞,
D∞ (T ) := (OEdur ⊗Zp T )for a GK∞ -representation T , and define
ϕ=1
T∞ (M ) := (OEd,ur ⊗AK M )for an étale ϕ-module M .

2.7.5

Various types of ϕ-modules with coefficients

We now introduce coefficients.
2.7.6 Definition. Let A be a p-adically complete Zp -algebra. A projective étale(ϕ, ΓK )-module of rank d with A-coefficients is a projective étale ϕ-module Mof rank d over AK,A equipped with a semi-linear action of ΓK , which commutes with ϕ, and which is furthermore continuous when M is endowed with
its canonical topology (i.e. the topology of Remark D.2).
If A is a finite type Z/pa -algebra, we can use Theorem 2.4.1 to give a usefulalternative description of étale (ϕ, ΓK )-modules with A-coefficients in terms ofétale ϕ-modules with coefficients in the ring W (C[ )A , as we now explain.
2.7.7 Definition. Let A be a p-adically complete Zp -algebra. An étale (ϕ, GK )module with A-coefficients (resp. an étale (ϕ, GKcyc )-module with A-coefficients,

52

CHAPTER 2. RINGS AND COEFFICIENTS

resp. an étale (ϕ, GK∞ )-module with A-coefficients) is by definition a finitely
generated W (C[ )A -module M equipped with an isomorphism∼

ϕM : ϕ∗ M −→ M
of W (C[ )A -modules, and a W (C[ )A -semi-linear action of GK (resp. GKcyc , resp.
GK∞ ), which is continuous and commutes with ϕM . We say that M is projective
if it is projective of constant rank as a W (C[ )A -module.
If A is a finite type Z/pa -algebra, then there is a functor from the categoryof finite projective étale (ϕ, ΓK )-modules with A-coefficients to the category offinite projective étale (ϕ, GK )-modules with A-coefficients, which takes an étale(ϕ, ΓK )-module M to W (C[ )A ⊗AK,A M , endowed with the extension of scalarsof ϕ, and the diagonal action of GK , with the action of GK on M being theaction inflated from ΓK .
Similarly, there is a functor from the category of finite projective étale ϕmodules over OE,A to the category of finite projective étale (ϕ, GK∞ )-modules
with A-coefficients, which takes an étale ϕ-module M to W (C[ )A ⊗OE,A M ,endowed with the extension of scalars of ϕ, and the diagonal action of GK∞ ,with the action of GK∞ on M being the trivial action.
2.7.8 Proposition. Let A be a finite type Z/pa -algebra for some a ≥ 1.
1. The functor M 7→ W (C[ )A ⊗AK,A M is an equivalence between the categoryof finite projective étale ϕ-modules over AK,A and the category of finite
projective (ϕ, GKcyc )-modules with A-coefficients.
It induces an equivalence of categories between the category of finite projective étale (ϕ, ΓK )-modules with A-coefficients and the category of finite
projective étale (ϕ, GK )-modules with A-coefficients.
A quasi-inverse functor is given by the composite of N 7→ N GKcyc and aquasi-inverse to the functor of Proposition 2.6.12.
2. The functor M 7→ W (C[ )A ⊗OE,A M is an equivalence of categories between
the category of finite projective étale ϕ-modules over OE,A and the categoryof finite projective étale (ϕ, GK∞ )-modules with A-coefficients. A quasiinverse functor is given by N 7→ N GK∞ and a quasi-inverse to the functorof Proposition 2.6.12.
Proof. This is a formal consequence of Theorem 2.4.1 and Proposition 2.6.12.
We begin with (2). Firstly, by Proposition 2.6.12 (and Remark 2.6.1) the functor
f=O
eE,A ⊗O M is an equivalence of categories between the category of
M 7→ M
E,A
finite projective étale ϕ-modules over OE,A and the category of finite projective
eE,A , so it is enough to show that M
f 7→ W (C[ )A ⊗ eétale ϕ-modules over O
OE,A
f is an equivalence of categories. By Example 2.1.3 and Remark 2.1.4, Kb∞
M
is a perfectoid field, and we have a canonical identification of Galois groups
GKb ∞ = GK∞ . The result then follows from the equivalence of categories given2.7. (ϕ, Γ)–MODULES

53

eE,A -modules
by Theorem 2.4.1, as we can think of ϕ as being an isomorphism of O∼
f −→ Mf.
ϕ∗ M
The same argument shows in (1) that the functor M 7→ W (C[ )A ⊗AK,A M isan equivalence of categories between the category of projective étale ϕ-modules
over AK,A , and the category of projective étale (ϕ, GKcyc )-modules with Acoefficients. Since ΓK = GK /GKcyc , this extends to the claimed equivalence of
categories, noting that by construction the continuity of the ΓK -action on M
is equivalent to the continuity of the GK -action on W (C[ )A ⊗AK,A M (sinceLemma 2.2.18 shows that the GK -action on W (C[ )A is continuous).

54

CHAPTER 2. RINGS AND COEFFICIENTSChapter 3

Moduli stacks of ϕ-modules
and (ϕ, Γ)-modules
In this chapter we build on the results of [EG21] (which in turn built on [PR09])
to construct our stacks of (ϕ, Γ)-modules, and various related stacks of ϕmodules. We show in particular that our stack XK,d of (ϕ, Γ)-modules is Indalgebraic.

3.1

Moduli stacks of ϕ-modules

In this section we put ourselves in the context of Situation 2.2.15; we also
remind the reader that the notion of étale ϕ-module is defined in Section 2.5.
We furthermore fix a finite extension E/Qp with ring of integers O and residue
field F; all of our coefficient rings from now on will be O-algebras. All of ourconstructions are compatible with replacing E by a finite extension, and we will
typically not comment on this, although see Remark 3.1.2 below.
If we fix integers a, d ≥ 1, then we may follow [EG21, §5] and define an fpqc
stack in groupoids Rad over Spec O/$a as follows: For any O/$a -algebra A,
we define Rad (A) to be the groupoid of étale ϕ-modules over AA which areprojective of rank d. If A → B is a morphism of O-algebras, and M is anobject of Rad (A), then the pull-back of M to Rad (B) is defined to be the tensor
product AB ⊗AA M .
A key point is that this definition does not require A+
A to be ϕ-stable, although (as far as we know) this hypothesis is required to make any deductions
about Rad beyond the fact that it is an fpqc stack (which relies just on Drinfeld’s
general descent results, as described in [Dri06] and [EG21, §5.1]).
Since we are ultimately interested in questions of algebraicity or of Indalgebraicity, from now on we regard Rad as an fppf stack over O/$a . By [Sta,Tag 04WV], we may also regard the stack Rad as an fppf stack over O, and as
a varies, we may form the 2-colimit R := lima Rad , which is again an fppf stack
−→
55

56

CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES

over O. In fact Rd lies over Spf O := lima Spec O/$a , the formal spectrum of−→O with respect to the $-adic, or equivalently p-adic, topology.
We now fix a polynomial F ∈ W (k)[T ] which is congruent to a positive
power of T modulo p (for example, an Eisenstein polynomial).
3.1.1 Definition. Suppose that A+
A is ϕ-stable. Let h be a non-negative integer, and let A be a p-adically complete O-algebra. A ϕ-module of F -height at
most h over A+
A is a pair (M, ϕM ) consisting of a finitely generated T -torsion
free A+-moduleM, and a ϕ-semi-linear map ϕM : M → M, with the furtherA
properties that if we writeΦM := 1 ⊗ ϕM : ϕ∗ M → M,then ΦM is injective, and the cokernel of ΦM is killed by F h .
A ϕ-module of finite F -height over A+
A is a ϕ-module of F -height at most hfor some h ≥ 0. A morphism of ϕ-modules is a morphism of the underlyingA+
A -modules which commutes with the morphisms ΦM .
We say that a ϕ-module of finite F -height is projective of rank d if it is a
finitely generated projective A+
A -module of constant rank d.
If we maintain the assumption that A+ is ϕ-stable, and if we fix integers
a, d ≥ 1 and an integer h ≥ 0, then we may again follow [EG21, §5] to defineaan fpqc stack in groupoids Cd,hover Spec O/$a as follows: For any O/$a a
algebra A, we define Cd,h (A) to be the groupoid of ϕ-modules of F -height at
most h over A+
A which are projective of rank d. If A → B is a morphism ofaa
Zp -algebras, and M is an object of Cd,h(A) then the pull-back of M to Cd,h(B)++
is defined to be the tensor product AB ⊗A M.
Aa
Just as for the stack Rad , we may and do also regard the stack Cd,hasa,an fppf stack over O, and we then, allowing a to vary, define Cd,h := lima Cd,h−→obtaining an fppf stack over O which in fact lies over Spf O. There are canonicalamorphisms Cd,h→ Rad and Cd,h → Rd given by tensoring with AA over A+A.
3.1.2 Remark. If E 0 /E is a finite extension with ring of integers O0 , then bydefinition we have (with obvious notation) Cd,h,O0 = Cd,h ×O O0 (in the casethat A+ is ϕ-stable, so that these stacks are defined) and Rd,O0 = Rd,O ×O O0(in general).
The following lemma provides a concrete interpretation of the A-valued
points of the stacks we have defined, when A is a $-adically complete O-algebra(rather than just an algebra over some O/$a ).
3.1.3 Lemma. If A is a $-adically complete O-algebra, then there is a canonical equivalence between the groupoid of morphisms Spf A → Rd and the groupoidof rank d étale ϕ-modules over AA . If A+
A is furthermore ϕ-stable, then there
is a canonical equivalence between the groupoid of morphisms Spf A → Cd,h andthe groupoid of ϕ-modules of rank d and F -height at most h over A+A.
Proof. This is immediate from Lemma D.5.

3.2. MODULI STACKS OF (ϕ, Γ)-MODULES

57

We now apply the results of [EG21, §5] to deduce various results about thestacks we have introduced. This requires the assumption that A+ is ϕ-stable.
3.1.4 Theorem. Suppose that A+ is ϕ-stable, and let a ≥ 1 be arbitrary.
a
1. The stack Cd,his an algebraic stack of finite presentation over Spec O/$a ,with affine diagonal.
a
2. The morphism Cd,h→ Rad is representable by algebraic spaces, proper, andof finite presentation.

3. The diagonal morphism ∆ : Rad → Rad ×O/$a Rad is representable by algebraic spaces, affine, and of finite presentation.
4. Rad is a limit preserving Ind-algebraic stack, whose diagonal is representable by algebraic spaces, affine, and of finite presentation.
Proof. Part (1) is [EG21, Thm. 5.4.9 (1)], and parts (2), (3) and (4) are provedin [EG21, Thm. 5.4.11], except for the claim that Rad is Ind-algebraic, whichis [EG21, Thm. 5.4.20].
3.1.5 Corollary. Suppose that A+ is ϕ-stable.
1. Cd,h is a p-adic formal algebraic stack of finite presentation over Spf O,
with affine diagonal.
2. Rd is a limit preserving Ind-algebraic stack, whose diagonal is representable by algebraic spaces, affine, and of finite presentation.
3. The morphism Cd,h → Rd is representable by algebraic spaces, proper, andof finite presentation.
4. The diagonal morphism ∆ : Rd → Rd ×Spf O Rd is representable by algebraic spaces, affine, and of finite presentation.
Proof. The first part is immediate from Theorem 3.1.4 (1) and Proposition A.13.
Everything else is immediate from Theorem 3.1.4.

3.2

Moduli stacks of (ϕ, Γ)-modules

In this section we begin the study of our main objects of interest, namely themoduli stacks of étale (ϕ, Γ)-modules. As in Chapter 2 we fix a finite extension
K/Qp . As in Section 3.1, we also fix a finite extension E of Qp with ring ofintegers O, which will serve as our ring of coefficients. As always, k denotes theresidue field of the ring of integers of K, and F denotes the residue field of O.
3.2.1 Definition. We let XK,d denote the moduli stack of projective étale(ϕ, ΓK )-modules of rank d. More precisely, if A is a p-adically complete Oalgebra, then we define XK,d (A) (i.e., the groupoid of morphisms Spf A →58

CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES

XK,d ) to be the groupoid of projective étale (ϕ, ΓK )-modules of rank d with Acoefficients, in the sense of Definition 2.7.6, with morphisms given by isomorphisms. If A → B is a morphism of complete O-algebras, and M is an objectof XK,d (A), then the pull-back of M to XK,d (B) is defined to be the tensor
product AK,B ⊗AK,A M .
It follows from the results of [Dri06], and more specifically from [EG21, Thm.
5.1.16], that XK,d is an fpqc stack over O. As in Remark 3.1.2, the definitionof XK,d behaves naturally with respect to change of the coefficient ring O.
One of the main results of this book is that XK,d is a Noetherian formal
algebraic stack. However, the proof of this is quite involved, and will only befully achieved at the conclusion of Chapter 5. In this section and the two that
follow it, we establish the preliminary result that XK,d is an Ind-algebraic stack.
We begin by discussing the moduli stacks of étale ϕ-modules over AK,A ,which will play an auxiliary role in our study of XK,d .
3.2.2 Definition. We let RK,d denote the moduli stack of rank d projectiveétale ϕ-modules, defined as in Section 3.1, taking A to be AK .
If A+
K is ϕ-stable, then Corollary 3.1.5 applies to RK,d ; this is in particular
the case if K/Qp is basic in the sense of Definition 2.1.12. Our first task is toestablish the same results for general K, which we will do by reducing to thebasic case.
3.2.3 Definition. If K/Qp is any finite extension, we set K basic := K ∩K0 (ζp∞ ).
By definition, K basic is basic, and (K basic )0 = K0 . Note that the naturaleK → Γ
e K basic is an isomorphism, and induces an isomorphism
restriction map ΓΓK → ΓK basic . By [Win83, Thm. 3.1.2], AK is a free AK basic -module of rank [K :K basic ] = [K(ζp∞ ) : K0 (ζp∞ )], and the inclusion AK basic ⊂ AK is ϕ-equivariant,so there is a natural morphism RK,d → RK basic ,d[K:K basic ] given by forgetting
the AK,A -algebra structure on an étale ϕ-module.
3.2.4 Remark. As noted in Remark 2.1.13, in order to ensure that A+
K is ϕstable, it suffices for K to be abelian over Qp . Thus, rather than relating thetheory for K to that for the field K basic introduced above, we could just as wellrelate it to any other subfield K 0 of K which is abelian over Qp , and for which
the natural map ΓK → ΓK 0 is an isomorphism; e.g. the field K 0 := K ∩ Qabp(where Qabp denotes the maximal abelian extension of K). It doesn’t matter(for our purposes) which particular K 0 we choose; K basic is simply a convenient
choice.
3.2.5 Lemma. The morphism RK,d → RK basic ,d[K:K basic ] is representable byalgebraic spaces, affine, and of finite presentation.
Proof. We can prove the statement after pulling back via a morphism Spec A →RK basic ,d[K:K basic ] , where A is an O/$a -algebra for some a ≥ 1. This morphism
corresponds to a projective étale ϕ-module over AK basic ,A of rank d[K : K basic ],3.2. MODULI STACKS OF (ϕ, Γ)-MODULES

59

and we need to show that the functor on A-algebras taking B to the set ofprojective étale ϕ-modules over AK,B of rank d, whose underlying étale ϕmodule over AK basic ,B coincides with MB , is representable by an affine schemeof finite presentation over Spec A.
Note that since MB is projective and in particular flat over AK basic ,B , wehave a natural inclusion i : MB ,→ AK,B ⊗AK basic ,B MB . The additional structure needed to make MB into a projective étale ϕ-module over AK,B is the dataof a morphism of étale ϕ-modules over AK basic ,B
f : AK,B ⊗AK basic ,B MB → MBsatisfying the conditions thatfi

1. the composite MB → AK,B ⊗AK basic ,B MB → MB is the identity morphism, and2. the kernel of f is AK,B -stable.
(We can then define the AK,B -module structure on MB via the formula λ · m :=f (λ ⊗ m). The first condition guarantees that this action is compatible withthe existing AK basic ,B -module structure on MB , and the second condition that
(λ1 λ2 ) · m = λ1 · (λ2 · m).)By [EG21, Prop. 5.4.8], the data of a morphism of étale ϕ-modules f :AK,B ⊗AK basic ,B MB → MB is representable by an affine scheme of finite presentation over Spec A, so it is enough to show that conditions (1) and (2) areclosed conditions, given by finitely many equations. To see this, we follow the
proof of [EG21, Prop. 5.4.8]. Exactly as in that argument, we can reduce tothe case that MB is free, and after choosing bases, any f is determined by thecoefficients of finitely many powers of T in the Laurent series expansions of theentries of the matrix given by f . Condition (1) is then evidently given by finitely
many equations in these coefficients.
To see that the same is true of condition (2), note that since MB is projective
and condition (1) implies in particular that f is surjective, we have a splitting
AK,B ⊗AK basic ,B MB = MB ⊕ ker(f ). The projection onto ker(f ) is given by(1 − i ◦ f ), so the condition that ker(f ) is AK,B -stable is the condition that forany λ in AK,B , and any m ∈ MB , we havef (λ(m − i(f (m)))) = 0.
This is evidently a closed condition, and since AK,B and MB are both finitely
generated AK basic ,B -modules, it is determined by finitely many equations, as
required.
3.2.6 Corollary. The stack RK,d is a limit preserving Ind-algebraic stack,whose diagonal is representable by algebraic spaces, affine, and of finite presentation.
Proof. This follows from Lemma 3.2.5, Corollary 3.1.5 (which establishes theclaimed properties for RK basic ,d[K:K basic ] ), and Corollary 3.2.9 below.

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The following series of results concerning morphisms of stacks culminates inCorollary 3.2.9, which was used in the proof of Corollary 3.2.6.
3.2.7 Lemma. Let X → Y be a morphism of stacks over a base scheme S,
which is representable by algebraic spaces. As usual, let ∆f : X → X ×Y X
denote the diagonal of f . If f is of finite type and quasi-separated, then ∆f isof finite presentation.
Proof. This can be checked after pulling back along an arbitrary morphism
T → Y, where T is a scheme, and hence reduced to the case of a morphism froman algebraic space to a scheme. In this case, the claim of the lemma is proved
in [Sta, Tag 084P].
3.2.8 Lemma. Let f : X → Y be a morphism of stacks over a base scheme S
which is representable by algebraic spaces, has affine diagonal, and is of finite
type. Suppose that the diagonal of Y is representable by algebraic spaces, affine,and of finite presentation. Then the diagonal of X is also representable byalgebraic spaces, affine, and of finite presentation.
Proof. We may factor the diagonal of X as
X → X ×Y X → X ×S X .
Since the morphism X ×Y X → X ×S X is pulled back from the diagonal
Y → Y ×S Y, it is enough to show that the relative diagonal ∆f : X → X ×Y X
is representable by algebraic spaces, affine, and of finite presentation.
Now ∆f is representable by algebraic spaces (since f is), and affine (byassumption). Since affine morphisms are quasi-compact, we see that f is quasiseparated, as well as being of finite type (by assumption); thus ∆f is of finite
presentation (by Lemma 3.2.7).
3.2.9 Corollary. Let X → Y be a morphism of stacks which is representableby algebraic spaces, affine, and of finite presentation. If Y is a limit preserving
Ind-algebraic stack, whose diagonal is representable by algebraic spaces, affine,and of finite presentation, then the same is true of X .
Proof. The claimed limit preserving property of X follows, by [EG21, Lem. 2.3.20 (3)],from that of Y and the fact that X → Y is of finite presentation, while theclaimed Ind-algebraic property of X follows from that of Y and the fact that
X → Y is representable by algebraic spaces. Finally, the claimed propertiesof the diagonal of X follow from those of Y by Lemma 3.2.8. (Note that byassumption the hypotheses of Lemma 3.2.8 hold, since morphisms of finite presentation are in particular of finite type, while affine morphisms are separated,
and thus have their diagonals being closed immersions, which in particular areagain affine.)
We now give a concrete description of RaK,d as an Ind-algebraic stack which
will be useful in Chapter 7. In order to state it, we introduce the notaaation CK
classifying rank d probasic ,d[K:K basic ],h for the stack over Spec O/$3.2. MODULI STACKS OF (ϕ, Γ)-MODULES

61

jective ϕ-modules over A+of T -height at most h; by Theorem 3.1.4,
K basic ,A
a
CK
is
an
algebraicstackof finite presentation over Spec O/$a .
basic ,d[K:K basic ],hWe begin with a lemma which is a variant of one of the steps appearing in
the proof of [EG21, Thm. 5.4.20].
3.2.10 Lemma. If T → RaK,d is a morphism whose source is a Noetherian
scheme, then there is a Noetherian scheme Z and a scheme-theoretically dominant and surjective morphism Z → T such that the composite Z → T →a
RaK,d → RaK basic ,d[K:K basic ] can be lifted to a morphism Z → CK
basic ,d[K:K basic ],hfor some sufficiently large value of h.
Proof. Since T is Noetherian, is is quasi-compact, and so admits a schemetheoretically dominant surjection from a Noetherian affine scheme (e.g. the disjoint union of the members of a finite cover of T by affine open subsets). Thus
we are reduced to the affine case.
Since T is affine, by [EG21, Prop. 5.4.7], we may find a scheme-theoreticallydominant surjection Z = Spec A → T such that the étale ϕ-module M overA corresponding to the composite Z → T → RK,d is free (of rank d) over
AK,A (rather than merely projective). Thus M is also free (of rank d[K :K basic ]) over AK basic ,A . We may then choose a ϕ-stable AK basic ,A -basis of M .
If we let M denote the A+-module spanned by this basis, then M is
K basic ,Aϕ-invariant, and has some height h. Thus M induces the desired morphisma
Z → CK
basic ,d[K:K basic ],h .
We now give the promised description of the Ind-algebraic stack structure
on RK,d .
3.2.11 Lemma. Fix a ≥ 1, and for each h ≥ 0, let RaK,d,K basic ,h denote thescheme-theoretic image of the base-changed morphisma
Cd[K:K
RK,d → RK,d ,basic ],h ×R basic
K,d[K:K basic ]so that RaK,d,K basic ,h is a closed algebraic substack of RaK,d . Then the canonicalmorphism limh RaK,d,K basic ,h → RaK,d is an isomorphism.
−→
Proof. We have to show that any morphism T → RaK,d whose source is a scheme
factors through the inductive limit. Since RaK,d is limit preserving, we may
assume that T is of finite type over O/$a . Let Z → T be as in the statementof Lemma 3.2.10. Then the composite
Z → T → RaK,d(3.2.12)aRK,d , and hence thelifts to a morphism Z → Cd[K:K
basic ],h ×R basic
K,d[K:K basic ]morphism (3.2.12) factors through RaK,d,K basic ,h . Since Z → T is schemetheoretically dominant, the original morphism T → RaK,d also factors through
RaK,d,K basic ,h , and hence through the inductive limit. Thus the lemma is proved.
(We remark that this argument is essentially identical to that used to prove [EG21,
Thm. 5.4.20].)

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CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES

We now turn to studying ΓK -actions on our ϕ-modules. To ease notation,
write Γ = ΓK from now on. We choose a topological generator γ of Γ, and letΓdisc := hγi; so Γdisc ∼= Z.
3.2.13 Remark. Since Γdisc is dense in Γ, while a projective étale ϕ-module M
is complete with respect to its canonical topology, in order to endow M withthe structure of an étale (ϕ, Γ)-module, it suffices to equip M with a continuous
action of Γdisc (where we equip Γdisc with the topology induced on it by Γ).
In order to study the properties of XK,d we will take advantage of Remark 3.2.13. Accordingly, we now consider the moduli stack of projective étale
ϕ-modules of rank d equipped with a semi-linear action of Γdisc . We don’t introduce particular notation for this stack, since the following proposition identifies
disc
it with a fixed point stack RΓK,d, which we now define.
There is a canonical action of Γdisc on Rd (that is, a canonical morphism
γ : Rd → Rd ): if M is an object of Rd (A), then γ(M ) is given by γ ∗ M :=AK,A ⊗γ,AK,A M . (Note that this is naturally a ϕ-module, because the actionof γ on AK,A commutes with ϕ.) Then we set
RΓd disc := Rd

×∆,Rd ×Rd ,Γγ

Rd ,where ∆ is the diagonal of Rd and Γγ is the graph of γ, so that Γγ (x) = (x, γ(x)).
3.2.14 Proposition. The moduli stack of projective étale ϕ-modules of rank dequipped with a semi-linear action of Γdisc is isomorphic to the fixed point stackRΓd disc .
Proof. Using the usual construction of the 2-fibre product, we see that RΓd disc∼consists of tuples (x, y, α, β), with x, y being objects of Rd , and α : x −→ y∼and β : γ(x) −→ y being isomorphisms. This is equivalent to the category
fibred in groupoids given by pairs (x, ι) consisting of an object x of Rd and∼an isomorphism ι : γ(x) −→ x. Thus an object of RΓd disc (A) is a projectiveétale ϕ-module of rank d with A-coefficients M , together with an isomorphism∼of ϕ-modules ι : γ ∗ M −→ M ; but this isomorphism is precisely the data of asemi-linear action of Γdisc = hγi on M , as required.
Since Rd is an Ind-algebraic stack, so is RΓd disc . More precisely, we have thefollowing lemma.
3.2.15 Lemma. RΓd disc is a limit preserving Ind-algebraic stack, whose diagonal
is representable by algebraic spaces, affine, and of finite presentation.
Proof. The description of RΓd disc as a fibre product shows that the forgetful
morphism RΓd disc → Rd (given by forgetting the Γdisc -action) is representableby algebraic spaces, affine, and of finite presentation, since these properties
hold for the diagonal of Rd , by Corollary 3.2.6. The claim of the lemma is
now seen to follow via another application of Corollary 3.2.6, together with
Corollary 3.2.9.

3.2. MODULI STACKS OF (ϕ, Γ)-MODULES

63

Restricting the Γ-action on an étale (ϕ, Γ)-module to Γdisc (and taking into
account Proposition 3.2.14), we obtain a morphism XK,d → RΓd disc , which by
Remark 3.2.13 is fully faithful. Thus XK,d may be regarded as a substack ofRΓd disc .
As already noted, the first step in proving that XK,d is a Noetherian formal
algebraic stack is to show that it is an Ind-algebraic stack. Although XK,d
is a substack of the Ind-algebraic stack RΓd disc , it is not a closed substack, butshould rather be thought of as a certain formal neighbourhood of Xd,red in RΓd disc(see Remark 7.2.18), and so even this statement will require additional work toprove. We begin with the following lemma, which allows us to reduce to thecase that K is basic.
3.2.16 Lemma. We have a 2-Cartesian diagram
XK,d


disc
RΓK,d

/ XK basic ,d[K:K basic ]/ RΓdisc
basic
K
,d[K:K basic ]where the horizontal arrows are the natural maps (forgetting the AK -module
structure), and the vertical arrows are the monomorphisms given by restricting
the Γ-action to Γdisc .
Proof. Unwinding the definitions, we need to show that if A is an O/$a -algebra
for some a ≥ 1, and M is an étale ϕ-module over AK,A with a semi-linear actionof Γdisc , then the action of Γdisc extends to a continuous action of Γ if and only
if the same is true of M regarded as a module AK basic ,A . Since AK,A is freeof finite rank over AK basic ,A , this is clear (for example, because it follows from
Lemma D.7 that the set of lattices in M regarded as an AK,A -module is cofinal
in the set of lattices in M regarded as an AK basic ,A -module).
As a consequence of Lemmas 3.2.5 and 3.2.16, we can deduce some propertiesof Xd from the corresponding properties of Rd .
3.2.17 Proposition.
1. The morphism XK,d → XK basic ,d[K:K basic ] is representable by algebraic
spaces, affine, and of finite presentation.
2. The diagonal of XK,d is representable by algebraic spaces, affine, and offinite presentation.
Proof. To make the proof easier to read, we write RK for RK,d and RK basic
for RK basic ,d[K:K basic ] . We factor the morphism RΓKdisc → RΓKdisc
basic asRΓKdisc := RK

×∆,RK ×RK ,Γγ

RK → RK→ RK basic

×∆,RK basic ×RK basic ,Γγ

×∆,RK basic ×RK basic ,ΓγRK

RK basic =: RΓKdisc
basic .

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CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES

By Lemma 3.2.5, the morphism RK → RK basic is representable by algebraic
spaces, affine, and of finite presentation. Thus so is the second arrow in thepreceding displayed expression. The first arrow is a base-change of a product of
two copies of the diagonal
RK → RK ×RK basic RK .
As in the proof of Lemma 3.2.8, we deduce from Lemma 3.2.5 that this diagonal
is representable by algebraic spaces, affine, and of finite presentation. Thus so isthe first arrow in the expression, and hence so is the morphism RΓKdisc → RΓKdisc
basic .
It follows from Lemma 3.2.16 that XK,d → XK basic ,d[K:K basic ] is then also
representable by algebraic spaces, affine, and of finite presentation. The claimed
properties of the diagonal of XK,d follow from the corresponding properties ofdisc
the diagonal of RΓK,dproved in Lemma 3.2.15, together with the fact that
Γdisc
XK,d → RK,d is a monomorphism.
From now on we will typically drop K from the notation, simply writing Xd ,Rd and so on. We conclude this section by showing that Xd is limit preserving,
using the material on lattices and continuity developed in Appendix D.
3.2.18 Lemma. We have γ(T ) − T ∈ pAK,A + T 2 A+
K,A . If K is basic thenγ(T ) − T ∈ (p, T )T A+.
K,A+Proof. Given the definitions of AK,A and A+
K,A in terms of AK and AK , itsuffices to prove the lemma for these latter rings (i.e.
P∞in the case when A =Zp ). We then reduce modulo p, and write γ(T ) = i=n ai T i , with ai ∈ k∞
and an 6= 0. Since the action of γ is continuous for the T -adic topology, we see inparticular that for m > 0 sufficiently large, we have γ(T )m = γ(T m ) ∈ T k∞ [[T ]].
s
nm+ . . . , we see that n > 0. Since γ p → 1 as s → ∞,Since γ(T )m = amnTs
we see that n = 1 and that ap1 = 1 for all sufficiently large s, which implies
that a1 = 1.
If K is basic then we have chosen T so that γ(T ) ∈ T A+
K,A , so that γ(T )−T ∈+(p, T )T AK,A by the above.

If K is basic, it follows from Lemma 3.2.18 that (D.24) holds, so that by
Lemma D.27 we can use the material on T -quasi-linear morphisms developedin Appendix D to study the action of γ.
3.2.19 Lemma. Xd is limit preserving.
Proof. By Proposition 3.2.17 it suffices to prove this in the case that K isbasic (since a morphism which is of finite presentation is in particular limit
preserving). Since Xd ,→ RΓd disc is fully faithful, and RΓd disc is limit preserving
by Corollary 3.1.5, it suffices to prove that Xd is limit preserving on objects.
Since RΓd disc is limit preserving on objects, we are reduced to showing that
if T = limi Ti is a limit of affine schemes, and Ti0 → RΓd disc is a morphism with←−the property that the composite
T → Ti0 → RΓd disc

3.3. WEAK WACH MODULES

65

factors through Xd , then for some i ≥ i0 , the composite
Ti → Ti0 → RΓd discfactors through Xd . This follows from the equivalence of conditions (1) and (7)of Lemma D.28, together with Lemma D.31.

3.3

Weak Wach modules

In this section we introduce the notion of a weak Wach module of height at
most h and level at most s. These will play a purely technical auxiliary role forus, and will be used only in order to show that Xd is an Ind-algebraic stack.
We suppose throughout this section that K is basic.
3.3.1 Remark. By Lemma D.28, if A is an O/$a -algebra for some a ≥ 1, andM is a rank d projective ϕ-module of T -height ≤ h over A, such that M[1/T ]is equipped with a semi-linear action of Γdisc , then this action extends to as
continuous action of Γ if and only if for some s ≥ 0 we have (γ p −1)(M) ⊆ T M.
3.3.2 Definition. Suppose that K is basic. A rank d projective weak Wachmodule of T -height ≤ h with A-coefficients is a rank d projective ϕ-module M
over A+
K,A , which is of T -height ≤ h, such that M[1/T ] is equipped with a semilinear action of Γ which is furthermore continuous when M[1/T ] is endowed with
its canonical topology (see Remark D.2).
s
If s ≥ 0, then we say that M has level ≤ s if (γ p − 1)(M) ⊆ T M.
Remark 3.3.1 shows that any projective weak Wach module is of level ≤ s forsome s ≥ 0.
A special role in the classical theory of (ϕ, Γ)-modules is played by the weak
Wach modules of level 0. However, this will not be important for us.
3.3.3 Definition. We let Wd,h denote the moduli stack of rank d projective
weak Wach modules of T -height ≤ h. (That this is an fpqc stack follows as in
Definition 3.2.1.) We let Wd,h,s denote the substack of rank d projective weak
Wach modules of T -height ≤ h and level ≤ s.
We recall from Section 3.1 that there is a p-adic formal algebraic stack Cd,hclassifying rank d projective ϕ-modules of T -height at most h. We consider
the fibre product RΓd disc ×Rd Cd,h , where the map RΓd disc → Rd is the canonicalmorphism given by forgetting the Γdisc action. By Proposition 3.2.14, this is themoduli stack of rank d projective ϕ-modules M of T -height at most h, equippedwith a semi-linear action of Γdisc on M[1/T ].
3.3.4 Lemma. RΓd disc ×Rd Cd,h is a p-adic formal algebraic stack of finite presentation over Spf O.
Proof. The map RΓd disc → Rd is representable by algebraic spaces and of finite
presentation, being a base-change of the diagonal of Rd , which is representable

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CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES

by algebraic spaces and of finite presentation by Corollary 3.1.5. Again by
Corollary 3.1.5, Cd,h is a p-adic formal algebraic stack of finite presentation, andthus RΓd disc ×Rd Cd,h is a p-adic formal algebraic stack of finite presentation, as
claimed.
Restricting the Γ-action on a weak Wach module to Γdisc , we obtain a morphism Wd,h → RΓd disc ×Rd Cd,h , which, by the evident analogue for weak Wachmodules of Remark 3.2.13, is fully faithful. Thus Wd,h may be regarded as a substack of RΓd disc ×Rd Cd,h . The following proposition records the basic propertiesof the stacks Wd,h and Wd,h,s .
3.3.5 Proposition.
1. For s ≥ 0, the morphism
Wd,h,s/ RΓdisc ×R Cd,hddis a closed immersion of finite presentation. In particular, each of thestacks Wd,h,s is a p-adic formal algebraic stack of finite presentation over
Spf O.
2. If s0 ≥ s, then the canonical monomorphism Wd,h,s ,→ Wd,h,s0 is a closed
immersion of finite presentation.
3. The canonical morphism lims Wd,h,s → Wd,h is an isomorphism. In par−→ticular, Wd,h is an Ind-algebraic stack.
Proof. Since RΓd disc ×Rd Cd,h classifies ϕ-modules M that are projective of rankd and of T -height ≤ h, which are endowed with a Γdisc -action on the underlying étale ϕ-module, in order to prove (1), we must show that the conditionsthat (γ p − 1)(M) ⊆ T M is a closed condition, and is determined by finitely
many equations. It suffices to check this after pulling back via an arbitrary morphism Spec A → RΓd disc ×Rd Cd,h , where A is an O/$a -algebra for some a ≥ 1.
This morphism gives rise to a projective A+
K,A ϕ-module M, such that M :=M[1/T ] is étale, and is furthermore endowed with a semi-linear Γdisc -action. Assalready stated above, we must now check that the condition (γ p −1)(M) ⊆ T M
is a finitely presented closed condition (in the sense that it holds after replacing
M by MB := A+
M for an A-algebra B if and only if Spec B → Spec A
K,B ⊗A+K,Afactors through a certain finitely presented closed subscheme of Spec A).
s
Note firstly that for n sufficiently large, we have (γ p − 1)(T n M) ⊆ T M.
Indeed, writingss
γ p − 1 = ((γ − 1) + 1)p − 1and expanding via the binomial theorem, it suffices to show that for n sufficiently
large, we have (γ − 1)m (T n M) ⊆ T M for all 0 ≤ m ≤ ps ; and this is immediatefrom Lemma D.19.
We next choose a finitely generated projective A+
K,A -module N such that F :=M ⊕ N is free, and write N := N[1/T ], F := F[1/T ]. Extend the morssphism γ p − 1 : M → M to the morphism f : F → F given by (γ p − 1, 0) :

3.4. Xd IS AN IND-ALGEBRAIC STACK

67s

M ⊕ N → M ⊕ N , so that the condition that (γ p − 1)(M) ⊆ T M is equivalent
to asking that f (F) ⊆ T F. By our choice of n above, the morphism F → F/T F
induced by f factors through a morphism F/T n F → F/T F; since F is finitely
generated, after enlarging n if necessary, it factors through F/T n F → T −n F/T F.
The vanishing of this morphism is obviously a closed condition; indeed it is givenby the vanishing of finitely many matrix entries, so is closed and of finite presentation, as required.
Since RΓd disc ×Rd Cd,h is a p-adic formal algebraic stack of finite presentation
over Spf O, it follows that so is each Wd,h,s . That Wd,h,s ,→ Wd,h,s0 is a closed
immersion of finite presentation follows immediately from a consideration of thecomposite
Wd,h,s ,→ Wd,h,s0 ,→ RΓd disc ×Rd Cd,h ,as we have just shown that both the composite and the second morphism areclosed immersions of finite presentation. Thus we have proved (1) and (2).
For (3), we need to show that every morphism Spec A → Wd,h , where A is
an O/$a -algebra for some a ≥ 1, factors through Wd,h,s for s sufficiently large.
As we already noted in Definition 3.3.2, this follows from Remark 3.3.1. Since
each Wd,h,s is an Ind-algebraic stack, so is Wd,h .

3.4

Xd is an Ind-algebraic stack

Continue to assume that K is basic. By definition, we have a 2-Cartesian
diagram/ RΓdisc ×R Cd,hWd,h(3.4.1)dd

/ RΓdiscXdd

If h0 ≥ h then the closed immersion Cd,h ,→ Cd,h0 is compatible with the morphisms from each of its source and target to Rd , and so we obtain a closed
immersionWd,h ,→ Wd,h0 .
(3.4.2)
By construction, the morphisms Wd,h → Xd are compatible, as h varies,
with the closed immersions (3.4.2). Thus we also obtain a morphism
lim Wd,h → Xd .
−→(3.4.3)hRoughly speaking, we will prove that Xd is an Ind-algebraic stack by showingthat it is the “scheme-theoretic image” of the morphism limh Wd,h → RΓd disc−→induced by (3.4.3). More precisely, choose s ≥ 0, and consider the composite
Wd,h,s → Wd,h → Xd → RΓd disc .

(3.4.4)

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CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES

This admits the alternative factorization
Wd,h,s → Wd,h → RΓd disc ×Rd Cd,h → RΓd disc .
Proposition 3.3.5 shows that the composite of the first two arrows is a closed
embedding of finite presentation, while Corollary 3.1.5 shows that the third
arrow is representable by algebraic spaces, proper, and of finite presentation.
Thus (3.4.4) is representable by algebraic spaces, proper, and of finite presentation.
a
Fix an integer a ≥ 1, and write Wd,h,s:= Wd,h,s ×Spf O Spec O/$a . Proposition 3.3.5 shows that Wd,h,s is a p-adic formal algebraic stack of finite presena
tation over Spf O, and so Wd,h,s
is an algebraic stack, and a closed substack ofWd,h,s .
a
3.4.5 Definition. We let Xd,h,s
denote the scheme-theoretic image of the composite(3.4.4)aWd,h,s,→ Wd,h,s −→ RΓd disc ,(3.4.6)

defined via the formalism of scheme-theoretic images for morphisms of Indalgebraic stacks developed in Appendix A.
More concretely, since RΓd disc is an Ind-algebraic stack, constructed as the 2colimit of a directed system of algebraic stacks whose transition morphisms areclosed immersions, the morphism (3.4.6), which is representable by algebraic
spaces, proper, and of finite presentation, factors through a closed algebraicasubstack Z of RΓd disc . We then define Xd,h,s
to be the scheme-theoretic imageaaof Wd,h,s in Z. Note that Xd,h,s is a closed algebraic substack of RΓd disc , and isindependent of the choice of Z.
3.4.7 Remark. As already observed above, the morphism (3.4.6) factors throughXd . However, since at this point we don’t know that Xd is Ind-algebraic, weacan’t directly define a scheme-theoretic image of Wd,h,s
in Xd . It might bepossible to do this using the formalism of [EG21]; since (3.4.6) is proper, thisascheme-theoretic image would then coincide with Xd,h,s. We don’t do this here;abut we do prove somewhat more directly, in Lemma 3.4.9 below, that Xd,h,s
isa substack of Xd .
As in Definition D.6, a lattice M in a projective étale ϕ-module M is a
finitely generated A+
K,A -submodule of M whose AK,A -span is M . Note that M
is not assumed to be projective.
3.4.8 Lemma. Suppose that M is a projective étale ϕ-module of rank d overa finite type Artinian O/$a -algebra A, and that M is endowed with an actionof Γdisc , such that the corresponding morphism Spec A → RΓd disc factors throughaXd,h,s. Then M contains a ϕ-invariant lattice M of T -height ≤ h, such thatps(γ − 1)(M) ⊆ T M.
Proof. Since an Artinian ring is a direct product of Artinian local rings, itsuffices to treat the case that A is local. Let the residue field of A be F0 , a3.4. Xd IS AN IND-ALGEBRAIC STACK

69

finite extension of F, and write O0 for the ring of integers in the compositumof E and W (F0 )[1/p], so that O0 has residue field F0 ; note that A is naturally
on O0 -algebra.
The projective étale ϕ-module M is in fact free of rank d, since AK,A isagain a local ring, and we fix an (ordered) basis for M as an AK,A -module.
Write MF0 := AK,F0 ⊗AK,A M ; the AK,A -basis of M gives rise to a corresponding
AK,F0 -basis of MF0 . Let CO0 be the category of Artinian local O0 /$a -algebras
for which the structure map induces an isomorphism on residue fields. By alifting of MF0 to an object Λ of CO0 , we mean a triple consisting of an étale
ϕ-module MΛ which is free of rank d, a choice of (ordered) AK,Λ -basis of MΛ ,and an isomorphism MΛ ⊗Λ F0 ∼= MF0 of étale ϕ-modules which takes the chosen
basis of MΛ to the fixed basis of MF0 . Let D be the functor CO0 → Sets taking Λto the set of isomorphism classes of liftings of MF0 to an étale ϕ-module MΛ
with Λ-coefficients, endowed with an action of Γdisc lifting that on MF0 . Notethat A is an object of CO0 , and that our originally chosen basis for M is classified
by a continuous morphism R → A.
The functor D is pro-representable by an object R of pro -CO0 , by the same
argument as in the proof of [EG21, Prop. 5.3.6]: namely, by Grothendieck’s
representability theorem, it suffices to prove the compatibility of D with fibre
products in CO0 , which is obvious. The universal lifting MR gives a morphism
Spf R → RΓd disc . The composite Spec A → Spf R → RΓd disc is of course just themorphism that classifies M .
We let D0 denote the subfunctor of D consisting of those lifts for which
there is a lattice MΛ of F -height ≤ h and level ≤ s, with MΛ [1/T ] = M . (Moresprecisely, we require that T h MΛ ⊆ (1 ⊗ ϕMΛ )(ϕ∗ MΛ ) and (γ p − 1)(MΛ ) ⊆0
T MΛ .) We claim that the functor D is pro-representable by a quotient S of R.
To see this, it is again enough to show that D0 preserves fibre products, and inturn it is enough to show that the property of an étale ϕ-module M with anaction of Γdisc admitting a ϕ-stable lattice M of T -height ≤ h, and such thats(γ p − 1)(M) ⊆ T M, is stable under taking direct sums and subquotients. Thisis obvious for direct sums, and the case of subquotients follows as in [EG21, Lem.
5.3.10] (which is the same result without the conditions on Γdisc ). More precisely,
once checks that if we have a short exact sequence 0 → M 0 → M → M 00 → 0of étale ϕ-modules with Λ-coefficients and actions of Γdisc , and M is a latticeof the appropriate kind in M , then the kernel and image of the map M → M 00
give the appropriate lattices M0 , M00 in M 0 , M 00 respectively. (The propertiesssthat (γ p − 1)(M0 ) ⊆ T M0 and (γ p − 1)(M00 ) ⊆ T M00 follow from a shorts(γ p −1)

diagram chase, using that the composite M → M → M/T M vanishes by
hypothesis.)
By the definition of D0 , the statement of the lemma is equivalent to thestatement that the map R → A factors through S. Let X = Wd,h,s ×RΓdisc Spf R,
da formal algebraic space, and let Spf T be the scheme-theoretic image of theΓdiscmorphism X → Spf R. Since the morphism Spec A → Rd
factors throughaXd,h,sby hypothesis, the morphism Spec A → Spf R factors through Spf T , so
it is enough to show that Spf T is a closed formal subscheme of Spf S. By

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CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES

Lemma A.32, it in turn suffices to show that if A0 is a finite type Artinian
local R-algebra for which R → A0 factors through a discrete quotient of R,
and for which the morphism Wd,h,s ×RΓdisc Spec A0 → Spec A0 admits a section,dthen R → A0 factors through S.
By replacing R with R ⊗W (F0 ) κ(A0 ) we can reduce to the case that A0 has
residue field F0 (cf. [EG21, Cor. 5.3.18]), so that R → A0 factors through S
if and only if the étale ϕ-module corresponding to Spec A0 → RΓd disc admitssa ϕ-stable lattice MA0 of T -height ≤ h for which (γ p − 1)(MA0 ) ⊆ T MA0 .
But a section to the morphism Wd,h,s ×RΓdisc Spec A0 → Spec A0 gives us admorphism Spec A0 → Wd,h,s , and the corresponding weak Wach module provides
the required lattice.
a3.4.9 Lemma. Each Xd,h,s
is a closed substack of Xd .
a
Proof. Since Xd,h,s
is a closed substack of RΓd disc , if it is a substack of Xd , it will inafact be a closed substack. Thus it suffices to show that Xd,h,s
is indeed a substackaof Xd . Since Xd,h,s is limit preserving, it is enough to check that if A is a finiteatype O/$a -algebra, then for any morphism Spec A → Xd,h,s, the composite
Γdiscamorphism Spec A → Xd,h,s → Rd
factors through Xd . Equivalently, if M
denotes the étale ϕ-module over A, endowed with a Γdisc -action, associated to
the given point Spec A → RΓd disc , then we must show that the Γdisc -action on
M is continuous.
By Lemma D.28, to see that the Γdisc -action on M is continuous, it sufficess
to produce a lattice M ⊆ M such that (γ p − 1)(M) ⊆ T M. We will produce
such a lattice by reduction to the Artinian case, as follows. Let {Ai }i∈I be theadirected system of Artinian quotients of A. Since
QA is of finite type over O/$ ,and so Noetherian, the natural map A → B := i Ai is injective. Lemma 3.4.8
shows that each base-changed module MAi admits a ϕ-invariant lattice Mi ofs
T -height ≤ h satisfying
the condition (γ p − 1)(Mi ) ⊆ T Mi .
Q
sWrite MB := i Mi ⊂ MB , and set M := M ∩ MB ⊂ MB . Since (γ p −ps1)(MB ) ⊆ T MB , we have (γ − 1)(M) ⊆ T M, so it only remains to checkthat M is a lattice. To see this, let M0 be a finite height lattice in M (i.e. alattice in M which is ϕ-stable, and for which the cokernel of Φ is killed by apower of T ; such a lattice exists by [EG21, Lem. 5.2.15]). By Lemma D.7 (3),it suffices to prove that there is an integer l ≥ 0 such that T l M0 ⊆ M ⊆ T −l M.
Accordingly, if for each i we let M0i denoted the image of Mi ⊗A Ai in MAi , weneed to show that we have T l M0i ⊆ Mi ⊆ T −l M0i .
The existence of such an l is established in the course of the proof of [EG21,
Lem. 5.3.14], although this is not explicitly recorded there. For the convenienceof the reader, we recall the argument in our present setting. Increasing h if
necessary, we may assume that M0 has T -height at most h. It follows that M0i
and Mi are each ϕ-stable lattices of height at most h in MAi . We claim that
we may take l = b(h + ap)/(p − 1)c.
To see this, let j be minimal such that T j Mi ⊆ M0i ; then we haveΦMi (ϕ∗ Mi ) ⊆ Mi ⊆ T −j M0i ⊆ T −h−j ΦM0i (ϕ∗ M0i ),3.4. Xd IS AN IND-ALGEBRAIC STACK

71

so that T h+j ϕ∗ Mi ⊆ ϕ∗ M0i . It follows from [EG21, Lem. 5.3.13] that h + j >(j − a)p, so that j < (h + ap)/(p − 1), and j ≤ l by definition. Thus T l Mi ⊆ M0i ,as claimed. Reversing the roles of Mi and M0i , we have T l M0i ⊆ Mi , and we aredone.
We now prove our first key structural result for Xd , in the case when K isbasic.
a3.4.10 Proposition. If K is basic, then the canonical morphism lim Xd,h,s→−→
Xd is an isomorphism. Thus Xd is an Ind-algebraic stack, and may in fact bewritten as the inductive limit of algebraic stacks of finite presentation, with thetransition maps being closed immersions.

Proof. Note firstly that if a0 ≥ a, h0 ≥ h and s0 ≥ s then the canonical moraa0phism Xd,h,s→ Xd,h0 ,s0 is a closed immersion by construction. By Lemma 3.4.9,
aeach Xd,h,s is a closed substack of Xd , so it remains to show that any moraphism T → Xd whose source is an affine scheme factors through some Xd,h,s, orequivalently, that the closed immersionaXd,h,s
×Xd T → T

(3.4.11)

is an isomorphism, for some choice of h and s.
Since Xd is limit preserving, by Lemma 3.2.19, we can reduce to the casewhere T = Spec A for a Noetherian O/$a -algebra A. If M denotes the étale(ϕ, Γ)-module corresponding to the morphism Spec A → Xd , then an application of [EG21, Prop. 5.4.7] shows that we may find a scheme-theoreticallydominant morphism Spec B → Spec A such that MB is free of rank d. If weashow that the composite Spec B → Spec A → Xd factors through Xd,h,s
for some
h and s, then we see that the morphism Spec B → Spec A factors through theaclosed subscheme Xd,h,s
×Xd Spec A of Spec A. Since Spec B → Spec A is schemetheoretically dominant, this implies that (3.4.11) is indeed an isomorphism, as
required.
Since MB is free, we may choose a ϕ-invariant free lattice M ⊆ MB , of height≤ h for some sufficiently large value of h. Since the Γdisc -action on M , and henceson MB , is continuous by assumption, Lemma D.28 shows that (γ p − 1)(M) ⊆T M for some sufficiently large value of s. Then M gives rise to a B-valued pointaof Wd,h,s, whose image in RΓd disc is equal to the étale ϕ-module MB . Thus theamorphism B → Xd corresponding to MB does indeed factor through Xd,h,s.
Finally, we drop our assumption that K is basic.
3.4.12 Proposition. Let K be an arbitrary finite extension of Qp . Then Xd
is an Ind-algebraic stack, and may in fact be written as the inductive limit ofalgebraic stacks of finite presentation over Spec O, with the transition maps
being closed immersions. Furthermore the diagonal of Xd is representable byalgebraic spaces, affine, and of finite presentation.
Proof. This is immediate from Propositions 3.2.17 and 3.4.10, together with
Corollary 3.2.9.

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CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES

3.5

Canonical actions and weak Wach modules

We now explain an alternative perspective on some of the above results, which
gives more information about the moduli stacks of weak Wach modules. The
results of this section are only used in Chapter 7, where we use them to establish
a concrete description of Xd in the case d = 1. For each s ≥ 0, we write Kcyc,sfor the unique subfield of Kcyc which is cyclic over K of degree ps . The following
lemma can be proved in exactly the same way as Lemma 4.3.3 below.
3.5.1 Lemma. Assume that K is basic. For any fixed a, h, there is a constant N (a, h) such that if N ≥ N (a, h), there is a positive integer s(a, h, N )with the property that for any finite type O/$a -algebra A, any finite projective ϕ-module M over A+
K,A of T -height at most h, and any s ≥ s(a, h, N ),there is a unique continuous action of GKcyc,s on Minf := Ainf,A ⊗A+ M
K,A
which commutes with ϕ and is semi-linear with respect to the natural actionof GKcyc,s on Ainf,A , with the additional property that for all g ∈ GKcyc,s wehave (g − 1)(M) ⊂ T N Minf .
It is possible to prove the following result purely in the world of (ϕ, Γ)modules (by following the proof of Lemma 4.3.3, interpreting the existence of asemilinear action of ΓKcyc,s in terms of linear maps between twists of ϕ-modules,
satisfying certain compatibilities), but we have found it more straightforward
to argue with Proposition 2.7.8.
3.5.2 Corollary. Assume that K is basic. For any fixed a, h, there is a constant N (a, h) such that if N ≥ N (a, h), there is a positive integer s(a, h, N )with the property that for any finite type O/$a -algebra A, any finite projective
ϕ-module M over A+
K,A of T -height at most h, and any s ≥ s(a, h, N ), theresis a unique semi-linear action of hγ p i on M which commutes with ϕ, and withpsthe additional property that (γ − 1)(M) ⊆ T N M.
In particular, this action gives M[1/T ] the structure of a projective étale(ϕ, ΓKcyc,s )-module.
Proof. By Lemma 3.5.1, there is a a unique continuous semi-linear action of GKcyc,son Minf := Ainf,A ⊗A+ M which commutes with ϕ, with the additional propK,Aerty that for all g ∈ GKcyc,s we have (g − 1)(M) ⊂ T N Minf . Write M :=f := W (C[ )A ⊗A
Minf . By Proposition 2.7.8, the GKcyc,s AK,A ⊗A+ M, M
inf,A
K,A
f endows M with the structure of a projective (ϕ, ΓK
action on M)-modulecyc,sfwith A-coefficients, and M with its GK-action is recovered as
cyc,sf = W (C[ )A ⊗A
M
M.
K,A
s

Since ΓKcyc,s is topologically generated by γ p , and since (g − 1)(M) ⊂ss
T N Minf for g ∈ GKcyc,s , we have (γ p − 1)(M) ⊂ T N Minf . We also have (γ p −1)(M) ⊆ M , and since M ∩ Minf = M (as is easily checked, by reducing to thecase that M is free, and then to the case that it is free of rank one), we have

3.5. CANONICAL ACTIONS AND WEAK WACH MODULES73s(γ p − 1)(M) ⊂ T N M, as required. For the uniqueness, note that if we have
two such actions, then taking their difference gives a nonzero ϕ-linear morphisms(γ p )∗ M → T N M, which is impossible for N sufficiently large (see e.g. the firstpart of the proof of Lemma 4.3.2 below).
Finally, the claim that this gives M[1/T ] the structure of a (ϕ, ΓKcyc,s )module is immediate from Remark 3.3.1.
We continue to assume that K is basic. For any s ≥ 1, we write Γs,discsΓ
to denote the subgroup of Γdisc generated by γ p . We write Rd s,disc in obvious
analogy to the notation RΓd disc introduced above. The result of Corollary 3.5.2
may be interpreted as constructing a morphism
Γ

a
Cd,h
→ Rd s,disc(3.5.3)for sufficiently large values of s (depending on a and h), lying over the morphismΓaCd,h→ Rd . Of course there is also a morphism RΓd disc → Rd s,disc given by
restricting the action of Γdisc to Γs,disc .
ΓSince the morphism Rd s,disc → Rd given by forgetting the Γs,disc -action isrepresentable by algebraic spaces and separated (indeed, even affine — it is abase-change of the morphism Rd → Rd ×O Rd giving the graph of the actionsof γ p , and this latter morphism is representable by algebraic spaces and affine,since the diagonal morphism of Rd is so), the diagonal morphismΓΓΓ

Rd s,disc → Rd s,disc ×Rd Rd s,disca
is a closed immersion. We may thus define a closed substack Zd,h,sof RΓd disc ×Rda
Cd,h via the following 2-Cartesian diagram:a
Zd,h,s/ RΓdisc ×R C aΓ
Rd s,disc/ RΓs,disc ×R RΓs,disc

d

d

d

d

d,h

d

in which the lower horizontal arrow is the diagonal, and the right-hand verticalarrow is the product of the restriction morphism and the morphism (3.5.3). In
aless formal language, the stack Zd,h,sparameterizes projective rank d ϕ-modules+
M over AK,A of T -height at most h, for which M[1/T ] is endowed with a Γdisc action extending the canonical action of Γs,disc given by Corollary 3.5.2.
a3.5.4 Proposition. Assume that K is basic. Then each Zd,h,s
is contained(as a closed algebraic substack) in Xd ×Rd Cd,h , and the natural morphismalima,s Zd,h,s→ Xd ×Rd Cd,h is an isomorphism.
−→a
Proof. That Zd,h,s
is a substack of Xd ×Rd Cd,h is immediate from Remark 3.3.1;
indeed, if we are given M over A+
K,A of T -height at most h, for which M[1/T ]74

CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES

is endowed with a Γdisc -action extending the canonical action of Γs,disc given bysaCorollary 3.5.2, then we have (γ p − 1)(M) ⊆ T N M ⊆ T M. Since Zd,h,s
is a
Γdisc
a
closed substack of Rd
×Rd Cd,h , it is a closed substack of Xd ×Rd Cd,h .
aTo see that the natural morphism lima,s Zd,h,s→ Xd ×Rd Cd,h is an isomor−→phism, we need to show that it is surjective, and so we need to show that for
any finite type O/$a -algebra A, and any projective ϕ-module M over A+
K,A ofT -height at most h, for which M[1/T ] is endowed with a continuous Γdisc -action,
there is some s ≥ 1 such that the restriction of this action to γs,disc agrees withthe canonical action of Γs,disc given by Corollary 3.5.2.
s0

By Remark 3.3.1, there is some s0 ≥ 1 such that (γ p − 1)(M) ⊆ T M.
Let N = N (a, h) be as in the statement of Lemma 3.5.1. By the equivalence ofsconditions (3) and (4) of Lemma D.28, there is some s ≥ s0 such that (γ p −1)(M) ⊆ T N M, as required.
For arbitrary K (not necessarily basic) we can deduce the following description of Xda as an Ind-algebraic stack, in the style of Lemma 3.2.11.
3.5.5 Lemma. Fix a ≥ 1. For each h, and each sufficiently large (dependingaon a and h) value of s, let XK,d,K
basic ,h,s denote the scheme-theoretic image ofthe base-changed morphismaaa
Zd[K:K
XK,d→ XK,d,basic ],h,s ×X basic
K,d[K:K basic ]aso that XK,d,K
basic ,h,s is a closed algebraic substack of XK,d . Then the naturalaamorphism limh,s XK,d,K
basic ,h,s → XK,d is an isomorphism.
−→
Proof. This is proved in the same way as Lemma 3.2.11, bearing in mind Proposition 3.5.4.

3.6

The connection with Galois representations

In Section 2.7.1 we recalled the relationship between étale (ϕ, Γ)-modules without coefficients (which is to say, with coefficients in Zp ) and p-adic representations of GK , and the similar relationship between étale ϕ-modules over OE andp-adic representations of GK∞ . In this section, we revisit those topics in thecontext of étale (ϕ, Γ)-modules (resp. étale ϕ-modules) and GK -representations(resp. GK∞ -modules) with coefficients.

3.6.1

Galois representations with coefficients

For a general p-adically complete Zp -algebra A, a projective étale (ϕ, Γ)-modulewith A-coefficients need not correspond to a family of GK -representations. It
will be useful, though, to have a version of such a correspondence in the casethat A is complete local Noetherian with finite residue field. In this case, followb K,A denote the mA -adic completion of AK,A , and we define
ing [Dee01] we let A
a formal étale (ϕ, Γ)-module with A-coefficients to be an étale (ϕ, Γ)-moduleb K,A in the obvious sense.
over A

3.6. THE CONNECTION WITH GALOIS REPRESENTATIONS

75

3.6.2 Remark. If A is a complete local Noetherian O-algebra, with finite residue
field, then the groupoid of formal étale (ϕ, Γ)-modules is equivalent to thegroupoid Xd (A) (which we remind the reader refers to the groupoid of morphisms Spf A → Xd ; here Spf A is taken with respect to the mA -adic topologyon A). Indeed, by definition, the latter groupoid may be identified with the
2-limit limi Xd (A/miA ), which is easily seen to be equivalent to the groupoid of←−
formal étale (ϕ, Γ)-modules, via an application of [GD71, Prop. 0.7.2.10(ii)].
b ur denote Ab ur ⊗b Zp A, where the completed tensor product is withWe let A
K,AK
b ur , and the mA -adic topology on A. The
respect to the usual topology on A
K
functors TA , DA defined byb ur ⊗A T )GKcyc ,DA (T ) = (AK,A
ϕ=1b ur ⊗ b
TA (M ) = (AK,A AK,A M )then give equivalences of categories between the category of finite projective
formal étale (ϕ, Γ)-modules with A-coefficients, and the category of finite freeA-modules with a continuous action of GK . (In fact the results of [Dee01] do
not require the (ϕ, Γ)-modules to be projective, but we will for the most part
only consider projective modules in this book.)
Continuing to assume that A is complete local Noetherian with finite residue
bE,A denote the mA -adic completion of OE,A , and we define O
bd:=field, we let O
E ur ,A
b
⊗
A,
where
the
completedtensorproductiswithrespect
to
the
usual
OEd
Zp
ur
topology on OEd
ur , and the mA -adic topology on A. Then the analogous statements to those of the previous paragraph, relating representations of GK∞ on
bE,A 1 via the functors
finite free A-modules and étale ϕ-modules over O
bd
D∞,A (T ) = (O⊗ T )GK∞ ,E ur ,A A
bd⊗ObE,A M )ϕ=1 ,T∞,A (M ) = (OE ur ,A
can be proved in exactly the same way as in [Dee01] (by passage to the limit
over A/mnA ).
We will also occasionally apply these statements in the case A = Fp . Theirvalidity in this case follows from their validity in the case when A is a finite
extension of Fp , and the fact that both Galois representations and projectiveétale (ϕ, Γ)-modules over Fp arise as base changes from such finite contexts; for
Galois representations, this follows from the compactness of GK and GK∞ , andfor étale (ϕ, Γ)-modules, it is an immediate consequence of Lemma 3.2.19.
We can use the equivalence between Galois representations and (ϕ, Γ)-modules
as a tool to deduce facts about the finite type points of Xd . More precisely,
if F0 /F is a finite extension, then the groupoid of points x ∈ Xd (F0 ) is canonically equivalent to the groupoid of Galois representations ρ : GK → GLd (F0 ).
1 One might refer to an étale ϕ-module over O
bE,A as a “formal étale ϕ-module over A”, butsince there are several different species of étale ϕ-modules under consideration throughout thebook, we avoid using this potentially ambiguous terminology.

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CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES

For this reason, we will often denote such a point x simply by the corresponding
Galois representation ρ.
Suppose now that F0 /F is a finite extension, and that x : Spec F0 → Xd is a
finite type point, with corresponding Galois representation ρ : GK → GLd (F0 ).
0d
Let (Xd )x be the category of Definition A.25 (with F there being Xd ). Let O =000
O ⊗W (F) W (F ) be the ring of integers in the finite extension E = W (F )E.
0

3.6.3 Proposition. There is a morphism Spf Rρ,O → Xd which is versal at00∼

the point x corresponding to ρ, an isomorphism Spf Rρ,O ×Xd Spf Rρ,O −→c ,O0c ,O0, where GL
denotes the completion of (GLd ) ,O0 along
GL
d,Rρ

,Z(ρ)

d,Rρ

,Z(ρ)

Rρ

the closed subgroup of (GLd )F0 given by the centraliser of ρ, and an isomorphism0
,O 0 ∼ cc ,O0 denotes the com−→ GLd,R,O0 ,1 , where GL
Spf Rρ,O ×(Xd) Spf Rρ
d,R
,1
d x

ρ

ρ

pletion of (GLd )R,O0 along the identity of (GLd )F0 .
ρ

Proof. The existence of the morphism follows from the theorem of Dee recalled
above, and the descriptions of the two fibre products are clear from its very
definition. To see that this morphism is versal, it suffices to show that if ρ :GK → GLd (A) is a representation with A a finite Artinian O-algebra, and if
ρB : GK → GLd (B) is a second representation, with B a finite Artinian Oalgebra admitting a surjection onto A, such that the base change ρA of ρB to
A is isomorphic to ρ (more concretely, so that there exists M ∈ GLd (A) with
ρ = M ρA M −1 ), then we may find ρ0 : GK → GLd (B) which lifts ρ, and isisomorphic to ρB . But this is clear: the natural morphism GLd (B) → GLd (A)is surjective, and so if M 0 is any lift of M to an element of GLd (B), then we
may set ρ0 = M 0 ρB (M 0 )−1 .
Because of the equivalence between (ϕ, Γ)-modules and Galois representations with Zp -coefficients, and because of the traditional notation ρ for Galois
representations, we will often denote a family of rank d projective étale (ϕ, Γ)modules over T , i.e. a morphism T → Xd , by ρT , or some similar notation.
We caution the reader that this notation is chosen purely for its psychologicalsuggestiveness; a family of (ϕ, Γ)-modules over a general base T does not admit
a literal interpretation in terms of Galois representations.

3.6.4

Galois representations associated to (ϕ, GK )-modules

At times it will be convenient to consider the Galois representations associatedto étale (ϕ, GK )-modules. Let A be a finite O/$a -algebra for some a ≥ 1, andlet M be a finite projective étale (ϕ, GK )-module (resp. étale (ϕ, GK∞ )-module)with A-coefficients. Then we may apply the equivalence of Proposition 2.7.8
to obtain an étale (ϕ, Γ)-module (resp. an étale ϕ-module) from M ; applying
the functor TA (resp. T∞,A ) of Section 3.6.1 to this latter object yields a GK representation (resp. GK∞ -representation) on a finite free A-module, which wedenote simply by TA (M ) (resp. T∞,A (M )).

3.6. THE CONNECTION WITH GALOIS REPRESENTATIONS

77

By passage to the limit over a, we can extend these functors to the casewhere A is a finite O-algebra, and in particular to the case that A = O0 is thering of integers in a finite extension E 0 /E. As in Section 3.6.1, we can and dothen further extend these functors to the case A = Fp .
While we have defined the functors TA (M ) (resp. T∞,A (M )) via our equivalences of categories, we note that both also admit a more direct description:
namely TA (M ) = M ϕ=1 (resp. T∞,A (M ) = M ϕ=1 ). Similarly, the compositesof the functors DA and D∞,A with the equivalences of Proposition 2.7.8 admit
the following simple description: if TA is a finite free A-module with a continuous action of GK (resp. GK∞ ), then the corresponding étale (ϕ, GK )-module(resp. (ϕ, GK∞ )-module) is given by W (C[ )A ⊗A TA (with ϕ acting on the first
factor in the tensor product, and GK (resp. GK∞ ) acting diagonally).

3.6.5

Galois representations over certain fields

There is one more context in which we will need to consider the correspondence
between Galois representations and étale (ϕ, Γ)-modules. Namely, suppose that
A is a complete Noetherian local Zp -algebra which is a domain, and which has
finite residue field. Let K denote the fraction field of A, and as usual, let mdenote the maximal ideal of A. We say that a representation ρ : GK → GLd (K)is continuous if there exists a finitely generated A-submodule L of Kd whichspans Kd over K (a “lattice”), such that GK preserves L and acts continuously(when L is given its m-adic topology).
b ur ⊗A K. We
b K,K := Ab K,A ⊗A K, and also write Ab ur := A
We write A
K,AK,K
b K,K .
have the obvious notion of a projective (ϕ, Γ)-module with coefficients in A
We say that such a (ϕ, Γ)-module D is étale if there exists a (not necessarilyb K,A contained in D such that the
projective) étale (ϕ, Γ)-module DA over A
b K,K ⊗ bevident morphism A
AK,A DA → D is an isomorphism.
Then, analogously to the case of A itself, we have functors TK , DK definedbyb ur ⊗K T )GKcyc ,DK (T ) = (AK,K
ϕ=1b ur ⊗ b
TK (M ) = (AK,KAK,K M )which give equivalences of categories between the category of finite projective
formal étale (ϕ, Γ)-modules with K-coefficients, and the category of finite dimensional K-vector spaces with a continuous action of GK . (This is proved bypassing to lattices on each side, and using the results of [Dee01]; note that thelattices involved need not be projective in general, and so we apply those resultsin their full generality.)This formalism is most often applied in the literature in the case when A =Zp , so that K = Qp . However, we will not consider étale (ϕ, Γ)-modules with
Qp -coefficients in this book. Rather we will apply the preceding formalism only
once, namely in our analysis of the closed Fp -points of Xd , which we make in
Section 6.6; and in this application, we will take A to be a complete local domainof characteristic p.

78

3.7

CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES(ϕ, GK )-modules and restriction

Our goal in this section is to define certain morphisms of stacks which are theanalogues, for families of (ϕ, Γ)-modules, of the restriction functors on Galois
representations ρ 7→ ρ|GK∞ and ρ 7→ ρ|GL (for any finite extension L of K).
3.7.1 Definition. We let RBK,d denote the moduli stack of rank d projectiveétale ϕ-modules over OE,A ; that is, the stack Rd constructed in Section 3.1 inthe case when AA = OE,A .
Recall that we also have the stack Xd of étale (ϕ, Γ)-modules of Definition 3.2.1. We now construct a morphism Xd → RBK,d which corresponds to
the restriction of Galois representations from GK to GK∞ .
3.7.2 Proposition. There is a canonical morphism Xd → RBK,d . If A is acomplete local Noetherian O-algebra with finite residue field, or equals Fp , thenthe morphism Xd (A) → RBK,d (A) is given by restriction of the corresponding
representation of GK to GK∞ .
Proof. If A is a finite type O/$a -algebra, for some a ≥ 1, then we define amorphism of groupoids Xd (A) → RBK,d (A) via the equivalences of categories of
Proposition 2.7.8; that is, given a finite projective étale (ϕ, Γ)-module with Acoefficients, we form the corresponding finite projective étale (ϕ, GK )-module,
which yields a finite projective étale (ϕ, GK∞ )-module by restricting the GK action to GK∞ , and thus gives an étale ϕ-module over OE,A . Since both Xd
and RBK,d are limit preserving (by Corollary 3.1.5 and Lemma 3.2.19), it follows
from [EG21, Lem. 2.5.4, Lem. 2.5.5 (1)] that this construction determines amorphism Xd → RBK,d .
Suppose now that A is a complete local Noetherian O-algebra with finite
residue field. Let M be a formal étale (ϕ, Γ)-module corresponding to a morbE,A corresponding tophism Spf A → Xd , and M∞ be the étale ϕ-module over O
the composite Spf A → Xd → RBK,d . The relationship between M and M∞ isexpressed as an isomorphism∼\\[W(C[ )A ⊗A
b K,A M −→ W (C )A ⊗O
bE,A M∞ .

(3.7.3)

Recall that the Galois representation associated to M is defined via TA (M ) :=ϕ=1
b ur ⊗ bb ur -linear mor(A, and that the evident (ϕ, GK )-equivariant A
K,A AK,A M )
K,A
phism
b ur ⊗A TA (M ) → A
b ur ⊗ b
A
K,AK,A AK,A M
is then an isomorphism (see for example [Dee01, Prop. 2.1.26]). Thus there isan induced natural (ϕ, GK )-equivariant isomorphism∼\\W(C[ )A ⊗A TA (M ) −→ W
(C[ )A ⊗A
b K,A M,\
where we recall that we write W(C[ )A for the mA -adic completion of W (C[ )A .
Similarly, we obtain a natural (ϕ, GK∞ )-equivariant isomorphism∼\\W(C[ )A ⊗A T∞,A (M∞ ) −→ W
(C[ )A ⊗ObE,A M∞ .

3.7. (ϕ, GK )-MODULES AND RESTRICTION

79

Combining these two isomorphisms with (3.7.3), we obtain a natural (ϕ, GK∞ )equivariant isomorphism∼\\W(C[ )A ⊗A TA (M ) −→ W
(C[ )A ⊗A T∞,A (M∞ ).

Each of TA (M ) and T∞,A (M∞ ) is a free A-module with trivial ϕ-action, and so if
we pass to ϕ-invariants in this isomorphism, and take into account Lemma 2.2.20,∼
we obtain an isomorphism TA (M )|GK∞ −→ T∞,A (M∞ ), which is what we
wanted to show. Finally, if A = Fp , the result follows by taking direct limits.
3.7.4 Proposition. The diagonal morphism ∆ : Xd → Xd ×RBK,d Xd inducedby the morphism of Proposition 3.7.2 is a closed immersion.
Proof. The product Xd ×RBK Xd can be described explicitly as follows: its
A-valued points are pairs (M1 , M2 ) of étale (ϕ, GK )-modules with a GK∞ equivariant isomorphism between them. The diagonal is defined by M 7→
(M, M ), with the isomorphism being the identity. To see that this defines aclosed immersion, we have to check that if we are given a pair (M1 , M2 ) of
(ϕ, GK )-modules with A-coefficients, together with a GK∞ -equivariant isomor∼
phism f : M1 −→ M2 , then the locus where f becomes GK -equivariant isclosed. That this is so follows from Corollary B.27. (That corollary shows that,for each g ∈ GK , there is an ideal Jg ⊆ A which cuts out
P the condition for the
given isomorphism to commute with g. The ideal J := g Jg then cuts out thecondition for the given isomorphism to be GK -equivariant.)
We now construct the restriction maps corresponding to finite extensionsof K.
3.7.5 Lemma. Let L/K be a finite extension. There is a canonical morphism XK,d → XL,d , which is representable by algebraic spaces, affine, andof finite presentation. If A is a complete local Noetherian O-algebra with finite
residue field, or equals Fp , then the morphism XK,d (A) → XL,d (A) is given by
restriction of the corresponding representation of GK to GL .
Proof. As in the case of Proposition 3.7.2, the morphism is defined first in thecase of O-algebras that are of finite type over O/$a for some a by applying
the equivalences of categories of Proposition 2.7.8: we send an étale (ϕ, GK )module M to an étale (ϕ, GL )-module ML via restricting the continuous GK action to a continuous GL -action. Since XK,d and XL,d are limit preserving(by Lemma 3.2.19), it follows from [EG21, Lem. 2.5.4, Lem. 2.5.5 (1)] that thisconstruction determines a morphism XK,d → XL,d .
Before verifying the various claimed properties of this morphism, we confirm
that it has the claimed effect on associated Galois representations. To this end,
we note that if A is a complete local Noetherian O-algebra, then, as in the
proof of Proposition 3.7.2, it follows from the definition that we have a natural(ϕ, GL )-equivariant isomorphism∼\\W(C[ )A ⊗A TA (M ) −→ W
(C[ )A ⊗A TA (ML ),80

CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES∼

and by Lemma 2.2.20, taking ϕ-invariants induces an isomorphism TA (M )|GL −→TA (ML ). We deduce the same statement if A = Fp by taking direct limits.
We now verify the claimed properties of the restriction morphism XK,d →XL,d . Suppose firstly that L/K is Galois, and let {gi } be a set of (finitely many)
coset representatives for GL in GK . For each i, we may give gi∗ M the structureof a (ϕ, GL )-module by letting each h ∈ GL act as gi−1 hgi acts on M ; then toextend the action of GL to an action of GK is to give an isomorphism of (ϕ, GL )modules gi∗ M → M for each index i, satisfying a slew of compatibilities. (Since
GL is open in GK , the continuity of the GK -action is automatic, given that the
GL -action that it is extending is continuous.)
We let Yi denote the stack classifying objects MA of XL,d (A), endowed with∼
an isomorphism gi∗ MA −→ MA . If we regard gi∗ as an automorphism of XL,d (A),then we may form its graph Γi , and we then have an isomorphism of stacks∼

Yi −→ XL,d (A) ×∆,XL,d (A)×XL,d (A),Γi XL,d (A).
(Here ∆ denotes the diagonal of XL,d ; cf. the definition of RΓd disc in Section 3.2
above, together with Proposition 3.2.14.) The projection onto the second factor
Yi → XL,d (A), which corresponds to forgetting the isomorphism, is a basechange of the diagonal ∆, and so is representable by algebraic spaces, affine andof finite presentation by Proposition 3.2.17.
We can rephrase our interpretation of objects of XK,d (A) as objects of∼
XL,d (A) endowed with isomorphisms gi∗ MA −→ MA satisfying certain compatibilities as the existence of a closed immersion
XK,d ,→ Y1 ×XL,d × · · · ×XL,d Yn(the point being that the compatibilities arise as base-changes of the double
diagonal, and so impose closed conditions). Since the morphisms Yi → XL,dare representable by algebraic spaces, affine and of finite presentation, it follows
that the same is true of the morphism XK,d → XL,d , as claimed.
Finally, let L/K be a general finite extension. Let M/K be the normal
closure of L/K, so that we have morphisms
XK,d → XL,d → XM,d .
By what we have already proved, both the second arrow and the composite of
the two arrows are representable by algebraic spaces and affine, and of finite
presentation. An affine morphism is separated, and thus has affine and finite
type diagonal. One sees (e.g. by Lemma 3.2.7) that the diagonal of an affinemorphism of finite type (and so, in particular, the diagonal of an affine morphismof finite presentation) is furthermore of finite presentation. A standard graphargument, in which we factor the first morphism as
XK,d ,→ XK,d ×XM,d XL,d → XL,d(the first arrow in this factorization being the graph of the first morphism, which
is a base change of the diagonal XL,d → XL,d ×XM,d XL,d , and the second arrow3.8. TENSOR PRODUCTS AND DUALITY

81

being the projection, which is a base change of the composite XK,d → XL,d →XM,d ), then shows that the first morphism is also representable by algebraic
spaces and affine, and of finite presentation.

3.8

Tensor products and duality

If M is a projective étale (ϕ, Γ)-module with A-coefficients, we define the dual(ϕ, Γ)-module byM ∨ := HomAK,A (M, AK,A );if A is a finite O-module, then there is a natural isomorphism T (M ∨ ) ∼= T (M )∨ .
Similarly, for each M , we let M (1) := M ⊗AK,A AK,A (1) denote the Tate twist,
where AK,A (1) denotes the free (ϕ, Γ)-module of rank 1 with a generator v onwhich Γ acts via the cyclotomic character and ϕ(v) = v. (Note that if A = O
then T (AK,A (1)) = O(1), the usual Tate twist on the Galois side.) We also
define the Cartier dual M ∗ := M ∨ (1), which again in the case that A is a
finite O-algebra is compatible with the usual notion for Galois modules. If
T → Xd corresponds to a family ρT , then we adopt the natural convention of∗writing ρT (1), ρ∨
T and ρT for the corresponding families.
Given two projective étale (ϕ, Γ)-modules M1 , M2 with A-coefficients ofrespective ranks d1 , d2 , we may form the tensor product (ϕ, Γ)-module M1 ⊗M2 (given by the tensor product on underlying AK,A -modules, and the tensor
products of the actions of each of ϕ and Γ). If A is a finite O-module, then we
have a natural isomorphism
T (M1 ⊗ M2 ) ∼= T (M1 ) ⊗ T (M2 ).
The tensor product induces a natural morphism
Xd1 ×O Xd2 → Xd1 d2 ;in particular, twisting by (the (ϕ, Γ)-module corresponding to) any character GK → O× gives an automorphism of each Xd . Given morphisms S → Xd1 ,T → Xd2 , denoted by ρS and ρ0T , we write ρS  ρ0T for the family corresponding
to the composite
S ×O T → Xd1 ×O Xd2 → Xd1 d2 .
If S = T then we write ρT ⊗ ρ0T for the composite
T → T ×O T → Xd1 ×O Xd2 → Xd1 d2 ,where the first map is the diagonal embedding.

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CHAPTER 3. STACKS OF ϕ-MODULES AND (ϕ, Γ)-MODULES

Chapter 4

Crystalline and semistable
moduli stacks
In this chapter we build on the results of Appendix F to define the potentially semistable and potentially crystalline substacks of XK,d . Using the results of the earlier chapters, and Kisin’s results on potentially semistable lifting
rings [Kis08], we show that these stacks are p-adic formal algebraic stacks, andcompute the dimensions of their special fibres.

4.1

Notation

Recall from Section 2.1.14 that the embedding SA ,→ Ainf,A depends on the∞
choice of a uniformizer π of K, and of a compatible family π 1/p of p-powerroots of unity in K. In this chapter we will want to consider all possible choices∞of π and π 1/p , and consequently we introduce some notation to do so. We∞[
write π [ ∈ OC
for the element determined by π 1/p . For each s ≥ 0 weswrite Kπ[ ,s for K(π 1/p ), and Kπ[ ,∞ for ∪s Kπ[ ,s . We write Sπ[ ,A for the imageof SA in Ainf,A via the homomorphism determined by u 7→ [π [ ], and OE,π[ ,A
for the image of OE,A in W (C[ )A under the corresponding homomorphism.
When π [ is fixed, we will often write u for [π [ ] when discussing Sπ[ ,A . We
write Eπ for the Eisenstein polynomial corresponding to π. We also have ab A,natural map θ : Ainf,A → OC,A (where the target is the tensor product OC ⊗which is the quotient of the source by the principal ideal generated by the kernelof the usual map θ : Ainf → OC ).
4.1.1 Remark. In contrast to many papers in the literature (e.g. [BMS18, Liu12])
we do not twist the embedding S → Ainf by ϕ. While such a twist is important in applications involving comparisons to Fontaine’s period rings, it is moreconvenient for us to use the embedding we have given here. Since ϕ is an automorphism of Ainf , it is in any case straightforward to pass back and forth
between these two conventions.
83

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CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS

4.2

Breuil–Kisin modules and Breuil–Kisin–Fargues
modules

4.2.1 Definition. Fix a choice of π [ . Let A be a p-adically complete O-algebra
which is topologically of finite type. We define a projective Breuil–Kisin module(resp. a projective Breuil–Kisin–Fargues module) of height at most h with Acoefficients to be a finitely generated projective Sπ[ ,A -module (resp. Ainf,A module) M, equipped with a ϕ-semi-linear morphism ϕ : M → M, with theproperty that the corresponding morphism ΦM : ϕ∗ M → M is injective, withcokernel killed by Eπh . If M is a Breuil–Kisin module then Ainf,A ⊗Sπ[ ,A M is
a Breuil–Kisin–Fargues module.
4.2.2 Remark. Note that our definition of a Breuil–Kisin–Fargues module is lessgeneral than the definition made in [BMS18], in that we require ϕ to take M
to itself; this corresponds to only considering Galois representations with nonnegative Hodge–Tate weights. This definition is convenient for us, as it allows
us to make direct reference to the literature on Breuil–Kisin modules. The restriction to non-negative Hodge–Tate weights is harmless in our main results,
as we can reduce to this case by twisting by a large enough power of the cyclotomic character (the interpretation of which on Breuil–Kisin–Fargues modules
is explained in [BMS18, Ex. 4.24]).
4.2.3 Definition. Let A be a p-adically complete O-algebra which is topologically of finite type. A Breuil–Kisin–Fargues GK -module with A-coefficients is
a Breuil–Kisin–Fargues module with A-coefficients, equipped with a continuous
semilinear action of GK which commutes with ϕ.
Note that if Minf is a Breuil–Kisin–Fargues GK -module with A-coefficients,
then W (C[ )A ⊗Ainf,A Minf is naturally an étale (ϕ, GK )-module in the sense of
Definition 2.7.7. The following definition is motivated by Theorem F.11 andCorollary F.23.
4.2.4 Definition. Let A be a p-adically complete O-algebra which is topologically of finite type over O (and recall then that Ainf,A is faithfully flat overSπ[ ,A , by Proposition 2.2.14). We say that a Breuil–Kisin–Fargues GK -modulewith A-coefficients Minf descends for π [ or descends to Sπ[ ,A if there is a Breuil–GK

Kisin module Mπ[ with Mπ[ ⊆ (Minf ) π[ ,∞ and for which the natural map
Ainf,A ⊗Sπ[ ,A Mπ[ → Minf is an isomorphism.
We say that Minf admits all descents if it descends for every choice of π [(for every choice of π), and if furthermore
1. The W (k) ⊗Zp A-submodule Mπ[ /[π [ ]Mπ[ of W (k)A ⊗Ainf,A Minf is independent of the choice of π and π [ .
2. The OK ⊗Zp A-submodule ϕ∗ Mπ[ /Eπ[ ϕ∗ Mπ[ of OC,A ⊗Ainf,A ,θ ϕ∗ Minf is
independent of the choice of π and π [ .

4.2. BREUIL–KISIN–FARGUES MODULES

85

If Minf admits all descents, then we say that it is crystalline if for each π [ , andeach g ∈ GK , we have(g − 1)(Mπ[ ) ⊆ ϕ−1 (µ)[π [ ]Minf(4.2.5)(where µ = [ε]−1 for some compatible choice of roots of unity ε = (1, ζp , ζp2 , . . . ) ∈[OC).
If L/K is a finite Galois extension, then we say that Minf admits all descents over L if the corresponding Breuil–Kisin–Fargues GL -module (obtainedby restricting the GK -action to GL ) admits all descents.
4.2.6 Remark. There is considerable redundancy and rigidity in Definition 4.2.4.
Note in particular that if for some π [ there exists a Sπ[ ,A -module Mπ[ withMinf = Ainf,A ⊗Sπ[ ,A Mπ[ , then since Sπ[ ,A → Ainf,A is faithfully flat, theSπ[ ,A -module Mπ[ is automatically finite projective (by [Sta, Tag 058S]).
We will find the following basic lemmas useful.
4.2.7 Lemma. Fix a choice of π. Suppose that Minf is a Breuil–Kisin–Fargues
GK -module with A coefficients, where A is a p-adically complete O-algebra whichis topologically of finite type over O. Then Minf descends for some choice of π [(for our fixed π) if and only if it descends for all such choices.
Proof. We can choose an element g ∈ GK with g(π [ ) = (π [ )0 , so that S(π[ )0 ,A =g(Sπ[ ,A ). Then if Mπ is a descent to Sπ,A , g(Mπ ) is a descent to S(π[ )0 ,A .
4.2.8 Lemma. Let A be a p-adically complete O-algebra which is topologicallyof finite type over O. Suppose that Minf is a Breuil–Kisin–Fargues GK -module.
Then if Minf descends for π [ , the Breuil–Kisin module Mπ[ is uniquely determined.
Proof. Assume to begin with that A is an O/$a -algebra of finite type, forGK
some a ≥ 1. By Proposition 2.7.8, the ϕ-module (W (C[ )A ⊗Ainf,A Minf ) π[ ,∞uniquely descends to an étale ϕ-module M over OE,π[ ,A . Thus if M1 , M2 are
two descents of Minf to Sπ[ ,A , then OE,π[ ,A ⊗Sπ[ ,A M1 and OE,π[ ,A ⊗Sπ[ ,A M2
coincide; they are both equal to M .
This allows us to show that M1 + M2 is also a descent of Minf . For this,we have to show that the natural map Ainf,A ⊗Sπ[ ,A (M1 + M2 ) → Minf is
an isomorphism. (Note that it will then follow automatically that M1 + M2
is projective over Sπ[ ,A , by Remark 4.2.6.) That it is a surjection is clear,since it is already a surjection when restricted to each summand. To see that
it is an injection, we may check that its composite with the injection Minf ,→W (C[ )A ⊗Ainf,A Minf is injective. But this composite may be factored as thecomposite of the embedding
Ainf,A ⊗Sπ[ ,A (M1 + M2 ) ,→ Ainf,A ⊗Sπ[ ,A M

86

CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS(obtained by tensoring the embedding M1 +M2 ,→ M with the extension Ainf,Aof Sπ[ ,A , which is flat by Proposition 2.2.14), and the isomorphismAinf,A ⊗Sπ[ ,A M ∼= Ainf,A ⊗Sπ[ ,A OE,π[ ,A ⊗OE,π[ ,A M∼
= W (C[ )A ⊗OE,π[ ,A M ∼= W (C[ )A ⊗Ainf,A Minf .
Since M1 + M2 is also a descent, we may replace M2 by M1 + M2 , and therefore
assume that M1 ⊆ M2 . Since Ainf,A ⊗Sπ[ ,A (M2 /M1 ) = 0, and as Sπ[ ,A →Ainf,A is faithfully flat (again by Proposition 2.2.14), we see that M2 /M1 = 0,
as required.
Suppose now that A is p-adically complete and topologically of finite type.
Let M1 and M2 be two descents of Minf . Since SA → Ainf,A is faithfully flat
(again by Proposition 2.2.14), we find that M1 /$a and M1 /$a both embed
into Minf /$a , for any a ≥ 1, and both provide descents for π [ of Minf /$a .
Applying the uniqueness result we’ve already proved (our coefficients now being
A/$a ), we find that M1 /$a and M2 /$a coincide as submodules of Minf /$a .
Passing to the inverse limit over a, we find that M1 and M2 coincide as submodules of Minf , proving the desired uniqueness in general.

4.3

Canonical extensions of GK∞ -actions

In this section we will consider a fixed choice of π [ , which we will accordinglysdrop from the notation. For each s ≥ 1, write Ks = K(π 1/p ). We now use avariant of the arguments of [CL11], which show that the trivial action of GK∞
on a Breuil–Kisin module with O/$a -coefficients can be extended to some GKs .
We begin with some preliminary lemmas, the first of which is an analogue for
Breuil–Kisin modules of [EG21, Lem. 5.2.14], and is proved in a similar way.
4.3.1 Lemma. Let A be a p-adically complete O-algebra which is topologicallyof finite type, and let M be a finite projective Breuil–Kisin module with Acoefficients of height at most h. Then M is a direct summand of a finite free
Breuil–Kisin module with A-coefficients of height at most h.
Proof. Since the map ΦM : ϕ∗ M → M is an injection with cokernel killed
by E h , there is a map ΨM : M → ϕ∗ M with ΨM ◦ ΦM and ΦM ◦ ΨM bothbeing given by multiplication by E h . Let P be a finite projective SA -module
such that F := M ⊕ P is a finite free SA module. Set Q := P ⊕ M ⊕ P, a finite
projective SA -module.
Note that M ⊕ Q ∼= F ⊕ F is a finite free SA -module, so it suffices to showthat Q can be endowed with the structure of a Breuil–Kisin module of height∼at most h. Since F is free, we can choose an isomorphism ΦF : ϕ∗ F −→ F, and4.3. CANONICAL EXTENSIONS OF GK∞ -ACTIONS

87

we then define ΦQ : ϕ∗ Q → Q as the compositeϕ∗ Q = ϕ∗ P ⊕ ϕ∗ M ⊕ ϕ∗ P∼−→ ϕ∗ M ⊕ ϕ∗ P ⊕ ϕ∗ P
= ϕ∗ F ⊕ ϕ∗ P
ΦF

→ F ⊕ ϕ∗ P∼−→ P ⊕ M ⊕ ϕ∗ PΨM

→ P ⊕ ϕ∗ M ⊕ ϕ∗ P

= P ⊕ ϕ∗ F
ΦF

→ P ⊕ F = Q.

Every morphism in this composite other than ΨM is an isomorphism. SinceΨM ◦ ΦM = ΦM ◦ ΨM = E h , we see that ΨM is an injection with cokernel killed
by E h , so the same is true of ΦQ , as required.
The following lemma is proved by a standard Frobenius amplification argument, which is in particular almost identical to the proof of [CEGS19, Lem.
4.1.9].
4.3.2 Lemma. Let Minf , Ninf be projective Breuil–Kisin–Fargues modules ofheight at most h, where A is a finite type O/$a -algebra. Suppose that N ≥e(a + h)/(p − 1). Letf : Minf → Ninf
be an Ainf,A -linear map, and suppose that for all m ∈ Minf , (f ◦ ΦMinf − ΦNinf ◦ϕ∗ f )(m) ∈ upN Ninf .
Then there is a unique Ainf,A -linear, ϕ-linear morphism f 0 : Minf → Ninfwith the property that for all m ∈ Minf , (f 0 − f )(m) ∈ uN Ninf ; in fact, we have(f 0 − f )(m) ∈ uN +1 Ninf .
Proof. We firstly prove uniqueness. Suppose that f 00 also satisfies the properties
that f 0 does, and write g = f 0 − f 00 , so that by assumption we have that im g ⊆uN Ninf and g ◦ ΦMinf = ΦNinf ◦ ϕ∗ g. We claim that this implies that im g ⊆uN +1 Ninf ; if this is the case, then by induction on N we have im g ⊆ uN Ninf
for all N , and so g = 0 (note that as Ninf is a finite projective Ainf,A -module,
it is u-adically complete and separated).
To prove the claim, note that since im g ⊆ uN Ninf we have im ϕ∗ g ⊆pN infu N . By [EG21, Lem. 5.2.6] (and its proof), and the assumption that Minf
has height at most h, we have im ΦMinf ⊇ ue(a+h−1) Minf . Since g ◦ ΦMinf =ΦNinf ◦ ϕ∗ g, it follows thatue(a+h−1) im g ⊆ im g ◦ ΦMinf = im ΦNinf ◦ ϕ∗ g ⊆ upN Ninfand since Ninf is u-torsion free, we have im g ⊆ upN −e(a+h−1) Ninf . Our assumption on N implies that pN − e(a + h − 1) ≥ N + 1, as required.

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CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS

We now prove the existence of f 0 . For any Ainf,A -linear map h : Minf → Ninf
we set δ(h) := ΦNinf ◦ϕ∗ h−h◦ΦMinf : ϕ∗ Minf → Ninf . We claim that for any s :ϕ∗ Minf → upN Ninf , we can find t : Minf → uN +1 Ninf with δ(t) = s. Given thisclaim, the existence of f 0 is immediate, taking s = δ(f ) and f 0 := f − t, sothat δ(f 0 ) = 0.
To prove the claim, we first prove the weaker claim that for any s as above we
can find h : Minf → uN +1 Ninf with im(δ(h) − s) ⊆ up(N +1) Ninf . Admitting thissecond claim, we prove the first claim by successive approximation: we set t0 =h, so that im(δ(t0 )−s) ⊆ up(N +1) Ninf . Applying the second claim again with N
replaced by N + 1, and s replaced by s − δ(t0 ), we find h : Minf → uN +2 Ninfwith im(δ(h) − (s − δ(t0 ))) ⊆ up(N +2) Ninf . Setting t1 = t0 + h, and proceeding
inductively, we obtain a Cauchy sequence converging (in the u-adically complete
finite Ainf,A -module HomAinf,A (Minf , Ninf )) to the required t.
Finally, we prove this second claim. If h : Minf → uN +1 Ninf then im ϕ∗ h ⊆p(N +1) infu
N , so it suffices to show that we can find h such that h ◦ ΦMinf = −s;but this is immediate, because the cokernel of ΦMinf is killed by ue(a+h−1) , andpN − e(a + h − 1) ≥ N + 1.
The following lemma is proved by a reinterpretation of some of the argumentsof [CL11, §2].
4.3.3 Lemma. For any fixed a, h, and any N ≥ e(a + h)/(p − 1), there isa positive integer s(a, h, N ) with the property that for any finite type O/$a algebra A, any projective Breuil–Kisin module M of height at most h, andany s ≥ s(a, h, N ), there is a unique continuous action of GKs on Minf :=Ainf,A ⊗SA M which commutes with ϕ and is semi-linear with respect to the natural action of GKs on Ainf,A , with the additional property that for all g ∈ GKs
we have (g − 1)(M) ⊂ uN Minf .
Proof. By definition, to give a semi-linear action of GKs on Minf is to give foreach g ∈ GKs a morphism of Breuil–Kisin–Fargues modules
βg : g ∗ Minf → Minf
with the property that for all g, h ∈ GKs , we have βgh = βh ◦ h∗ βg . Now,
the requirement that (g − 1)(M) ⊆ uN Minf implies that if we have two suchmorphisms βg , βg0 then (βg − βg0 )(g ∗ Minf ) ⊆ uN Minf , so that βg0 = βg by theuniqueness assertion of Lemma 4.3.2.
We now use the existence part of Lemma 4.3.2 to construct the required
action. Since the homomorphism SA ,→ Ainf,A takes u to the Teichmüller liftnof (π 1/p )n , we can and do choose s(a, h, N ) sufficiently large that for all s ≥s(a, h, N ) and g ∈ GKs , we have g(u) − u ∈ upN Ainf,A ; thus for all λ ∈ SA , wehave g(λ) − λ ∈ upN Ainf,A .
We claim that for each g ∈ GKs , we may define an Ainf,A -linear map
αg : g ∗ Minf → Minf
with the following two properties:

4.3. CANONICAL EXTENSIONS OF GK∞ -ACTIONS

89• (αg ◦ Φg∗ Minf − ΦMinf ◦ ϕ∗ αg )(ϕ∗ g ∗ Minf ) ⊆ upN Minf .
• For each m ∈ M, we have αg (1 ⊗ m) − m ∈ uN Minf .
It suffices to construct such maps in the case when M is free; indeed, byLemma 4.3.1, we may write F = M ⊕ P where F, P are respectively free andprojective Breuil–Kisin modules of height at most h, and given a morphism αg :g ∗ Finf → Finf satisfying our requirements (where we write Finf = Ainf,A ⊗SA F),
we may obtain the desired morphism αg : g ∗ Minf → Minf by projection onto
the corresponding direct summands.
If then F is a free Breuil–Kisin module of height at most h, we choose abasis f1 , . . . , fd for F, and then, for each g ∈ GKs , we define an Ainf,A -linearmap
αg : g ∗ Finf → Finf
(which depends on our choice of basis) byXXαg (λ i ⊗ fi ) =λi fi .
ii

P
We now check that αg has the claimed properties. If ΦF (1 ⊗ fi ) = j θi,j fj (soPthat θi,j ∈ SA ), then for any i λi ⊗ fi ∈ ϕ∗ g ∗ Finf , we compute thatXX
λ i ⊗ fi ) =λi (g(θi,j ) − θi,j )fj ∈ upN Finf ,(αg ◦ Φg∗ Finf − ΦFinf ◦ ϕ∗ αg )(iwhile for anyi,jPi λi fi ∈ F (note then that each λi ∈ SA ), we compute thatX
 X
αg 1 ⊗ (λ i fi ) −
λi fi
i

i

X
X
X
= αg (g(λi ) ⊗ gi ) −λi gi =(g(λi ) − λi )gi ∈ uN Finf ;iii

in both computations we have used the fact that, by our choice of s, if λ ∈ SA ,then g(λ) − λ ∈ upN Finf ⊆ uN Finf .
Returning to the general case of a projective Breuil–Kisin module M ofheight at most h, it follows from Lemma 4.3.2 that there is a unique morphismof Breuil–Kisin–Fargues modules
βg : g ∗ Minf → Minf
with im(βg − αg ) ⊆ uN Ainf,A ⊗SA M. Since we have αgh = αh ◦ h∗ αg , it follows
from this uniqueness that βgh = βh ◦ h∗ βg , so we have constructed an actionof GKs on Minf . It remains to check that (g − 1)(M) ⊆ uN Minf ; for this, note
that, for any m ∈ M, we have(g − 1)m = βg (1 ⊗ m) − m = (βg − αg )(1 ⊗ m) + αg (1 ⊗ m) − m ∈ uN Minf ;here we have used the fact im(βg − αg ) ⊆ uN Ainf,A ⊗SA M, together with thesecond of the properties satisfied by the maps αg .

90

4.4

CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS

Breuil–Kisin–Fargues GK -modules and canonical actions

In an earlier draft of this book, we claimed to show that for Breuil–Kisin–Fargues
GK -modules admitting descents for all π [ , the actions of the GKπ[ ,s ⊆ GK (forsufficiently large values of s) are necessarily the canonical actions considered in
Section 4.3. We then used this claim to establish finiteness properties of thestacks of Breuil–Kisin–Fargues GK -modules that we consider later in this chapter. We are grateful to Dat Pham for pointing out a mistake in our argument,
and we do not know whether this is in fact automatic. However, the following
result of Caruso–Liu (which motivated our consideration of canonical actions in
the first place) shows that the Breuil–Kisin–Fargues GK -modules coming frompotentially semistable GK -representations do necessarily have this property; so
we can (and do) impose this as an extra condition in the definition of our stacks.
The following lemma is essentially the content of [CL11, §3] (although they
are concerned with getting precise bounds on the size of s, which is unimportantfor us). Recall that, by Theorem F.11, to any O-lattice in a semistable Erepresentation of GK with Hodge–Tate weights in [0, h] (for some integer h ≥ 0),
there corresponds a Breuil–Kisin–Fargues GK -module Minf of height at most hwhich admits all descents.
4.4.1 Proposition. For any fixed K, a, h and N ≥ e(a + h)/(p − 1) there is aconstant s0 (K, a, h, N ) ≥ s(a, h, N ) (where s(a, h, N ) is as in Lemma 4.3.3)
with the following property: for any Breuil–Kisin–Fargues GK -module Minfcorresponding as above to a semistable GK representation with Hodge–Tateweights in [0, h], and for any choice of π [ with corresponding descent Mπ[ ,if s ≥ s0 (K, a, h, N ), then the restriction to GKπ[ ,s of the action of GK on
Minf ⊗O O/$a agrees with the action obtained from Mπ[ by Lemma 4.3.3.
Proof. This is essentially the content of [CL11, Prop. 3.3.1]. By the statementof Lemma 4.3.3, it is enough to check that if s is sufficiently large, then for
all g ∈ GKπ[ ,s we have (g − 1)(Mπ[ ⊗O O/$a ) ⊂ uN Ainf ⊗S Mπ[ ⊗O O/$a ,where the action of g is via the given action of GK on Minf . This is immediatefrom [CL11, Lem. 3.2.1] (taking L = Mπ[ , and noting that in their notation,b is an extension of scalars of L to a subring of Ainf,O/$a ; this extension ofLscalars involves a twist by ϕ, but since ϕ is a continuous automorphism of Ainf ,this is harmless).

4.5

Stacks of semistable and crystalline Breuil–Kisin—Fargues modules

We now define moduli stacks parameterizing the Breuil–Kisin–Fargues GK modules admitting all descents that were introduced in Definition 4.2.4. As
discussed in Section 4.4, we also impose a condition that the GK -actions agree

4.5. STACKS OF SEMISTABLE AND CRYSTALLINE BKF MODULES 91
with the canonical actions introduced in Section 4.3. To this end, we make thefollowing definition.
4.5.1 Definition. For each K, a, h and N ≥ e(a + h)/(p − 1), we fix a constants0 (K, a, h, N ) as in the statement of Proposition 4.4.1. Let A be a p-adicallycomplete O-algebra which is topologically of finite type, and let Minf be a projective Breuil–Kisin–Fargues GK -module with A-coefficients which admits alldescents. Then we say that the GK -action on Minf is canonical if for any a ≥ 1,
any π [ , and any s ≥ s0 (K, a, h, N ), the restriction to GKπ[ ,s of the action of GK
on Minf ⊗O O/$a agrees with the action obtained from Mπ[ by Lemma 4.3.3.
4.5.2 Remark. The point of this definition is that it is immediate from Proposition 4.4.1 that the Breuil–Kisin–Fargues modules coming from lattices insemistable GK -representations have canonical GK -actions.
We now define our moduli stacks of Breuil–Kisin–Fargues GK -modules.
a4.5.3 Definition. For any h ≥ 0 we let Cd,ss,hdenote the limit preservingacategory of groupoids over Spec O/$ determined by decreeing, for any finiteatype O/$a -algebra A, that Cd,ss,h(A) is the groupoid of rank d projective Breuil–Kisin–Fargues GK -modules with A-coefficients, which are of height at most h,which admit all descents, and whose GK -action is canonical.
aadenote the limit preserving subcategory of groupoids of Cd,ss,hWe let Cd,crys,hconsisting of those Breuil–Kisin–Fargues GK -modules Minf which are furtheraamore crystalline. We let Cd,ss,h := lima Cd,ss,h, and let Cd,crys,h := lima Cd,crys,h.
−→−→4.5.4 Remark. This definition uniquely determines limit preserving categories
fibred in groupoids over O by [EG21, Lem. 2.5.4]. We will see shortly that theinductive limits Cd,ss,h and Cd,crys,h are in fact p-adic formal algebraic stacks(Theorem 4.5.20).
If A is a p-adically complete O-algebra, then as usual we write Cd,ss,h (A)(resp. Cd,crys,h (A)) to denote the groupoid of morphisms from Spf A (definedusing the p-adic topology on A) to Cd,ss,h (resp. Cd,crys,h ). More concretely, wehaveaCd,ss,h (A) = lim Cd,ss,h(A/$a ),←−aand similarly for Cd,crys,h (A). We will typically apply this convention in the casewhen A is furthermore topologically of finite type over O. In this case each of thequotients A/$a is of finite type over O/$a (by the definition of topologically ofafinite type), and so the objects of Cd,ss,h(A/$a ) admit a concrete interpretationas Breuil–Kisin–Fargues GK -modules over A. The objects of Cd,ss,h (A) and ofCd,crys,h (A) then similarly admit a concrete interpretation.
4.5.5 Lemma. Let A be a p-adically complete O-algebra which is topologicallyof finite type. Then Cd,ss,h (A) is the groupoid of rank d projective Breuil–Kisin–
Fargues GK -modules with A-coefficients, which are of height at most h, which
admit all descents, and whose GK -action is canonical. In addition, Cd,crys,h (A)is the subgroupoid consisting of those Breuil–Kisin–Fargues GK -modules Minf
which are furthermore crystalline.

92

CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKSProof. This follows from Lemma D.5.
The following lemma is crucial: it shows that Breuil–Kisin–Fargues modules
which admit all descents give rise to (ϕ, Γ)-modules.
4.5.6 Lemma. There is a natural morphism Cd,ss,h → Xd , which on Spf Apoints (for p-adically complete O-algebras A that are topologically of finite type)is given by extending scalars to W (C[ )A .
Proof. As indicated in the statement of the lemma, this morphism is defined,
for finite type O/$a -algebras, via Minf 7→ W (C[ )A ⊗Ainf,A Minf , with the target
object being regarded as an A-valued point of Xd via the equivalence of Proposition 2.7.8. Since both Cd,ss,h and Xd are limit preserving (the former by its very
construction, and the latter by Lemma 3.2.19), it follows from [EG21, Lem. 2.5.4,
Lem. 2.5.5 (1)] that this construction determines a morphism Cd,ss,h → Xd .
4.5.7 Remark. The proof of Lemma 4.5.6 illustrates a general principle (which
was also applied in the proofs of the results in Section 3.7), namely that to construct a morphism between limit preserving stacks over Spec O/$a , or over Spf O,
it suffices to define the intended morphism on finite type O/$a -algebras (witha either fixed or varying, depending on which case we are in).
Similarly, if P is any property of such a morphism that is preserved underbase-change, and that is tested by pulling back over A-valued points, then to
test for the property P , it suffices to consider such pull-backs in the case when
A is of finite type over O/$a .
We will apply these principles consistently throughout the remainder of thissection.
For any h ≥ 0 and any choice of π [ , we write Cπ[ ,d,h for the moduli stack ofrank d projective Breuil–Kisin modules over Sπ[ ,A of height at most h, and Rπ[ ,dfor the corresponding stack of rank d projective étale ϕ-modules. We also write
Cπa[ ,d,h and Raπ[ ,d for their base-change over O/$a , for any a ≥ 1.
Recall that, by Proposition 3.7.2, we have a natural morphism XK,d → Rπ[ ,d ,which for each s ≥ 1 can be factored, via Lemma 3.7.5, as
XK,d → XKπ[ ,s ,d → Rπ[ ,d .
4.5.8 Proposition. For any fixed a, h, and any N and s(a, h, N ) as in Lemma 4.3.3,
athen for any s ≥ s(a, h, N ) there is a canonical morphism Cπa[ ,d,h → XK,dπ [ ,sobtained from the canonical action of Lemma 4.3.3. This morphism fits into acommutative triangle
Cπa[ ,d,h
z
a
XK
[ ,d
π ,s


/ Ra [

π ,d4.5. STACKS OF SEMISTABLE AND CRYSTALLINE BKF MODULES 93
Proof. As discussed in Remark 4.5.7, it is enough to show that if A is a finite type O/$a -algebra, and M is a finite projective étale Breuil–Kisin modulewith A-coefficients, then we can canonically extend the action of GKπ[ ,∞ on
W (C[ )A ⊗Sπ[ ,A M to an action of GKπ[ ,s , if s ≥ s(a, h, N ). Such an extension
is provided by Lemma 4.3.3.
a4.5.9 Lemma. The morphism Cπa[ ,d,h → XK
,d of Proposition 4.5.8 is repreπ [ ,ssentable by algebraic spaces, proper, and of finite presentation.

Proof. This follows by a standard graph argument. Namely, we write this morphism as the compositeaaCπa[ ,d,h → Cπa[ ,d,h ×Ra [ XK
→ XK
.
[ ,d
[ ,d
π ,d

π ,s

π ,s

The first arrow is a closed immersion, being the base change of the diagonal
morphism Xd → Xd ×Rπ[ ,d Xd , which is a closed immersion by Proposition 3.7.4.
The second arrow is representable by algebraic spaces and proper, because itis the base change of the morphism Cπa[ ,d,h → Raπ[ ,d of Theorem 3.1.4. The
composite morphism is thus representable by algebraic spaces and proper. Sucha morphism is in particular of finite type and separated (and so also quasiseparated), and hence Lemma 4.5.10 below allows us to conclude that it is alsoof finite presentation (once we take into account Lemma 3.2.19, which showsthat its target is limit preserving).
4.5.10 Lemma. Let Z ,→ X be a morphism of stacks over a locally Noetherianbase scheme S which is representable by algebraic spaces, of finite type, andquasi-separated. If X is limit preserving, then this morphism is furthermore offinite presentation.
Proof. It is enough to show, for any morphism T → X whose source is an affine
S-scheme T , that the pulled back morphism (of algebraic spaces) T ×X Z → T
is of finite presentation. Since X is limit preserving, the morphism T → X
factors through a morphism of S-schemes T → T 0 , where T 0 is affine and offinite presentation over S. Thus, replacing T by T 0 , we may assume that T isof finite presentation over S, and consequently Noetherian.
The base-changed morphism T ×X Z → T is thus a finite type and quasiseparated morphism from an algebraic space to a Noetherian scheme, and hence
is of finite presentation by [Sta, Tag 06G4]. This proves the lemma.
4.5.11 Definition. The morphisms of Lemma 3.7.5 and Proposition 4.5.8 allow
us to define a stack Cπa[ ,s,d,h by the requirement that it fits into a 2-Cartesian
diagram/ C a[
Cπa[ ,s,d,h(4.5.12)
π ,d,h

a
XK,d

/ Xa



Kπ[ ,s ,d94

CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKSThe stack Cπa[ ,s,d,h classifies rank d projective Breuil–Kisin modules M of heightat most h over Sπ[ ,A equipped with an extension of the canonical GKπ[ ,s -action
on W (C[ )A ⊗Sπ[ ,A M to an action of GK (making it an étale (ϕ, GK )-module).
Since the lower horizontal arrow in (4.5.12) is representable by algebraic
spaces and of finite presentation, by Lemma 3.7.5, and since Cπa[ ,d,h is analgebraic stack of finite presentation over O/$a , by Theorem 3.1.4, we seethat Cπa[ ,s,d,h is also an algebraic stack of finite presentation over O/$a . Since
the right hand vertical arrow in this diagram is representable by algebraic spaces,
proper, and of finite presentation, by Lemma 4.5.9, so is the left hand verticalarrow.
4.5.13 Lemma. The diagonal of Cπa[ ,s,d,h is affine and of finite presentation.
Proof. As we already noted, the morphism Cπa[ ,s,d,h → XK,d is representable byalgebraic spaces and proper (and so in particular is both separated and of finitetype); the claim of the lemma thus follows from Proposition 3.2.17, together
with Lemma 4.5.14 below.
4.5.14 Lemma. Suppose that f : X → Y is a morphism of stacks over a basescheme S which is representable by algebraic spaces, separated, and of finite
type. Suppose also that the diagonal ∆Y : Y → Y ×S Y is affine and of finite
presentation. Then the diagonal ∆X : X → X ×S X is affine and of finite
presentation.
Proof. The diagonal morphism
X → X ×S X
may be factored as the composite of the relative diagonal
X → X ×Y X

(4.5.15)

X ×Y X → X ×S X ,(4.5.16)

and the morphism
which is a base change of the diagonal morphism ∆Y . Our assumption on ∆Y ,then, implies that the morphism (4.5.16) is affine and of finite presentation.
Since X → Y is representable by algebraic spaces and separated, its diagonal (4.5.15) is a closed immersion, and thus affine. Since it is furthermore offinite type, the morphism (4.5.15) is also of finite presentation, by Lemma 3.2.7.
Thus ∆X is the composite of morphisms that are affine and of finite presentation, and the lemma follows.
4.5.17 Proposition. For each π [ and each s ≥ s0 (K, a, h, N ), there are nataaaural closed immersions Cd,crys,h→ Cd,ss,h→ Cπa[ ,s,d,h . In particular, Cd,ss,haand Cd,crys,hare algebraic stacks of finite presentation over O/$a , and haveaffine diagonals.

4.5. STACKS OF SEMISTABLE AND CRYSTALLINE BKF MODULES 95
Proof. We have already observed that Cπa[ ,s,d,h is of finite presentation over O/$a ,and has affine diagonal by Lemma 4.5.13. Once we construct the claimed closed
immersions, the claims of finite presentation will follow from Lemma 4.5.10;
and since closed immersions are in particular monomorphisms, the diagonalsof Cd,ss,h and Cd,crys,h are then obtained via base change from the diagonalof Cπa[ ,s,d,h , and are in particular affine. We thus focus on constructing theseclosed immersions. Throughout the proof, we take into Remark 4.5.7, whichallows us to restrict our attention to the points of the various stacks in questionthat are defined over finite type O/$a -algebras.
aa→ XK,d, given by extending
Lemma 4.5.6 constructs a morphism Cd,ss,h[scalars to W (C ), and it follows from Lemma 4.2.8 that there is a natural moraphism Cd,ss,h→ Cπa[ ,d,h , defined via Minf 7→ Mπ[ . The composite morphisms
a
a
a
a
a
a
Cd,ss,h → Cπ[ ,d,h → XK
and Cd,ss,h→ XK,d→ XK
coincide by defini[ ,d
[ ,d
π ,s

π ,s

tion (see Definitions 4.5.1 and 4.5.3). Thus these morphisms induce a morphismaCd,ss,h→ Cπa[ ,s,d,h .

(4.5.18)

To see that (4.5.18) is a monomorphism, it is enough to note that if A is
any finite type O/$a -algebra A, and Minf is any Breuil–Kisin–Fargues module over A which admits all descents and whose GK -action is canonical, cora
responding to an A-valued point of Cd,ss,h, then Minf = Ainf,A ⊗Sπ[ ,A Mπ[ isdetermined by Mπ[ , and the GK -action on Minf is determined by the GK -action
on W (C[ )A ⊗Ainf,A Minf .
We now show that (4.5.18) is a closed immersion. It is enough to do thisafter pulling back to some finite type O/$a -algebra A, where we need to showthat the conditions that Minf = Ainf,A ⊗Sπ[ ,A Mπ[ is GK -stable and admits alldescents, and that the GK -action is canonical, are closed conditions. We begin
with the first of these. It is enough to show that for each g ∈ GK , the condition
that g(Minf ) ⊂ Minf is closed. This condition is equivalent to the vanishing of
the composite morphismg ∗ Minf → Minf [1/u] → Minf [1/u]/Minf ,aand this is a closed condition by Lemma B.28. Let Cd,Gdenote the closed
Kainf
substack of Cπ[ ,s,d,h for which M is GK -stable.

We next show that for each choice of uniformizer π 0 , and each (π 0 )[ , the condition that Minf admits a descent to SA,(π0 )[ is a closed condition. To this end,
note that by Proposition 2.7.8, Minf [1/u] descends uniquely to S(π0 )[ ,A [1/u], soaawe have a morphism Cd,G
→ Ra(π0 )[ ,d . Then the fibre product Cd,G×R a 0 [KK(π ) ,da0C(π0 )[ ,d,h is the moduli space of pairs (M, M ) where M := Mπ [ is as above (anaobject classified by Cd,G,K), and M0 is a Breuil–Kisin module for S(π0 )[ ,A ofheight at most h which satisfies

W (C[ )A ⊗S(π0 )[ ,A M0 = W (C[ )A ⊗Sπ[ ,A M.

96

CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKSaConsider the substack Cd,G
0 [ of this fibre product satisfying the condition
K ,(π )
that
Ainf,A ⊗S(π0 )[ ,A M0 = Minf .
(4.5.19)

Note that this is exactly the condition that Minf admits a descent for (π 0 )[ , andthat M0 is this descent (which is uniquely determined by Lemma 4.2.8), so theaaprojection Cd,G
0 [ → Cd,GK is a monomorphism, and we need to show that
K ,(π )aaa
it is a closed immersion. Since the projection Cd,G×Ra 0 [ C(π0 )[ ,d,h → Cd,GKK(π ) ,daais proper (being a base change of the proper morphism C(π0 )[ ,d,h → R(π 0 )[ ,d ),and since proper monomorphisms are closed immersions, it is enough to show
a
a
a
a
C(π
that Cd,G
0 )[ ,d,h ; that is, we must0 [ is a closed substack of Cd,GK ×R 0 [K ,(π )(π ) ,dshow that (4.5.19) is a closed condition. This again follows from Lemma B.28.
We now consider the closed substack of Cπa[ ,s,d,h for which Minf is GK -stable
and admits a descent for each (π 0 )[ . We need to show that for each (π 0 )[ , thefurther conditions thatMπ[ /[π [ ]Mπ[ = M(π0 )[ /[(π 0 )[ ]M(π0 )[andϕ∗ Mπ[ /Eπ[ ϕ∗ Mπ[ = ϕ∗ M(π0 )[ /E(π0 )[ ϕ∗ M(π0 )[are closed conditions.
The arguments in both cases are very similar (and in turn are similar tothe proof of Lemma B.28), so we only give the argument in the second case,leaving the first to the reader. Each side is a projective OK ⊗Zp A-submoduleof the projective OC ⊗Zp A-module OC ⊗Ainf,A ,θ ϕ∗ Minf , and indeed spans
this module after extension of scalars to OC . By symmetry, it is enough toshow that for each element m ∈ ϕ∗ M(π0 )[ /E(π0 )[ ϕ∗ M(π0 )[ , the condition thatm ∈ ϕ∗ Mπ[ /Eπ[ ϕ∗ Mπ[ is closed.
Write P := ϕ∗ Mπ[ /Eπ[ ϕ∗ Mπ[ , and choose a finite projective OK ⊗Zp Amodule Q such that F := P ⊕ Q is free. We can think of m as an element of
OC ⊗OK P and thus as an element of OC ⊗OK F , and it is enough to check thatthe condition that m ∈ F is closed. Choosing a basis for F , we reduce to thecase that F is one-dimensional, and thus to showing that the condition that anelement of OC ⊗Zp A lies in OK ⊗Zp A is closed. Since OC is a torsion-free andthus flat OK -module, OC /$a is flat and thus free as a module for the Artinianring OK /$a . Thus OC ⊗Zp A is a free OK ⊗Zp A-module, and the result follows
by choosing a basis.
We now need to show that the condition that the GK -action is canonicalis a closed condition. Explicitly, by Lemma 4.3.3, we need to show that foreach b ≤ a, each N ≥ e(b + h)/(p − 1), each s ≥ s0 (K, b, h, N ), each π [ , andeach g ∈ GKπ[ ,s , the condition that(g − 1)(Mπ[ ⊗O O/$b ) ⊂ uN (Minf ⊗O O/$b )is closed. This follows from Lemma B.29.

4.5. STACKS OF SEMISTABLE AND CRYSTALLINE BKF MODULES 97aa
Finally, it remains to show that the monomorphism Cd,crys,h→ Cd,ss,his aclosed immersion; that is, for each g ∈ GK and each π [ we need to show thatthe condition that(g − 1)(Mπ[ ) ⊆ ϕ−1 (µ)[π [ ]Minf

is a closed condition. It is enough to show that for each m ∈ Mπ[ , the condition
that (g − 1)(m) ∈ ϕ−1 (µ)[π [ ]Minf is closed. Since ϕ−1 (µ)[π [ ]Minf is a finite
projective Ainf,A -module, this follows from another application of Lemma B.29.
4.5.20 Theorem. Cd,ss,h is a p-adic formal algebraic stack of finite presentation, as is its closed substack Cd,crys,h . In addition, both of these stacks haveaffine diagonal, and each of the morphisms Cd,ss,h → XK,d and Cd,crys,h → XK,dis representable by algebraic spaces, proper, and of finite presentation.
Proof. Since Cd,crys,h is a closed substack of Cd,ss,h , it suffices to prove the statements of the lemma for Cd,ss,h . (Here we use the fact that a closed immersion
is a monomorphism to see that the diagonal of Cd,crys,h is obtained via basechange from the diagonal of Cd,ss,h , while we use Lemma 4.5.10 to transfer finite
presentation properties from Cd,ss,h to Cd,crys,h ).
By Proposition A.13, to see that Cd,ss,h is a p-adic formal algebraic stack ofafinite presentation, it is enough to show that each Cd,ss,his an algebraic stack ofafinite presentation over O/$ , which was proved in Proposition 4.5.17. Sinceaeach Cd,ss,hhas affine diagonal, by the same proposition, we see as well that
Cd,ss,h has affine diagonal.
That Cd,ss,h → XK,d is representable by algebraic spaces, proper, and offinite presentation also follows from Proposition 4.5.17. Indeed, for each a theaaamorphism Cd,ss,h→ XK,d factors as Cd,ss,h→ Cπa[ ,s,d,h → XK,d, where the firstmorphism is a closed immersion of finite type algebraic stacks, so it is enoughato prove the same properties for the morphism Cπa[ ,s,d,h → XK,d; as explainedin Definition 4.5.11, this follows from Lemma 4.5.9.

4.5.21

Potentially crystalline and potentially semistable
stacksWe now consider the corresponding potentially semistable and potentially crystalline versions of these moduli stacks of Breuil–Kisin–Fargues modules.
L/K

4.5.22 Definition. For a Galois extension L/K, define stacks Cd,ss,h and
L/K

L/K,a

Cd,crys,h as follows: for each a ≥ 1 we let Cd,ss,h denote the limit preserving
category of groupoids over Spec O/$a determined by decreeing, for any finite
L/K,atype O/$a -algebra A, that Cd,ss,h (A) is the groupoid of Breuil–Kisin–Fargues
GK -modules with A-coefficients, which are of height at most h, which admit alldescents over L, and whose GL -actions are canonical.
L/K,aa
We let Cd,crys,h denote the limit preserving subcategory of groupoids of Cd,ss,hconsisting of those Breuil–Kisin–Fargues GK -modules Minf for which the actionof GL is crystalline.

98

CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS
L/K

L/K,a

L/K

L/K,a

We let Cd,ss,h := lima Cd,ss,h , and let Cd,crys,h := lima Cd,crys,h .
−→−→The following lemma is proved in exactly the same way as Lemma 4.5.5.
4.5.23 Lemma. Let A be a p-adically complete O-algebra which is topologicallyL/Kof finite type over O. Then Cd,ss,h (A) is the groupoid of Breuil–Kisin–Fargues
GK -modules with A-coefficients, which are of height at most h, and which adL/K
mit all descents over L, and whose GL -actions are canonical; and Cd,crys,h (A)is the subgroupoid of those Breuil–Kisin–Fargues GK -modules Minf which arefurthermore crystalline.
L/K

4.5.24 Proposition. For any Galois extension L/K, any d, and any h, Cd,ss,hL/K

is a p-adic formal algebraic stack of finite presentation, and Cd,crys,h is a closed
substack, which is thus again a p-adic formal algebraic stack of finite presenL/K
L/K
tation. Both Cd,ss,h and Cd,crys,h have affine diagonal, and the natural morL/K

L/K

phisms Cd,ss,h → XK,d and Cd,crys,h → XK,d are representable by algebraic spaces,
proper, and of finite presentation.
L/K

L/L

Proof. We have a natural morphism Cd,ss,h → Cd,ss,h (given by restricting theaction of GK to GL ; the target is of course just the stack denoted Cd,ss,h above,L/K
but for L instead of K), and a natural morphism Cd,ss,h → XK,d , and thus anatural morphism
L/K
L/L
Cd,ss,h → Cd,ss,h ×XL,d XK,d(4.5.25)
We claim that (4.5.25) is a closed immersion. Given this, the proposition
L/L
follows. Indeed, by Theorem 4.5.20, Cd,ss,h is a p-adic formal algebraic stackL/Lof finite presentation, and the morphism Cd,ss,h → XL,d is representable byalgebraic spaces, proper, and of finite presentation. Furthermore, XK,d → XL,dis representable by algebraic spaces and of finite presentation by Lemma 3.7.5.
L/K
L/K
That Cd,crys,h is a closed substack of Cd,ss,h follows from Proposition 4.5.17. The
claims on the diagonals are proved by appeal to Lemma 4.5.14 (cf. the proof of
Lemma 4.5.13).
It remains to prove the claim. It suffices to prove this after pulling back via amorphism Spec A → XL,d for some finite type O/$a -algebra A. Unwinding thedefinitions, we have to show that if Minf is a Breuil–Kisin–Fargues GL -modulewith A-coefficients, with a compatible action of GK on W (C[ )A ⊗Ainf,A Minf ,then the condition that Minf is GK -stable is a closed condition. It suffices toshow that for each g ∈ GK , the condition that g(Minf ) ⊆ Minf is a closed
condition; as in the proof of Proposition 4.5.17, this follows from Lemma B.28,
applied to the composite morphismg ∗ Minf → Minf [1/u] → Minf [1/u]/Minf .
We end this section with the following lemma, which will be used in the
proof of Proposition 4.8.10.

4.5. STACKS OF SEMISTABLE AND CRYSTALLINE BKF MODULES 99
4.5.26 Lemma. Let R be a complete local Noetherian O-algebra with residueb := C L/K ×X
field F, together with a morphism Spf R → XK,d , and let CK,dd,ss,hSpf R. Then there is a projective morphism of schemes C → Spec R whosebmR -adic completion is isomorphic to C.
L/K

Proof. Note that since Cd,ss,h → XK,d is proper and representable by algebraic
b is a formal algebraic space, and the morphism C
b → Spf R
spaces, we see that C
b is a properis proper and representable by algebraic spaces. In particular, CmR -adic formal algebraic space over Spf R.
b we will use an argument of Kisin, see
To show that we can algebraize C,[Kis09b, Prop. 2.1.10], which proves a similar statement for moduli of Breuil–
L/K
Kisin modules. In order to use this, we show that we can realise Cd,ss,h as aclosed substack of a product of moduli stacks of Breuil–Kisin modules.
Let π, π 0 be two choices of uniformizers of L, chosen such that π/π 0 is not apth power, and choose compatible systems of p-power roots π [ , (π 0 )[ . Note thatthe closure of the subgroup of GL generated by GLπ[ ,∞ and GL(π0 )[ ,∞ is just GL ,because its fixed field is contained in Lπ[ ,∞ ∩ L(π0 )[ ,∞ = L, so that a continuous
action of GL is determined by its restrictions to GLπ[ ,∞ and GL(π0 )[ ,∞ .
Suppose firstly that L = K. We have a natural morphism
Cd,ss,h → Cπ[ ,d,h ×Spf O C(π0 )[ ,d,h ,(4.5.27)

which we claim is a monomorphism. To see this, it is enough to note that Minf =Ainf,A ⊗Sπ[ ,A Mπ[ is determined by Mπ[ , while the GK -action on Minf is determined by the actions of GKπ[ ,∞ and GK(π0 )[ ,∞ , which are in turn determinedby the conditions that they act trivially on Mπ[ and M(π0 )[ respectively.
We may factor (4.5.27) as the composite
Cd,ss,h → (Cπ[ ,d,h ×Spf O C(π0 )[ ,d,h )×(Rπ[ ,d ×Spf O R(π0 )[ ,d ) XK,d → Cπ[ ,d,h ×Spf O C(π0 )[ ,d,h .
Since the composite is a monomorphism, the first morphism is a monomorphism.
This first morphism is also proper (since the morphism Cd,ss,h → XK,d obtainedby composing it with the projection to XK,d is proper, by Theorem 4.5.20,
while this projection is separated, being a base-change of the product of themorphisms C(π)[ ,d,h → Rπ[ ,d and C(π0 )[ ,d,h → R(π0 )[ ,d , each of which is proper,
and so in particular separated, by Theorem 3.1.4) and so it is in fact a closed
b is a closed algebraic subspace of
immersion. Thus C(Cπ[ ,d,h ×Rπ[ ,d Spf R) ×Spf R (C(π0 )[ ,d,h ×R(π0 )[ ,d Spf R).
By the Grothendieck Existence theorem for algebraic spaces [Knu71, Thm.
V.6.3] (and the symmetry between π [ and (π 0 )[ ), we are therefore reduced toshowing that the morphism Cπ[ ,d,h ×Rπ[ ,d Spf R → Spf R can be algebraized toa projective morphism. In the case h = 1, this is [Kis09b, Prop. 2.1.10] (bearing
in mind the main result of [BL95], as in the proof of [Kis09b, Prop. 2.1.7]), andthe case of general h can be proved in exactly the same way, as explained in the

100 CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS
proof of [Kis10, Prop. 1.3]; the key point is that Cπ[ ,d,h ×Rπ[ ,d Spf R inherits anatural very ample (formal) line bundle from the affine Grassmannian.
We now consider the case of a general finite Galois extension L/K, where
we argue as in the proof of Proposition 3.7.5. Let {gi }i=1,...,n be a set of coset
L/K
representatives for GL in GK . By definition, to give an object of Cd,ss,h (A) (forL/Lsome A) is the same as giving an object of Cd,ss,h (A), together with an extensionof the action of GL on the underlying Breuil–Kisin–Fargues module Minf
A to anaction of GK . Now, for each i we can give gi∗ MinfthestructureofaBreuil–A
Kisin–Fargues module by letting h ∈ GL act as gi−1 hgi acts on Minf;itfollows
A
L/L∗
inf
from the definitions that gi MA is also an object of Cd,ss,h (A). Then to give an
extension of the action of GL on Minf
A to an action of GK is to give for each i
L/L
∗
inf ∼
inf
an isomorphism gi MA −→ MA of objects of Cd,ss,h (A), satisfying a slew ofcompatibilities.
L/L
We let Yi denote the stack classifying objects Minf
A of Cd,ss,h (A), endowed∗
infwith an isomorphism gi∗ Minf
A → MA . If we regard gi as an automorphism
L/L
of Cd,ss,h (A), then we may form its graph Γi , and we then have an isomorphismof stacks∼
L/L
L/L
Yi −→ Cd,ss,h (A) ×∆,C L/L (A)×C L/L (A),Γi Cd,ss,h (A).
d,ss,hd,ss,h(Here ∆ denotes the diagonal; cf. the definition of RΓd disc in Section 3.2 above,
together with Proposition 3.2.14.) The projection onto the second factor Yi →L/L
Cd,ss,h (A), which corresponds to forgetting the isomorphism, is a base-changeof the diagonal ∆, and so is affine, since ∆ is affine by Theorem 4.5.20. In
particular, Yi ×XL,d Spf R admits a natural ample formal line bundle, pulled
L/Lback from the natural very ample formal line bundle on Cd,ss,h ×XL,d Spf R,
whose existence we established above.
L/K
We can rephrase our interpretation of objects of Cd,ss,h (A) as objects of
L/Linf
Cd,ss,h (A) endowed with isomorphisms gi∗ Minf
A → MA satisfying compatibilities
as the existence of a closed immersion
L/K

Cd,ss,h ,→ Y1 ×C L/L × · · · ×C L/L Ynd,ss,hd,ss,h

L/K(the compatibilities that cut out Cd,ss,h arise as base-changes of the double diagL/K

onal, and so impose closed conditions). It follows that Cd,ss,h ×XL,d Spf R inherits
a natural ample formal line bundle. As noted in the proof of Lemma 3.7.5, since
the morphism XK,d → XL,d is affine, it has affine diagonal, so that the naturalmorphism
L/K

L/K

Cd,ss,h ×XK,d Spf R → Cd,ss,h ×XL,d Spf R
L/K

is affine. It follows that that Cd,ss,h ×XK,d Spf R in turn inherits a natural ample
formal line bundle (see e.g. [Sta, Tag 0892]), and the result then follows fromanother application of [Knu71, Thm. V.6.3].

4.6. INERTIAL TYPES

4.6

101

Inertial types

In this section we examine how to extract the inertial type of the Galois representation associated to a Breuil–Kisin–Fargues GK -module; in Section 4.7 westudy the same problem for Hodge–Tate weights, and in Section 4.8 we use
these results to define our moduli stacks of potentially semistable and potentially crystalline representations of fixed inertial and Hodge type.
In contrast to the rest of the book, in this and the following sections wewill need to consider Kisin modules with coefficients with p inverted. To thisend, we will write A◦ and B ◦ for p-adically complete flat O-algebras which
are topologically of finite type over O, and write A = A◦ [1/p], B = B ◦ [1/p].
We hope that this change of notation will not cause any confusion. We will
freely use that if A◦ is topologically of finite type over O, then A := A◦ [1/p] is
Noetherian and Jacobson ([FK18, §0, Prop. 9.3.2, 9.3.10]), and the residue fieldsof the maximal ideals of A are finite extensions of K ([FK18, §0, Cor. 9.3.7]).
Let L/K be a finite Galois extension with inertia group IL/K , and suppose
now that E is large enough that it contains the images of all embeddings L ,→Qp , that all irreducible E-representations of IL/K are absolutely irreducible,
and that every irreducible Qp -representation of IL/K is defined over E. Write lfor the residue field of L, and write L0 = W (l)[1/p].
Let MA◦ be a Breuil–Kisin–Fargues GK -module with A◦ -coefficients which
admits all descents over L, and write MA◦ := MA◦ ,π[ /[π [ ]MA◦ ,π[ for themodule considered in Definition 4.2.4 (1) (for some choice of π [ , with π auniformizer of L; note that by the definition of MA◦ admitting all descents
over L, the quotient MA◦ is actually well-defined as a W (l) ⊗Zp A◦ -submoduleof W (k)A ⊗Ainf,A MA◦ , independently of the choice π or π [ ). Then MA◦ has anatural W (l) ⊗Zp A-semilinear action of Gal(L/K), which is defined as follows:
if g ∈ Gal(L/K), then g(MA◦ ,π[ ) = MA◦ ,g(π[ ) (see the proof of Lemma 4.2.7),
so the morphism g : MA◦ ,π[ → g(MA◦ ,π[ ) = MA◦ ,g(π[ ) induces a morphismg : MA◦ ,π[ /[π [ ]MA◦ ,π[ → MA◦ ,g(π[ ) /[g(π [ )]MA◦ ,g(π[ ) ,and the source and target are both canonically identified with MA◦ .
This action of Gal(L/K) on MA◦ induces an L0 ⊗Qp A-linear action of IL/K
on the projective L0 ⊗Qp A-module MA◦ ⊗A◦ A. Fix a choice of embedding
σ : L0 ,→ E, and let eσ ∈ L0 ⊗Qp E be the corresponding idempotent. Theneσ (MA◦ ⊗A◦ A) is a projective A-module of rank d, with an A-linear actionof IL/K . (Indeed, since MA◦ is a finite projective SA -module of rank d, MA◦
is a finite projective W (k) ⊗Zp A◦ -module of rank d, and MA◦ ⊗A◦ A is a finite
projective L0 ⊗Qp A-module of rank d. It follows that eσ (MA◦ ⊗A◦ A) is a finite
projective A-module of rank d.)
Note that up to canonical isomorphism, this module does not depend on
the choice of eσ : indeed the induced action of ϕ commutes with IL/K andinduces isomorphisms between the eσ (MA◦ ⊗A◦ A) (with σ varying), because
the cokernel of ϕ is killed by Eπ[ , which is a unit in our setting (in which [π [ ] = 0

102 CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS
and p is a unit).
4.6.1 Definition. Let MinfA◦ be as above. Then we write
WD(MinfA◦ ) := eσ (MA◦ ⊗A◦ A),
a projective A-module of rank d with an A-linear action of IL/K .
4.6.2 Remark. The point of this definition is that if A◦ is a finite flat O-algebra,
then it computes inertial types in the following sense: writing A := A◦ [1/p],
and MA◦ := W (C[ )A◦ ⊗Ainf,A◦ Minf
A◦ , we have a potentially semistable representation of GK on a free A-module given by VA (MA◦ ) := TA◦ (MA◦ ) ⊗A◦ A.
As explained in Section F.24, the inertial type Dpst (VA (MA◦ ))|IK is then given
by WD(MinfA◦ ) with its action of IL/K .
4.6.3 Proposition. Let A◦ be a p-adically complete flat O-algebra which is
topologically of finite type over O, and write A = A◦ [1/p]. Let MinfA◦ and MA◦
be as above, and fix σ : L0 ,→ E. Then WD(MinfA◦ ) is a finite projective Amodule of rank d with an action of IK , whose formation is compatible with basechanges A◦ → B ◦ of p-adically complete flat O-algebras which are topologicallyof finite type over O.
Proof. Writing u for [π [ ], the compatibility of formation with base change reduces to the observation that the natural map SA ⊗A B → SB induces anisomorphism
SB /uSB ∼= (SA /uSA ) ⊗A B.
Let τ be a d-dimensional E-representation of IL/K .
4.6.4 Definition. In the setting of Proposition 4.6.3, we say that MinfA◦ hasinertial type τ if Zariski locally on Spec A, WD(Minf)
isisomorphic
to
the
base
◦
A
change of τ to A.
4.6.5 Corollary. In the setting of Proposition 4.6.3, we can decompose Spec A
as the disjoint union of open and closed subschemes Spec Aτ , where Spec Aτ
is the locus over which MinfA◦ has inertial type τ . Furthermore, the formationof this decomposition is compatible with base changes A◦ → B ◦ of p-adicallycomplete flat O-algebras which are topologically of finite type over O.
Proof. By Proposition 4.6.3 (more precisely, by the compatibility with basechange), we can define Spec Aτ to be the locus over which MinfA◦ has inertial
type τ . That Spec A is actually the disjoint union of the Spec Aτ follows easily
from our assumptions on E, and standard facts about the representation theoryof finite groups in characteristic zero. For lack of a convenient reference, wesketch a proof as follows.
Since E has characteristic zero, the representation P := ⊕r r is a projective
generator of the category of E[IL/K ]-modules, where r runs over a set of representatives for the isomorphism classes of irreducible E-representations of IL/K .

4.7. HODGE–TATE WEIGHTS

103

Our assumption that E is large enough that each r is absolutely irreducible
Q furthermore shows that EndIL/K (r) = E for each r, so that EndIL/K (P ) = r E.
Standard Morita theory then shows that the functor M 7→ HomIL/K (P, M )induces anof E[IL/K ]-modules and the catQequivalence between the category
Q
egory of r E-modules. Of course, a r E-module is just given by a tuple(Nr )r of E-vector spaces, and in this optic, the functor HomIL/K (P, –) can bewritten as
LM 7→ HomIL/K (r, M ) r , with a quasi-inverse functor being given by(Nr ) 7→ r r ⊗E Nr . It is easily seen (just using the fact that HomIL/K (P, –)induces an equivalence of categories) that M is a finitely generated projective
A-module, for some E-algebra A, if and only if each HomIL/K (r, M ) is a finitely
generated projective A-module. Writing the various τ in the form ⊕r rnr , weare done.

4.7

Hodge–Tate weights

Let A◦ be a p-adically complete flat O-algebra which is topologically of finite
type over O, and let MinfA◦ be a Breuil–Kisin–Fargues GK -module of height at
most h with A◦ -coefficients, which admits all descents. We now explain how
to interpret the condition that MinfA◦ has a fixed Hodge type, following [Kis08,
Lem. 2.6.1, Cor. 2.6.2] (but bearing in mind the corrections to these results
explained in [Kis09a, A.4]). We would like to thank Mark Kisin for a helpful
conversation about these results, and for some suggestions regarding the proofof Proposition 4.7.2.
Fix some choice of π [ , and write MA◦ for Mπ[ ,A◦ , and u for [π [ ]. For each 0 ≤
i ∗
i
◦
◦
i ≤ h we define Fili ϕ∗ MA◦ = Φ−1MA◦ (E(u) MA ), and we set Fil ϕ MA =∗ϕ MA◦ for i < 0.
4.7.1 Remark. The point of this definition is that it captures the Hodge filtration. Indeed, if A◦ is a finite flat O-algebra, then writing A = A◦ [1/p],
and MA◦ := W (C[ )A◦ ⊗Ainf,A◦ Minf
A◦ , we have a representation of GK on afree A-module given by VA (MA◦ ) := TA◦ (MA◦ ) ⊗A◦ A. As explained in Section F.24 (or see the proof of [Kis08, Cor. 2.6.2]), there is a natural identificationof DdR (VA (MA◦ )) with (ϕ∗ MA◦ /E(u)ϕ∗ M)⊗A◦ A, under which Fili DdR (VA (MA◦ ))is identified with
(Fili ϕ∗ MA◦ /E(u) Fili−1 ϕ∗ MA◦ ) ⊗A◦ A;
in particular, this latter module is a finite projective K ⊗Qp A-module.
The following proposition shows that the final conclusion of the preceding
remark holds for more general choices of A◦ ; the proof uses that particular case(i.e. the case that A◦ is a finite flat O-algebra) as an input.
4.7.2 Proposition. Let A◦ be a p-adically complete flat O-algebra which is
topologically of finite type over O, and write A = A◦ [1/p]. Let MinfA◦ and MA◦
be as above. For each 0 ≤ i ≤ h,(Fili ϕ∗ MA◦ /E(u) Fili−1 ϕ∗ MA◦ ) ⊗A◦ A

104 CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS
is a finite projective K ⊗Qp A-module, whose formation is compatible with basechanges A◦ → B ◦ of p-adically complete flat O-algebras which are topologicallyof finite type over O.
Proof. For any morphism of O-algebras A◦ → B ◦ , we write MB ◦ := SB ◦ ⊗SA◦
MA◦ . We begin by showing (following the proof of [Kis08, Lem. 2.6.1]) that forany p-adically complete O-algebra A◦ which is topologically of finite type over O,
both MA◦ / im ΦMA◦ and ϕ∗ MA◦ / Filh ϕ∗ MA◦ are finite projective OK ⊗Zp A◦ modules, whose formation is compatible with arbitrary base changes A◦ → B ◦(with B ◦ also topologically of finite type).
Indeed, that MA◦ / im ΦMA◦ is a finite projective OK ⊗Zp A◦ -module whose
formation is compatible with base change follows as in the proof of [Kis08,
Lem. 2.6.1] from the fact that ΦM remains injective after any base change(because E(u) remains a non-zero-divisor after any base change), and the resultfor ϕ∗ MA◦ / Filh ϕ∗ MA◦ follows from this and the short exact sequence
0 → ϕ∗ MA◦ / Filh ϕ∗ MA◦ → MA◦ /E(u)h MA◦ → MA◦ / im ΦMA◦ → 0.
We claim that for each 0 ≤ i ≤ h, (Filh ϕ∗ MA◦ /E(u)i Filh−i ϕ∗ MA◦ ) ⊗A◦
A is a finite projective K ⊗Qp A-module whose formation is compatible withbase change. Admitting the claim, the proposition follows from the short exact
sequence
0 −→ Fili ϕ∗ MA◦ /E(u) Fili−1 ϕ∗ MA◦
E(u)h−i−→ Filh ϕ∗ MA◦ /E(u)h−i+1 Fili−1 ϕ∗ MA◦−→ Filh ϕ∗ MA◦ /E(u)h−i Fili ϕ∗ MA◦ −→ 0

(4.7.3)(because after tensoring with A, the second and third terms are projective andcompatible with base change, so that the sequence splits, and the first term isalso projective and compatible with base change).
To prove the claim, we firstly consider for each 0 ≤ i ≤ h the finite OK ⊗Zp
A◦ -moduleϕ∗ MA◦ /(E(u)i ϕ∗ MA◦ + Filh ϕ∗ MA◦ ).
Since this is the cokernel of the morphism
E(u)i ϕ∗ MA◦ → ϕ∗ MA◦ / Filh ϕ∗ MA◦ ,we see that its formation is compatible with base change. We have a short exact
sequence of finite type A◦ -modules
0 → Filh ϕ∗ MA◦ /E(u)i Filh−i ϕ∗ MA◦ → ϕ∗ MA◦ /E(u)i ϕ∗ MA◦→ ϕ∗ MA◦ /(E(u)i ϕ∗ MA◦ + Filh ϕ∗ MA◦ ) → 0,

(4.7.4)in which the second and third terms are compatible with base change, and weneed to prove that after inverting p, the first term is projective and is of formation compatible with base change. To this end, note firstly that as discussed

4.7. HODGE–TATE WEIGHTS

105

above, if A◦ is a finite flat O-algebra, then each(Fili ϕ∗ MA◦ /E(u) Fili−1 ϕ∗ MA◦ ) ⊗A◦ A
is a finite projective K ⊗Qp A-module, and it follows from (4.7.3) and an easy
induction that the same is true of (Filh ϕ∗ MA◦ /E(u)i Filh−i ϕ∗ MA◦ ) ⊗A◦ A.
To simplify notation, we write (4.7.4) as
0 → K(A◦ ) → M1 → M2 → 0,
and for any A◦ -algebra C, we write K(C) for the kernel of the surjection M1 ⊗A◦
C → M2 ⊗A◦ C. In particular for B ◦ as in the statement of the proposition, if
as usual we write B := B ◦ [1/p], then we have
K(B) = (Filh ϕ∗ MB ◦ /E(u)i Filh−i ϕ∗ MB ◦ ) ⊗B ◦ B;
so we need to show that K(A) is projective, and that K(B) = K(A) ⊗A B.
Let m be a maximal ideal of A, so that A/mi is a finite E-algebra for each i.
bm is a flat A-algebra, since the Artin–Rees Lemma shows that
Then since A
bm coincides with m-adic completion for finite type modules,
tensoring with A
and since the formation of kernels commutes with limits, we see thatbm = K(Abm ) = lim K(A/mi ).
K(A) ⊗A◦ A←−
i

Since A/mi is a finite E-algebra, we see that K(A/mi ) is a finite projective(K ⊗Qp A)/mi -module of rank bounded independently of i. It follows thatbm -module. Since Am is Noethebm is a finite projective OK ⊗Z A
K(A) ⊗A◦ A
pb
rian, Am is a faithfully flat Am -algebra, so that K(A) ⊗A◦ Am is a finite projective OK ⊗Zp Am -module. Since this holds for all m, we see that K(A) is a finite
projective OK ⊗Zp A-module, as claimed.
It remains to show that we have K(B) = K(A) ⊗A B. We have a naturalsurjective morphism of finite projective K ⊗Qp B-modules
K(A) ⊗A B → K(B),(4.7.5)which we need to show is an isomorphism. Note firstly that if B = A/m forsome maximal ideal m of A, then this follows from the previous paragraph. Now
suppose that B is general. Since the kernel of (4.7.5) is in particular a finite
projective B-module, is enough to prove that for any maximal ideal mB of B,(4.7.5) becomes an isomorphism after tensoring with B/mB . Suppose that mB
lies over a maximal ideal m of A, so that B/mB is a finite field extension of A/m;
we need to show that the induced surjection
K(A/m) ⊗A/m B/mB → K(B/mB )is an isomorphism. Each side is determined by the de Rham filtration on thecorresponding GK -representation, which is compatible with the extension ofscalars, so we are done.

106 CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKSIn what follows, we will work in the relative setting of an extension L/K. To
this end, we let A◦ be a p-adically complete flat O-algebra, with A := A◦ [1/p],
let L/K be a finite Galois extension, and let MinfA◦ be a Breuil–Kisin–Fargues
GK -module of rank d and height at most h with A◦ -coefficients, which admits alldescents over L. Fix some choice of π [ a uniformizer of L, write MA◦ for Mπ[ ,A◦ ,and u for [π [ ]. Applying Proposition 4.7.2, with L in place of K, we obtain aprojective L ⊗Qp A-module (ϕ∗ MA◦ /E(u)ϕ∗ MA◦ ) ⊗A◦ A, which is filtered byprojective submodules. This filtered module has a natural action of Gal(L/K),
which is semi-linear with respect to the action of Gal(L/K) on L ⊗Qp A induced
by its action on the first factor. Since L/K is a Galois extension, the tensor
product L⊗Qp A is an étale Gal(L/K)-extension of K⊗Qp A, and so étale descentallows us to descend (ϕ∗ MA◦ /E(u)ϕ∗ MA◦ ) ⊗A◦ A to a filtered module over
K ⊗Qp A; concretely, this descent is achieved by taking Gal(L/K)-invariants.
This leads to the following definition.
4.7.6 Definition. In the preceding situation, we write∗∗
DdR (Minf
A◦ ) := (ϕ MA◦ /E(u)ϕ MA◦ ) ⊗A◦ A

Gal(L/K),

and more generally, for each i ≥ 0, we write
i ∗
i−1 ∗
Fili DdR (Minfϕ MA◦ ) ⊗A◦ A
A◦ ) := (Fil ϕ MA◦ /E(u) Fil

Gal(L/K)

inf(and for i < 0, we write Fili DdR (MinfA◦ ) := DdR (MA◦ )). The property of beinga finite rank projective module is preserved under étale descent, and so we findthat DdR (MinfA◦ ) is a rank d projective K ⊗Qp A-module, filtered by projective
submodules.
∼
Q Since A is an E-algebra, we have the product decomposition K ⊗Qp A −→σ:K,→E A, and so, if we write eσ for the idempotent corresponding to the factor
labeled by σ in this decomposition, we find thatY
DdR (Minfeσ DdR (MinfA◦ ) =A◦ ),σ:K,→E

where each eσ DdR (MinfA◦ ) is a projective A-module of rank d. For each i, we
write
i
inf
Fili eσ DdR (MinfA◦ ) = eσ Fil DdR (MA◦ ).
Each Fili eσ DdR (MinfA◦ ) is again a projective A-module.
The base-change property proved in Proposition 4.7.2 shows that the vari+1
ious quotients Fili eσ DdR (Minfeσ DdR (MinfA◦ )/ FilA◦ ) are again projective Amodules. Phrased more geometrically, then, we see that each eσ DdR (MinfA◦ ) gives
rise to a vector bundle over Spec A which is endowed with a filtration by subbundles. We may thus decompose Spec A into a disjoint union of open and closed
subschemes over which the ranks of the various subbundles Fili eσ DdR (MinfA◦ )(or equivalently, the ranks of the various constituents
i+1
Fili eσ DdR (Minfeσ DdR (MinfA◦ )/ FilA◦ )

4.7. HODGE–TATE WEIGHTS

107

of the associated graded bundle) are constant.
To encode this rank data, and the corresponding decomposition of Spec A,
it is traditional to use the terminology of Hodge types, which we now recall.
4.7.7 Definition. A Hodge type λ of rank d is by definition a set of tuples ofintegers {λσ,j }σ:K,→Qp ,1≤j≤d with λσ,j ≥ λσ,j+1 for all σ and all 1 ≤ j ≤ d − 1.
If D := (Dσ )σ:K,→E is a collection of rank d vector bundles over Spec A,
labeled (as indicated) by the embeddings σ : K ,→ E, then we say that D has
Hodge type λ if Fili Dσ has constant rank equal to #{j | λσ,j ≥ i}.
4.7.8 Corollary. In the preceding context, if λ is a Hodge type of rank d, thenwe let Spec Aλ denote the open and closed subscheme of Spec A over which thetuple eσ DdR (Minf)
of filtered vector bundles is of Hodge type λ. We have a◦Acorresponding decompositionaSpec A =Spec Aλ ,(4.7.9)λlabeled by the set of Hodge types λ of rank d. This decomposition is compatible
with base changes A◦ → B ◦ of p-adically complete flat O-algebras which are
topologically of finite type over O.
Proof. This follows from the preceding discussion.

4.7.10 Remark. A priori, the tuple of filtered vector bundles eσ DdR (MinfA◦ ) σ:K,→E
on Spec A, and hence the decomposition (4.7.9) of Spec A, depends on the descent Mπ[ ,A◦ , and hence on the choice of π [ . However, Theorem 4.7.13 below
implies in fact that the decomposition (4.7.9) is independent of the choice of π [ ;see Remark 4.7.14 for a more detailed explanation of this.
As in Section 4.6, suppose now that E is large enough that it contains theimages of all embeddings L ,→ Qp , that all irreducible E-representations of IL/K
are absolutely irreducible, and that every irreducible Qp -representation of IL/K
is defined over E. We can then immediately combine Corollaries 4.6.5 and 4.7.8,
obtaining the following result.
4.7.11 Corollary. In the preceding situation, we have a decompositionaSpec A =Spec Aλ,τ ,(4.7.12)λ,τlabeled by the set of Hodge types λ of rank d, and the set of inertial types τof IL/K . This decomposition is compatible with base changes A◦ → B ◦ of padically complete flat O-algebras which are topologically of finite type over O.
The Breuil–Kisin–Fargues GK -module MinfA◦ is of Hodge type λ and inertial
type τ if and only if Aλ,τ = A.
The following key theorem relates the constructions of this section, and theprevious one, to the p-adic Hodge theory of GK -representations. If B is a
finite E-algebra, then we say that a sub-O-algebra B ◦ ⊆ B is an order of B if
B ◦ [1/p] = B, and B ◦ is a finite O-algebra.

108 CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS
4.7.13 Theorem. Suppose that A◦ is a finite flat O-algebra, let M be a projective étale (ϕ, GK )-module with A◦ -coefficients, and write VA (M ) = TA◦ (M )[1/p].
Let L/K be a finite Galois extension. Then VA (M )|GL is semistable with Hodge–
Tate weights in [0, h] if and only if there is an order (A◦ )0 of A := A◦ [1/p]◦ 0that contains A◦ , and a Breuil–Kisin–Fargues GK -module Minf
(A◦ )0 with (A ) coefficients, which is of height at most h, which admits all descents over L, whose
GL -action is canonical, and which satisfies M(A◦ )0 = W (C[ )(A◦ )0 ⊗Ainf,(A◦ )0Minf(A◦ )0 .
Furthermore VA (M )|GL is crystalline if and only if Minf
(A◦ )0 is crystalline as◦ 0
a Breuil–Kisin–Fargues GL -module with (A ) coefficients.
In either case, the Hodge type is determined by applying Definition 4.7.7 toarising from Definition 4.7.6, and the inertial
the tuple eσ DdR (Minf
)
◦
0
(A )
σ:K,→E
type of VA (M ) is given by Definition 4.6.4.
Proof. Suppose firstly that (A◦ )0 and Minf
(A◦ )0 exist. Then if we simply forgetthe (A◦ )0 -coefficients and consider Minf
asa Breuil–Kisin–Fargues GK -module(A◦ )0with Zp -coefficients, the theorem follows from Corollaries F.23 and F.25. Conversely, suppose that VA (M )|GL is semistable with Hodge–Tate weights in [0, h].
If we show that (A◦ )0 and Minf
(A◦ )0 exist, then (as discussed in Remarks 4.6.2
and 4.7.1) it again follows from Corollary F.25 that the inertial and Hodge
types are given by the claimed recipes; so it suffices to prove this existence. We
begin with a preliminary reduction.
Q
The ring A is a product of Artinian local E-algebras,say A = i Ai ; thenQ
if A◦i denotes the image of A◦ in Ai , the product i A◦i is an order of A containing A◦ . Thus it is no loss of generality to replace A◦ by this product, andhence, by working one factor at a time, to assume that A is local. The residue
field E 0 of A is then a finite extension of E, and A is naturally an E 0 -algebra.
The compositum OE 0 A◦ is an order in A containing A◦ which is furthermore
an OE 0 -algebra. Replacing A◦ by this compositum, we may thus assume thatA◦ is an OE 0 -algebra. Then, relabelling E 0 as E if necessary, we can and do
assume that E 0 = E . Thus, reformulating the preceding discussion slightly, we
have Ared = E and A◦red = OE .
By Corollary F.23 there is a unique Breuil–Kisin–Fargues GK -module Minf
with Zp -coefficients which is of height at most h and admits all descents over L,
and satisfies
M = W (C[ ) ⊗Ainf Minf .
For each π [ we have by definition a descent Mπ[ of Minf . Fix some choice of π [ ,write u for [π [ ], and write S for Sπ[ and M for Mπ[ .
We now follow the proof of [Kis08, Prop. 1.6.4]. In particular, we work forthe most part with the Breuil–Kisin module M, rather than the Breuil–Kisin–
Fargues module Minf . It is possible that by using [BMS18, Prop. 4.13], we
could make our arguments with Minf itself, but since this does not seem likely
to significantly simplify the proof, and would make it harder for the reader tocompare to the arguments of [Kis08], we have not attempted to do this.

4.7. HODGE–TATE WEIGHTS

109

Note firstly that M ⊂ Minf ⊂ M are stable under the action of A◦ on M .
Indeed, since A◦ is local, it is enough to check stability under the action of (A◦ )× .
In the case of Minf it is immediate from the unicity of Minf that we have Minf =aMinf for any a ∈ (A◦ )× , and similarly for M it follows from Lemma 4.2.8.
In particular M is naturally a finite SA◦ -module, which is projective as an
S-module. While M need not be a projective SA◦ -module, OE,A◦ ⊗SA◦ M
is a projective OE,A◦ -module, because after the faithfully flat base extension
OE,A◦ ,→ W (C[ )A◦ it is identified with M . It follows from [Kis08, Lem. 1.6.1]
that M[1/p] is a projective SA◦ [1/p]-module, necessarily of rank d, and thatM[1/u] is a projective SA◦ [1/u]-module, again of rank d.
In fact, it will be useful to note that M[1/p] is actually free of rank d. To
see this, it suffices to prove it after base-changing toY(SA◦ )[1/p]red = (SA◦red )[1/p] = SO [1/p] ∼O[[u]][1/p].
=
σ:W (k),→O

Since O[[u]][1/p] is a PID (by the Weierstrass preparation theorem), any finitely
generated projective module over (SA◦red )[1/p] of constant rank
Q is necessarily
free. For later use, we note that since (OE,A◦ )red = O ⊗Zp OE = σ:W (k),→O OE ,and OE is a local ring (indeed a discrete valuation ring, with uniformizer p),
any projective module of constant rank over OE,A◦ is also necessarily free.
Now let M0O denote the image of M under the projection M[1/u] → M[1/u]⊗A
E (the map A → E being the projection from A to its residue field), andlet M0O ⊂ MO be the canonical inclusion (with finite cokernel) of M0O into thecorresponding finite projective SO -module. (The existence of MO follows fromthe structure theory of S-modules, see [BMS18, Prop. 4.3]. Concretely, we have
MO = M0O [1/u] ∩ M0O [1/p].) The module MO is again a Breuil–Kisin moduleof height at most h.
Choose an SO -basis for MO , and lift it to an SA◦ [1/p]-basis of M[1/p],
and let (A◦ )00 be the A◦ -subalgebra of A generated by these matrix coefficients.
These coefficients have bounded powers of p in their denominators (when we
think of A as equalling A◦ [1/p]) and lie in O after passing to the quotient of A
by its maximal ideal (equivalently its nilradical); in other words these entries
lie in A◦ + p−N nil(A◦ ) for some N ≥ 0, and this A◦ -submodule of A generates
a finite A◦ -subalgebra of A. Thus (A◦ )00 is an order in A. Let M(A◦ )00 be theS(A◦ )00 -submodule of M[1/p] spanned by this basis. Then M(A◦ )00 is ϕ-stableby construction. Again by construction we have that
M(A◦ )00 [1/p] = (A◦ )00 ⊗A◦ M[1/p],
and that
O ⊗(A◦ )00 OE,(A◦ )00 ⊗S(A◦ )00 M(A◦ )00 = OE ⊗S MO = OE ⊗S M0O(where the first tensor product is with respect to the surjection (A◦ )00 →(A◦ )00red = O). Thus OE,(A◦ )00 ⊗S(A◦ )00 M(A◦ )00 and OE,(A◦ )00 ⊗SA◦ M are two
rank d projective OE,(A◦ )00 -modules which coincide after inverting p, and also

110 CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS
after reducing modulo nil (A◦ )00 ). In fact, they are both free of rank d, by our
above observation that a projective OE,A◦ -module of constant rank is necessarily free (applied now with (A◦ )00 in place of A◦ ). If we choose bases for eachof them which coincide modulo nil (A◦ )00 , then we may regard each of thesebases as a basis of OE,(A◦ )00 ⊗SA◦ M[1/p] over OE,(A◦ )00 [1/p], and so theydiffer
by a change-of-basis matrix lying in 1 + nil (A◦ )00 )Md OE,(A◦ )00 [1/p] . Just as
we argued above in the construction of (A◦ )00 , we see that we may enlarge (A◦ )00
to an order (A◦ )0 such that the entries of this change-of-basis matrix lie in (A◦ )0 .
Consequently, if we set M(A◦ )0 := S(A◦ )0 ) ⊗S(A◦ )00 M(A◦ )00 , then we find that
OE,(A◦ )0 ⊗SA◦ M = OE,(A◦ )0 ⊗S(A◦ )00 M(A◦ )0 .
Extending scalars further, we find that
M(A◦ )0 := W (C[ )(A◦ )0 ⊗W (C[ )A◦ M= W (C[ )(A◦ )0 ⊗SA◦ M = W (C[ )(A◦ )00 ⊗S(A◦ )0 M(A◦ )0 .
We claim that M(A◦ )0 is a Breuil–Kisin module of height at most h, i.e. thatthe cokernel of ΦM(A◦ )0 is killed by E(u)h . This cokernel is, as an S-module,a successive extension of copies of the cokernel of ΦMO (the injectivity of Φ
implies that the formation of the cokernel of Φ is exact), and is in particularp-torsion free (by [Kis09b, Lem. 1.2.2 (2)]). After inverting p the cokernel is abase change of the cokernel of ΦM , and is therefore killed by E(u)h , as required.
We now set Minf
(A◦ )0 := Ainf,(A◦ )0 ⊗S(A◦ )0 M(A◦ )0 , a Breuil–Kisin–Fargues
GK -module of height at most h with (A◦ )0 -coefficients which by construction
satisfies M(A◦ )0 = W (C[ )(A◦ )0 ⊗Ainf,(A◦ )0 Minf
(A◦ )0 . It remains to prove thatinf
M(A◦ )0 admits all descents over L and that the GL -action is canonical. Applying Corollary F.23 again to M(A◦ )0 regarded as an étale (ϕ, GK )-module with
Zp -coefficients, we see that there is a Breuil–Kisin–Fargues GK -module (Minf )0of height at most h with Zp -coefficients which admits all descents over L andsatisfies M(A◦ )0 = W (C[ ) ⊗Ainf (Minf )0 . Write (M)0 for the descent of (Minf )0 ,for our particular choice of π [ ; then by the uniqueness of Breuil–Kisin modules
with Zp -coefficients ([Kis06, Prop. 2.1.12]) we have (M)0 = M(A◦ )0 , so that ininf
fact (Minf )0 = Minf
(A◦ )0 . Thus M(A◦ )0 admits all descents over L to Breuil–Kisin
modules with Zp -coefficients, and by another application of Lemma 4.2.8, these
Breuil–Kisin modules are (A◦ )0 -stable. Finally, the GL -action is canonical by
Proposition 4.4.1, and we are done.
4.7.14 Remark. It follows from Theorem 4.7.13 that the decomposition (4.7.9)
is independent of the choice of π [ . Indeed, the condition that Fili eσ DdR (MinfA◦ )has given constant rank can be checked after base changing to all A/m, for ma maximal ideal of A, so it follows from Proposition 4.7.2 and Theorem 4.7.13
that the direct factor Aλ of A is characterized as follows: for any A◦ -algebra B ◦which is finite and flat as an O-algebra, the canonical morphism A → B ◦ [1/p]factors through Aλ if and only if the GK -representation VB (MinfB ◦ ) has Hodge
type λ.

4.8. STACKS OF POTENTIALLY SEMISTABLE REPRESENTATIONS 111

4.8

Moduli stacks of potentially semistable representations

We are now in a position to define the main objects of interest in this chapter,
which are the closed substacks of Xd classifying representations which are potentially crystalline or potentially semistable of fixed Hodge and inertial types.
In this section L/K will denote a finite Galois extension with inertia group IL/K ;we will always assume (without specifically recalling this) that E is large enoughthat all irreducible E-representations of IL/K are absolutely irreducible, andthat every irreducible Qp -representation of IL/K is defined over E. For any
L/K

L/K

such L/K we have the stacks Cd,ss,h and Cd,crys,h that we defined in DefiniL/K,fl

L/K,fltion 4.5.22, and we write Cd,ss,h and Cd,crys,h respectively for their flat parts (inthe sense recalled in Appendix A).
4.8.1 Definition. We say that a Hodge type λ is effective if λσ,i ≥ 0 for eachσ and i, and that λ is bounded by h if λσ,i ∈ [0, h] for each σ and i.
4.8.2 Proposition. Let L/K be a finite Galois extension. Then the stacks
L/K,fl
L/K,fl
L/K,fl,λ,τCd,ss,h and Cd,crys,h are scheme-theoretic unions of closed substacks Cd,ss,h
L/K,fl,λ,τ

and Cd,crys,h , where λ runs over all effective Hodge types that are boundedby h, and τ runs over all d-dimensional E-representations of IL/K . These latter closed substacks are uniquely characterised by the following property: if A◦
L/K,fl
L/K,flis a finite flat O-algebra, then an A◦ -point of Cd,ss,h (resp. Cd,crys,h ) is a point
L/K,fl,λ,τ

L/K,fl,λ,τof Cd,ss,h(resp. Cd,crys,h ) if and only if the corresponding Breuil–Kisin–
Fargues module MinfA◦ has Hodge type λ in the sense given by applying Definition 4.7.7 to the tuple eσ DdR (MinfA◦ ) σ:K,→E arising from Definition 4.7.6, andinertial type τ in the sense of Definition 4.6.4; or, more succinctly (and writing
A := A◦ [1/p], as usual), if and only if Aλ,τ = A.
L/K,fl,λ,τ
L/K,fl,λ,τor Cd,crys,h is a p-adic formal algebraic stack
Finally, each stack Cd,ss,hof finite presentation which is flat over Spf O and whose diagonal is affine, andthe natural morphisms to Xd are representable by algebraic spaces, proper, andof finite presentation.
L/K,flProof. We give the proof for Cd,ss,h , the crystalline case being formally idenL/K,fltical. Since Cd,ss,h is a flat p-adic formal algebraic stack of finite presentation
over Spf O, we can choose a smooth surjection
L/K,flSpf B ◦ → Cd,ss,h(4.8.3)where B ◦ is topologically of finite type over O.
◦
As usual, we write B := B`[1/p]. By Corollary 4.7.11 we may write Spec B
as a disjoint union Spec B = λ,τ B λ,τ , and so correspondingly factor B as aQ
product B = λ,τ B λ,τ . If we let B λ,τ,◦ denote the image of B ◦ in B λ,τ , then112 CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKSQ
we obtain an induced injection B ◦ ,→ λ,τ B λ,τ,◦ , which induces a schemetheoretically dominant morphisma
Spf B λ,τ,◦ → Spf B ◦ .
(4.8.4)λ,τ
∼L/K,flWrite R = Spf B ◦ ×C L/K,fl Spf B ◦ , so that [Spf B ◦ /R] −→ Cd,ss,h . Then wed,ss,hclaim that each Spf B λ,τ,◦ is R-invariant. Granting this, if we write Rλ,τ for the
L/K,fl,λ,τ
restriction of R to Spf B λ,τ,◦ , and then define Cd,ss,h:= [Spf B λ,τ,◦ /Rλ,τ ], it
L/K,fl,λ,τfollows from the discussion of closed substacks in Appendix A that Cd,ss,h
L/K,flembeds as a closed sub-formal algebraic stack of Cd,ss,h . Furthermore, the induced morphism
a L/K,fl,λ,τ
L/K,flCd,ss,h→ Cd,ss,h(4.8.5)λ,τinduces the morphism (4.8.4) after pull-back via the morphism (4.8.3), whichis representable by algebraic spaces, smooth, and surjective. Since this lattermorphism is scheme-theoretically dominant, so is (4.8.5).
We now verify that Spf B λ,τ,◦ is R-invariant, i.e. that R0 := Spf B λ,τ,◦ ×C L/K,fld,ss,hSpf B ◦ and R1 := Spf B ◦ ×C L/K,fl Spf B λ,τ,◦ coincide as closed sub-formal alged,ss,hbraic spaces of R = Spf B ◦ ×C L/K,fl Spf B ◦ (and thus that both coincide withd,ss,hRλ,τ := Spf B λ,τ,◦ ×C L/K,fl Spf B λ,τ,◦ ).
d,ss,h

L/K,flSince Spf B ◦ → Cd,ss,h is representable by algebraic spaces and smooth, it
is in particular representable by algebraic spaces, flat, and locally of finite type,
and thus so are each of the projections R0 → Spf B λ,τ,◦ and R1 → Spf B λ,τ,◦ .
It follows from Lemmas A.3 and A.4 that any affine formal algebraic space
which is étale over either of R0 or R1 is of the form Spf A◦ , where A◦ is $adically complete, topologically of finite type, and flat over O. Thus, to showthat R0 and R1 coincide as subspaces of R, it suffices to show that if A◦ is any$-adically complete, topologically of finite type, and flat O-algebra endowedwith a morphism Spf A◦ → R, then this morphism factors through R0 if andonly if it factors through R1 . Unwinding the definitions of R, R0 , and R1 as
fibre products, this amounts to showing that if we are given a pair of morphism
L/K,flSpf A◦ ⇒ Spf B ◦ which induce the same morphism to Cd,ss,h , then one factors
through Spf B λ,τ,◦ if and only if the other does.
The pair of morphisms Spf A◦ ⇒ Spf B ◦ correspond to a pair of morphisms
f0 , f1 : B ◦ ⇒ A◦ . The fact that these morphisms induce the same morphism
L/K,flto Cd,ss,h may be rephrased as saying that f0∗ Minf and f1∗ Minf are isomorphic Breuil–Kisin–Fargues modules over A◦ . Now, applying the base change
statements from Propositions 4.6.3 and 4.7.2, we find that both f0∗ WD(MinfB◦ )inf
and f1∗ WD(Minf)
areisomorphic,
asI-representations,
to
WD(M
),
and
K
B◦
A◦
∗
inf
that both f0∗ DdR (MinfB ◦ ) and f1 DdR (MB ◦ ) are isomorphic, as filtered K ⊗Qp B∗
infmodules, to DdR (MinfA◦ ). Thus either, and hence both, of f0 WD(MB ◦ ) and4.8. STACKS OF POTENTIALLY SEMISTABLE REPRESENTATIONS 113
inf
f1∗ WD(MinfB ◦ ) are of inertial type τ if and only if WD(MA◦ ) is of inertial type τ ,∗inf∗while either, and hence both, of f0 DdR (MB ◦ ) and f1 DdR (MinfB ◦ ) are of Hodge
)
is
of
Hodge
type
λ.
Consequently
f0 factors
type λ if and only if DdR (Minf◦Athrough Spf B λ,τ,◦ if and only f1 does. This shows that Spf B λ,τ,◦ is indeed
R-invariant.
L/K,fl,λ,τWe now show that Cd,ss,h:= [Spf B λ,τ,◦ /Rλ,τ ] satisfies the required propL/K,flerty, i.e. that for any finite flat O-algebra A◦ , a morphism Spf A◦ → Cd,ss,h
L/K,fl,λ,τfactors through Cd,ss,hif and only if Aλ,τ = A. By construction, such a factorization occurs if and only if the induced morphism Spf A◦ ×C L/K,fl Spf B ◦ →d,ss,hSpf B ◦ factors through Spf B λ,τ,◦ .
L/K,flSince the surjection Spf B ◦ → Cd,ss,h is representable by algebraic spaces,smooth, and surjective, the fibre product Spf A◦ ×C L/K,fl Spf B ◦ is a formal alged,ss,hbraic space, and the surjection Spf A◦ ×C L/K,fl Spf B ◦ → Spf A◦ is representabled,ss,hby algebraic spaces, smooth, and surjective. We may thus find an open cover of
Spf A◦ ×C L/K,fl Spf B ◦ by affine formal algebraic spaces Spf C ◦ , with C ◦ beingd,ss,ha faithfully flat $-adically complete A◦ -algebra which is topologically of finite
type. (We are again applying Lemmas A.3 and A.4.) If we write C := C ◦ [1/p],
then the base-change property of Corollary 4.7.11 shows that the hypothesizedfactorization occurs if and only if C = C λ,τ , and also that C λ,τ = Aλ,τ ⊗A C,
so that (since C is faithfully flat over A) C λ,τ = C if and only if Aλ,τ = A. In
L/K,fl
L/K,fl,λ,τ
conclusion, we have shown that Spf A◦ → Cd,ss,h factors through Cd,ss,hif
λ,τ
and only if A = A, as required.
L/K,fl,λ,τis immediate from Proposition 4.8.6
Finally, the uniqueness of Cd,ss,hbelow; the properties of being of finite presentation,of having affine diagonal,
L/K,fl,λ,τ
and of the morphism Cd,ss,h→ Xd being representable by algebraic spaces,
proper, and of finite presentation, are immediate from Proposition 4.5.24 andLemma 4.5.10.
4.8.6 Proposition. Suppose that Y and Z are two Noetherian formal algebraic
stacks, both lying over Spf O and both flat over O, and both embedded as closed
substacks of a stack X over Spec O (in the usual sense that the inclusions of
substacks Y, Z ,→ X are representable by algebraic spaces, and induce closed
immersions when pulled back over any scheme-valued point of X ). Suppose alsothat each of Yred and Zred are locally of finite type over F, and suppose further
that, for any finite flat O-algebra A◦ , a morphism Spf A◦ → X factors through Y
if and only if it factors through Z. Then Y and Z coincide.
Proof. The O-flat part W := (Y ×X Z)fl of the 2-fibre product Y ×X Z embeds
as a closed substack of each of Y and Z, and it suffices to show that each of
those embeddings are isomorphisms. The stack W inherits all of the hypotheses
shared by Y and Z, including the fact that, for any finite flat O-algebra A◦ , theA◦ -valued points of W coincide with the A◦ -valued points of each of Y and Z.
Thus, replacing the pair Y, Z by the pairs W, Y and W, Z in turn, we may
reduce to the case when Y is actually a closed substack of Z.

114 CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS
Since Z is Noetherian, lies over Spf O, and is flat over O, we can find asmooth surjection U → Z, whose source is a disjoint union of affine formal
algebraic spaces Spf C ◦ , where C ◦ is a flat I-adically complete Noetherian Oalgebra, for some ideal I which contains $ and defines the topology on C ◦ .
The assumption on Zred further implies that C ◦ /I is a finite type F-algebra.
Then Spf C ◦ ×Z Y is a closed sub-formal algebraic space of Spf C ◦ , and (since it
is also O-flat) it follows from Lemmas A.3 and A.4 that it is of the form Spf B ◦ ,where B ◦ is an O-flat quotient of C ◦ , again endowed with the I-adic topology.
Since the property of being an isomorphism can be checked fppf locally on thetarget, we may replace the closed embedding Y ,→ Z by Spf B ◦ ,→ Spf C ◦ .
In summary, we have a surjection C ◦ → B ◦ of flat, I-adically complete Oalgebras, with C ◦ /I (and hence also B ◦ /I) being of finite type over F, and withthe further property that any morphism C ◦ → A◦ of O-algebras in which A◦ is
finite flat over O factors through B ◦ ; we then have to prove that this surjection
is an isomorphism.
Write B = B ◦ [1/p], C = C ◦ [1/p]; since B ◦ and C ◦ are flat over O, it
is enough to check that the surjection C → B is also injective. Now, we
can embed C into the product of its localizations at all maximal ideals, andsince C is Noetherian, it in fact embeds into the product of the completionsof these localizations. It therefore embeds into the product of all of its local
Artinian quotients A. We claim that any such quotient is in fact obtained froma morphism C ◦ → A◦ , where A◦ is a finite flat O-algebra; since any such morphism C ◦ → A◦ factors through B ◦ by assumption, it follows that any such A
is a quotient of B, so that the surjection C → B is indeed injective.
It remains to prove the claim. Let A◦ be the image of the composite◦
C → C → A; we need to show that A◦ is a finite O-algebra. This follows from Lemma 4.8.7 below.
The following lemma is no doubt standard, but we recall a proof for the sakeof completeness.
4.8.7 Lemma. If R → A is a morphism of O-algebras, with R being p-adicallycomplete and Noetherian (e.g. a complete local O-algebra with finite residue
field), and A being a finite-dimensional E-algebra, then the image of R in A is
finite over O.
Proof. Since R is Noetherian and p-adically complete, the Artin–Rees lemmashows that the same is true of its image in A (see e.g. [AM69, Prop. 10.13]).
Thus, by replacing R by its image in A and A itself by the E-span of this image,
we are reduced to checking the following statement: if M is a p-adically complete
and separated torsion free O-module, and if M [1/p] is finite-dimensional, thenM is finite over O. There are many ways to see this, of course; here is one.
Since V := M [1/p] is finite dimensional, we may find a finite spanning set
for this vector space contained in M ; if L denotes its O-span, then L is a latticein V . Since L is compact with respect to its p-adic topology, the p-adic topology
on M induces the p-adic topology on L, and in particular L is closed in the padic topology on M . Thus M/L is a p-adically separated submodule of V /L.

4.8. STACKS OF POTENTIALLY SEMISTABLE REPRESENTATIONS 115
Note that V /L consists entirely of p-power torsion elements, and thus that thesame is true of M/L. T
TThis implies that n pn (M/L)[pn+1 ] ⊆ n pn (M/L) = 0. On the other∼hand, pn (M/L)[pn+1 ] ⊆ (M/L)[p] ⊆ (V /L)[p] −→ L/pL, and so {pn (M/L)[pn+1 ]is a decreasing sequence of finite
sets, which thus eventually stabilizes. We conclude that pn (M/L)[pn+1 ] = 0 for some sufficiently large value of n; equivalently, (M/L)[pn ] = (M/L)[pn+1 ], and thus in fact (M/L)[pn ] = (M/L)[p∞ ] =M/L. Consequently L ⊆ M ⊆ p−n L, showing that M itself is finite over O, asclaimed.
4.8.8 Definition. Let τ be an inertial type, and let λ be a Hodge type. To
begin with, assume in addition that λ is effective, and choose h ≥ 0 such that λis bounded by h, as well as a finite Galois extension L/K for which the kernelcrys,λ,τof τ contains IL . We then define XK,dto be the scheme-theoretic image of
L/K,fl,λ,τthe morphism Cd,crys,hss,λ,τ→ XK,d , and XK,dto be the scheme-theoretic image

L/K,fl,λ,τ

of the morphism Cd,ss,h→ XK,d . (The characterization of these closed
substacks of XK,d given in Theorem 4.8.12 will show that they are well-defined
substacks of XK,d , independently of the auxiliary choices of h and L/K thatwere used in their definition.)
In general (i.e. if λ is not effective), then we choose an integer h0 so that λ0σ,i :=
ss,λ,τ
λσ,i + h0 ≥ 0 for all σ and all i, and define XK,d to be the unique substackof XK,d with the property that if B is a O/$a -algebra for some a ≥ 1, thenss,λ,τ
a morphism ρ : Spec B → XK,d factors through XK,d if and only if the mor0

ss,λ0 ,τphism ρ ⊗ h : Spec B → XK,d factors through XK,d . (Here we are using
the notation of Section 3.8; more precisely, we are twisting by the étale (ϕ, Γ)0crys,λ,τ
module corresponding to h .) We define XK,din the same way. The point◦of this definition is that if A is a finite flat O-algebra, then a representation
ρ : GK → GLd (A◦ [1/p]) is potentially semistable (resp. crystalline) of Hodge0
type λ and inertial type τ if and only if ρ ⊗ h is potentially semistable (resp.
crystalline) of Hodge type λ0 and inertial type τ ; again, Theorem 4.8.12 below
shows that these stacks are defined independently of the choices made in thisdefinition, and in particular independently of the choice of h0 .
04.8.9 Remark. Rather than introducing the twist by h , it would perhaps be
more natural to work throughout with Breuil–Kisin and Breuil–Kisin–Fargues
modules which are not necessarily ϕ-stable, as in [BMS18]. However, it would
still frequently be useful to make the corresponding Tate twists (see for example
the proof of [BMS18, Lem. 4.26]), and in particular we would need to make such
twists in order to use the results of [PR09, EG21], so we do not see any advantage
in doing so.
The rest of this section is devoted to proving some fundamental properties
ss,λ,τ
crys,λ,τof the stacks XK,dand XK,d . We begin with an analysis of versal rings. To
this end, fix a point Spec F0 → XK,d for some finite extension F0 of F, giving rise
to a continuous representation ρ : GK → GLd (F0 ). Let O0 = W (F0 ) ⊗W (F) O,

116 CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS
the ring of integers in a finite extension E 0 of E having residue field F0 . Then0
the versal morphism Spf Rρ,O → XK,d of Proposition 3.6.3 induces morphisms
crys,λ,τ,O 0

ss,λ,τ,O 0→ XK,d and Spf Rρ
Spf Rρ
rings are as in Section 1.12).

→ XK,d (where these deformation
crys,λ,τ,O 0

4.8.10 Proposition. The morphisms Spf Rρ

ss,λ,τ,O 0→ XK,d and Spf Rρ

→ss,λ,τ,O 0
crys,λ,τcrys,λ,τ,O 0
and Spf Rρ→ XK,d→
XK,d factor through versal morphisms Spf Rρ
ss,λ,τ
XK,d respectively.

Proof. By definition, we can and do assume that λ is effective. We give thecrys,λ,τ,O 0
proof for Spf Rρ, the argument in the semistable case being identical.
0b := C L/K,fl,λ,τ ×X
Spf R,O .
We begin by introducing the fibre product Cd,crys,hρ

K,d

L/K,fl,λ,τ

Since Cd,crys,h → XK,d is proper and representable by algebraic spaces, we seeb is a formal algebraic space, and that the morphismthat Cb → Spf R,OC
ρ

0(4.8.11)is proper and representable by algebraic spaces. The latter condition can beb isreexpressed by saying that (4.8.11) is an adic morphism; thus we see that C
,O 0a proper mR,O0 -adic formal algebraic space over Spf Rρ .
ρ

If we let Spf R denote the scheme-theoretic image of the morphism (4.8.11),
0then by Propositions 3.4.12, 4.8.2 and Lemma A.30, the versal morphism Spf Rρ,O →crys,λ,τXK,d induces a versal morphism Spf R → XK,d, and so the assertion of thecrys,λ,τ,O 0

proposition may be rephrased as the claim that R and Rρ

coincide as0quotients of Rρ,O .
L/K,flSince Cd,crys,h is flat over Spf O, it follows from Lemma A.31 that R is an O00crys,λ,τ,O 0
flat quotient of Rρ,O . Since Rρ
is also an O-flat quotient of Rρ,O ,◦0

we are reduced by Proposition 4.8.6 to showing that if A is any finite flat O 0
algebra, then an O0 -homomorphism Rρ,O → A◦ factors through R if and only
crys,λ,τ,O 0

if it factors through Rρ

.
crys,λ,τ,O 0, thenTo this end, note that if the morphism factors through Rρ
by Theorem 4.7.13 there is an order (A◦ )0 ⊇ A◦ in A◦ [1/p] such that the in0
b Consequently the morphism
duced morphism Spf(A◦ )0 → Spf Rρ,O lifts to C.
0

Spf(A◦ )0 → Spf Rρ,O factors through Spf R, and since Spf(A◦ )0 → Spf A◦ is0scheme-theoretically dominant, the morphism Spf A◦ → Spf Rρ,O also factors
through Spf R.
0
It remains to prove the converse, namely that if Spf A◦ → Spf Rρ,O faccrys,λ,τ,O 0

tors through Spf R, then it factors through Rρ
R⊗

,O 0
Rρ

crys,λ,τ,O 0Rρ

.

If we write S :=(a quotient of R), then equivalently, we wish to showthat any morphism Spf A◦ → Spf R in fact factors through Spf S.

4.8. STACKS OF POTENTIALLY SEMISTABLE REPRESENTATIONS 117
To do this, we will apply Lemma A.32, not in the context of O0 -algebras,but rather in the context of E 0 -algebras. That is, we will invert p, and workwith the rings R[1/p] and S[1/p] (or rather on certain completions of theseb but since the latter isrings). We would also like to invert p on the object C,
a formal algebraic space, doing so directly would lead us into considerations ofrigid analytic geometry that we prefer to avoid. Thus, we begin by observing
L/K
that by Lemma 4.5.26 (together with [Knu71, Thm. V.6.3], to pass from Cd,ss,h
L/K,flb can be promoted to a projective scheme over
to its closed substack C), C
d,crys,h0Spec R. That is, there is a projective morphism of schemes C → Spec Rρ,O ,with scheme-theoretic image equal to Spec R, whose mR,O0 -adic completion isρb Write C[1/p] := C ⊗O0 E 0 . Since scheme-theoretic dominanceisomorphic to C.
is preserved by flat base-change, the morphism C[1/p] → Spec R[1/p] is properand scheme-theoretically dominant.
Returning to our argument, we want to show, for any finite flat O0 -algebra A◦ ,that a morphism R → A◦ necessarily factors through S, or, equivalently, thatthe induced morphism R[1/p] → A := A◦ [1/p] factors through S[1/p]. Since
A is a product of Artinian local E-algebras, we can check this assertion factorby-factor; replacing A by one of these factors (and A◦ by the image of A◦\ dein this factor) we may thus assume that A is local. We then let R[1/p]\ denote the completion of R at the kernel of its map to A, and we let S[1/p]note the corresponding completion of S[1/p]; by Artin–Rees, we also have that\ = R[1/p]\ ⊗R[1/p] S[1/p], and so the map R[1/p] → A factors throughS[1/p]\ → A factors through S[1/p].
\
S[1/p] if and only if the induced morphism R[1/p]
To show the desired factorization, we see from Lemma A.32 that it suffices to\ → B is a morphism to a finite Artinian E-algebra for whichshow that if R[1/p]\
CB → Spec B admits a section, then this morphism factors through S[1/p].
0\ → B; byWrite B ◦ to denote the image of the composite Rρ,O → R[1/p]◦Lemma 4.8.7, we see that B is an order in B. Let Z ,→ CB ◦ denote thescheme-theoretic closure of the section Spec B → CB in CB ◦ . This is proper
over Spec B ◦ , flat over O, and irreducible of dimension one. Thus Z is finite overSpec B ◦ , and hence Z = Spec(B ◦ )0 for some order (B ◦ )0 in B containing B ◦ .
Thinking of Z as a section of C over (B ◦ )0 , we find that the induced morphism
L/K,fl,λ,τSpf(B ◦ )0 → XK,d lifts to a morphism Spf(B ◦ )0 → Cd,crys,h . Theorem 4.7.13
0crys,λ,τ,O 0

then implies that the morphism Rρ,O → B ◦ factors through Rρ, and\ → B does indeed factor through S[1/p],\ as required.
thus the morphism R[1/p]4.8.12 Theorem. Let τ be an inertial type, and let λ be a Hodge type. Thencrys,λ,τ
ss,λ,τ
the closed substacks XK,dand XK,d of XK,d are p-adic formal algebraic
stacks which are of finite type and flat over Spf O, and are uniquely determined as O-flat closed substacks of XK,d by the following property: if A◦ is a
ss,λ,τ
crys,λ,τfinite flat O-algebra, then XK,d(A◦ ) (resp. XK,d(A◦ )) is precisely the sub-118 CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKS
groupoid of XK,d (A◦ ) consisting of GK -representations which become potentiallysemistable (resp. potentially crystalline) of Hodge type λ and inertia type τ after
inverting p.
Proof. We can and do assume that λ is effective. By Propositions 3.4.12, 4.8.2
crys,λ,τ
ss,λ,τ
and A.21, XK,dand XK,d of XK,d are p-adic formal algebraic stacks which
are of finite type and flat over Spf O.
ss,λ,τ
We now verify the claimed description of the A◦ -valued points of XK,d(A◦ )crys,λ,τ
and XK,d(A◦ ), for finite flat O-algebras A◦ . Since the argument is identicalin either case, we give it in the semistable case. Any such algebra A◦ is a
product of finitely many finite flat local O-algebras, and so we immediately
reduce to verifying the claim in the case when A◦ is furthermore local. SinceA◦ is then a complete local finite flat O-algebra, the claimed description follows
from Proposition 4.8.10. Indeed, the residue field F0 is a finite extension of F,
and if as above we set O0 = W (F0 ) ⊗W (F) O, then the morphism Spf A◦ → XK,dss,λ,τ
factors through XK,d if and only if it factors through the versal morphism
ss,λ,τ,O 0

ss,λ,τ

Spf Rρ→ XK,d of Proposition 4.8.10. In turn, this happens if and only
if the corresponding GK -representation is potentially semistable of Hodge type λss,λ,τ,O 0
and inertial type τ , by the defining property of Rρ
.
It remains to show the claimed uniqueness statement. We prove in Corollary 5.5.18 below that XK,d is a Noetherian formal algebraic stack, and in Theorem 5.5.12 below that Xd,red is of finite presentation over F. The reader caneasily confirm that Chapter 5 contains no citation to the present chapter, andthus that those results are independent of the present ones; in particular, it issafe to invoke them here. These results imply that any closed substack Y of
XK,d is Noetherian (so that it makes sense to speak of Y being flat over O), andthat Yred is of finite type over F. The claimed uniqueness then follows from
Proposition 4.8.6.
We will make use of the following corollary in Chapter 6.
4.8.13 Corollary. For any Hodge type λ, and for any a ≥ 1, the corresponding
crys,λ,O 0amorphism Spf Rρ
/$a → XK,dis effective, i.e. is induced by a morphism
crys,λ,O 0Spec Rρa/$a → XK,d.
crys,λ,O 0

Proof. By Proposition 4.8.10, the morphism Spf Rρcrys,λ,O

tors through a versal morphism Spf Rρ

0a/$a → XK,dfac-

crys,λ/$a → XK,d×Spf O Spec O/$a .

crys,λSince XK,d
is a p-adic formal algebraic stack (by Theorem 4.8.12) the basecrys,λchange XK,d×Spf O Spec O/$a is an algebraic stack, and so the result follows
from [Sta, Tag 07X8].

Finally, we can compute the dimensions of our potentially crystalline andsemistable stacks. It is presumably possible to develop the dimension theory of
p-adic formal algebraic stacks in some generality, but we do not need to do so,
as we can instead work with the special fibres, which are algebraic stacks.

4.8. STACKS OF POTENTIALLY SEMISTABLE REPRESENTATIONS 119
crys,λ,τcrys,λ,τ:= XK,d×Spf O Spec F
ss,λ,τ
ss,λ,τ
and X K,d := XK,d ×Spf O Spec F are equidimensional of dimension
4.8.14 Theorem. The algebraic stacks X K,d

X#{1 ≤ i < j ≤ d|λσ,i > λσ,j }.

σcrys,λ,τIn particular, if λ is regular, then the algebraic stacks X K,dequidimensional of dimension [K : Qp ]d(d − 1)/2.

ss,λ,τ

and X K,dare

Proof. Once again, we give the argument
P in the crystalline case, the semistablecase being identical. Write dλ :=σ #{1 ≤ i < j ≤ d|λσ,i > λσ,j }. The
crys,λ,τ
×Spf O Spec F is of finite type over F, and we need toalgebraic stack XK,dcrys,λ,τshow that it is equidimensional of dimension dλ . Let x : Spec F0 → XK,dbe afinite type point corresponding to a Galois representation ρ, with corresponding
crys,λ,τcrys,λ,O 0
versal morphism Spf Rρ
/$ → X K,d. We have the fibre productcrys,λ,O 0

Spf Rρ

/$ ×

crys,λ,O 0\
crys,λ,τ
(X K,d)Spf Rρ

∼ c,/$ −→ GLcrys,λ,O 0
/$,1
d,R
ρ

x

c crys,λ,O0where GLdenotes the completion of (GLd )Rcrys,λ,O0 /$ along the
d,R
/$,1
ρ

ρ

identity element in its special fibre. By [EG19, Lem. 2.40], it is therefore enoughcrys,λ,O 0
crys,λ,O 0
to recall that since Rρ
is equidimensional of dimension dλ +1, Rρ
/$is equidimensional of dimension dλ (see [BM14, Lem. 2.1]).

120 CHAPTER 4. CRYSTALLINE AND SEMISTABLE MODULI STACKSChapter 5

Families of extensions
In this chapter we extend the theory of the Herr complex to the context of(ϕ, Γ)-modules with coefficients, and use it to develop a theory of families ofextensions of (ϕ, Γ)-modules. We then inductively construct families which cover
the underlying reduced stack XK,d,red , and employ obstruction theory to deducethat XK,d is a Noetherian formal algebraic stack.

5.1

The Herr complex

In this chapter we consider the Herr complex; our approach is informed by
the papers [Her98, Her01, CC98, Liu08a, Pot13, KPX14], and in particular wefollow [Pot13, KPX14] in considering it as an object of the derived category.
Our main technical result is to show that it is a perfect complex, which we will
do by showing that it satisfies the following well-known criterion.
5.1.1 Lemma. Let A be a Noetherian commutative ring, and let C • be an objectof D(A). Then the following two conditions are equivalent:
1. There is a quasi-isomorphism F • → C • where F • is a complex of flat Amodules, concentrated in a finite number of degrees; and the cohomology
groups of C • are all finite A-modules.
2. C • is perfect.
Proof. This follows from [Sta, Tag 0658], [Sta, Tag 0654] and [Sta, Tag 066E].

Let A be a p-adically complete O-algebra, and let M be an A-module with
commuting A-linear endomorphisms ϕ, Γ (in our main applications, M will be
a projective étale (ϕ, Γ)-module with A-coefficients, but it will be convenientin our arguments to be able to consider more general possibilities, such as subquotients of base changes of such modules). Then the Herr complex of M is by
121

122

CHAPTER 5. FAMILIES OF EXTENSIONS

definition the complex of A-modules C • (M ) in degrees 0, 1, 2 given byM(ϕ−1,γ−1)/ M ⊕ M (γ−1)⊕(1−ϕ) / M.

We have the following useful interpretation of the cohomology groups of theHerr complex.
5.1.2 Lemma. Let M1 , M2 be projective étale (ϕ, Γ)-modules with A-coefficients.
Then for i = 0, 1 there are natural isomorphisms
H i C • (M1∨ ⊗ M2 ) ∼= ExtiAK,A ,ϕ,Γ (M1 , M2 ).
Proof. This is straightforward; in particular the case i = 0 is immediate. Wheni = 1, any extension M of M1 by M2 splits on the level of the underlying
projective AK,A -modules, and such a splitting is unique up to an element of
HomAK,A (M1 , M2 ) = M1∨ ⊗ M2 . Given such a splitting of M , we obtain twoelements f, g of M1∨ ⊗ M2 by writing
ϕM2 ϕM2 ◦ f
γM2 γM2 ◦ g
ϕM =, γM =
.
0
ϕM1
0
γM1
The condition that ϕM , γM commute shows that (f, g) determines a class in
H 1 C • (M1∨ ⊗ M2 ) (this class is easily seen to be well-defined, by our above
remark about the ambiguity in the choice of splitting). We leave the verificationthat this gives the claimed bijection to the sufficiently enthusiastic reader.
In order to show that the Herr complex is a perfect complex, we will need todevelop a little of the theory of the ψ-operator on (ϕ, Γ)-modules. We will only
need this in the case that A is an F-algebra, and we mostly work in this context
from now on. Note that for any K, if A is an F-algebra, then A+
K,A is (ϕ, Γ)+p0estable, and (AK,A ) is (ϕ, Γ)-stable; indeed we have ϕ(T ) = T and γ(T ) − T ∈e
T 2 A+
K,A by Lemma 3.2.18, and the stability under the action of Γ follows by
an identical argument to the proof of Lemma 3.2.18 (that is, it follows from theecontinuity of the action of Γ).
We claim that for any K, 1, ε, . . . , εp−1 is a basis for E0K as a ϕ(E0K )-vector
space. To see this, note that since the extension E0K /ϕ(E0K ) is inseparable ofdegree p, while E0K /E0Qp is a separable extension, it is enough to show the result
for E0Qp . In this case we have E0Qp = Fp ((ε − 1)), so ϕ(E0Qp ) = Fp ((εp − 1)) andPp−1
the claim is clear. Then for any x ∈ E0K we can write x = i=0 εi ϕ(xi ), andwe define ψ : E0K → E0K by ψ(x) = x0 . By definition ψ is continuous, Fp -linear,e and satisfies ψ ◦ ϕ = id. Since ψ commutes with ∆, we havecommutes with Γ,an induced map ψ : EK → EK , which again is continuous, Fp -linear, commuteswith Γ, and satisfies ψ ◦ ϕ = id.
If A is an Fp -algebra, then since ψ is continuous we can extend scalars
from Fp to A and complete to obtain continuous A-linear maps ψ : A0K,A →5.1. THE HERR COMPLEX

123e
A0K,A and ψ : AK,A → AK,A . Again, these are continuous and commute with Γand Γ, and satisfy ψ ◦ ϕ = id. We havei0
A0K,A = ⊕p−1
i=0 ε ϕ(AK,A ),and if x ∈ A0K,A and we write x =Pp−1 ii=0 ε ϕ(xi ), then ψ(x) = x0 .

5.1.3 Proposition. Let A be an F-algebra, and let M be a projective étale(ϕ, Γ)-module with A-coefficients. Then there is a continuous and open A-linear
surjection ψ : M → M such that ψ commutes with Γ, and we haveψ(ϕ(a)m) = aψ(m),ψ(aϕ(m)) = ψ(a)mfor any a ∈ AK,A , m ∈ M .
Proof. We define ψ : M → M to be the composite
Φ−1

ψ⊗1

M
M →
AK,A ⊗AK,A ,ϕ M → AK,A ⊗AK,A M = M.

The relations ψ(ϕ(a)m) = aψ(m), ψ(aϕ(m)) = ψ(a)m are immediate from thedefinitions. That ψ is continuous follows from the continuity of Φ−1
M and thecontinuity of ψ on AK,A , while the surjectivity of ψ is immediate from therelation ψ(ϕ(m)) = m. That ψ is open follows from Lemma 5.1.4 below.
5.1.4 Lemma. Let A be an F-algebra, let M be a projective étale (ϕ, Γ)-modulewith A-coefficients, and let M be a ϕ-stable lattice in M . Then there is an
integer h ≥ 0 such that for any integer n, we haveψ(T h+np M) ⊆ T n M ⊆ ψ(T np M).
Proof. Since ψ ◦ ϕ = id, for any n we haveψ(T np ϕ(M)) = T n M.
This implies that
T n M = ψ(T np ϕ(M)) ⊆ ψ(T np M).
For the other direction, choose h such thatT h M ⊆ ΦM (ϕ∗ M) ⊆ M.
Then we haveψ(T h M) ⊆ ψ(ΦM (ϕ∗ M)) = M.
It follows thatψ(T h+np M) = T n ψ(T h M) ⊆ T n M,
as required.

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CHAPTER 5. FAMILIES OF EXTENSIONS

5.1.5 Lemma. Let A be a Noetherian F-algebra, and let M be a projective étale(ϕ, Γ)-module with A-coefficients. Then M contains a (ϕ, Γ)-stable lattice.
Proof. By [EG21, Lem. 5.2.15], M contains a ϕ-stable lattice M. Let ΓM be theA+
K,A -submodule of M generated by the elements γm with γ ∈ Γ, m ∈ M. We
claim that ΓM is a lattice; note that since Γ and ϕ commute, ΓM is ϕ-stable,
and it is Γ-stable because A+
K,A is Γ-stable. To see that it is a lattice, it isenough (since A+
isNoetherian)to show that it is contained in a lattice (asK,Ait certainly spans M ). In particular, it is enough to show that ΓM ⊆ T −n M
for some n ≥ 0. This follows easily from the compactness of Γ; for example, if
e1 , . . . , em are generators of the finitely generated A+
K,A -module M, then we havema continuous map j : Γ → M , γ 7→ (γe1 , . . . , γed ), and Γ = ∪n j −1 (T −n Mm )is an open cover of Γ. Since this has a finite subcover we are done.
5.1.6 Lemma. Let A be an F-algebra. Then for any i ∈ Z we have γ(T i ) ∈×
T i (A+K,A ) .
Proof. This follows from Lemma 3.2.18. Indeed, we can write γ(T ) = T + λiiiiwith λ ∈ T 2 A+
K,A , so for any i ≥ 1, we have γ(T ) = (T + λ) , and γ(T ) − T ∈
+
i+1 +
i
i
×
i
−i
T AK,A , whence γ(T ) ∈ T (AK,A ) . Since γ(T )γ(T ) = 1, the result thenalso holds for i ≤ 0.
5.1.7 Corollary. Let A be a Noetherian F-algebra, and let M be a projectiveétale (ϕ, Γ)-module with A-coefficients. Then M contains a lattice M0 such thatfor all m ≥ 0, T −m M0 is (ψ, Γ)-stable.
Proof. By Lemma 5.1.5 there is a (ϕ, Γ)-stable lattice M, and by Lemma 5.1.4,
the lattice T −n M is ψ-stable for all n  0. It is also γ-stable for all n ≥ 0
by Lemma 5.1.6, so we may take M0 = T −n M for any sufficiently large valueof n.
If M is an étale (ϕ, Γ)-module with A-coefficients, then we write
M 0 := A0K,A ⊗AK,A M,ean étale (ϕ, Γ)-modulewith A-coefficients.
5.1.8 Lemma. Let A be an F-algebra, and let M be a projective étale (ϕ, Γ)module with A-coefficients. Then we have a decomposition of ϕ(A0K,A )-modules
p−1 i
M 0 = ker(ψ) ⊕ ϕ(M 0 ), and we may write ker(ψ) = ⊕i=1
ε ϕ(M 0 ).
Proof. Since ψ ◦ φ is the identity on M 0 , the endomorphism φ ◦ ψ of M 0 is
idempotent, and so M 0 decomposes as the direct sum of its kernel and image.
Since φ is injective, while ψ is surjective, we may express this decomposition as
M 0 = ker(ψ) ⊕ ϕ(M 0 ), as claimed.
∼
As M 0 is étale, we have A0K,A ⊗ϕ(A0K,A ) ϕ(M 0 ) −→ M 0 . Since A0K,A =
p−1 i
p−1 i
i
0
0
0
0
0
⊕p−1
i=0 ε ϕ(AK,A ), we find that M = ⊕i=0 ε ϕ(AK,A )ϕ(M ) = ⊕i=0 ε ϕ(M ).
p−1
Now, for 1 ≤ i ≤ p − 1 we have ψ(εi ϕ(M 0 )) = ψ(εi )M 0 = 0, so ⊕i=1 εi ϕ(M 0 ) ⊆5.1. THE HERR COMPLEX

125

i
0
0
0
ker(ψ). Since we have ⊕p−1
i=1 ε ϕ(M ) ⊕ ϕ(M ) = ker(ψ) ⊕ ϕ(M ), it follows that
this inclusion is an equality, as required.

The proof of the following result follows the approach of [CC98], although
one difference in our situation is that because we are working in characteristic p,
we do not need to reduce to the case that K/Qp is unramified.
5.1.9 Proposition. Let A be a Noetherian F-algebra. If M is a projective étale(ϕ, Γ)-module with A-coefficients, then (1 − γ) is bijective on ker(ψ).
Proof. We begin with some preliminaries on the action of γ on ε ∈ (A0K,A )+ .
We have γ(ε) = εχ(γ) , and we write χ(γ) = 1 + pN u where N ≥ 1 and u ∈ Z×p.
pn−1
n+N −1
×Then for each n ≥ 1 we can write χ(γ) = 1+p
un with un ∈ Zp , andif 0 ≤ r ≤ n + N − 1 and i ∈ Z then we have
γpn−1(εi ) = εi ϕr (εipn+N −1−run).

(5.1.10)Note that since un is a p-adic unit, (εun − 1)/(ε − 1) is a unit in (A0K,A )+ , andwe can write
(γ p

n−1

− 1)(ε − 1) = ε(εun − 1)pn+N −1∈ (ε − 1)pn+N −1((A0K,A )+ )× . (5.1.11)

It follows that for any r ∈ Z, we have
(γ p

n−1

− 1)((ε − 1)r ) ∈ (ε − 1)r+p

n+N −1−1(A0K,A )+ .

(5.1.12)

Indeed if r ≥ 0 then we may write
(γ

pn−1

r

− 1)((ε − 1) ) = (γ

pn−1

r−1
X
n−1
− 1)((ε − 1)) ·(γ p (ε − 1)r−1−j ) · (ε − 1)j ,
j=0
n−1

so (5.1.12) follows from (5.1.10) and the fact that γ p (ε−1) ∈ (ε−1)(A0K,A )+ ,which in turn follows from (5.1.11). The case r ≤ 0 then follows from the resultfor −r by writing
n−1

(γ

pn−1

(γ p
− 1)((ε − 1)−r ),− 1)((ε − 1) ) = − pn−1(γ
(ε − 1)−r )(ε − 1)−rrn−1n+N −1unand noting that (γ p (ε − 1))/(ε − 1) = (ε1+p− 1)/(ε − 1) is a unit.
Since both ψ and γ commute with the action of ∆, in order to prove theproposition it suffices to show that (1 − γ) is bijective on the kernel of ψ : M 0 →i0
M 0 . By Lemma 5.1.8, this kernel equals ker(ψ) = ⊕p−1
i=1 ε ϕ(M ). It follows
i0
from (5.1.10) that γ preserves ε ϕ(M ) for any integer i, so it is enough to provethat if (i, p) = 1 that (1 − γ) acts invertibly on εi ϕ(M 0 ).
In fact, we will prove the stronger statement that for each n ≥ 1, eachn−1
0 ≤ t ≤ N − 1, and each (i, p) = 1, the operator (1 − γ p ) acts invertiblyn−1
on εi ϕn+t (M 0 ) (note that (1 − γ p ) acts on this space by (5.1.10)).

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CHAPTER 5. FAMILIES OF EXTENSIONS
Writing
p−1 j
p−1 i+pε ϕ(M 0 )) = ⊕j=0ε
εi ϕn+t−1 (M 0 ) = εi ϕn+t−1 (⊕j=0n+t−1

ϕn+t (M 0 ),(5.1.13)
we see that if t > 0 then the statement for some pair (n, t) (and all i) implies
n−1
n−2
n−2
the statement for (n, t − 1). Writing (1 − γ p ) = (1 − γ p )(1 + γ p
+···+
n−2
γ p (p−1) ), and again using (5.1.13), we see also that the statement for (n, t)
implies the statement for (n−1, t). We can therefore assume that n is arbitrarily
large and that t = N − 1.
Let M be a (ψ, Γ)-stable lattice in M (which exists by Corollary 5.1.7),
let M0 = (A0K,A )+ ⊗AK,A M, and choose n large enough that pn+N −1 ≥ 3 andn−1j− 1)(M0 ) ⊆ (ε − 1)2 M0 .

(5.1.14)− 1)((ε − 1)r M0 ) ⊆ (ε − 1)r+2 M0(5.1.15)(γ p
It follows that
(γ p

n−1

for all r ∈ Z, because if m ∈ M0 then
(γ p

n−1

−1)((ε−1)r m) = (γ pn−1−1)((ε−1)r )·γ pn−1(m)+(ε−1)r (γ pn−1−1)(m),

so that (5.1.15) follows from (5.1.12) and (5.1.14), together with our assumptionthat pn+N −1 ≥ 3.
By (5.1.10), we have
(γ p

n−1

− 1)(εi ϕn+N −1 (x)) = εi ϕn+N −1 (εiun γ pn−1(x) − x),so it is enough to check that the map f : M 0 → M 0 given by
f (x) = εiun γ pn−1(x) − xis invertible.
Let α = εiun − 1, so that α/(ε − 1) is a unit in (A0K,A )+ . Then for any r ∈ Z
and x ∈ (ε − 1)r M0 , it follows from (5.1.15) thatεiun pn−11
f − 1 (x) =(γ− 1)(x) ∈ (ε − 1)r+1 M0 .
αα
In particular the sumg(x) :=

∞ 
X
j=0

1
1− f
α

j
(x)

converges. Since f, g are additive by definition, we have




1
1
1 − f ◦ g(x) = g ◦ 1 − f (x) = g(x) − x,ααso that the function g : M → M is a left and right inverse to α1 f . Thus f isinvertible, as required.

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127

Suppose that A is an F-algebra. Using the ψ operator, we can give an alternative description of the Herr complex, which will be important in establishingthat it is a perfect complex. Let Cψ• (M ) be the complex in degrees 0, 1, 2 givenbyM(ψ−1,γ−1)/ M ⊕ M (γ−1)⊕(1−ψ) / M.

We have a morphism of complexes C • (M ) → Cψ• (M ) given byM(ϕ−1,γ−1)(−ψ,1)1


M/ M ⊕ M (γ−1)⊕(1−ϕ) / M(ψ−1,γ−1)(5.1.16)−ψ/ M ⊕ M (γ−1)⊕(1−ψ) / M(That this is a morphism of complexes follows from the facts that ψ ◦ ϕ = id,and that ψ commutes with γ.)
5.1.17 Proposition. Let A be a Noetherian F-algebra, and let M be a projective étale (ϕ, Γ)-module with A-coefficients. Then the morphism of complexesC • (M ) → Cψ• (M ) defined by (5.1.16) is a quasi-isomorphism.
Proof. The cokernel of (5.1.16) is the zero complex, while its kernel is the complex
0/ ker(ψ) (γ−1) / ker(ψ),

which is acyclic by Proposition 5.1.9.
In fact, we require a slightly stronger statement than the quasi-isomorphismof the preceding lemma. Namely, we need a statement that takes into accounttopologies, which will make use of the following lemma.
5.1.18 Lemma. If f : X → Y is a continuous open map of topological spaces,
and if Y 0 ⊆ Y is the inclusion of a subspace into Y (i.e. Y 0 is a subset of Y endowed with the induced topology), then the base-changed map X 0 := f −1 (Y 0 ) →Y 0 is again open, when X 0 is endowed with the topology induced by that of X.
Proof. Let U 0 be an open subset of X 0 ; then we may write U 0 = U ∩ X 0 , forsome open subset U of X. Thus
f (U 0 ) = f (U ∩ X 0 ) = f U ∩ f −1 (Y 0 ) = f (U ) ∩ Y 0 .
Since f is open, we conclude that f (U 0 ) is an open subset of Y 0 , as required.
We now have the following strengthening of Proposition 5.1.17, which takes
into account the topologies on the complexes.
5.1.19 Proposition. Let A be a countable Noetherian F-algebra and let M
be a projective étale (ϕ, Γ)-module with A-coefficients. Then the morphism ofcomplexes C • (M ) → Cψ• (M ) defined by (5.1.16) induces topological isomorphisms
on each of the associated cohomology modules.

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CHAPTER 5. FAMILIES OF EXTENSIONS

Proof. Proposition 5.1.17 shows that (5.1.16) is a quasi-isomorphism. By definition, the induced map on H 0 is the identity map, and is therefore a homeomorphism. Since ψ is continuous and open, we see that the maps C i (M ) → Cψi (M )are continuous and open for each i; in particular, the isomorphism on H 2 isinduced from the continuous open map −ψ : C 2 (M ) → Cψ2 (M ), and is therefore
a homeomorphism. (Here and below we use the standard fact that a quotientmorphism of topological groups is necessarily open.)
The case of H 1 requires a little more work. Since the maps C 0 (M ) → Cψ0 (M )and H 1 C • (M ) → H 1 Cψ• (M ) are isomorphisms, we see that the continuous
morphism C 1 (M ) → Cψ1 (M ) induces a continuous isomorphism of the modulesof cocycles Z 1 (M ) → Zψ1 (M ), and it suffices to show that this map is open
(since this will imply that the induced bijection on H 1 is both continuous andopen, and thus is an isomorphism of topological groups).
e1 (M ) denote the preimage of Z 1 (M ) in C 1 (M ), and if we let
If we let Zψ1
K ⊆ C (M ) = M ⊕ M denote ker(ψ) ⊕ 0, then we have inclusions Z 1 (M ), K ⊆Ze1 (M ), and hence a continuous morphism
Z 1 (M ) ⊕ K → Ze1 (M ),(5.1.20)

which is in fact a bijection. Each of Z 1 (M ), K, and Ze1 (M ) is closed subgroupof C 1 (M ) = M ⊕ M . Lemma D.3 shows that this latter topological group is
Polish, and thus so is any of its closed subgroups. Corollary C.3 then showsthat (5.1.20) is in fact a homeomorphism, while Lemma 5.1.18 shows that themorphism Ze1 (M ) → Zψ1 (M ) is open. Consequently, we find that the morphism
Z 1 (M ) → Zψ1 (M ) is open, as required.
5.1.21 Lemma. Let A be a Noetherian F-algebra, and let X be an A-module
subquotient of a finitely generated projective A[[T ]]-module, endowed with its
natural subquotient topology. If the topology on X is discrete, then X is finitely
generated as an A-module.
Proof. Write X = Y /Z, where Y is an A-submodule of a finitely generated
projective A[[T ]]-module M. Since the topology on X is discrete, Z is open
in Y , and thus Z ⊇ U ∩ Y , for some open neighbourhood U of zero in M. Sucha neighbourhood U contains T n M for some n  0, and so we find that X is asubquotient of the finitely generated A-module M/T n M, as required.
5.1.22 Theorem. Let A be a Noetherian O/$a -algebra such that A/$ is countable, and let M be a projective étale (ϕ, Γ)-module with A-coefficients.
1. The Herr complex C • (M ) is a perfect complex of A-modules, concentrated
in degrees [0, 2].
2. If either (i) B is a finite A-algebra; or (ii) B is a finite type A-algebra andA itself is of finite type over O/$a ; then there is a natural isomorphismin the derived category∼•C • (M ) ⊗L
A B −→ C (M ⊗AK,A AK,B ).

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129

In particular, there is a natural isomorphism∼

H 2 (C • (M )) ⊗A B −→ H 2 (C • (M ⊗AK,A AK,B )).
Proof. Since A is Noetherian, AK,A is a flat A-algebra (being a localisation•of the power series ring A+K,A ), so C (M ) is a complex of flat A-modules. By[Sta, Tag 07LU] (for part (1), taking R = O/$a ), and [Pil20, Prop. 2.2.2] (forpart (2), again taking R = O/$a ), it suffices to prove the result in the casethat A is an F-algebra, which we assume from now on. We begin with (1). By
Lemma 5.1.1, we need only check that the cohomology groups of C • (M ) are
finitely generated A-modules. In order to do this, we will make two truncationarguments.
Firstly, by Lemma 5.1.5, we can choose a (ϕ, Γ)-stable lattice M ⊆ M .
We claim that for every n ≥ 1 the Herr complex C • (T n M) is acyclic; consequently, the natural morphism of complexes C • (M ) → C • (M/T n M) is a quasiisomorphism.
To see the claim, it suffices to show that (1 − ϕ) : T n M → T n M is an
isomorphism; indeed, the exactness of C • (T n M) is a formal consequence of this.
We begin by checking injectivity. If m ∈ T n M and (1−ϕ)(m) = 0, then we havem = ϕ(m) ∈ T pn M so that in particular we have m ∈ T n+1 M. By induction,
we see that m ∈ T n M for all n, so that m = 0, as required.
We now prove surjectivity. If m ∈ T n M, then we have seen in the previous
2n+2
paragraph that ϕ(m) ∈ T n+1 M, ϕ
M, and so on. Since M is T P (m) ∈ T
adically complete, we can set x = i≥0 ϕi (m) ∈ T n M; then (1 − ϕ)(x) = m, as
required.
We now turn to our other truncation argument. Let M0 be as in Lemma 5.1.7,
0
so that for each n0 ≤ 0, T n M0 is a (ψ, Γ)-stable lattice in M . In particu0
lar, for each n0 ≤ 0, Cψ• (T n M) is a subcomplex of Cψ• (M ). We claim that if0n0 ≤ −2 then Cψ• (M/T n M0 ) is acyclic, so that the natural morphism of com0
plexes Cψ• (T n M0 ) → Cψ• (M ) is a quasi-isomorphism.
0
As above, it is enough to show that (1 − ψ) is bijective on M/T n M0 . Notefirstly that if r ≤ n0 then ψ(T r M0 ) ⊆ T r+1 M0 . Indeed, for each integer r ≤ 0
we haveψ(T r M0 ) ⊆ ψ(T pbr/pc M0 ) = T br/pc ψ(M0 ) ⊆ T br/pc M0 ,and the claim follows since if r ≤ −2 then br/pc ≥ r + 1.
We now show that (1 − ψ) is injective on M/T −n M0 . Suppose that m ∈ M
0with (1−ψ)(m) ∈ T n M0 . Then m ∈ T r M0 for some r  0. If r ≥ n0 then we are0
done, and if not then since ψ(m) ∈ T r+1 M0 and (1 − ψ)(m) ∈ T n M ⊆ T r+1 M,
0
we have m ∈ T r+1 M. It follows by induction that m ∈ T n M0 , as required.
For surjectivity, take m ∈ M , and choose r  0 such that m ∈ T r M0 . If
r < n0 then ψ(m) ∈ T r+1 M0 , so by induction we see that there is some s ≥ 0
Ps−1 i0such that ψ s (m) ∈ T n M0 . If we set x =i=0 ψ (m), then (1 − ψ)(x) =0m − ψ s (m) ∈ m + T n M, as required.

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CHAPTER 5. FAMILIES OF EXTENSIONS
We now consider the quasi-isomorphisms0Cψ• (T n M0 ) → Cψ• (M ) ← C • (M ) → C • (M/T n M).
Proposition 5.1.19 shows that the middle quasi-isomorphism induces a topological isomorphism on cohomology modules, and thus altogether we obtainmorphisms0
H i Cψ• (T n M0 ) → H i C • (M/T n M)
which are isomorphisms of A-modules, and continuous with respect to the natural topologies on the source and target. Since the target is endowed with thediscrete topology, we find that the source is also endowed with the discrete
topology. By Lemma 5.1.21, it follows that these cohomology modules are indeed finitely generated over A, and hence so are the cohomology modules ofC • (M ), as required.
We now turn to (2), where we follow the proof of [KPX14, Thm. 4.4.3]. Sinceboth (i) and (ii) in particular require B to be a finite type A-algebra, we seethat B is Noetherian, and B is countable, so that in particular C • (M ⊗AK,A
AK,B ) is a perfect complex of B-modules by part (1) (replacing A by B, and M
by M ⊗AK,A AK,B ). Since C • (M ) is a perfect complex of A-modules, C • (M )⊗LAB
is also a perfect complex of B-modules, and the natural map AK,A ⊗A B → AK,B
induces a morphism•
C • (M ) ⊗L
A B → C (M ⊗AK,A AK,B ).

(5.1.23)

In case (i), when B is in fact finite as an A-module, the natural map AK,A ⊗A
B → AK,B is an isomorphism (indeed, we have A[[T ]] ⊗A B = B[[T ]], because
A[[T ]] ⊗A B is finitely generated over A[[T ]] and thus T -adically complete andseparated by the Artin–Rees lemma), so (5.1.23) is certainly an isomorphism inthis case.
We now turn to case (ii), when B is only assumed to be a finite type Aalgebra, but A, and hence also B, is furthermore of finite type over O/$a . Thisimplies in particular that A is Jacobson, so that if m is any maximal ideal of B,
then B/m is finite as an A-module. From this, and the already-proved case (i),we conclude that (5.1.23) is an isomorphism if we replace B by B/m.
It follows that we have a chain of quasi-isomorphisms
∼

∼

L
•
L
•
(C • (M ) ⊗L
A B) ⊗B B/m −→ C (M ) ⊗A B/m −→ C (M ⊗A B/m)∼−→ C • (M ⊗AK,A AK,B ) ⊗L
B B/m,
where the final quasi-isomorphism comes from replacing A by B, B by B/m,and M by M ⊗AK,A AK,B in (5.1.23). (That this is indeed a quasi-isomorphism
follows by case (i), since B/m is a finite B-module, while we have (M ⊗AK,A
AK,B ) ⊗AK,B AK,B/m = M ⊗AK,A AK,B/m = M ⊗AK,A (AK,A ⊗A B/m) =M ⊗A B/m).
Thus (5.1.23) becomes a quasi-isomorphism after applying ⊗L
B B/m, for any
maximal ideal m of B. It follows from [KPX14, Lem. 4.1.5] that (5.1.23) is5.1. THE HERR COMPLEX

131

a quasi-isomorphism, as required. Finally, the compatibility of the formationof H 2 with base change is an immediate consequence of this isomorphism inthe derived category, together with the vanishing of all of the higher degree
cohomology groups.
5.1.24 Corollary. Let A be a p-adically complete Noetherian O-algebra suchthat A/$ is countable, and let M be a projective étale (ϕ, Γ)-module. Then the
Herr complex C • (M ) is a perfect complex concentrated in degrees [0, 2].
Proof. This follows from [Sta, Tag 0CQG] and Theorem 5.1.22. (We apply thebase-change statement of part (2) of the theorem to the surjective (and hence
finite) morphisms A/$a+1 → A/$a .)We also note the following rather technical corollary Theorem 5.1.22, which
we will use in our discussion of families of extensions below.
5.1.25 Corollary. If A is of finite type over O/$a for some a ≥ 1, and M is a
projective étale (ϕ, Γ)-module, then the Herr complex C • (M ) can be represented
by a complex C 0 → C 1 → C 2 of finite projective A-modules in degrees [0, 2].
Proof. Theorem 5.1.22 shows that C • (M ) is perfect, and so by [Sta, Tag 0658],
the present corollary will follow provided that we show that C • (M ) has toramplitude in [0,2]. Since any A-module is a filtered direct limit of finite Amodules, it suffices to show that C • (M )⊗L A M has cohomological amplitude
lying in [0, 2] for any finite A-module M . If we let B = A ⊕ M , thought ofas an A-algebra by declaring M to be a square-zero ideal, then it is equivalent
to show that C • (M )⊗L A B has cohomological amplitude lying in [0, 2]. But
Theorem 5.1.22 shows that this latter complex is isomorphic (in the derived
category of B-modules) to C • (M ⊗AK,A AK,B ), whose cohomological amplitudelies in [0,2] by construction.
We now explain the comparison between the Herr complex and Galois cohomology, and the relationship to Tate local duality. These results are essentially
due to Herr and are proved in [Her98, Her01] but we follow [KPX14] and formulate them as statements in the derived category.
Exactly as in [KPX14, Defn. 2.3.10], if M1 , M2 are projective étale (ϕ, Γ)modules with A-coefficients, then there is a cup productC • (M1 ) ⊗A C • (M2 ) → C • (M1 ⊗AK,A M2 ).
More precisely, the cup product arises from the following generalities. If wehave two complexes of A-modules C • and D• , then the tensor product C • ⊗ D•
has differential given byd(x ⊗ y) = dx ⊗ y + (−1)i x ⊗ y
if x ∈ C i , y ∈ Dj . If f • : C • → C • , then we write Fib(f |C • ) := Cone(f )[−1],which by definition has Fib(f )i = C i ⊕ C i−1 and di ((x, y)) = (di (x), −di−1 (y) −f i (x)).

132

CHAPTER 5. FAMILIES OF EXTENSIONSAs a special case of [KPX14, Lem. 2.3.9], if f1 : C1• → C1• , f2 : C2• → C2• ,then we have a natural morphismFib(1 − f1 |C1• ) ⊗ Fib(1 − f2 |C2• ) → Fib(1 − f1 ⊗ f2 |C1• ⊗ C2• )(5.1.26)
Then by definition we have C • (M ) = Fib 1 − γ| Fib(1 − ϕ|M ) , so that bymultiple applications of (5.1.26) we have morphismsC • (M1 ) ⊗A C • (M2 ) = Fib 1 − γ| Fib(1 − ϕ|M1 ) ⊗A Fib 1 − γ| Fib(1 − ϕ|M2 )→ Fib 1 − γ| Fib(1 − ϕ|M1 ) ⊗A Fib(1 − ϕ|M2 )→ Fib 1 − γ| Fib 1 − ϕ|(M1 ⊗A M2 )
 Fib 1 − γ| Fib 1 − ϕ|(M1 ⊗AK,A M2 )= C • (M1 ⊗ M2 )whose composite defines the cup product.
5.1.27 Lemma. If A is a finite type O/$a -algebra, then there is an isomorphism
H 2 (C • (AK,A (1))) ∼= A,compatible with base change.
Proof. Since by Theorem 5.1.22 (2) the formation of H 2 (C • (M )) is compatible with base change, the result follows from the case A = O/$a , which isimmediate from Theorem
5.1.29 below (that is, from the natural isomorphism
H 2 GK , (O/$a )(1) ∼= O/$a ).
If M is a projective étale (ϕ, Γ)-module over a finite type O/$a -algebra,
then we define the Tate duality pairing between the Herr complexes of M andof its Cartier dual M ∗ as the following composite of the cup product, truncation,and the isomorphism of Lemma 5.1.27:
C • (M ) × C • (M ∗ ) → C • (AK,A (1))) → H 2 (C • (AK,A (1)))[−2] ∼= A[−2].
5.1.28 Proposition. Let A be a finite type O/$a -algebra, and let M be a
projective étale (ϕ, Γ)-module.
1. The Tate duality pairing induces a quasi-isomorphism∼

C • (M ) −→ RHomA (C • (M ∗ ), A))[−2].
2. If A is a finite extension of F, then the Euler characteristic χA (C • (M ))is equal to −[K : Qp ]d.
Proof. For part (1), by [KPX14, Lem. 4.1.5], it is enough to treat the case that A
is a field, in which case A is a finite extension of F. Then both parts follow from
Theorem 5.1.29 below (that is, from the corresponding statements for Galois
representations).

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133

Finally we recall the relationship between the Herr complex and Galois cohomology. As in [Pot13, §2] it is possible to upgrade the following theorem
to an isomorphism in the derived category, but as we do not need this we do
not give the details here. If A is a complete local Noetherian O-algebra withfinite residue field, and M is a formal projective étale (ϕ, Γ)-module with Acoefficients, then we can define the Herr complex C • (M ) exactly as for étale(ϕ, Γ)-modules, so that by definition we have C • (M ) = limn C • (MA/mnA ). By←−[Sta, Tag 0CQG], C • (M ) is a perfect complex, concentrated in degrees [0, 2].
5.1.29 Theorem. If A is a complete local Noetherian O-algebra with finite
residue field, and T is a finitely generated projective A-module with a continuous
action of GK , then there are isomorphisms of A-modules∼

H i (GK , T ) −→ H i (C • (DA (T )))which are functorial in T and compatible with cup products and duality.
Proof. This is [Dee01, Prop. 3.1.1], which is deduced from the results of [Her01,
Her98] by passage to the limit.
5.1.30 Remark. In accordance with our general convention of writing ρT fora family T → Xd , if T is an affine scheme of finite type over O/$a , then we
write H 2 (GK , ρT ) for the pull-back to T of the cohomology group H 2 (C • (M ))on Xd , where M is the étale (ϕ, Γ)-module corresponding to the morphism
T → Xd . By Theorem 5.1.22, H 2 (GK , ρT ) is a coherent sheaf, whose formationis compatible with arbitrary finite type base-change, and so in particular by
Theorem 5.1.29 its specialisations at Fp -points coincide with the usual Galois
cohomology groups. This compatibility with base-change also allows us to extend the definition of H 2 (GK , ρT ) to arbitrary (not necessarily affine) schemes Tof finite type over O/$a .
5.1.31 Lemma. Let T be a scheme of finite type over O/$a , and let ρT , ρ0T
be families of Galois representations over T . Then sections of H 2 (GK , ρT ⊗(ρ0T )∨ (1)) are in natural bijection with homomorphisms ρT → ρ0T .
5.1.32 Remark. Here by a homomorphism ρT → ρ0T we mean a homomorphismof the corresponding (ϕ, Γ)-modules.
Proof of Lemma 5.1.31. This follows from Proposition 5.1.28 and Lemma 5.1.2.

5.1.33

Obstruction theory

We now show that the Herr complex provides Xd with a nice obstruction theory
in the sense of Definition A.34.
Let A be a p-adically complete O-algebra, and let x : Spec A → Xd be amorphism, corresponding to a projective (ϕ, Γ)-module M . We wish to consider
the problem of deforming M to a square zero thickening of A. Specifically, if
0 → I → A0 → A → 0

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CHAPTER 5. FAMILIES OF EXTENSIONS

is a square zero extension, then we let Lift(x, A0 ) be the set of isomorphismclasses of projective (ϕ, Γ)-modules M 0 with A0 -coefficients which have the property that M 0 ⊗A0 A ∼= M.
For any such thickening A0 , we define a corresponding obstruction class asfollows. The underlying AK,A -module of M has a unique (up to isomorphism)f, and we may lift ϕ, γ to semi-linear
lifting to a projective AK,A0 -module M
fendomorphisms ϕ,e γe of M . (Indeed, AK,A0 is a square zero thickening of AK,A ,and a finite projective module P over any ring R deforms uniquely through any
square zero extension R0 → R, as its deformations are controlled by
H 1 (Spec R, ker(R0 → R) ⊗ EndR (P )),which vanishes (as Spec R is affine). To see that ϕ, γ lift, think of them as
AK,A -linear maps ϕ∗ M → M , γ ∗ M → M .)However, there is no guarantee that we can find lifts ϕ,e γe which commute. To
measure the obstruction to the existence of such lifts, let ad M = HomAK,A (M, M )be the adjoint of M . This naturally has the structure of a (ϕ, Γ)-module; indeed, we have a natural identification ϕ∗ ad M = HomAK,A (ϕ∗ M, ϕ∗ M ), and we
define Φad M : ϕ∗ ad M → ad M by(Φad M (f ))(x) = ΦM (f (Φ−1
M (x))).
We define the action of Γ in the analogous way. Then we let ox (A0 ) be the image
in H 2 (C • (ad M )) ⊗A I = H 2 (C • (ad M ⊗A I)) of
ϕe
eγ ϕ
e−1 γ
e−1 − 1 ∈ ad M ⊗A I = C 2 (ad M ⊗A I).
5.1.34 Lemma. The cohomology class ox (A0 ) is well-defined independently of
the choice of ϕ,e γe, and vanishes if and only if Lift(x, A0 ) 6= 0.
Proof. Any other liftings ϕ
e0 , γe0 are obtained from our given liftings ϕ,
e γ
e by
0
0
setting ϕe = (1 + X)ϕ,e γe = (1 − Y )e
γ , for some X, Y ∈ ad M ⊗A I. A simple
computation shows that
ϕ
e0 γ
e0 (ϕ
e0 )−1 (eγ 0 )−1 − ϕe
eγ ϕ
e−1 γ
e−1 = (1 − γ)X + (1 − ϕ)Y,which shows that the cohomology class ox (A0 ) is well-defined, and that it vanishes if and only if we can choose ϕ
e0 , γe0 so that ϕ
e0 γ
e0 (ϕ
e0 )−1 (eγ 0 )−1 = 1, which is0
in turn equivalent to Lift(x, A ) 6= 0, as required.
If F is an A-module, we let A[F ] := A ⊕ F be the A-algebra with multiplication given by(a, m)(a0 , m0 ) := (aa0 , am0 + a0 m).
This is a square zero thickening of A, and Lift(x, A[F ]) 6= 0, because we havethe trivial lifting given by M ⊗A A[F ].
5.1.35 Lemma. Suppose that F is a finitely generated A-module. Then there∼
is a natural isomorphism of A-modules Lift(x, A[F ]) −→ H 1 (C • (ad M ) ⊗L
A F ).

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135

Proof. We begin by constructing the isomorphism on the level of sets. Notethat C • (ad M ) ⊗L F is computed by C • (ad M ⊗A F ). Liftings of M to A[F ] aredetermined by the corresponding liftings ϕ,e γ
e of ϕ, γ, and we obtain a class in
H 1 (C • (ad M ⊗A F )) by taking the image of of(ϕϕe −1 − 1, γeγ −1 − 1) ∈ ad M ⊗A F ⊕ ad M ⊗A F = C 1 (ad M ⊗A F ).
An elementary calculation shows that (ϕϕe −1 − 1, γeγ −1 − 1) is in the kernel of1
2
C (ad M ⊗A F ) → C (ad M ⊗A F ) if and only if ϕeeγ = γeϕ,e so it only remains
to check that the lifting given by ϕ,e γe is trivial if and only if the corresponding
cohomology class vanishes.
To see this, note that the endomorphisms of the trivial lifting are of theform 1 + X, for some X ∈ ad M ⊗ F . The corresponding ϕ,e γe are given byϕe = (1 + X)ϕ(1 + X)−1 , γe = (1 + X)γ(1 + X)−1 , which is equivalent to(ϕϕe −1 − 1, γeγ −1 − 1) = ((1 − ϕ)X, (1 − γ)X), which by definition is equivalentto (ϕϕe −1 − 1, γeγ −1 − 1) being a coboundary, as required.
To compare the A-module structures, recall that by definition the A-module
structure on Lift(x, A[F ]) is defined as follows (see [Sta, Tag 07Y9]). If r ∈ A,
then we have a homomorphism fr : A[F ] → A[F ] given by fr (a, f ) = (a, rf ),f of M , we let rM
f be the base change of M
f via fr . Explicitly,and given a lifting M
this means that we replace (ϕe − ϕ) and (eγ − γ) by r(ϕe − ϕ) and r(eγ − γ).
Similarly, the addition map Lift(x, A[F ]) × Lift(x, A[F ]) → Lift(x, A[F ])comes from the obvious identification Lift(x, A[F ])×Lift(x, A[F ]) = Lift(x, A[F ×F ]) together with base change via the homomorphism A[F × F ] → A[F ] givenby (a, f1 , f2 ) 7→ (a, f1 + f2 ). With obvious notation, this amounts to settingϕe1  ϕe2 := ϕe1 + ϕe2 − ϕ, γ
e1  γe2 := γe1 + γe2 − γ.
On the other hand, by definition the A-module structure on H 1 (C • (ad M )⊗LF ) is given by the obvious A-module structure on the pairs (ϕϕe −1 − 1, γeγ −1 −
1). This is obviously the same as the A-module structure that we have justexplicated on Lift(x, A[F ]).
5.1.36 Proposition. Xd admits a nice obstruction theory in the sense of Definition A.34.
Proof. Since Xd is limit preserving by Lemma 3.2.19, it follows from Theorem 5.1.22 and Lemmas 5.1.34 and 5.1.35 that the Herr complex C • (ad M )provides the required obstruction theory.

5.2

Residual gerbes and isotrivial families

In this section we briefly discuss the notion of isotrivial families of (ϕ, Γ)-modules
over reduced F-schemes; i.e. of families which are pointwise constant. The
language of residual gerbes (see Appendix E) provides a convenient frameworkfor doing this. In the following discussion we allow F0 to denote any algebraic
extension of F.
We have seen that Xd is a quasi-separated Ind-algebraic stack, which by
Proposition 3.4.12 may be written as the inductive limit of algebraic stacks

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with closed immersions (and so, in particular, monomorphisms) as transition
morphisms. Thus the discussion at the end of Appendix E applies, and inparticular, for each point x ∈ |Xd |, the residual gerbe Zx at x in Xd exists.
Furthermore, if x is a finite type point, then the canonical monomorphism Zx ,→Xd is an immersion.
If we consider an F0 -valued point x : Spec F0 → Xd (by abuse of notation
we use x to denote both this point, and its image x ∈ |Xd |, which is a finite
type point of Xd ), then we may base-change the residual gerbe to Xd at xx
(which is a gerbe over F) over F0 via the composite Spec F0 −→ Xd → Spec F;
equivalently, we may regard x as a point of the base-change (Xd )F0 , and thenconsider the residual gerbe of (Xd )F0 at this point. This residual gerbe is then of
the form [Spec F0 /G], for a finite type affine group scheme G over F0 . (By [Sta,Tag 06QG] the group G may be described as the fibre product x ×Xd x, andhence its claimed properties follow from the fact that by Proposition 3.4.12, thediagonal of Xd is affine and of finite type.) In fact, we have the following more
precise result regarding G.
5.2.1 Lemma. If x is an F0 -valued point of (Xd )F0 , then Aut(x) is an irreducible smooth closed algebraic subgroup of GLd/F0 .
Proof. As already noted, it follows from Proposition 3.4.12 that Aut(x) is a
finite type affine group scheme over F0 . Let D denote the étale (ϕ, Γ)-module
over F0 corresponding to x, and let R := EndAK,F0 ,ϕ,Γ (D). Then, if ρ : GK →GLd (F0 ) denotes the Galois representation corresponding to x, we see that also
R = EndGK (ρ), and hence that R is an F0 -subalgebra of Md (F0 ). Furthermore,
if A is any finite type F0 -algebra, and if DA denotes the base-change of D over A,
then the natural morphism R ⊗F0 A → EndAK,F0 ,ϕ,Γ (DA ) is an isomorphism by
Theorem 5.1.22 (2) and Lemma 5.1.2 (note that A is automatically flat over F0 ).
Thus\
Aut(x)(A) := AutAK,A ,ϕ,Γ (DA ) = (R ⊗F0 A)× = (R ⊗F0 A) GLd (A),so that Aut(x), as a scheme over F0 , is precisely the open subscheme R ∩ GLd ,where we think of R as an affine subspace of the affine space Md . In particular,
we see that Aut(x) is an open subscheme of an affine space, and thus smoothand irreducible.
Suppose that S is a reduced F0 -scheme of finite type, and that S → Xd isa morphism such that every closed point of S maps to some fixed F0 -valued
point x ∈ |Xd |. We can think of this morphism as classifying a family of Galois
representations ρS over S whose fibre at each closed point is isomorphic to the
fixed representation ρ : GK → GLd (F0 ) classified by x. If G := AutGK (ρ), thenLemma E.5 shows that the morphism S → X factors through the residual gerbe[Spec F0 /G], and thus corresponds to an étale locally trivial G-bundle E overS. (A priori, the G-bundle E over S is fppf locally trivial. However, since G
is smooth, by Lemma 5.2.1, we see that E is in fact smooth locally trivial. Bytaking an étale slice of a smooth cover over which E trivializes, we see that E5.3. TWISTING FAMILIES

137

is in fact étale locally trivial.) We can then describe the family ρS concretely
as a twist by E of the constant family S × ρ, i.e. ρS = E ×G ρ.

5.3

Twisting families

We now begin our study of the dimensions of certain families of (ϕ, Γ)-representations,and the behaviours of these dimensions under the operations of twisting by families of 1-dimensional representations, and forming extensions of families.
We will need in particular to be able to twist families of representations
by unramified characters. Given an Fp -algebra A and an element a ∈ A× ,we have a (ϕ, Γ)-module Ma whose underlying AK,A -module is free of rank 1,
generated by some v ∈ Ma for which ϕ(v) = av and γ(v) = v. If A = F thenthe corresponding representation of GK is the unramified character ura taking
a geometric Frobenius to a. The universal instance of this construction comesby taking a = x ∈ A = F[x, x−1 ]; the corresponding (ϕ, Γ)-module Mx is thenclassified by a morphism which we denote urx : Gm := Spec F[x, x±1 ] → X1 ,which evidently factors through X1,red .
Given any morphism T → Xd,red , with T a reduced finite type F-scheme,
corresponding to a family ρT of GK -representations over T , we may consider
the family ρT  urx over T × Gm , as in Section 3.8. We refer to this operationon (ϕ, Γ)-modules as unramified twisting.
5.3.1 Definition. We say that ρT is twistable if whenever ρt ∼= ρt0 ⊗ ura where×t, t0 ∈ T (Fp ) and a ∈ Fp , then a = 1. We say that it is essentially twistable if×for each t ∈ T (Fp ), the set of a ∈ Fp for which there exists t0 ∈ T (Fp ) withρt ∼= ρt0 ⊗ ura is finite.
5.3.2 Lemma. If the dimension of the scheme-theoretic image of T in Xd,red
is e, then the dimension of the scheme-theoretic image of T × Gm in Xd,red is at
most e + 1. If T contains a dense open subscheme U such that ρU is essentiallytwistable, then equality holds.
5.3.3 Remark. Since a twistable representation is essentially twistable, we seethat if T contains a dense open subscheme U such that ρU is twistable, thenequality holds in Lemma 5.3.2.
Proof of Lemma 5.3.2. We may assume that T is irreducible. Write f for themorphism T → Xd,red and g for the morphism T × Gm → Xd,red . By [Sta,Tag 0DS4], we may, after possibly replacing T by a nonempty open subscheme,
assume that for each Fp -point t of T we have dim Tf (t) = dim T − e.
Let v = (t, λ) be an Fp -point of T ×Gm . Then (T ×Gm )g(v) contains Tf (t) ×{λ}, so that dim(T × Gm )g(v) ≥ dim Tf (t) = dim T − e = dim(T × Gm ) − (e + 1),
so the first claim follows from another application of [Sta, Tag 0DS4].
For the last part, we may replace T by U , and then the hypothesis that ρT
is essentially twistable means that we can identify (T × Gm )g(v) with a finite
union of fibres Tf (t0 ) (indexed by the finitely many a for which there is a t0 and

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CHAPTER 5. FAMILIES OF EXTENSIONS

an isomorphism ρt ∼= ρt0 ⊗ ura ), so that equality holds in the above inequality,
and the result again follows from [Sta, Tag 0DS4].
Recall that for each embedding σ : k ,→ F, we have a corresponding fundamental character ωσ : IK → F× , which corresponds via local class field theory(normalised to take uniformizers to geometric Frobeneii) to the composite of σ×and the natural map OK→ k × . It is well-known, and easy to check, that
Q e(K/Qp )= |IK . If F0 is an algebraic extension of F, then we can and doσ ωσ
identify the embeddings k ,→ F and k ,→ F0 .
Let n = (nσ )σ:k,→F be a tuple of integers 0 ≤ nσ ≤ p − 1. The characters ωσ×can all be extended to GK , and the restriction to IK of any character GK → Fp
Q −nσ
is equal to σ ωσ
for some n (which is unique unless the character isQunramified). We write ψn : GK → F× for a fixed choice of an extension of σ ωσ−nσ
to GK . If nσ = 0 for all σ, or nσ = p − 1 for all σ (these are exactly the cases
corresponding to unramified characters), then we fix the choice ψn = 1. We canand do make our choice so that if n, n0 are such that (ψn ψn−10 )|IK = |IK , then−1
in fact ψn ψn0 = .
Somewhat abusively, we will also write ψn for the constant family of (ϕ, Γ)modules D(ψn ) over any Fp -scheme T .
5.3.4 Remark. The irreducible representations α : GK → GLd (Fp ) are easilyclassified; indeed, since the wild inertia subgroup must act trivially, they aretamely ramified, so that the restriction of α to IK is diagonalizable. Considering
the action of Frobenius on tame inertia, it is then easy to see that α is induced
from a character of the unramified extension of K of degree d. It follows thateach irreducible α is absolutely irreducible.
As we have just seen, there are only finitely many such characters up tounramified twist, so that in particular there are only finitely many such α up tounramified twist. It follows that there are up to unramified twist only finitely
many irreducible (ϕ, Γ)-modules over Fp of any fixed rank.
It is convenient to extend the notation ρT to algebraic stacks. To this end,
if T is a reduced algebraic stack of finite type over Fp , we will denote a morphism
T → (Xd,red )Fp by ρT , and will sometimes abusively refer to ρT as a family of
GK -representations parameterized by T .
5.3.5 Definition. We say that a representation ρ : GK → GLd (Fp ) is maximally nonsplit of niveau 1 if it has a unique filtration by GK -stable Fp -subspaces
such that all of the graded pieces are one-dimensional representations of GK .
We say that the family ρT is maximally nonsplit of niveau 1 if ρt is maximallynonsplit of niveau 1 for all t ∈ T (Fp ), and that ρT is generically maximallynonsplit of niveau 1 if there is a dense open substack U of T such that ρU is
maximally nonsplit of niveau 1.
In particular, we say that an algebraic substack T of (Xd,red )Fp is maximallynonsplit of niveau 1 if ρt is maximally nonsplit of niveau 1 for all t ∈ T (Fp ),and that T is generically maximally nonsplit of niveau 1 if there is a dense open
substack U of T such that U is maximally nonsplit of niveau 1.

5.3. TWISTING FAMILIES

139

5.3.6 Remark. We will eventually see in Theorem 6.5.1 below that (Xd,red )Fp
itself is an algebraic stack of finite type over Fp , and is generically maximallynonsplit of niveau 1.
Our next goal is to prove a structure theorem (Proposition 5.3.8) for families which are generically maximally nonsplit of niveau 1. We begin with thefollowing technical lemma.
5.3.7 Lemma. Let T be a reduced scheme of finite type over an algebraicallyclosed field k, let M be a coherent sheaf on T ×k Gm , and for each t ∈ T (k), let∼
Mt denote the restriction of M to Gm −→ t ×k Gm ,→ T ×k Gm . Suppose thatfor each point t ∈ T (k), Mt is a length one skyscraper sheaf supported at a single
point of Gm . Then there is a dense open subscheme U of T such that, over U ,the composite Supp(M ) ,→ T ×k Gm → T pulls back to an isomorphism, while
the pullback of M over U ×k Gm is locally free of rank one over its support.
Proof. Let Fitt (M ) ⊆ OT ×Gm denote the Fitting ideal sheaf of M , and write
Z := Spec OT ×Gm /Fitt (M ); recall that Supp(M ) is a closed subscheme of Z,
supported on the same underlying closed subset of T ×k Gm . For each t ∈ T (k),we find that the fibre Zt of Z over t is equal to Spec OGm /Fitt (Mt ) (recall
that the formation of Fitting ideals is compatible with base-change), which,by assumption, is a single reduced point of Gm . Thus the composite Z →T ×k Gm → T is a morphism with reduced singleton fibres, and hence we mayfind a dense open subscheme U of T such that this morphism restricts to an
isomorphism over U . In particular, the pull-back of Z over U is reduced (sinceU is), and thus coincides with the pull-back of Supp(M ) over U . This showsthat the morphism Supp(M ) → T pulls back to an isomorphism over U .
If we use the inverse of the isomorphism just constructed to identify U with
an open subscheme of Supp(M ), then our assumption on the nature of Mt fort ∈ T (k) implies that the fibre of M over each k-point of U is one-dimensional.
Since U is reduced, we find that the restriction of M to U is furthermore locally
free of rank one, as claimed.
5.3.8 Proposition. If T is a reduced finite type algebraic stack over Fp , and ρT
is generically maximally nonsplit of niveau 1, then there exist:• a dense open substack U of T ;• tuples ni , 1 ≤ i ≤ d;
• morphisms λi : U → Gm , 1 ≤ i ≤ d;
• morphisms xi : U → (Xi,red )Fp , 0 ≤ i ≤ d, corresponding to families ofi-dimensional representations ρi over U;
such that xd is the restriction of ρT to U, and for each 0 ≤ i ≤ d − 1, we haveshort exact sequences of families over U
0 → ρi → ρi+1 → urλi+1 ⊗ ψni+1 → 0.

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CHAPTER 5. FAMILIES OF EXTENSIONS

In particular, for each t ∈ U, we have
urλ1 (t) ⊗ ψn1∗0ur
λ2 (t) ⊗ ψn2

ρt ∼
=
..

.

...
...
..
.

∗
∗
..
.

0

0

urλd (t) ⊗ ψnd

...




.


(5.3.9)5.3.10 Remark. We cannot necessarily attain (5.3.9) over all of T . For example,
there are families of two-dimensional representations of GK which are generically maximally nonsplit of niveau one, but which also specialise to irreducible
representations. (The existence of such families follows from Theorem 6.5.1
below.)
Proof of Proposition 5.3.8. We begin by assuming that T is a scheme, say T =T . Without loss of generality, we can replace by T by a dense open subscheme,
and thus assume that ρT is maximally nonsplit of niveau 1, i.e. that ρt is maximally nonsplit of niveau 1 for each t ∈ T (Fp ), and also assume that T is adisjoint union of finitely many integral open and closed subschemes. Replacing
T by each of these subschemes in turn, we may assume that T itself is integral.
For each n, we may form the family−1ρT  (ur−1x ⊗ ψn )(1)over T ×Fp Gm (where as above x denotes the variable on Gm , so that Gm :=Spec Fp [x, x−1 ]), and then consider (in the notation of Remark 5.1.30)
−1
Hn := H 2 GK , ρT  (ur−1x ⊗ ψn )(1) ,a coherent sheaf on T ×Fp Gm . Since the formation of H 2 (GK , –) is compatible
with arbitrary finite type base-change, we see that for any point t ∈ T (Fp ), thepull-back of Hn over t ×Fp Gm ∼= Gm admits the description−1(Hn )t ∼= H 2 GK , ρt ⊗ (ur−1x ⊗ ψn )(1) ,and that the fibre (Hn )(t,x0 ) of this pull-back at a point x0 ∈ Gm (Fp ), admits
the description−1∼(Hn )(t,x0 ) ∼= H 2 GK , ρt ⊗ (ur−1x0 ⊗ ψn )(1) = HomGK (ρt , urx0 ⊗ ψn )(the second isomorphism following from Tate local duality). The assumptionthat ρt is maximally non-split shows that this fibre is non-zero for exactly one
choice of n and one choice of x0 , and is then precisely one-dimensional. In particular, we see, for this distinguished choice of n, that (Hn )t is set-theoreticallysupported at x0 . Now let ξ0 : Spec Fp []/(2 ) ,→ Gm be the (unique up toscalar) non-trivial tangent vector to x0 , and let (Hn )(t,ξ0 ) denote the pull-backof (Hn )t over ξ0 . A similar computation using Tate local duality and the maximal non-splitness of ρt shows that (Hn )(t,ξ0 ) is also one-dimensional; thus (Hn )t

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141

is in fact a skyscraper sheaf of length one supported at x0 . For all other choicesof n, we see that (Hn )t vanishes (since
L all its fibres do).
Applying Lemma 5.3.7 to M := n Hn , and replacing T by an appropriately
chosen dense open subscheme, we find that Supp(M ) maps isomorphically to T ,and that M is locally free of rank one over its support. Taking into account thedefinition of M as a direct sum, and the irreducibility of T , we find that in fact
M = Hn for exactly one choice nd of n, and that Hn = 0 for all other possible
choices. Let λ : T → Gm denote the composite of the inverse isomorphism∼
T −→ Supp(M ) with the projection Supp(M ) → Gm . Again using the factthat the formation of H 2 commutes with base-change, we find that the pullback MT of M to T ∼= Supp(M ) may be identified with
−1H 2 ρT  (ur−1λ ⊗ ψnd )(1) .
We have already remarked that MT is locally free of rank one; replacing T by anon-empty open subscheme once more, we may in fact assume that MT is free ofrank one, and so choose a nowhere zero section of MT , which by Lemma 5.1.31
we may and do regard as a surjection
ρT  urλ ⊗ ψnd .
The kernel of this surjection is a rank (d − 1) family of projective étale (ϕ, Γ)modules, so the result for T follows by induction on d.
We now return to the general case that T is a reduced finite type algebraic
stack over Fp . Since Xd is Ind-algebraic by Proposition 3.4.12, we can form
the scheme-theoretic image T 0 of the morphism T → (Xd,red )Fp , which is again
a reduced finite type algebraic stack over Fp . The family ρT 0 is again generically maximally nonsplit of niveau 1 (since by the stacky version of Chevalley’s
theorem (see [Ryd11, App. D]), the image of T in T 0 is constructible, and so
contains a dense open subset of T 0 ), and since the morphism T → T 0 is schemetheoretically dominant, any dense open substack of T 0 pulls back to a dense
open substack of T . It therefore suffices to prove the result for T 0 . Replacing T
by T 0 , then, we can and do assume that T is a closed substack of (Xd,red )Fp .
Let T → T be a smooth cover by a (necessarily) reduced and finite type Fp scheme (we can ensure that T is finite type over Fp , since T is so), and let ρTbe the induced family. We wish to deduce the result for T from the result for T ,which we have already proved. We thus replace T by a dense open subscheme(and correspondingly replace T by a dense open substack, namely the image ofthis open subscheme of T ), so that the family ρT is maximally non-split, and sothat the morphisms λi : T → Gm and xi : T → (Xi,red )Fp exist. We will thenshow that these morphisms factor through the smooth surjection T → T . This
amounts to showing that we can factor the morphism of groupoids
T ×T T → T × T(5.3.11)through a morphism of groupoids T ×T T → T ×Gm T or T ×T T → T ×(Xi,red )F T
prespectively.

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Since T is a substack of X , the natural morphism induces an isomorphism∼
T ×T T −→ T ×X T . Thus for any test scheme T 0 , a T 0 -valued point of T ×T T
consists of a pair of morphisms f0 , f1 : T 0 ⇒ T , and an isomorphism of families∼f0∗ ρT −→ f1∗ ρT(5.3.12)(where fi∗ ρT denotes the pull-back of ρT to T 0 via fi ).
In the first case, in which we consider one of the morphisms λi , we needto show (employing the notation just introduced) that the morphism f0 × f1 :T 0 × T 0 → T × T necessarily factors through T ×Gm T (this latter fibre productformed using the morphisms λi ). More concretely, this amounts to checking thatλi ◦ f0 = λi ◦ f1 . Since the source of (5.3.11) is reduced and of finite type overFp , it suffices to check this for reduced test schemes T 0 of finite type over Fp ,and thus to check that these morphisms coincide at the Fp -valued points t0 of
T 0 . This follows from the assumed isomorphism (5.3.12), and the uniqueness ofthe filtration at each Fp -valued point of the family.
In the second case, in which we consider one of the morphisms xi , the morphism T ×(Xi,red )F T → T × T is no longer a monomorphism, and so we have topactually construct a morphism of groupoids T ×T T → T ×(Xi,red )F T (ratherpthan simply verify a factorization). So, given f0 , f1 : T 0 → T, and an isomorphism (5.3.12), we have to construct a corresponding isomorphism∼f0∗ (ρT )i −→ f1∗ (ρT )i ,(5.3.13)

where (ρT )i denotes the family of i-dimensional subrepresentations of ρT corresponding to the morphism xi . We will define the desired isomorphism (5.3.13)
simply to be the restriction of the isomorphism (5.3.12). For this definition to
make sense, we have to show that (5.3.12) does in fact restrict to an isomorphism (5.3.13). In fact, it suffices to check that (5.3.12) induces an embeddingf0∗ (ρT )i ,→ f1∗ (ρT )i ; reversing the roles of f0 and f1 then allows us to promote
this to an isomorphism.
Again, it suffices to check this on reduced test schemes T 0 of finite typeover Fp , and then it suffices to show that the composite(5.3.12)f0∗ (ρT )i ,→ f0∗ ρT −→ f1∗ ρT → f1∗ ρT /f1∗ (ρT )ivanishes. This can be checked on Fp -valued points, where it again follows fromthe uniqueness of the filtrations that define the (ρT )i .

5.4

Dimensions of families of extensions

Suppose that we have a morphism T → (Xd,red )Fp , with T a reduced finite typeFp -scheme, corresponding to a family ρT of GK -representations over T . Fix arepresentation α : GK → GLa (Fp ).
5.4.1 Lemma. If Ext2GK (α, ρt ) is of constant rank r for all t ∈ T (Fp ), thenH 2 (GK , ρT ⊗ α∨ ) is locally free of rank r as an OT -module.

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143

Proof. Since the formation of H 2 is compatible with arbitrary finite type basechange (by Theorem 5.1.22 (2)), we see that H 2 (GK , ρT ⊗ α∨ ) is a coherent
sheaf on T of constant fibre rank r. Since T is reduced, the lemma follows.
By Corollary 5.1.25, if T is affine, then we can and do choose a good complex(that is, a bounded complex of finite rank locally free OT -modules)
CT0 → CT1 → CT2
computing H • (GK , ρT ⊗ α∨ ). Suppose that we are in the context of the preceding lemma, i.e. that Ext2GK (α, ρT ) is locally free of some rank r. It follows
that the truncated complex
CT0 → ZT1
is again good (here ZT1 := ker(CT1 → CT2 )). As in Remark 5.1.30, we write
H 1 (GK , ρT ⊗ α∨ )for the cohomology group of this complex in degree 1; its formation is compatible with arbitrary finite type base-change, and its specialisations to finite typepoints of T agree with the usual Galois cohomology.
In particular, if we choose another integer n, we may use the surjection
ZT1 → H 1 (GK , ρT ⊗ α∨ ) (together with Lemma 5.1.2) to construct a universal
family of extensions
0 → ρT → E → α⊕n → 0(5.4.2)parameterized by the vector bundle V corresponding to the finite rank locally
free sheaf (ZT1 )⊕n , giving rise to a morphism
V → (Xd+an,red )Fp .

(5.4.3)(See [CEGS19, §4.2] for a similar construction.)
Write Gα := AutGK (α), thought of as an affine algebraic group over Fp .
5.4.4 Proposition. Maintain the notation and assumptions above; so we assume in particular that Ext2GK (α, ρt ) is of constant rank r for all t ∈ T (Fp ).
1. If e denotes the dimension of the scheme-theoretic image of T in (Xd,red )Fp ,then the dimension of the scheme-theoretic image, with respect to the morphism (5.4.3), of V in (Xd+an,red )Fp is bounded above by
e + n([K : Qp ]ad + r) − n2 dim Gα .
2. Suppose that:(a) n = 1;(b) α is one-dimensional;(c) ρT is generically maximally nonsplit of niveau 1;

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CHAPTER 5. FAMILIES OF EXTENSIONS(d) if K = Qp , then after replacing T by a dense open subscheme, sothat the tuples ni exist, and so that the morphisms λi and xi as in
the statement of Proposition 5.3.8 are defined on T , at least one of
the following conditions holds at each t ∈ T (Fp ):i. urλd−1 (t) ⊗ ψnd−1 6= α(1).
ii. urλd (t) ⊗ ψnd = α.
iii. urλd (t) ⊗ ψnd = α(1).
Then equality holds in the inequality of (1). Furthermore the family ofextensions EV corresponding to V is generically maximally nonsplit of
niveau 1.

3. Suppose that we are in the setting of (2), and that after replacing T by adense open subscheme, the following conditions hold:(a) For each t ∈ T (Fp ) the representation ρt is maximally nonsplit, andfurthermore the unique quotient character of ρt (which exists by theassumption that ρT is generically maximally nonsplit of niveau 1) isequal to α(1).
(b) Furthermore, if we write
0 → rt → ρt → α(1) → 0
for the corresponding filtration of ρt , and write γt for the uniquequotient character of rt then for each t ∈ T (Fp ) either
i. we have γt 6= α(1), orii. the cyclotomic character  is trivial, we have γt = α(1) = α, andthe extension of α by γt = α(1) induced by ρt is très ramifiée.
Then, after replacing V by a dense open subscheme, we have that for allt ∈ V (Fp ) the extension of α by α(1) induced by the extension of α by ρtis très ramifiée.
5.4.5 Remark. See Remark 5.4.10 below for a further discussion of condition (2d).
Proof of Proposition 5.4.4. Writing T as the union of its irreducible components, we obtain a corresponding decomposition of V into the union of its irreducible components, and we can prove the theorem one component at a time.
Thus we may assume that T (and hence V ) is irreducible. Replacing T by anon-empty affine open subscheme, we may furthermore assume that it is affine,so that we may find a good complex CT• supported in degrees [0, 2] that is
isomorphic in the derived category to the Herr complex, as above.
Let f : T → (Xd,red )Fp be the given morphism, and let g : V → (Xd+an,red )Fp
denote the morphism (5.4.3). In order to relate the scheme-theoretic imageof g to the scheme-theoretic image of f , we will also consider (in a slightly
indirect manner) the scheme-theoretic image of the morphism V → (Xd,red )Fp ×(Xd+an,red )Fp (the first factor being the composite of f and the projection from

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145

V to T , and the second factor being g). If (Xd,red )Fp were a scheme or an
algebraic space, then the dimension of this latter scheme-theoretic image would
be greater than or equal to the dimension of the former; but since (Xd,red )Fp isa stack, we have to be slightly careful in comparing the dimensions of these two
images.
Let v : Spec Fp → V be an Fp -point, and let t denote the composite
Spec Fp → V → T , so that t is an Fp -point of T . We also write f (t) forthe composite f ◦ t, and g(v) for the composite g ◦ v. We write Tf (t) for the fibre
product T ×(Xd,red )F Spec Fp (with respect to the morphisms f : T → (Xd,red )Fpp

and f (t) : Spec Fp → (Xd,red )Fp ), and write Vg(v) and V(f (t),g(v)) for the evident
analogous fibre products.
We begin with (1). In order to prove the required bound on the dimensionof the scheme-theoretic image of g, it suffices (e.g. by [Sta, Tag 0DS4]) to showthat?

dim Vg(v) ≥ dim V − e − n([K : Qp ]ad + r) + n2 dim Gα
for v lying in some non-empty open subscheme of V . As already alluded toabove, what we will actually be able to do is to estimate the dimension of thefibre V(f (t),g(v)) , and so our first job is to compare the dimension of this fibrewith that of Vg(v) .
Let ρf (t) denote the Galois representation corresponding to f (t), and let
Gt := Aut(ρf (t) ). Then, by the discussion of Section 5.2, the morphism t :Spec Fp → (Xd,red )Fp induces an immersion[Spec Fp /Gt ] ,→ (Xd,red )Fp ,which in turn induces a monomorphism[Spec Fp /Gt ] ×(Xd,red )F Vg(v) ,→ Vg(v) .
p(5.4.6)Since Spec Fp is a Gt -torsor over [Spec Fp /Gt ], we see that V(f (t),g(v)) is a Gt torsor over [Spec Fp /Gt ] ×(Xd,red )F Vg(v) , and hence thatpdim V(f (t),g(v)) ≤ dim Gt + dim Vg(v) .

(5.4.7)

Thus it suffices to prove the inequality?

dim V(f (t),g(v)) ≥ dim V − e − n([K : Qp ]ad + r) + n2 dim Gα + dim Gt .
For t lying in some non-empty open subscheme of T , we have that e = dim T −dim Tf (t) , and so, replacing T by this non-empty open subscheme and V by its
preimage, we have to show that?

dim V(f (t),g(v)) ≥ dim V −dim T +dim Tf (t) −n([K : Qp ]ad+r)+n2 dim Gα +dim Gt
for v lying in some non-empty open subscheme of V . In fact we will show that
this inequality holds for every Fp -point v of V .

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The right hand side of our putative inequality can be rewritten as
nrk ZT1 + dim Tf (t) − n([K : Qp ]ad + r) + n2 dim Gα + dim Gt ,while the local Euler characteristic formula shows that
H 0 (GK , ρf (t) ⊗ α∨ ) − H 1 (GK , ρf (t) ⊗ α∨ ) + r = −[K : Qp ]ad.
Thus we may rewrite our desired inequality as?
dim V(f (t),g(v)) ≥ n rk ZT1 − dim H 1 (GK , ρf (t) ⊗ α∨ )+ n dim H 0 (GK , ρf (t) ⊗ α∨ ) + dim Tf (t) + n2 dim Gα + dim Gt .
There is a canonical isomorphism V(f (t),g(v)) ∼= Tf (t) ×T Vg(v) , and so, writing
S := (Tf (t) )red , we find that dim V(f (t),g(v)) = dim S ×T Vg(v) . If we let ρS denotethe pull-back of the family ρT over S, then the Galois representations attached
to all the closed points of S are isomorphic to ρf (t) . The discussion of Section 5.2
shows we may find an étale cover S 0 → S such that the pulled-back family ρS 0
is constant. We may replace S by S 0 without changing the dimension we aretrying to estimate, and thus we may furthermore assume that ρS is the trivial
family with fibres ρf (t) .
If we let CS0 → ZS1 denote the pull-back of the good complex CT0 → ZT1 to
S, then this complex is also the pull-back to S of the good complex Ct0 → Zt1
corresponding to the representation ρt . If we let W denote the vector space
H 1 (GK , ρt ⊗ α∨ )⊕n , thought of as an affine space over Fp , then there is a
projection S ×T V → S ×Fp W , whose kernel is a (trivial) vector bundle which
we denote by V 0 .
The affine space W parameterizes a universal family of extensions 0 →ρf (t) → ρW → α⊕n → 0, giving rise to a morphism h : W → (Xd+an,red )Fp , andgthe composite morphism S ×T V → V −→ (Xd+an,red )Fp admits an alternative
factorization ash

S ×T V → S ×Fp W → W −→ (Xd+an,red )Fp .
Thus
S ×T Vg(v) = S ×T V ×W Wh(w)(where w denotes the image of v in W under the projection), and hence
dim S ×T Vg(v) = rk V 0 + dim S + dim Wh(w)= n rk ZT1 − dim H 1 (GK , ρf (t) ⊗ α∨ ) + dim S + dim Wh(w) ,so that our desired inequality becomes equivalent to the inequality?

dim Wh(w) ≥ n dim H 0 (GK , ρf (t) ⊗ α∨ ) + n2 dim Gα + dim Gt .

(5.4.8)

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147

There is an action of Gt × AutGK (α⊕n ) on
W = H 1 (GK , ρf (t) ⊗ α∨ )⊕n ∼= Ext1GK (α⊕n , ρt ),which lifts to an action on the family ρW . There is furthermore a (unipotent)
action of H 0 (GK , ρt ⊗ α∨ )⊕n on ρW (lying over the trivial action on W ). Altogether, we find that the morphism h factors through [W/ H 0 (GK , ρt ⊗ α∨ )⊕n o(Gt × AutGK (α⊕n )) ], implying that
dim Wh(w) ≥ n dim H 0 (GK , ρt ⊗ α∨ ) + n2 dim Gα + dim Gt ,as required.
We now turn to (2), so that we assume that n = 1, that α is one-dimensional,
and that ρT is generically maximally nonsplit of niveau 1. By Proposition 5.3.8,
after possibly replacing T with a non-empty open subscheme, we can and do
assume that ρT is maximally nonsplit of niveau one, that we can write ρT as anextension of families
0 → rT → ρT → β T → 0(5.4.9)×

where the family β T arises from twisting a character β : GK → Fp via amorphism T → Gm , and that if K = Qp , then (2d) holds for all finite typepoints t of T . (Indeed, β T is the family corresponding to the character urλd ⊗ψndin the notation of Proposition 5.3.8.)
We want to show that EV is generically maximally nonsplit of niveau 1.
Assume that this is the case; we now show that we then have equality in thebound of part (1). After replacing V with a non-empty open subscheme, we canand do suppose that EV is maximally nonsplit of niveau 1, so that in particular,
for every finite type point v of V , the extension
0 → ρf (t) → Eg(v) → α → 0
is maximally nonsplit. From the proof of part (1), we see that it is enoughto show that (after shrinking V as we have) equality holds in both (5.4.7)
and (5.4.8).
We begin by considering (5.4.7). For equality to hold here, it is enough tocheck that the immersion (5.4.6) induces an isomorphism on underlying reduced
substacks∼([Spec Fp /Gt ] ×(Xd,red )F Vg(v) )red −→ (Vg(v) )red .
p

We can check this on the level of Fp -valued points, for which it suffices toshow that the representation Eg(v) has a unique d-dimensional subrepresentation,
namely ρf (t) ; but this is immediate from Eg(v) being maximally non-split of
niveau 1.
Similarly, to show that equality holds in (5.4.8), it is enough to show thatthe morphism[W/ H 0 (GK , ρf (t) ⊗ α∨ ) o Gt × AutGK (α) ] → (Xd+a,red )Fp

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is a monomorphism. To see this, note that it follows from the definition of“maximally nonsplit” that any automorphism of Eg(v) induces automorphismsof ρf (t) and α, so that we have
W ×(Xd+a,red )F W = W ×Fp H 0 (GK , ρt ⊗ α∨ ) o Gt × AutGK (α) .
pHence it is enough to show that the morphism [W/W ×(Xd+a,red )F W ] →p(Xd+a,red )Fp is a monomorphism; this follows from Lemma A.33.
In order to prove (2), it remains to prove that EV is generically maximallynonsplit of niveau 1. We need to show that after possibly shrinking V , for eachv ∈ V the image in Ext1GK (α, β f (t) ) of the element of Ext1GK (α, ρf (t) ) corresponding to Eg(v) is nonzero. To this end, note that, after possibly shrinking T(and remembering that T is reduced), we may assume that H 2 (GK , β T ⊗ α∨ )is locally free of some constant rank. We can therefore repeat the constructionthat we carried out after Lemma 5.4.1, and find a good complex
CT0 (β) → ZT1 (β)
whose cohomology in degree 1, which we denote by HT1 (β), is compatible withbase change and computes H 1 (GK , β t ⊗ α∨ ) = Ext1GK (α, β t ) at finite typepoints t of T . By [Sta, Tag 064E] we have a morphism of complexes(CT0 → ZT1 ) → (CT0 (β) → ZT1 (β)),
compatible with the natural maps on the cohomology groups induced by thecorresponding morphism of Herr complexes.
The kernel of ZT1 → HT1 (β) is a coherent sheaf, so after possibly shrinking
the reduced scheme T , we can suppose that it is a vector subbundle of ZT1 . By
definition, we see that if we delete from V the corresponding subbundle thenthe required condition holds. It therefore suffices to show that we are deleting aproper subbundle of V . If this were not the case, then (considering the fibre of
V and the subbundle under consideration over some closed point t), we would
have an exact sequence
0 → Ext1GK (α, β t ) → Ext2GK (α, rt ) → Ext2GK (α, ρt ) → Ext2GK (α, β t ) → 0;
equivalently, by Tate local duality we would have an exact sequence
0 → HomGK (β t , α(1)) → HomGK (ρt , α(1)) → HomGK (rt , α(1)) →Ext1GK (β t , α(1)) → 0.
Assume for the sake of contradiction that this is the case. Since ρt is maximallynonsplit, the map HomGK (β t , α(1)) → HomGK (ρt , α(1)) is an isomorphism; so
it suffices to show that we cannot have an isomorphism HomGK (rt , α(1)) →Ext1GK (β t , α(1)).
Since rt is maximally nonsplit, HomGK (rt , α(1)) is at most 1-dimensionalwith equality if and only if urλd−1 (t) ⊗ ψnd−1 = α(1); while by the local Euler

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149

characteristic formula, Ext1GK (β t , α(1)) has dimension at least [K : Qp ], withequality if and only if β t 6= α and β t 6= α(1). It follows that we must have K =Qp , urλd−1 (t) ⊗ ψnd−1 = α(1), β t 6= α and β t 6= α(1); but this case was excluded
by assumption (2d), and we have our a contradiction.
Finally, suppose that we are in the situation of (3). We wish to argue in thesame way as part (2), but rather than deleting the split locus, we need to delete
the peu ramifiée locus. To be more precise, note that the surjection ρT → α(1)T(where α(1)T denotes the constant rank one family given by spreading out α(1)over T ) induces a composite morphism ZT1 → H 1 (GK , ρT ⊗ α∨ ) → H 1 (GK , T )(where T denotes the spreading out of the mod p cyclotomic character  over T ),which in turn induces a morphism V → Y of total spaces of vector bundles
over T , with Y denoting the total space of the trivial bundle over T having
fibre H 1 (GK , ). Now Y contains a trivial subbundle Yp.r. of codimension 1,
parameterizing the peu ramifiée classes. The preimage of Yp.r. is a closed subscheme Vp.r. of V , and we wish to show that Vp.r. is a proper closed subschemeof V .
For this, it is enough to show that for each t (perhaps after replacing T by anon-empty open subset), there is some extension of α by ρf (t) with the property
that the induced extension of α by β t = α(1) is très ramifiée. We have an exact
sequence
Ext1GK (α, ρt ) → Ext1GK (α, β t ) → Ext2GK (α, rt ),so we are done if Ext2GK (α, rt ) = 0. Since rt is maximally nonsplit, this meansthat we are done unless rt admits α(1) as its unique quotient character.
By hypothesis, this implies that  is trivial, and we can assume that theextension of β t = α(1) = α by α(1) = α induced by ρt is très ramifiée. Write ct
for the corresponding class in H 1 (GK , ). As in Remark 5.4.10 below, it isenough to show that there is a très ramifiée extension of α by α(1), corresponding
to a class d ∈ H 1 (GK , ) such that the cup product of ct and d vanishes. If thisis not the case, then ct must be an unramified class in H 1 (GK , 1) = H 1 (GK , );
but the extension of K cut out by ct considered as a class in H 1 (GK , 1) is anextension given by adjoining a pth root of a uniformizer, so is in particular aramified extension, and we are done.
5.4.10 Remark. The curious looking condition Proposition 5.4.4 (2d) is not
merely an artefact of our arguments. Indeed, if we fix characters α, β : GQp →×

Fp , with β 6= α, β 6= α(1), and let

α(1)
r=
0

∗
β



be a non-split extension, then any extension of α by r induces the trivial extension of α by β. To see this, note that if we fix an extension of α by β, then thecondition that we can form an extension of α by r realising this extension of α−1by β is that the cup product of the corresponding classes in H 1 (GQp , αβ (1))−1and H 1 (GQp , βα ) vanishes. But by Tate local duality, this cup product is a

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perfect pairing of 1-dimensional vector spaces, so one of the two classes mustvanish, as required.

5.5

Xd is a formal algebraic stack

We now use the theory of families of extensions to show that Xd is a formal algebraic stack. The key ingredient is Theorem 5.5.12, which we prove by induction
on d. We begin by setting up some useful terminology, which is motivated by thegeneralisations of the weight part of Serre’s conjecture formulated in [GHS18].
Recall that the notion of a Serre weight was defined in Section 1.12. Since F isassumed to contain k, we can and do identify embeddings k ,→ Fp and k ,→ F.
5.5.1 Definition. If k is a Serre weight, and ρ : GK → GLd (Fp ) is maximallynonsplit of niveau 1, then we say that ρ is of weight k if we can write
χ1 ∗ . . . ∗ 0 χ2 . . . ∗ ρ∼=  ..
.. 
..
.
.
. 
0 ...
0 χd
where
• χi |IK = 1−i−kσ,d+1−i, andσ:k,→F ωσQ−1• if (χi+1 χ−1, then kσ,d−i − kσ,d+1−i = p − 1 for all σ if andi )|IK = −1−1
only if χi+1 χi =  and the element of Ext1GK (χi , χi+1 ) = H 1 (GK , )determined by ρ is très ramifiée (and otherwise kσ,d−i − kσ,d+1−i = 0 for
all σ).

Note that each maximally nonsplit representation of niveau one is of weight k
for a unique k. For each k, there exists a ρ which is maximally nonsplit of
niveau 1 and weight k; the existence of such a ρ follows by an easy induction,and is in any case an immediate consequence of Theorem 5.5.12 below.
05.5.2 Definition. We say that a weight k 0 is a shift of a weight k if kσ,d= kσ,d00
for all σ, and if for each 1 ≤ i ≤ d − 1, we either have kσ,i− kσ,i+1= kσ,i − kσ,i+1
0
0
for all σ, or we have kσ,i− kσ,i+1= p − 1 and kσ,i − kσ,i+1 = 0 for all σ. (Notein particular that any weight is a shift of itself.)

5.5.3 Definition. We say that a representation ρ : GK → GLd (O) or ρ : GK →GLd (Zp ) is crystalline, or crystalline of weight λ, if ρ[1/p] is crystalline, resp.
crystalline of weight λ. We say that ρ is a crystalline lift of ρ : GK → GLd (F)(resp. of ρ : GK → GLd (Fp )).
The motivation for Definitions 5.5.1 and 5.5.2 is the following lemma.
5.5.4 Lemma. If ρ is maximally nonsplit of niveau 1 and weight k, if k 0 is a
shift of k, and λ is a lift of k 0 , then ρ has a crystalline lift (defined over Zp )of weight λ. Furthermore, this lift can be chosen to be ordinary in the sense of
Definition 6.4.1 below.

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151

Proof. Write ρ as in Definition 5.5.1, and assume (by replacing E by a finite unramified extension if necessary) that both ρ and the filtration in Definition 5.5.1
are defined over F. Note firstly that by [GHS18, Lem. 5.1.6] (which uses the
opposite conventions for the sign of the Hodge–Tate weights to those of thisQ e(K/Qp )book), together with the observation made above that σ ωσ
= |IK , wecan find crystalline characters ψi : GK → O× such that ψ i = χi and ψi has
Hodge–Tate weights given by λσ,d+1−i .
It is enough to prove that ρ has a crystalline lift which is a successive extension of characters ψi of the form 0

ψ1 ∗ . . . ∗ 0 ψ20 . . . ∗  ..
..  ,
..
 .
.
. 
0 ...
0 ψd0
where each ψi0 is an unramified twist of ψi ; note that any such representation
is by definition ordinary in the sense of Definition 6.4.1. We prove this by
induction on d, the case d = 1 being trivial. In the general case, write ρ as anextension
0 → r → ρ → χd → 0,
and suppose inductively that r has a lift of the required form. Write r : GK →GLd−1 (O) for this lift; then we inductively seek a lift of ρ the form
0 → r → ρ → ψd0 → 0,

(5.5.5)where ψd0 is an unramified twist of ψd . We insist also that0ψd−1
(ψd0 )−1 6= ;(5.5.6)00
note that this could only fail if kσ,d= kσ,d−1
for all σ, and in this case we are
only ruling out a single unramified twist. Then since ρ is maximally nonsplit,
we have HomGK (r, ψd ) = 0, and it follows as in the proof of Lemma 6.3.1 thatsuch a ρ, if it exists, is automatically crystalline.
It is therefore enough to show that there is a lift of the form (5.5.5) satisfying (5.5.6). As in the proof of [GHLS17, Thm. 2.1.8], we have an exact
sequence



δ
2
0 −1
H 1 GK , (ψd0 )−1 ⊗ r → H 1 (GK , χ−1⊗rd ⊗ r) → H GK , (ψd )(5.5.7)

Thus, if c ∈ H 1 (GK , χ−1d ⊗ r) is the class determined by ρ, it is enough toshow that we can choose the unramified twist ψd0 so that
δ(c) = 0. Now,

H 2 GK , (ψd0 )−1 ⊗ r is dual to H 0 GK , r∨ ⊗ (ψd0 ) ⊗ E/O , and since r is maxi0
mally nonsplit, we see that this latter group is equal to H 0 GK , (ψd−1)−1 ψd0  ⊗
E/O . In other words, to show that δ(c) = 0 in (5.5.7), it is enough to show0
the corresponding statement where r is replaced by ψd−1
; so we need only showthat we can choose ψd0 so that the extension of χd by χd−1 lifts to a crystalline

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CHAPTER 5. FAMILIES OF EXTENSIONS

0
extension of ψd0 by ψd−1
. This is an immediate consequence of the weight partof Serre’s conjecture for GL2 , as proved in [GLS15]; see in particular [GLS15,
Thm. 6.1.18] and the references therein (and note that these results also holdfor p = 2, by [Wan17]).

5.5.8 Remark. We emphasize that the preceding lemma applies only to ρ thatare maximally split of niveau 1. In Theorem 6.4.4 below we prove an analogous
result for arbitrary ρ.
Suppose that ρT is generically maximally nonsplit of niveau 1, where T issome integral finite type Fp -stack. We say that ρT is generically of weight k
if there is a dense open substack U of T such that every Fp -point of ρU is
maximally nonsplit of niveau 1, and is of weight k.
5.5.9 Lemma. If ρT is generically maximally nonsplit of niveau 1, then ρT isgenerically of weight k for a unique Serre weight k.
Proof. Consider a dense open substack U ⊆ T , tuples ni and morphisms λi :U → Gm as in Proposition 5.3.8. We inductively determine the kσ,i as follows.
We let kσ,d be the unique choice with p − 1 ≥ kσ,d ≥ 0 not all equal to p − 1and satisfyingY−kψ n1 | I K =ωσ σ,d .
σ:k,→Fp

Inductively, if kσ,i+1 has been determined, then we demand that kσ,i satisfies p−
1 ≥ kσ,i − kσ,i+1 ≥ 0 and(ψnd−i ψn−1)|IK = 
d+1−i

Y

k

ωσσ,i

−kσ,i+1.

σ:k,→Fp

This determines kσ,i , except in the case that (ψnd−i ψn−1)|IK = . In thisd+1−icase, we consider the character λd+1−i λ−1
d−i : U → Gm . Since U is integral, thischaracter is either constant or has open image. After possibly shrinking U, we
may therefore assume that either λd+1−i (t) 6= λd−i (t) for all t ∈ U, in whichcase we set kσ,d+1−i − kσ,d−i = 0 for all σ, or that the character is trivial.
If the character is trivial, we may consider the locus in U over which theextension between χi+1 and χi is peu ramifiée. The argument used in in the
proof of part (3) of Proposition 5.4.4 shows that this is a constructible subset
of U; since U is integral, after shrinking U we can thus suppose that it is either
equal to U, or that it is empty. In the former case we set kσ,d+1−i − kσ,d−i = 0
for all σ, and in the latter we take kσ,d+1−i − kσ,d−i = p − 1 for all σ. It
follows from the construction that every Fp -point of ρU is maximally nonsplitof niveau 1 and weight k, as required.
It will be convenient to use the following variant of the notation establishedin Proposition 5.3.8. If ρT is generically maximally nonsplit of niveau one and5.5. Xd IS A FORMAL ALGEBRAIC STACK

153

weight k, then we set νi := λd+1−i , and write ωk,i for the character d−i ψnd+1−i ,so thatY−k
ωk,i |IK =ωσ σ,i ,σ:k,→Fpand (5.3.9) can be rewritten asurνd (t) ⊗ ωk,d
∗

0
ur
⊗
−1 ωk,d−1
νd−1 (t)∼ρt = 
..

.

∗
∗
..
.

...
...
..
.




.


urν1 (t) ⊗ 1−d ωk,1
(5.5.10)
We have the eigenvalue morphism ν : U → (Gm )d given by (ν1 , . . . , νd ).
0

...

0

5.5.11 Definition. If k is a Serre weight, we let (Gm )dk denote the closed
subgroup scheme of (Gm )d parameterizing tuples (x1 , . . . , xd ) for which xi =xi+1 whenever kσ,i − kσ,i+1 = p − 1 for all σ.
Note that (Gm )dk is closed under the simultaneous multiplication action of
Gm on (Gm )dk (i.e. the action a · (x1 , . . . , xd ) := (ax1 , . . . , axd )).
It follows from the definitions that the eigenvalue morphism ν is valuedin (Gm )dk .
We have seen in Proposition 3.4.12 that Xd is an Ind-algebraic stack, whichis in fact the 2-colimit of algebraic stacks with respect to transition morphismsthat are closed immersions. We let Xd,red be the underlying reduced substackof Xd ; it is then a closed substack of Xd with the same structure: namely, itis the 2-colimit of closed algebraic stacks (which can even be assumed to be
reduced) with respect to transition morphisms that are closed immersions.
5.5.12 Theorem.
1. The Ind-algebraic stack Xd,red is an algebraic stack, of finite presentation
over F.
2. We can write (Xd,red )Fp as a union of closed algebraic substacks of finite
presentation over Fp
[ k
small
(Xd,red )Fp = Xd,red,F∪
Xd,red,F ,ppkwhere:small• Xd,red,F
is empty if d = 1, and otherwise is non-empty of dimensionpstrictly less than [K : Qp ]d(d − 1)/2.
k• each Xd,red,F

is a closed irreducible substack of dimension [K :p

Qp ]d(d − 1)/2, and is generically maximally nonsplit of niveau one
and weight k. The corresponding eigenvalue morphism is dominant(i.e. has dense image in (Gm )dk ).

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CHAPTER 5. FAMILIES OF EXTENSIONS

3. If we fix an irreducible representation α : GK → GLa (Fp ) (for somea ≥ 1), then the locus of ρ in Xd,red (Fp ) for which dim HomGK (ρ, α) ≥ r(for any r ≥ 1) is (either empty, or) of dimension at most
[K : Qp ]d(d − 1)/2 − dr (a2 + 1)r − a /2e.
small
Furthermore, the locus of ρ in Xd,red,F(Fp ) for which dim HomGK (ρ, α) ≥pr is of dimension strictly less than this.

4. If we fix an irreducible representation α : GK → GLa (Fp ) (for somea ≥ 1), then the locus of ρ in Xd,red (Fp ) for which dim Ext2GK (α, ρ) ≥ r
is of dimension at most[K : Qp ]d(d − 1)/2 − r.
5.5.13 Remark. Note that in parts (3) and (4), the locus of points in question
corresponds to a closed substack of (Xd,red )Fp , by upper-semicontinuity of fibre
dimension.
k

5.5.14 Remark. Since the Xd,red,F are irreducible, have dimension equal topthat of (Xd,red )Fp , and have pairwise disjoint open substacks (corresponding tomaximally nonsplit representations of niveau 1 and weight k), they are in fact
distinct irreducible components of (Xd,red )Fp .
5.5.15 Remark. We can, and will, be much more precise about the structure of
Xd,red . Namely, in Chapter 6 we combine Theorem 5.5.12 with additional arguments to show that Xd,red is equidimensional of dimension [K : Qp ]d(d − 1)/2;
kaccordingly, the irreducible components of (Xd,red )Fp are precisely the Xd,red,F ,pand in particular are in bijection with the Serre weights k. We also show thatthese irreducible components are all defined over F.
5.5.16 Remark. Note that since the eigenvalue morphism has dense image, itkfollows that Xd,red contains a dense open substack with the property that νi (t) 6=
νi+1 (t) unless kσ,i − kσ,i+1 = p − 1 for all σ.
5.5.17 Remark. As will be evident from the proof, the upper bound of Theorem 5.5.12 (3) is quite crude when [K : Qp ] > 1, although it is reasonably sharpin the case K = Qp . However, it suffices for our purposes, and indeed we will
only use the (even cruder) consequence Theorem 5.5.12 (4) in the rest of thebook.
Proof of Theorem 5.5.12. Recall that a closed immersion of reduced algebraic
stacks that are locally of finite type over Fp which is surjective on finite-type
points is necessarily an isomorphism. As recalled above, Xd,red is an inductivealimit of such stacks (indeed, by Lemma A.9, we have Xd,red = lim Xd,h,s,red,−→awhere the Xd,h,s are as in Section 3.4), and so if we produce closed algebraick

small
substacks Xd,red,Fand Xd,red,F of Xd , the union of whose Fp -points exhaustsppthose of Xd,red , then Xd,red,Fp will in fact be an algebraic stack which is the5.5. Xd IS A FORMAL ALGEBRAIC STACK

155

k

small
union of its closed substacks Xd,red,Fand Xd,red,F . Thus (1) is an immediateppconsequence of (2) (where the “union” statement in (2) is now to be understoodon the level of Fp -points).
Claim (4) follows from (3) (with α replaced by α ⊗ ) by Tate local duality
and the easily verified inequalitydr (a2 + 1)r − a /2e ≥ r.
Thus it is enough to prove (2) and (3), which we do simultaneously by induction
on d.
As recalled in Remark 5.3.4, there are up to twist by unramified characters
only finitely many irreducible Fp -representations of GK of any fixed dimension.
Accordingly, we let {αi } be a finite set of irreducible continuous representations
αi : GK → GLdi (Fp ), such that any irreducible continuous representation of GK
over Fp of dimension at most d arises as an unramified twist of exactly one ofthe αi . We let the 1-dimensional representations in this set be the characters ψndefined in Section 5.3.
Each αi corresponds to a finite type point of Xdi ,red , whose associated residual gerbe is a substack of Xdi ,red of dimension −1: the morphism Spec Fp →Xdi ,red corresponding to αi factors through a monomorphism [Spec Fp /Gm ] →Xdi ,red . It follows from Lemma 5.3.2 that for each αi there is an irreducible
closed zero-dimensional algebraic substack of Xdi ,red of finite presentation over Fp
whose Fp -points are exactly the unramified twists of αi .
kIn particular, if d = 1, then we let X1,red be the zero-dimensional stack constructed in the previous paragraph, whose Fp -points are the unramified twistsof ψk ; this satisfies the required properties by definition, so (2) holds when d = 1.
For (3), note that if r > 0 then we must have a = 1, and then the locus where
HomGK (ρ, α) is non-zero (equivalently, 1-dimensional) is exactly the closed substack of dimension −1 corresponding to α, so the required bound holds.
We now begin the inductive proof of (2) and (3) for d > 1. In fact, it will
be helpful to simultaneously prove another statement (2’), which is as follows:k,fixedfor each k, there is a closed irreducible algebraic substack Xd,red,F of (Xd,red )Fppof finite presentation over Fp and dimension [K : Qp ]d(d − 1)/2 − 1, which is
generically maximally nonsplit of niveau 1 and weight k, and furthermore has
the property that the corresponding character ν1 is trivial. The discussion of theprevious paragraph also proves (2’) when d = 1. The point of hypothesis (2’) isk,fixedthat we can use Proposition 5.4.4 (2) to construct Xd,red,F by using (2) and (2’)
pkin dimension (d − 1), and then construct Xd,red,F from it using Lemma 5.3.2.
p

We now prove the inductive step, so we assume that (2), (2’) and (3) hold ink,fixeddimension less than d. We begin by constructing the closed substacks Xd,redk

and Xd,red,F . Let k d−1 be the Serre weight in dimension (d − 1) obtained bypdeleting the first entry in k. We set α := 1−d ωk,1 . If kσ,1 − kσ,2 = p − 1 forall σ then we say that we are in the très ramifiée case, and we let T denote

156k

CHAPTER 5. FAMILIES OF EXTENSIONS,fixedkd−1d−1; otherwise, we let T denote Xd−1,red,F. In the latter case, it follows
Xd−1,red,Fppfrom our inductive hypothesis (more precisely, from the assumption that (2)and (3) hold in dimension (d − 1)), together with Tate local duality, that after
replacing T with an open substack we can and do assume that for each Fp point t of T , we have Ext2GK (α, rt ) = 0, where rt is the (d − 1)-dimensional
representation corresponding to t. If we are in the très ramifiée case then it
follows similarly that after replacing T with an open substack we can and do
assume that for each Fp -point t of T , Ext2GK (α, rt ) is 1-dimensional. In eithercase we can in addition assume that rt is maximally nonsplit of niveau one andweight k d−1 .
k,fixedWe let T be an irreducible scheme which smoothly covers T , and we let Xd,red,Fbe the irreducible closed substack of (Xd,red )Fp constructed as the schemetheoretic image of V in the notation of Proposition 5.4.4. Part (2) of thatk,fixedproposition, together with the inductive hypothesis, implies that Xd,red,F haspthe claimed dimension. (Note in particular that if K = Qp , then condition (2d)of Proposition 5.4.4 holds. Indeed, if we are in the très ramifiée case thencondition (2d)(iii) holds, and otherwise the inductive hypothesis that the image of the eigenvalue morphism is dense in (Gm )d−1
implies that condition
k
d−1

k

(2d)(i) holds after shrinking T .) We then let Xd,red,F be the substack obtainedpk,fixedfrom Xd,red by twisting by unramified characters, which has the claimed dimension by Lemma 5.3.2 (and the fact that by Proposition 5.4.4 (2), there is ak,fixeddense open substack of Xd,red whose Fp -points correspond to representations
which are of niveau 1 and are maximally nonsplit, so that in particular thecorresponding family is twistable).
k,fixedk

By construction, we see that the stacks Xd,red,F and Xd,red,F satisfy theppproperties required of them, except possibly that in the très ramifiée case, we
k,fixedkneed to check that Xd,red,F (and consequently Xd,red,F ) is generically maxpp
imally nonsplit of niveau 1 and weight k. More precisely, in this case weneed to check the condition that the final extension is not just nonsplit, butis (generically) très ramifiée. This follows from Proposition 5.4.4 (3). Indeed
kd−1
by our inductive assumption that Xd−1,red,Fis generically maximally nonsplitpof niveau 1 and weight k d−1 , and the image of the eigenvalue morphism isdense, we see that we are in case (3)(b)(i) unless  is trivial and furthermore wehave kσ2 − kσ3 = p − 1 for all σ, in which case we satisfy (3)(b)(ii).
small
To complete the proof of (2), we need only construct Xd,red,F. By Propop
sition 5.4.4, Tate local duality, upper semi-continuity of the fibre dimension,and the inductive hypothesis, we see that for each 1 ≤ i ≤ d, each αi (of dimension ai , say), and each s ≥ 0 there is a finitely presented closed algebraic
substack Xs,αi ,Fp of (Xd,red )Fp , whose Fp -points contain all the representationsof the form 0 → ρd−ai → ρ → αi → 0 for which dimFp Ext2GK (αi , ρd−ai ) = s,p

5.5. Xd IS A FORMAL ALGEBRAIC STACK

157

and whose dimension is at most[K : Qp ](d − ai )(d − ai − 1)/2 − ds((ai 2 + 1)s − ai )/2e + [K : Qp ]ai (d − ai ) + s − 1;small
is of dimension
furthermore, the locus where ρd−ai is an Fp -point of Xd−ai ,red,Fp
strictly less than this. These stacks are only nonzero for finitely many valuesof s. For fixed ai , we see that as a function of s, this quantity is maximisedby s = 0, as well as by s = 1 when ai = 1. (To see this, we have to maximise thequantity s − ds((ai 2 + 1)s − ai )/2e. Suppose firstly that ai > 1. Then if s = 0
we have 0, while if s > 0 we have s − ds((ai 2 + 1)s − ai )/2e ≤ s − s((ai 2 +1)s − ai )/2 ≤ s − s(ai 2 + 1 − ai )/2 ≤ s − 3s/2 < 0. Meanwhile if ai = 1, thenfor s = 0 we have 0, for s = 1 we have 1 − d1/2e = 1, while for s > 1 we haves − ds(2s − 1)/2e ≤ s − s(2s − 1)/2 ≤ s − 3s/2 < 0.) It follows that as a functionof ai the bound is maximised at ai = 1 and s = 0 or 1, when it is equal to[K : Qp ]d(d − 1)/2 − 1, and it is otherwise strictly smaller.
By Lemma 5.3.2, it follows that the locus in (Xd,red )Fp of representations of
the form 0 → ρd−ai → ρ → α0 → 0, with α0 an unramified twist of αi for which
dimFp Ext2GK (α0 , ρd−ai ) = s, is of dimension at most [K : Qp ]d(d − 1)/2, withequality holding only if ai = 1 and s = 0 or 1. Furthermore, the locus of thosesmall
representations for which ρd−ai is an Fp -point of Xd−a
is of dimension
i ,red,Fp
strictly less than [K : Qp ]d(d−1)/2. Putting this together, we see that (2) holds
small
in dimension d if we take Xd,red,F
to be the union of the twists by unramifiedpcharacters of the substacks Xs,αi for which dim αi > 1 or s > 1, together withthe union of the twists by unramified characters of the substacks of the Xs,αi ,Fp
small
for which dim αi = 1, s = 0 or 1, and ρd−1 is an Fp -point of Xd−1,red,F.
p
Finally we prove (3) in dimension d. In the case r = 0, there is nothing
to prove. If dim HomGK (ρ, α) ≥ r ≥ 1, then we may place ρ in a short exact
sequence
0 → θ → ρ → α⊕r → 0,

where θ is of dimension d − ra < d. We may apply part (2) so as to find that
Xd−ar,red,Fp has dimension at most [K : Qp ](d − ar)(d − ar − 1)/2. Let Us be thelocally closed substack of Xd−ar,red,Fp over which dim H 2 (GK , θ ⊗ α∨ ) = s; by
the inductive hypothesis, this locus has dimension at most [K : Qp ](d − ar)(d −ar − 1)/2 − s (a2 + 1)s − a /2, and over this locus we may construct a universal
family of extensions
0 → θ → ρUs → α⊕r → 0.
The locus of ρ we are interested in is contained in the scheme-theoretic image ofthis family in (Xd,red )Fp , and Proposition 5.4.4 shows that this scheme-theoretic
image has dimension bounded above by[K : Qp ](d − ar)(d − ar − 1)/2 − s((a2 + 1)s − a)/2 + r([K : Qp ]a(d − ar) + s) − r2=[K : Qp ]d(d − 1)/2 − (r(a2 + 1)r − a)/2 − (r − s)2 /2 − (ar(ar − 1))/2≤[K : Qp ]d(d − 1)/2 − (r(a2 + 1)r − a)/2.

158

CHAPTER 5. FAMILIES OF EXTENSIONS

Since this conclusion holds for each of the finitely many values of s (andsince the dimension is an integer, allowing us to take the floor of this upperbound), we are done.
5.5.18 Corollary. Xd is a Noetherian formal algebraic stack.
Proof. Theorem 5.5.12 shows that Xd,red is an algebraic stack that is of finite presentation over F (and hence quasi-compact and quasi-separated). Together with Proposition 3.4.12, which shows that Xd is isomorphic to the inductive limit of a sequence of finitely presented algebraic stacks with respect
to transition morphisms that are closed immersions, this implies that Xd is indeed a (locally countably indexed) formal algebraic stack, by Proposition A.12.
By Remark A.20, Xd is Ind-locally of finite type over O, and it then follows
from Proposition 5.1.36 and Theorem A.35 that Xd is locally Noetherian, andhence Noetherian (as we have already seen that it is quasi-compact and quasiseparated).

Chapter 6

Crystalline lifts and thefiner structure of Xd,red
Up to this point, we have shown that Xd is a Noetherian formal algebraic stack.
In this chapter, we will additionally show that (Xd )red is equidimensional ofdimension [K : Qp ]d(d−1)/2, and enumerate its irreducible components. In thecourse of doing this, we will also prove a result on the existence of crystalline lifts
which is of independent interest. We furthermore determine the closed pointsof (Xd )red , and describe the maximal substack of Xd over which the universal(ϕ, Γ)-module gives rise to a continuous GK -representation.

6.1

The fibre dimension of H 2 on crystalline deformation rings

Let ρ : GK → GLd (F0 ) be a continuous representation, for some finite extension
F0 of F, and let O0 be the ring of integers in a finite extension E 0 /E, with residue
crys,λ,O 0
field F0 . Let λ be a Hodge type, and consider the lifting ring R := Rρ. Let
M denote the universal étale (ϕ, Γ)-module over R (note that there is a universal
étale (ϕ, Γ)-module over R, and not merely a universal formal étale (ϕ, Γ)module, as a consequence of Corollary 4.8.13). Let α◦ : GK → GLa (O0 ) be a
representation with α : GK → GLa (F0 ) being absolutely irreducible, and let N
be the base change to R of the étale (ϕ, Γ)-module over O0 which corresponds
to α◦ .
Let C • (N ∨ ⊗ M ) denote the Herr complex associated to N ∨ ⊗ M , as in Section 5.1; this is a perfect complex of R-modules, with cohomology concentrated
in degrees [0, 2], as a consequence of [Sta, Tag 0CQG] and Theorem
5.1.22 (ap
plied to the quotients R/maR ). Write Ext2 := H 2 C • (N ∨ ⊗M ) . For each r ≥ 1,
write
Xr := {x ∈ Spec R | dim κ(x) ⊗R Ext2 ≥ r}.
159

CHAPTER 6. CRYSTALLINE LIFTS AND Xd,red

160

The theorem on upper semi-continuity of fibre dimension for coherent sheaves
shows that Xr is a closed subset of Spec R.
6.1.1 Theorem. If Ext2 is $-power torsion, then for each r ≥ 1, Xr is ofcodimension ≥ r + 1 in Spec R.
Proof. The assumption that Ext2 is $-power torsion implies that Ext2 is (settheoretically) supported on the closed subscheme Spec(R/$)red of Spec R. Write
M and N to denote the pull-backs of M and N respectively, over (R/$)red , write2∨
Ext := Ext2 C • (N ⊗ M ) , and define2

X r := {x ∈ Spec(R/$)red | dim κ(x) ⊗(R/$)red Ext ≥ r}.
Since Ext2 is compatible with base change, and is supported on Spec(R/$)red ,while R is O-flat, it suffices to show that the codimension of X r in Spec(R/$)red
is at least r. Since Spec(R/$)red is equidimensional of dimension [K : Qp ]d(d −1)/2 + d2 , it suffices in turn to show that X r is of dimension at most ([K :Qp ]d(d − 1)/2) + d2 − r.
0
By Proposition 3.6.3, we have a versal morphism Spf Rρ,O → Xd . Consider0

the fibre product Spf Rρ,O ×Xd Xd,red . This is a closed formal algebraic subspace0of Spf Rρ,O , and so by Lemma A.3 is of the form Spf S for some quotient S
0of Rρ,O . Its description as a fibre product shows that the morphism Spf S →Xd,red is versal. Since Xd,red is an algebraic stack, this morphism is effective, i.e.
arises from a morphism
Spec S → Xd,red(6.1.2)([Sta, Tag 0DR1]), which is furthermore flat, by [Sta, Tag 0DR2]. Let x0 =Spec F0 , thought of as the closed point of Spec S, and endowed with a morphism
x0 → Xd,red corresponding to the representation ρ. Since Spf S → Xd,red isversal, it follows from the Artin Approximation Theorem that we may find a
finite type F-scheme U , equipped with a point u0 with residue field F0 , suchthat the complete local ring of U at u0 is isomorphic to S, and such that (6.1.2)
may be promoted to a smooth morphism
U → Xd,red ,(6.1.3)in the sense that on u0 (= Spec F0 ), the morphism (6.1.3) induces the given
bU,u ∼
morphism x0 (= Spec F0 ) → Xd,red , and on Spec O
0 = Spec S, the morphism (6.1.3) induces the morphism (6.1.2). (See [Sta, Tag 0DR0].)
It follows from Proposition 3.6.3, by pulling back over Xd,red , that Spf S×Xd,redc d,S , a certain completion of (GLd )S , and thus thatSpf S is isomorphic to GL
c d,F0 , a certain completion of (GLd )F0 . This
Spf S ×Xd,red x0 is isomorphic to GL
latter fibre product may be identified with the completion of Spec S ×Xd,red
x0 along its closed subspace x0 ×Xd,red x0 , and so we deduce that the morphism (6.1.2) has relative dimension d2 at the point x0 in its domain. Thus themorphism (6.1.3) has relative dimension d2 at the point u0 in its domain. Since6.2. TWO GEOMETRIC LEMMAS

161

this latter morphism is also smooth, its relative dimension is locally constanton its domain [Sta, Tag 0DRQ], and thus (since S is a local ring, so that Spec S
is connected), we see that (6.1.2) in fact has relative dimension d2 .
f denote the universal étale (ϕ, Γ)-module over S, and Ne denote the
Let M

2
2
•
∨
g := H C (Ne ⊗Mf) , and write
base change of N to S. Write Ext2g ≥ r}.
Yr := {x ∈ Spec S | dim κ(x) ⊗S Ext
We claim that Yr has dimension at most ([K : Qp ]d(d − 1)/2) + d2 − r. In
order to see this, we note that (Yr )Fp is the pull-back to Spec S via the flat
morphism (6.1.2) of the locus considered in Theorem 5.5.12 (4) which has dimension at most [K : Qp ]d(d − 1)/2 − r. The required bound on the dimensionof Yr follows, for example, from [Sta, Tag 02RE].
0
The composite Spf R/$ → Spf Rρ,O → Xd is effective by Corollary 4.8.13,
i.e. arises from a morphism Spec R/$ → Xd . Thus the composite Spf(R/$)red →Spf R/$ → Xd arises from a morphism Spec(R/$)red → Xd , and hence factors
0through Xd,red . Consequently, we find that the surjection Rρ,O → (R/$)red
factors through S. The closed immersion Spec(R/$)red ,→ Spec S then induces
a closed immersion X r ,→ Yr for each r ≥ 0. The upper bound on the dimension of Yr computed in the preceding paragraph then provides the desired upperbound on the dimension of X r , completing the proof of the theorem.

6.2

Two geometric lemmas

In this section we establish two lemmas which will be used in the proof of
Theorem 6.3.2 below. We begin with the following simple lemma.
6.2.1 Lemma. Suppose that 0 → F → G → H → 0 is a short exact sequence of coherent sheaves on a reduced Noetherian scheme X, with F and G
being locally free, and H being torsion, in the sense that its support does notcontain any irreducible component of X. Then there is an effective Cartierdivisor D which contains the scheme-theoretic support of H, and whose settheoretic support coincides with the set-theoretic support of H, with the property
that if f : T → X is any morphism which meets D properly, in the sense thatthe pull-back f ∗ OX → f ∗ OX (D) of the canonical morphism OX → OX (D) is
injective, then Li f ∗ H = 0 if i > 0.
Proof. Replacing X by each of its finitely many connected components in turn,
we see that it is no loss of generality to assume that X is connected, and we
do so; this ensures that each locally free sheaf on X is of constant rank. If η
is any generic point of X, then Hη = 0 by assumption, and so Fη → Gη is
an isomorphism. In particular, F and G are of the same rank, say r, and theinduced morphism∧r F → ∧r G
(6.2.2)

CHAPTER 6. CRYSTALLINE LIFTS AND Xd,red

162

is an injection of invertible sheaves. (Indeed, it follows from [Bou98, III, §7,
Prop. 3] that a map of locally free sheaves of rank r is injective if and only if themap on ∧r is injective.) If we let D denote the support of the cokernel of (6.2.2),
then D is an effective Cartier divisor, and twisting by ∧r F ∨ identifies (6.2.2)
with the canonical section OX ,→ OX (D). The scheme-theoretic support of H
is contained in D, and the set-theoretic supports of H and of D coincide (thisis a special case of the general fact that the Fitting ideal is contained in theannihilator, and has the same radical as it).
Now suppose that f : T → X meets D properly in the sense described in
the statement of the theorem. Then we see that∧r f ∗ F → ∧r f ∗ G
is injective, and thus that f ∗ F → f ∗ G is also injective. Consequently
0 → f ∗F → f ∗G → f ∗H → 0
is short exact, and so Li f ∗ H = 0 for i > 0, as claimed.
We now introduce some notation related to the second of the lemmas (Lemma 6.2.7
below).
6.2.3 Hypothesis. Let X be a Noetherian scheme, and suppose that we have ashort exact sequence of coherent sheaves
0 → F → G → H → 0,

(6.2.4)such that• G is locally free;• H has the property that, for each r ≥ 1, the locus
Xr := {x ∈ X | dim κ(x) ⊗OX H ≥ r}
is of codimension ≥ r + 1 in X.
e → X is a morphism of Noetherian
In the setting of Hypothesis 6.2.3, if π : X∗eeschemes, then we write G := π G and H := π ∗ H, and we let Fe denote the image
e so that we again have a shortof the pulled-back morphism π ∗ F → π ∗ G = G,exact sequencee → 0.
0 → Fe → Ge → H(6.2.5)6.2.6 Remark. If E is a locally free coherent sheaf on a Noetherian scheme X,
and s : OX → E is a morphism, then we can think of E as being the sheaf ofsections of a vector bundle over X, and think of s as being a section of this vector
bundle. We can then speak of the zero locus of s; it is the closed subscheme of Xrwhich locally, if we choose an isomorphism E ∼and write s = (a1 , . . . , ar ),= OX
is cut out by the ideal sheaf (a1 , . . . , ar ); more globally, it is cut out by the ideal
sheaf which is the image of the composites⊗id

E ∨ = OX ⊗OX E ∨ −→ E ⊗OX E ∨ → OX ,

6.2. TWO GEOMETRIC LEMMAS

163

where the second arrow is the canonical pairing.
We note that s is nowhere-vanishing, i.e. the zero locus of s is empty, if andonly if s is a split injection (so that s gives rise to a morphism of vector bundles,
rather than merely of sheaves). We also note that if E has rank r, then the
Hauptidealsatz shows that the zero locus of a section of E has codimension at
most r around each of its points.
6.2.7 Lemma. Suppose that we are in the setting of Hypothesis 6.2.3, that thee → X is a surjective morphism whose domain is a Noetherian scheme,
π : X
and that Fe is locally free. Suppose further that OX → F is a morphism withthe property that the pulled-back morphism OXe → Fe is a nowhere-vanishingsection. Then the composite OX → F → G is nowhere-vanishing.
Proof. The statement may be checked by working locally at each of the points
e by its pull-back overof X; more precisely, we may replace X by Spec OX,x and X
Spec OX,x . We have the stratification of X by the closed subsets Xr (where in
the case r = 0, we declare X0 := X), and if x ∈ X, then we set r(x) :=dim κ(x) ⊗OX H, or equivalently, the maximal value of r for which x ∈ Xr ;we then prove the theorem by induction on r(x). In fact, we assume that thestatement is true after localizing at points x for which r(x) < r for all exact
sequences (6.2.4) which satisfy the hypotheses of the lemma, and prove it for
our given exact sequence after localizing at an x for which r(x) = r.
∼
In the case r = 0, the sheaf H is zero, so that F −→ G. Thus the inducedemorphism OXe → G gives a nowhere-vanishing section. Since π is surjective,
this ensures that the induced morphism OX → G is also nowhere-vanishing, as
required.
We now consider the case r ≥ 1. Since we have replaced X by Spec OX,x ,we may assume that G is free. By Nakayama’s lemma and our assumptionthat r(x) = r, we may choose a surjection
r
OX
→ H,

(6.2.8)rwhich we may then lift to a morphism OX→ G. If we let K denote the kernelof (6.2.8), then we obtain a morphism of short exact sequences

0

/K

/ Or

X

/H

/0

0


/F


/G

/H

/0

(6.2.9)e denote the image of π ∗ K
Pulling back this diagram back via π, and letting K
r
in OX
,
we
obtainacorresponding
morphismofshortexact sequences
e
0

e
/K

/ Or

e
/H

/0

0


/ Fe


/ Ge

e
/H

/0

e
X

(6.2.10)

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CHAPTER 6. CRYSTALLINE LIFTS AND Xd,red

The morphism of short exact sequences (6.2.9) induces a short exact sequencer
0 → K → F ⊕ OX
→ G → 0,(6.2.11)which is furthermore split (since G is free and we are over an affine scheme), sothat we may writer ∼F ⊕ OX(6.2.12)= K ⊕ G.
Correspondingly, the short exact sequence (6.2.10) induces a short exact sequencee → Fe ⊕ Or → Ge → 0,
0→KeX
which can be thought of as being obtained from (6.2.11) by pulling back via
π and then taking the quotient by the (naturally identified) copies of L1 π ∗ Hsitting inside π ∗ K and π ∗ F. Since (6.2.11) is split, so is this latter short exact
sequence, and so we also obtain an isomorphismr ∼ eeFe ⊕ OXe =K⊕G(6.2.13)which can be thought of as being obtained via by pulling back the isomorphism (6.2.12) along π, and then taking the quotient of each side by the appropriate copy of L1 π ∗ H. Since all the other summands appearing in the isomore
phism (6.2.13) are locally free, we see that the same is true of K.
Suppose, by way of obtaining a contradiction, that the composite OX →F → G is not nowhere-vanishing; then we see that its image lies in mx G. Thus
the pulled back morphism OXe → Ge vanishes at each point x
e lying over x (andthere is at least one such point, since π is surjective by assumption). Since
the pulled-back morphism OXe → Fe is nowhere-vanishing, so is the inducedrmorphism OXe → Fe ⊕ OX
e . A consideration of (6.2.13) then shows that thee which is pulled-back from the induced morphism
induced morphism OXe → K,
OX → K, must be nowhere-vanishing.
Thus, if we consider the exact sequencer
0 → K → OX
→ H → 0,it satisfies Hypothesis 6.2.3, and we are given a morphism OX → K whose
e is a nowhere-vanishing section of K.
e Applying the inductive
pull-back to Xrhypothesis to this situation, we find that the composite OX → OX
is a sectionof a rank r locally free sheaf whose zero locus is contained in Xr (because
if we localise at a point not in Xr , then the section is nowhere vanishing by
the inductive hypothesis), and is therefore of codimension at least r + 1 byhypothesis. This zero locus contains x (since this section factors through K,rr
and H ∼/K has fibre dimension exactly r at x, so the map OX⊗OX κ(x) →= OX
H ⊗OX κ(x) is an isomorphism), and hence is non-empty. On the other hand,as we noted above, the zero locus of a section of a rank r locally free sheaf, if
it is non-empty, has codimension at most r. This contradiction completes theproof of the lemma.

6.3. CRYSTALLINE LIFTS

6.3

165

Crystalline lifts

Given a d-tuple of labeled Hodge–Tate weights λ, and a d0 -tuple of labeled
Hodge–Tate weights λ0 , we say that λ0 is slightly greater than λ (and that λ isslightly less than λ0 ) if for each σ : K ,→ Qp we have λ0σ,d ≥ λσ,1 + 1, and theinequality is strict for at least one σ.
This is not standard terminology, but it will be convenient for us; it ismotivated by the following well-known result. Here, as throughout the book,we slightly abuse terminology and refer to lattices in crystalline representations
as themselves being crystalline.
6.3.1 Lemma. Let 0 → ρ◦1 → ρ◦ → ρ◦2 → 0 be an extension of Zp -valued representations of GK , with ρ◦1 and ρ◦2 being crystalline of Hodge–Tate weights λ1 ,λ2 respectively. If λ1 is slightly less than λ2 , then ρ◦ is crystalline.
Proof. This follows easily from the formulae in [Nek93, Prop. 1.24]. Indeed,
if we let ρ1 , ρ2 be the Qp -representations corresponding to ρ◦1 , ρ◦2 , and we set
11
V := ρ1 ⊗ ρ∨
2 , then we need to show that hf (V ) = h (V ). Since the Hodge–Tateweights of V are all negative, and for at least one embedding σ : K ,→ Qp theσ-labeled Hodge–Tate weights are all less than −1, we have h1 (V ) − h1f (V ) =h2 (V ) = h0 (V ∨ (1)) = 0, as required.
Arguing inductively, the following theorem will allow us to construct crystalline lifts of any given ρ. In particular it implies Theorem 1.2.2 from theintroduction, in the more refined form of Theorem 6.4.4 below.
6.3.2 Theorem. Suppose given a representation ρd : GK → GLd (Fp ) that
admits a lift ρ◦d : GK → GLd (Zp ) which is crystalline with labeled Hodge–Tateweights λ. Let 0 → ρd → ρd+a → α → 0 be any extension of GK -representations
over Fp , with α : GK → GLa (Fp ) irreducible, and let α◦ : GK → GLa (Zp ) be
any crystalline lifting of α with labeled Hodge–Tate weights λ0 , which we assume
to be slightly greater than λ.
Then we may find a lifting of the given extension to an extension◦
0 → θd◦ → θd+a
→ α◦ → 0of GK -representations over Zp , where θd◦ : GK → GLd (Zp ) again has the property that the associated p-adic representation θd : GK → GLd (Qp ) is crystalline◦
with labeled Hodge–Tate weights λ. Furthermore, θd+a
is crystalline, and wecrys,λ◦may choose θd to lie on the same irreducible component of Spec Rρdthat ρ◦ddoes.
Proof of Theorem 6.3.2. We may and do choose our field of coefficients E to belarge enough that the various Galois representations are defined over O or F as
crys,λ, and we let X denote the component
the case may be. We write R := Rρcrys,λof the crystalline lifting scheme Spec Rρ
on which ρ◦d lies. As recalled in
Section 1.12, X is reduced. Over X we have a good complex C • supported

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CHAPTER 6. CRYSTALLINE LIFTS AND Xd,red

in degrees [0, 2] computing Ext•GK (α◦ , ρ◦ ) for the universal deformation ρ◦ ; indeed, by a straightforward variant of Corollary 5.1.25, we can choose a goodcomplex supported in degrees [0, 2] and quasi-isomorphic to the Herr complex
for ρ◦ ⊗ (α◦ )∨ . Our assumption on the Hodge–Tate weights implies that Ext2 is
supported (set-theoretically) on the special fibre X of X. Indeed, the formationof Ext2 is compatible with base change, and at any closed point x of the genericfibre of R, we have Ext2GK (α◦ , ρ◦x ) = HomGK (ρ◦x , α◦ (1)) by Tate local duality,
and this space vanishes by the assumption that λ0 is slightly greater than λ.
As usual, we let B 2 denote the image of C 1 in C 2 ; it is a subsheaf of thelocally free sheaf C 2 , and so is torsion free.1 By [Sta, Tag 0815], we may find ae → X, whose centre lies (set-theoretically) in the special fibre of X,
blow up π : X
such that the torsion-free quotient of π ∗ B 2 becomes locally free (recall that Ext2e • denote theis supported on the special fibre of X); in other words, if we let C•
e 2 (thatpull-back by π of C , the corresponding OXe -module of 2-coboundaries B
e 1 in C
e 2 ) is locally free. Thus, if we let Z
e1 denote the O e is, the image of C
X
•
1
e
e
module of 1-cocycles for C , then Z is also locally free; so in particular, thee0 → Ze1 is a good complex.
complex C
The given extension ρd+a is classified by a class in Ext1GK (α, ρd ), which is
to say, a class in H 1 of the complex κ ⊗ C • . (Here we write κ to denote Fthought of as the residue field of R.) Lifting this class to an element of κ ⊗ C 1 ,and then to an element of C 1 , we find ourselves in the following situation: wehave a morphism c : R → C 1 , whose image under the coboundary lies in mR C 2(reflecting the fact that we have a cochain which becomes a cocycle at the closed
point).
c
Consider the composite b : R −→ C 1 → B 2 , which pulls back to a section
eb : O e → B
e 2 . It follows from Lemma 6.2.7 above (applied with F = B 2 ,X
2
G = C , and H = Ext2 ), together with Theorem 6.1.1, that the lifted section eb
must have non-empty zero locus, which (being closed) must contain a point xelying over the closed point x ∈ X. The section c itself pulls back to a section
e 1 , whose value at the point x
e1 . In other words,ec : OXe → C
e lies in the fibre of Z
e0 → Ze1 ], giving rise
the fibre of ec at xe is a one-cocycle in the complex κ(ex) ⊗ [C101e → Ze ] lifting the original class in Ext1G (α, ρd )to a class e ∈ H κ(ex) ⊗ [CK
that classifies ρd+a .
e is flat over Zp (because X is), we may find a morphism fe : Spec Zp →Since Xe passing through the point xe p ) (in the sense that the closed point ofXe ∈ X(Fe). The morphism fe determines (and, by the valuative criterionSpec Zp maps to xof properness, is determined by) a morphism f : Spec Zp → X, lifting the closed
point x ∈ X, which in turn corresponds to a representation θd◦ : GK → GLd (Zp ),e • ) is the cokernel of an inclusion
as in the statement of the theorem. Now H 2 (Ce 2 ,→ Ce 2 ), and it is torsion (because itof locally free sheaves (the inclusion B
is pulled back from H 2 (C • ), which is set-theoretically supported in the special
1 Since X is reduced, torsion free is a reasonable notion for a finitely generated module,
equivalent to the associated primes lying among the minimal primes.

6.4. POTENTIALLY DIAGONALIZABLE CRYSTALLINE LIFTS

167

fibre) and so Lemma 6.2.1 above shows that there is an effective Cartier divisor
e with the property that
D contained (set-theoretically) in the special fibre of Xefor any morphism to X that meets D properly, the higher derived pull-backs ofe • ) under this morphism vanish. Since the domain of fe is p-torsion free, it
H 2 (Ce properly, and in particular meets D properly; thus
meets the special fibre of X
we infer thate• ) = 0
Li fe∗ H 2 (Cfor i > 0. From this vanishing we deduce thate • ) = fe∗ H 1 (Ce• )Ext1GK (α◦ , θd◦ ) = H 1 (f ∗ C • ) = H 1 (fe∗ C
e 0 → Ze1 ]) = H 1 fe∗ [Ce0 → Ze1 ] .
= fe∗ H 1 [C(The first of these identifications is the general base-change property of theperfect complex C • , the second follows from the fact that f = π ◦ fe, the thirde • ) for i > 0 and the base-change spectral
follows from the vanishing of Li fe∗ H 2 (Csequencee • ) =⇒ H p+q (fe∗ Ce• )E2p,q := L−p fe∗ H q (C(see e.g. [Sta, Tag 0662]), the fourth holds by definition of H 1 , and the fifth
follows from right-exactness of fe∗ .)e, we may find
Since fe identifies the closed point of Spec Zp with the point x1 e∗ e 01
ea class e ∈ H f [C → Z ] lifting the class e, which is then identified with an◦element of Ext1GK (α◦ , θd◦ ). This element classifies an extension 0 → θd◦ → θd+a→◦◦◦
α → 0 which by construction lifts the given extension 0 → ρd → ρd+a → α◦ →0, and so by Lemma 6.3.1 satisfies the conclusions of the theorem.

6.4

Potentially diagonalizable crystalline lifts

We now use Theorem 6.3.2 to prove Theorem 1.2.2. There are at least two differences between the proof of Theorem 1.2.2 and previous work on the problem(in particular the results of [Mul13, GHLS17].) One is that, rather than working
only with lifts which are extensions of inductions of characters, we utilise extensions of more general potentially diagonalizable representations. The other
difference is that our argument exploits our control of the support of H 2 . In
previous work, the only tool used to deal with classes in H 2 was twisting thevarious irreducible representations by unramified characters; at points where H 2
has dimension greater than 1, this seems to be insufficient.
In fact, we prove various refinements of this result, allowing us to produce
potentially diagonalizable lifts; such lifts are important for automorphy lifting
theorems. We can also control the Hodge–Tate weights of these potentially diagonalizable lifts; for example, we can insist that the gaps between the weights arearbitrarily large, which is useful in applications to automorphy lifting. Alternatively, we can produce lifts whose Hodge–Tate weights lift some Serre weight,

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CHAPTER 6. CRYSTALLINE LIFTS AND Xd,red

proving a conjecture which is important for the formulation of general Serreweight conjectures (see the discussion after [GHS18, Rem. 5.1.8]).
We begin by recalling some definitions and some basic lemmas about extensions of crystalline representations. Let λ be a regular d-tuple of Hodge–Tateweights.
6.4.1 Definition. A crystalline representation of weight λ is ordinary if it has
a GK -invariant decreasing filtration whose associated graded pieces are all onedimensional, such that the σ-labeled Hodge–Tate weight of the ith graded pieceis λσ,i . In other words, we can write the representation in the form
χ1 ∗ . . . ∗ 0 χ2 . . . ∗ 


 ..
.. 
.
.
.
.
. 
0 ...
0 χd
where χi is a crystalline character whose σ-labeled Hodge–Tate weight is λσ,d+1−i .
If ρ◦1 , ρ◦2 : GK → GLd (Zp ) are two crystalline representations, then we writeρ◦1 ∼ ρ◦2 , and say that ρ◦1 connects to ρ◦2 , if and only if the following conditions
hold: ρ◦1 and ρ◦2 have the same labeled Hodge–Tate weights, have isomorphic
reductions modulo mZp , and determine points on the same irreducible component of the corresponding crystalline lifting rings. The following definition was
originally made in [BLGGT14, §1.4].
6.4.2 Definition. A representation ρ◦ : GK → GLd (Zp ) is potentially diagonalizable if there is a finite extension K 0 /K and crystalline characters χ1 , . . . , χd :×
GK 0 → Zp such that ρ◦ |GK 0 ∼ χ1 ⊕ · · · ⊕ χd .
6.4.3 Lemma. Let 0 → ρ◦1 → ρ◦ → ρ◦2 → 0 be an extension of Zp -valued
representations of GK . If ρ◦ is crystalline, and ρ◦1 and ρ◦2 are potentially diagonalizable, then ρ◦ is also potentially diagonalizable.
Proof. We may choose K 0 /K such that ρ◦1 |GK 0 and ρ◦2 |GK 0 both connect to directsums of crystalline characters, and such that ρ|GK 0 is trivial. It then follows frompoints (5) and (7) of the list before [BLGGT14, Lem. 1.4.1] that ρ◦ |GK 0 connects
to the direct sum of the union of the sets of crystalline characters for ρ◦1 |GK 0and ρ◦2 |GK 0 , as required.
Note that it follows in particular from Lemma 6.4.3 that ordinary representations are potentially diagonalizable.
6.4.4 Theorem. Let K/Qp be a finite extension, and let ρ : GK → GLd (Fp )be a continuous representation. Then ρ admits a lift to a crystalline representation ρ◦ : GK → GLd (Zp ) of some regular labeled Hodge–Tate weights λ.
Furthermore:
1. ρ◦ can be taken to be potentially diagonalizable.

6.4. POTENTIALLY DIAGONALIZABLE CRYSTALLINE LIFTS

169

2. If every Jordan–Hölder factor of ρ is one-dimensional, then ρ◦ can betaken to be ordinary.
3. ρ◦ can be taken to be potentially diagonalizable, and λ can be taken to bea lift of a Serre weight.
4. ρ◦ can be taken to be potentially diagonalizable, and λ can be taken to havearbitrarily spread-out Hodge–Tate weights: that is, for any C > 0, we canchoose ρ◦ such that for each σ : K ,→ Qp , we have λσ,i − λσ,i+1 ≥ C foreach 1 ≤ i ≤ n − 1.
Proof. We prove all of these results by induction on d, using Theorem 6.3.2. We
begin by proving (1), and then explain how to refine the proof to give each of(2)-(4).
Suppose firstly that ρ is irreducible (this is the base case of the induction).
×0
K
Then ρ is of the form IndG
GK 0 ψ for some character ψ : GK 0 → Fp , where K /K
is unramified of degree d. We can choose (for example by [GHS18, Lem. 7.1.1])
×K
a crystalline lift ψ : GK 0 → Zp of ψ such that ρ◦ := IndG
GK 0 ψ has regular◦0
Hodge–Tate weights. Then ρ is crystalline (because K /K is unramified), and
it is potentially diagonalizable, because ρ◦ |G0K is a direct sum of crystalline
characters.
For the inductive step, we may therefore suppose that we can write ρ as
an extension 0 → ρ1 → ρ → ρ2 → 0, with ρ2 irreducible, and we may assume
that we have regular potentially diagonalizable lifts ρ◦1 , ρ◦2 of ρ1 , ρ2 respectively.
Twisting ρ◦1 by a crystalline character with sufficiently large Hodge–Tate weights
and trivial reduction modulo p (which exists by [GHS18, Lem. 7.1.1]), we may
suppose that the Hodge–Tate weights of ρ◦1 are slightly less than those of ρ◦2 .
By Theorem 6.3.2, there is a lift θ1◦ of ρ1 with θ1◦ ∼ ρ◦1 , and a lift ρ◦ of ρ which
is an extension
0 → θ1◦ → ρ◦ → ρ◦2 → 0.
Since ρ◦1 is potentially diagonalizable, so is θ1◦ , and it follows from Lemmas 6.3.1
and 6.4.3 that ρ◦ is crystalline and potentially diagonalizable. This completes
the proof of (1).
We claim that if every Jordan–Hölder factor of ρ is one-dimensional, thenthe lift that we produced in proving (1) is actually automatically ordinary.
Examining the proof, we see that it is enough to show that if ρ◦1 is ordinaryand θ1◦ ∼ ρ◦1 then θ1◦ is also ordinary; this is immediate from [Ger19, Lem.
3.3.3(2)].
To show (3) and (4) we have to check that we can choose the Hodge–Tateweights of the lifts of the irreducible pieces of ρ appropriately. This requires
substantially more effort in the case of (3), but that effort has already been madein the proof of [GHS18, Thm. B.1.1] (which proves the case that ρ is semisimple).
Indeed, examining that proof, we see that it produces lifts satisfying all of
the conditions we need; we need only note that the “slightly less” condition is
automatic, except in the case that for each σ we have (in the notation of theCHAPTER 6. CRYSTALLINE LIFTS AND Xd,red

170

proof of [GHS18, Thm. B.1.1]) hσ + xσ = Hσ + 1, in which case we can takehσ + xσ = Hσ + p for each σ instead.
Finally, to prove (4) it is enough to check the case that ρ is irreducible (because we can choose the Hodge–Tate weights of the character that we twisted ρ◦1by to be arbitrarily large). For this, note that in our application of [GHS18,
Lem. 7.1.1], we can change any labeled Hodge–Tate by any multiple of (pd − 1),so we can certainly arrange that the gaps between labeled Hodge–Tate weights
are as large as we please.
6.4.5 Remark. To complete the proof of Theorem 1.2.2, it is enough to note thatby definition the lifts produced in Theorem 6.4.4 (3) have all their Hodge–Tateweights in the interval [0, dp − 1].
6.4.6 Remark. The proof of Theorem 6.4.4 (3) actually proves the slightly
stronger statement that there is a Serre weight such that if λ is any lift ofthat Serre weight, then ρ admits a potentially diagonalizable crystalline lift ofweight λ. (Note that in the case that K/Qp is ramified, there may be many suchchoices of a lift of a given Serre weight, because by definition, the choice of a liftof λ depends on a choice of a lift of each embedding k ,→ Fp to an embedding
K ,→ Qp . However, it is noted in the first paragraph of the proof of [GHS18,
Thm. B.1.1] that in the case that ρ is semisimple, there is a crystalline lift ofweight λ for any choice of lift λ, so the same is true in our construction.)
As a consequence of Theorem 6.4.4, we can remove a hypothesis madein [EG14, App. A], proving in particular the following result on the existence of
globalizations of local Galois representations.
6.4.7 Corollary. Suppose that p - 2d. Let K/Qp be a finite extension, and let
ρ : GK → GLn (Fp ) be a continuous representation. Then there is an imaginary
CM field F and a continuous irreducible representation r : GF → GLn (Fp ) suchthat
• each place v|p of F satisfies Fv ∼= K,
• for each place v|p of F , either r|GFv ∼= ρ or r|GFvc ∼= ρ, and• r is automorphic, in the sense that it may be lifted to a representationr : GF → GLn (Qp ) coming from a regular algebraic conjugate self dual
cuspidal automorphic representation of GLn /F .
Proof. This is immediate from [EG14, Cor. A.7] (since [EG14, Conj. A.3] is a
special case of Theorem 6.4.4).

6.5

The irreducible components of Xd,red

We now complete the analysis of the irreducible components of Xd,red that webegan in Section 5.5. Recall that Theorem 5.5.12 shows that Xd,red is an algebraic stack of finite presentation over F, and has dimension [K : Qp ]d(d − 1)/2.

6.5. THE IRREDUCIBLE COMPONENTS OF Xd,red

171

Furthermore, for each Serre weight k, there is a corresponding irreducible comk
ponent Xd,red,F of (Xd,red )Fp , and the components for different weights k arepdistinct (see Remark 5.5.14).
We are now finally in a position to prove the following result.
6.5.1 Theorem. Xd,red is equidimensional of dimension [K : Qp ]d(d − 1)/2,
and the irreducible components of (Xd,red )Fp are precisely the various closedksubstacks Xd,red,F of Theorem 5.5.12; in particular, (Xd,red )Fp is maximallypknonsplit of niveau 1. Furthermore each Xd,red,F can be defined over F, i.e. ispk

the base change of an irreducible component Xd,red of Xd,red .
Proof. By Theorem 5.5.12, it is enough to prove that each irreducible componentof (Xd,red )Fp is of dimension of at least [K : Qp ]d(d − 1)/2. Indeed, this showskthat the irreducible components of (Xd,red )Fp are precisely the Xd,red,F ; topsee that these may all be defined over F, we need to show that the actionof Gal(Fp /F) on the irreducible components of (Xd,red )Fp is trivial. This follows
immediately by considering its action on the maximally nonsplit representationsof niveau 1 (since the action of Gal(Fp /F) preserves the property of being
maximally nonsplit of niveau 1 and weight k).
In order to see that each irreducible component of (Xd,red )Fp is of dimensionof at least [K : Qp ]d(d − 1)/2, it suffices to show that every finite type point xof (Xd,red )Fp is contained in an irreducible substack of (Xd,red )Fp of dimension
at least [K : Qp ]d(d − 1)/2; so it suffices in turn to show that each x is contained in an equidimensional substack of dimension [K : Qp ]d(d − 1)/2. But
by Theorem 6.4.4, there is a regular Hodge type λ such that x is contained incrys,λXd
×Spf O Spec F (for O sufficiently large), and this stack is equidimensionalof dimension [K : Qp ]d(d − 1)/2 by Theorem 4.8.14, as required.
We end this section with the following result, showing that in contrast to
crys,λ,τ
ss,λ,τ
its substacks Xd
and Xd, the formal algebraic stack Xd is not a p-adic
formal algebraic stack.
6.5.2 Proposition. Xd is not a p-adic formal algebraic stack.
Proof. Assume that Xd is a p-adic formal algebraic stack, so that its special
fibre X d := Xd ×O F is an algebraic stack, which is furthermore of finite typeover F (since Xd is a Noetherian formal algebraic stack, by Corollary 5.5.18,
and Xd,red is of finite type over F, by Theorem 5.5.12). Since the underlying
reduced substack of X d is Xd,red , which is equidimensional of dimension [K :Qp ]d(d − 1)/2, we see that X d also has dimension [K : Qp ]d(d − 1)/2.
Consider a finite type point x : Spec F0 → Xd,red , corresponding to a representation ρ : GK → GLd (F0 ). By Proposition 3.6.3 there is a corresponding versal morphism Spf Rρ /$ → X d , and applying [EG19, Lem. 2.40] asin the proof of Theorem 4.8.14, we conclude that Rρ /$ must have dimension d2 + [K : Qp ]d(d − 1)/2. However, it is known that there are representations ρ for which Rρ /$ is formally smooth of dimension d2 + [K : Qp ]d2 (seeCHAPTER 6. CRYSTALLINE LIFTS AND Xd,red

172

for example [All19, Lem. 3.3.1]; indeed, a Galois cohomology calculation showsthat the dimension is always at least d2 + [K : Qp ]d2 , which is all that we need).
Thus we must have d2 = d(d − 1)/2, a contradiction.
0

6.5.3 Remark. A similar argument shows that the versal morphisms Spf Rρ,O →0

Xd and Spf Rρ,O /$ → Xd ⊗O F of Proposition 3.6.3 are not effective.

6.6

Closed points

Recall that if Y is an algebraic stack, then |Y| denotes its underlying topologicalspace. The points of |Y|, which we also refer to as the points of Y, are theequivalence classes of morphisms Spec k → Y, with k a field; two morphisms
Spec k → Y and Spec l → Y are deemed to be equivalent if we may find a fieldΩ and embeddings k, l ,→ Ω such that the pull-backs to Spec Ω of the two given
morphisms coincide. A point of Y is called finite type if it is representable bya morphism Spec k → Y which is locally of finite type. If Y is locally of finite
type over a base-scheme S, then a point of Y is of finite type if and only if its
image in S is a finite type point of S in the usual sense [Sta, Tag 01T9].
We say that a point of Y is closed if the corresponding point of |Y| is closed
in the topology on |Y|. Closed points are necessarily of finite type. However, if
X is not quasi-DM, then finite type points of X need not be closed, even if X
is Jacobson (in contrast to the situation for schemes [Sta, Tag 01TB]).
If Y is locally of finite type over an algebraically closed field k, then the map
Y(k) → |Y| (taking a k-valued point of Y to the point it represents) identifies
Y(k) with the set of finite type points of Y. (This is a standard consequence of
the Nullstellensatz, applied to a scheme chart of Y.)
Throughout this section, we fix a value of d ≥ 1, and write X rather than Xd .
In order to describe the finite type and closed points of (Xred )Fp , we introduce
the following terminology.
6.6.1 Definition. Let ρ, θ : GK → GLd (Fp ) be continuous representations.
Then we say that θ is a partial semi-simplification of ρ if there are short exact
sequences
0 → ri → ρi → r0i → 0
for i = 1, . . . such that ρ1 = ρ, ρi+1 = ri ⊕ r0i , and ρn = θ for n sufficiently
large.
Note in particular that the semi-simplification of ρ is a partial semi-simplificationof ρ. We also consider the following related notion.
6.6.2 Definition. Let ρ, θ : GK → GLd (Fp ) be continuous representations.
Then we say that θ is a virtual partial semi-simplification of ρ if there is a short
exact sequence of the form
0 → r → r ⊕ ρ → θ → 0.

6.6. CLOSED POINTS173

It follows from an evident induction on the integer n in Definition 6.6.1 thatif θ is a partial semi-simplification of ρ, then it is in particular a virtual partial
semisimplification; however, the converse does not hold in general. (See theintroduction to [Zwa00].)
Our promised description of the finite type and closed points of (Xred )Fp isgiven by the following theorem.
6.6.3 Theorem.
1. The morphism X (Fp ) = Xred (Fp ) → |(Xred )Fp | that sends each morphism
Spec Fp → X to the point of (Xred )Fp that it represents is injective, andits image consists precisely of the finite type points. Thus the finite typepoints of (Xred )Fp are in natural bijection with the isomorphism classes ofcontinuous representations ρ : GK → GLd (Fp ).
2. A finite type point of |(Xred )Fp | is closed if and only if the associated Galois
representation ρ is semi-simple.
3. If x ∈ |(Xred )Fp | is a finite type point, corresponding to the Galois representation ρ, then the closure {x} contains a unique closed point, whose
corresponding Galois representation is the semi-simplification ρss of ρ.
More generally, if x and y are two finite type points of |(Xred )Fp |, corresponding to Galois representations ρ and θ respectively, then y lies in{x} if and only if θ is a virtual partial semi-simplification of ρ.
Proof. Since Fp -valued points of X coincide with Fp -valued points of Xred , andso also with Fp -valued points of (Xred )Fp , the claim of (1) is a particular caseof the more general consequence of the Nullstellensatz that we recalled above.
Suppose now that we have a short exact sequence
0 → r → r ⊕ ρ → θ → 0.

(6.6.4)

We now follow the proof of [Rie86, Prop. 3.4] to show that the point y corresponding to θ lies in the closure of the point x corresponding to ρ. Denotethe morphism r → r ⊕ ρ in (6.6.4) by (f, g). Then for t in a sufficiently small
open neighbourhood U of 0 ∈ A1 , the morphism (f + t1r , g) : r → r ⊕ ρ is aninjection with projective cokernel (to see this one can for example consider asplitting of(6.6.4) on the level of vector spaces); so we have a morphism (where
we as usual abuse notation by writing families of (ϕ, Γ)-modules as families ofGalois representations) U → X given by t 7→ ρt := (r ⊕ ρ)/ im(f + t1r , g). Afterpossibly shrinking U further we may assume that f + t1r is an automorphismof r for all Fp -points t 6= 0 of U , so that ρt ∼= ρ; while for t = 0 we have ρt ∼= θ.
Thus θ is indeed in the closure of ρ, as claimed. This proves the “if” directionof (3), and the “only if” direction of (2).
The “if” direction of (2) follows from the “only if” direction of (3), and so it
remains to prove this latter statement. To this end, we fix ρ, θ : GK → GLd (Fp ),and assume that the point y corresponding to θ lies in the closure of the point

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CHAPTER 6. CRYSTALLINE LIFTS AND Xd,red

x corresponding to ρ. In order to avoid discussing deformation theory overFp , we extend our coefficients O if necessary so that ρ and θ are both definedover F. We may and do also assume that x and y are distinct; i.e. that ρ andθ are not isomorphic (as representations defined over Fp , or equivalently, as
representations defined over F). Let x0 and y0 denote the images of x and yrespectively in Xred ; then x0 and y0 are again distinct. Let Z denote the closureof {x0 }, thought of as a reduced closed substack of Xred ; then y0 is a point of Z.
Let R be the framed deformation ring of θ over O; then R is a versal ring to
X at y. Let Spf S = Spf R ×X Z; then S is a versal ring to Z at y0 . Since Z isalgebraic, the versal ring S is effective [Sta, Tag 07X8], and the corresponding
morphism Spec S → Z is furthermore flat [Sta, Tag 0DR2]. The morphism
Spec F → Z corresponding to x0 is quasi-compact and scheme-theoreticallydominant; thus the base-changed morphism
Spec S ×Z,x0 Spec F → Spec S(6.6.5)is again scheme-theoretically dominant.
Let η be a point in the image of (6.6.5), and let Spec T be the Zariski
closure of η in Spec S. Then T is a quotient of R which is an integral domain.
If K denotes the fraction field of T , then the morphism Spec K → Xred factors
through the morphism Spec F → Xred corresponding to x0 , and so the Galois
representation GK → GLd (K) classified by the point η of Spec T is isomorphic
to ρ (see Section 3.6.5 for the definition of this Galois representation). Thus, if L
denotes the splitting field of ρ, then we see that the deformations of θ classified
by T all factor through the finite group Gal(L/K). It follows immediatelyfrom [Zwa00, Thm. 1] that θ is a virtual partial semi-simplification of ρ, asrequired.

6.7

The substack of GK -representations

We now briefly explain the relationship between our stacks and the stacks of
GK -representations constructed by Wang-Erickson in [WE18]. Write XdGal forthe formal algebraic stack characterised by the following property: if A is an
O-algebra in which p is nilpotent, then XdGal (A) is the groupoid of continuous
morphisms
ρ : GK → GLd (A)(where A has the discrete topology, and GK its natural profinite topology).
That this is a formal algebraic stack follows from [WE18, Thm. 3.8, Rem. 3.9].
Equivalently, we can think of XdGal (A) as the groupoid of rank d projective Amodules TA with a continuous action of GK (where each TA has the discrete
topology).
For any finite type O/$a -algebra A, and any TA ∈ XdGal (A), we set
DA (TA ) := W (C[ )A ⊗A TA ,

6.7. THE SUBSTACK OF GK -REPRESENTATIONS

175

which naturally has the structure of a rank d projective (ϕ, GK )-module with
A-coefficients (with ϕ acting on the first factor in the tensor product, and GK
acting diagonally). (In the case that A is actually a finite O/$a -algebra, thisagrees with the construction given in Section 3.6.4.)
By Proposition 2.7.8, for each finite type O/$a -algebra A the assignment
TA 7→ DA (TA ) gives a functor XdGal (A) → Xd (A). Since both XdGal and Xd arelimit preserving (for XdGal this follows easily from the definition, since any ρ
as above factors through a finite quotient of GK ), this defines a morphism ofstacks
XdGal → Xd .
(6.7.1)
We now show that this morphism is in fact a monomorphism.
6.7.2 Theorem. The morphism (6.7.1) is a monomorphism, and is furthermore
versal at finite type points in the sense of Definition 7.1.1 below.
Proof. The versality statement follows from the fact that for any finite Artinian
Zp -algebra A, the A-valued points of XdGal and Xd coincide. (Over such a ring A,étale (ϕ, Γ)-modules do arise from Galois representations.)
To prove the monomorphism claim, we need to show that for any finite
type O/$a -algebra A, the functor XdGal (A) → Xd (A) is fully faithful; that is,
we need to show that given T1 , T2 ∈ XdGal (A), we have?

HomGK ,A (T1 , T2 ) = Homϕ,GK (DA (T1 ), DA (T2 )).
Writing T := T1∨ ⊗A T2 , it suffices to show that?

T GK = DA (T )ϕ=1,GK ,or even that?

T = DA (T )ϕ=1 .
Since T is a finite projective A-module, we have
DA (T )ϕ=1 := (W (C[ )A ⊗A T )ϕ=1 = (W (C[ )A )ϕ=1 ⊗A T,
and the result follows from Lemma 2.2.19.
6.7.3 Remark. In [WE18, §4], Wang-Erickson explains how to associate étale
ϕ-modules to GK∞ -representations with open kernel. A similar argument would
allow us to associate projective étale (ϕ, Γ)-modules to objects of XdGal (A).
However, if we directly followed the strategy of [WE18], the resulting étale(ϕ, Γ)-modules would have as coefficients the rings AK ⊗Zp A, rather than therings AK,A that we use throughout this book. Thus this approach would not
obviously yield Theorem 6.7.2, which is why we’ve preferred to make a more
direct argument in terms of étale (ϕ, GK )-modules.

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CHAPTER 6. CRYSTALLINE LIFTS AND Xd,red

6.7.4 Remark. It follows immediately from Theorem 6.7.2 that XdGal is largest
substack of Xd over which the universal (ϕ, Γ)-module can be realised as a GK representation.
In fact, it should be possible to strengthen Theorem 6.7.2. For example,[WE18, Thm. 3.8] shows that we can writeaGal
XdGal =
Xd,D,
(6.7.5)D

where D runs over the isomorphism classes of d-dimensional semisimple Fp Gal
representations of GK , and Xd,D(A) is the groupoid of those ρ the semisimplification of whose reductions modulo $ is D. In view of Theorem 6.6.3, we expect
Gal
that the monomorphism (Xd,D)red → Xd,red (induced by the monomorphism ofTheorem 6.7.2) will actually be a closed immersion, and hence (given the verGal
sality statement of Theorem 6.7.2) that Xd,D
will be identified with the formal
completion of Xd along the image of this closed immersion. In fact, this image will contain a unique closed point (corresponding to the semisimple GaloisGal
representation D), and we imagine that Xd,D
could also be identified with thecoherent completion (in the sense [AHR20, Defn. 2.1]) of Xd a this closed point.
However, we don’t pursue these ideas further here.
6.7.6 Remark. As well as considering GK -representations, it is also possible toconsider a larger substack of Xd over which our (ϕ, Γ)-modules can be realised as“Weil–Deligne”-representations. (Here the notion of Weil–Deligne representations is not quite the usual one, but rather is given by representations of certain
discretizations of GK , as described in [EG20, §1.2] and the references therein.)In the case K = Qp and d = 2, this locus is discussed in [EG20, §1]; roughly
speaking, the difference between it and XdGal is that it contains families ofsemisimple representations given by direct sums of unramified twists of fixed
irreducible representations (the key point being that the universal unramified
character does not correspond to a GK -representation, but does correspond toa representation of the Weil group WK , see Remark 7.2.19 below).

Chapter 7

The rank one caseIn this chapter we describe some of our key constructions explicitly in the casewhere d = 1. More precisely, following the notation of Chapter 3, we give
explicit descriptions of the stacks Rd , RΓd disc , and Xd , all in the case when d = 1(and K is arbitrary).

7.1

Preliminaries

While it may be possible to find an explicit description of the stacks we areinterested in directly from their definitions, this is not how we proceed. Rather,
we use the relationship between étale ϕ-modules (resp. étale (ϕ, Γ)-modules)
and Galois representations to construct certain affine formal algebraic spaces
U , V , and W , along with morphisms U → R1 , V → RΓ1 disc , and W → X1 , eachsatisfying the conditions of Lemma 7.1.8 below. An application of this lemmathen yields a description of each of R1 , RΓ1 disc , and X1 .
7.1.1 Definition. Let S be a locally Noetherian scheme, let X and Y be stacks
over S and let f : X → Y be a morphism. We say that f is versal at finite typepoints if for each morphism x : Spec k → X , with k a finite type OS -field, themorphism f is versal at x, in the sense of Definition A.26.
7.1.2 Remark. As the discussion of versality in Appendix A should make clear,this definition is related to the various properties considered in [EG21, Def. 2.4.4]
and the surrounding discussion. As noted in [EG21, Rem. 2.4.6], it is somewhat
complicated to define the notion of versality at literal points of a stack, since
the points of a stack are (by definition) equivalence classes of morphisms fromthe spectrum of a field to the stack, and it is not immediately clear in general
whether or not the versality condition would be independent of the choice ofequivalence class representative. In the preceding definition we obviate thispoint, by directly requiring versality at all (finite type) representatives of all(finite type) points.

177

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CHAPTER 7. THE RANK ONE CASE

7.1.3 Lemma. If X → Y is a morphism of stacks over a locally Noetherianbase S which is versal at finite type points, and if Z → Y is a morphism ofstacks, then the induced morphism X ×Y Z → Z is again versal at finite typepoints.
Proof. Let xe : Spec k → Xe := X ×Y Z with Spec k of finite type over S; from thedefinition of the 2-fibre product Xe, we see that giving the morphism x
e amounts
to giving the pair of morphisms x : Spec k → X and z : Spec k → Z (theresult of composing x
e with each of the projections), as well as an isomorphism∼y −→ y 0 , where y and y 0 are the k-valued points of X obtained respectively bycomposing x with the morphism X → Y and by composing z with the morphism
Z → Y. It is then a straightforward diagram chase, working from the variousddefinitions, to deduce the versal property of the morphism (Xe)xe → Zbz (where
we use the notation of Definition A.25) from the versal property of the morphism
Xbx → Yby .
The following result is standard, but we indicate the proof for the sake ofcompleteness.
7.1.4 Lemma. If f : X → Y is a morphism of algebraic stacks, each of finite
type over a locally Noetherian scheme S, then f is versal at finite type points if
and only if f is smooth.
Proof. The “if” direction follows from an application of the infinitesimal lifting
property of smooth morphisms; see e.g. the implication “(4) =⇒ (1)” of [EG21,
Lem. 2.4.7 (4)]. For the converse, choose a smooth surjective morphism U → X
with U a scheme (necessarily locally of finite type over S); then by the direction
already proved, this morphism is also versal at finite type points, and hence sois the composite morphism U → X → Y. To show that f is smooth, it suffices
to show the same for this composite. It follows from the implication “(1) =⇒(4)” of [EG21, Lem. 2.4.7 (4)] that f is smooth in a neighbourhood of eachfinite type point of U ; but since these points are dense in U , the desired result
follows. (We have cited [EG21] in this argument purely for our own convenience;of course the present lemma is just a version of Grothendieck’s result relatingsmoothness and formal smoothness.)
We now strengthen the previous result to cover the case where X and Y are
not necessarily assumed to be algebraic, but merely the map between them isassumed to be representable by algebraic stacks.
7.1.5 Lemma. If f : X → Y is a morphism of limit preserving stacks over a
locally Noetherian base S, and f is representable by algebraic stacks, then f isversal at finite type points if and only if f is smooth.
Proof. If f is smooth, then by Lemma A.2 (2) it satisfies the infinitesimal lifting property, which immediately implies that it is versal at finite type points.
Conversely, suppose now that f is versal at finite type points.

7.1. PRELIMINARIES179

It follows from [EG21, Cor. 2.1.8] that f is limit preserving on objects, andsince it is also representable by algebraic stacks, it is locally of finite presentation,
by Lemma A.2 (1). We now have to verify that for any test morphism Z → Y
whose source is a scheme, the induced morphism
X ×Y Z → Z,

(7.1.6)

which is again locally of finite presentation, is in fact smooth. Of course it
suffices to do this for affine schemes, and then, since Y is limit preserving, foraffine schemes that are of finite type over S. Since f is representable by algebraic
stacks, we find that the source of (7.1.6) is an algebraic stack. Furthermore,since this morphism is locally of finite presentation, its source is again locallyof finite type over S. Finally, Lemma 7.1.3 shows that (7.1.6) is versal at finite
type points. The claimed result then follows from Lemma 7.1.4.
We may apply the previous lemma to obtain a criterion for representing a
stack as the quotient of a sheaf by a smooth groupoid. We first make anotherdefinition.
7.1.7 Definition. We say that a morphism X → Y of stacks over a locally
Noetherian scheme S is surjective on finite type points if for each morphismx : Spec k → Y, with k a finite type OS -field, we may find a field extension l ofk and a morphism Spec l → X which makes the diagram
Spec l/X

Spec k/Ycommutative.
7.1.8 Lemma. Suppose that Z is a stack over a locally Noetherian scheme S,
and that f : U → Z is a morphism from a sheaf to Z which is representableby algebraic spaces1 and surjective on finite type points (in the sense of Definition 7.1.7). Suppose furthermore that both U and Z are limit preserving, andthat f is versal at finite type points. Then the morphism f is in fact smooth andsurjective, and, if we let R := U ×Z U , then the induced morphism [U/R] → Z
is an isomorphism.
Proof. It follows from Lemma 7.1.5 that the morphism f is smooth. We claimthat it is furthermore surjective; by definition, this means that if T → Z is
any test morphism from an affine scheme, we must show that the morphism ofalgebraic spaces g : T ×Z U → T induced by f is surjective. Since Z is limit
preserving, we may assume that T is of finite type over S; also, since g is a basechange of the smooth morphism f , it is itself smooth, and so in particular has
1 Since the source of the morphism is a sheaf, this is equivalent to being representable byalgebraic stacks.

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CHAPTER 7. THE RANK ONE CASE

open image. Our assumption that f is surjective on finite type points implies
furthermore that the image of g contains all the finite type points of T ; thus g
is indeed surjective.
Since smooth morphisms are in particular flat, the second assertion of thelemma follows from Lemma A.33.
7.1.9 Remark. If we omit the assumption in Lemma 7.1.8 that the morphism
U → Z be representable by algebraic spaces, then the morphism [U/R] → Z
need not be an isomorphism.
An example illustrating this is given by taking Z to be a scheme Z, locally offinite type over a locally Noetherian base scheme S, choosing a closed subscheme
Y of Z which is not also open, and defining
U to be the formal scheme obtained`b
as the disjoint union U := (Z \ Y ) Z,
where Zb denotes the completion of Z
along Y . We let U → Z be the obvious morphism; this is then a surjective
monomorphism, which is not an isomorphism, although it is versal at every
finite type point, and (since it is a monomorphism) we have that R := U ×Z U
coincides with the diagonal copy of U inside U ×S U (so that [U/R] = U ).
We will find the following lemmas useful in verifying the hypotheses ofLemma 7.1.8 in our application.
7.1.10 Lemma. Let f : X → Y and g : Y → Z be morphisms of stacks over a
locally Noetherian scheme S. Assume that f is representable by algebraic stacks
and locally of finite type, and that both g and the morphism Z → S are limit
preserving on objects. Then f is an isomorphism if and only if the induced
morphism
Spec A ×Z X → Spec A ×Z Y
(7.1.11)
is an isomorphism for every morphism Spec A → Z with A a locally of finite
type Artinian local OS -algebra.
Proof. The “only if” direction is evident, and so we focus on the “if” direction.
We begin by reducing to the case Z = S. We first recall a basic fact about
fibre products: if T → Y is any morphism from a scheme to the stack Y,
then the base-change T ×Y X may be described as an iterated fibre product
T ×(T ×Z Y) (T ×Z X ) (where we regard T as lying over Z via the composite
T → Y → Z, and as lying over T ×Z Y via the graph of the given morphism
T → Y).
In order to show that f is an isomorphism, we have to show that for any
morphism T → Y, the base-changed morphism T ×Y X → T is an isomorphism.
By the fact about fibre products just recalled (and remembering that the basechange of an isomorphism is an isomorphism), it suffices to show that for any
morphism T → Z whose source is a scheme, the base-changed morphism
T ×Z X → T ×Z Y

(7.1.12)

is an isomorphism. Since Z → S is limit preserving on objects, we may and do
assume that T is locally of finite type over S. We will then replace both Z and7.1. PRELIMINARIES181

S by T (mapping identically to itself), and the morphism f by (7.1.12). Notethat the projection T ×Z Y → T is again limit preserving on objects (beinga base-change of g, which has this property). We now have to reinterpret our
original hypothesis on A-valued points of Z (for locally of finite type Artinian
local OS -algebras A) in this new context. To this end, note that if Spec A → T
is locally of finite type with A being Artinian local, then Spec A is also locallyof finite type over S. The fibre product of (7.1.12) with Spec A over T may
then be identified with the morphism (7.1.11), and is thus an isomorphism (byassumption). Putting all this together, we see that we have reduced to the casewhen Z = S; so we return to our original notation, but assume in addition
that Z = S (mapping to itself via the identity).
What we have to show is that if T → Y is any morphism, thenT ×Y X → T(7.1.13)is an isomorphism. Since Y → S is limit preserving, we may assume that T is
locally of finite type over S. Any finite type Artinian local T -algebra is thenalso finite type over S, and so, replacing both Y and S by T (and again applying
the fibre product fact recalled above, with T replaced by Spec A) we find that
we may make another reduction, to the case when Y = S. The representability
assumption on f then implies that X is an algebraic stack. In this case thelemma is standard, but we recall a proof for completeness.
Our hypothesis that f induces an isomorphism on A-valued points implies
that the morphism f is versal at finite type points, and so by Lemma 7.1.4, it issmooth. It also implies that f contains all finite type points in its image; since
the image of f is constructible, we see that f is in fact surjective. This same
argument implies that the diagonal morphism X → X ×S X is surjective, andthus that f is universally injective. Since f is smooth, it is flat and locally offinite type. Since it is also universally injective, it is an open immersion. Since
it is surjective, it is an isomorphism, as claimed.
7.1.14 Lemma. Let Y be a stack over a locally Noetherian scheme S, let Z bea closed substack of Y, and let f : X → Y be a morphism of stacks. Suppose thatboth f and the morphism Y → S are limit preserving on objects; then f factors
through Z if and only if and only if for any finite local Artinian OS -algebra A,
and any morphism Spec A → Y over S, the morphism Spec A×Y X → Y factors
through Z.
Proof. The “only if” direction is trivial. For the converse, consider the basechange via f of the closed immersion Z ,→ Y; this is a closed immersion g :X ×Y Z ,→ X of stacks over Y, which we must show is an isomorphism. The
hypothesis implies that for any morphism Spec A → Y whose source is a finite
type local Artinian OS -algebra, the pull-back of gSpec A ×Y (X ×Y Z) → Spec A ×Y X
is an isomorphism; it then follows from Lemma 7.1.10 that g is an isomorphism.

182

CHAPTER 7. THE RANK ONE CASE

The next lemma lets us compute fibre products over stacks in certain situations. We recall that if Y is a stack (over some base scheme S), then the inertia
stack IY is the stack over S whose T -valued points (for any S-scheme T ) consistof pairs (y, α), where y is a T -valued point of Y, and α is an automorphism of y.
It is a group object in the category of stacks lying over Y via morphisms that are
representable by sheaves. We also remind the reader that there is a canonicalisomorphism∼
IY −→ Y ×∆Y ,Y×S Y,∆Y Y,
where the subscripts ∆Y indicate that both copies of Y are regarded as lying overthe product Y ×S Y via the diagonal morphism ∆Y : Y → Y ×S Y, and where,
under this identification, the forgetful morphism IY → Y may be identified withprojection onto the first copy of Y.
7.1.15 Lemma. Let X → Y be a morphism from a sheaf to a stack (both oversome base-scheme S), and suppose that the natural morphism X×Y X → X×S X
factors through the diagonal copy of X lying in the target (so that the groupoid
X ×Y X is in fact a group object in the category of sheaves over X). Then there∼
is a canonical isomorphism X ×Y X −→ X ×Y IY of group objects over X.
Proof. By assumption, we have a commutative diagram/X

X ×Y XY∆Y∆X/ X ×S X# / Y ×S Y

in which the outer rectangle is 2-Cartesian. Since X is a sheaf, the diagonal ∆X :X → X ×S X is a monomorphism, and so one immediately checks that the lefthand trapezoid is also 2-Cartesian. Thus we obtain the required isomorphism∼∼∼

X ×Y X −→ X ×Y×S Y Y −→ X ×Y (Y ×Y×S Y Y) −→ X ×Y IY .

7.2

Moduli stacks in the rank one caseWe now give concrete descriptions of our various stacks in the rank one case.
Before doing this, we prove a result that describes the stack structure on familiesof rank one objects. (It encodes the fact that the automorphisms of a rank one
object are simply the scalars.)
7.2.1 Lemma. If Z denotes either of the stacks R1 or X1 , then there is a
canonical isomorphism∼b m −→Z ×O G
IZ(7.2.2)b m is the $-adic completion of Gm over O).
of group objects over Z (where G

7.2. MODULI STACKS IN THE RANK ONE CASE

183

Proof. For definiteness we give the proof in the case of R1 ; the proof in the X1case is identical. We begin by defining the morphism (7.2.2): if A is any $adically complete O-algebra, and M is an étale ϕ-module over AK,A , then (7.2.2)
is defined on A-valued points of the source lying over M by mapping an elementa ∈ A× to the automorphism of M given by multiplication by a.
Since R1 has affine diagonal which is of finite presentation, the morphism (7.2.2)
is a morphism between finite type group objects over R1 , each of whose structure morphisms is representable by algebraic spaces, indeed affine, and of finite
presentation. Furthermore, R1 itself is limit preserving over O. Thus to showthat (7.2.2) is an isomorphism, it suffices (by Lemma 7.1.10) to show that it
induces an isomorphism on A-valued points for A an Artinian local O-algebraof finite type. Returning to the notation of the preceding paragraph (but now
assuming that A is Artinian local), we have to show that any automorphism of
M is given by multiplication by an element of A× . Since M is of rank 1, we
find that
HomAK,A ,ϕ (M, M ) ∼= (M ∨ ⊗AK,A M )ϕ=1 ∼= (AK,A )ϕ=1 ∼=A(the final isomorphism following from Lemma 2.2.19, since AK,A ⊆ W (C[ )A by
Proposition 2.2.12). The lemma follows.

7.2.3

Local Galois theory

We briefly recall the local Galois theory that is relevant to our computation.
If L/K is an algebraic extension, we write ILab for the image of IL in Gab
L (notethat this is not the abelianization of IL ). We have short exact sequences
1 → IK → GK → FrobZ
K →1(7.2.4)

1 → IKcyc → GKcyc → FrobfKZ → 1,

(7.2.5)bandbwhere f := [k∞ : k].
These induce corresponding short exact sequencesZab
1 → IK→ Gab
K → FrobK → 1(7.2.6)fZ
ab
1 → IK→ Gab
Kcyc → FrobK → 1.
cyc

(7.2.7)b

andb

If G is any of the various profinite Galois groups appearing in the preceding
discussion, we let O[[G]] denote the corresponding completed group ring over O.
This is a pro-Artinian ring, which gives rise to the affine formal algebraic spaceb m (the $-adic
Spf O[[G]]. We endow this space with the trivial action of G
completion of Gm over O).

184

7.2.8

CHAPTER 7. THE RANK ONE CASE

Galois lifting rings for characters

If F0 /F is a finite extension, and ρ : GKcyc → (F0 )× is a continuous character,then we can write O0 = O ⊗W (F) W (F0 ), and consider the universal lifting O0

algebra Rρ,O . If x : Spf F0 → R1 is the corresponding finite type point, then0

we have a versal morphism Spf Rρ,O → R1 ; indeed it follows exactly as in the
proof of Proposition 3.6.3 that we have an isomorphism
00∼b m ) ,O0 ,Spf Rρ,O ×R1 Spf Rρ,O −→ (GR(7.2.9)ρb m ) ,O0 denotes the completion of (Gm ) ,O0 along (Gm )F0 .
where (GRR
ρ

7.2.10ρ

Descriptions of the rank one stacks

We can now establish our explicit description of R1 .
7.2.11 Proposition. There is an isomorphism
h

i
∼
ab
b m /Gb m −→Spf O[[IK
]]
×
G
R1
cyc
b m -action is taken to be(where, in the formation of the quotient stack, the Gtrivial).
Proof. We begin by constructing a morphismabb m → R1 .
Spf O[[IK]] × G
cyc(7.2.12)For this, we choose a lift of σcyc ∈ GKcyc of FrobfK , splitting the short exact sequences (7.2.5) and (7.2.7). If A is any discrete Artinian quotient ofabab]] → A to a contin]], then we extend the continuous morphism O[[IK
O[[IK
cyc
cyc
ab
uous morphism O[[GKcyc ]] → A by mapping σcyc to 1 ∈ A. We may view thislatter morphism as a rank 1 representation of GKcyc with coefficients in A; it
thus gives rise to a rank one projective étale ϕ-module MA over AK,A , andabtherefore to a morphism Spf O[[IK]] → R1 . We extend this to the morcyc
phism (7.2.12) by unramified twisting; more precisely, we have an induced morab
b m → R1 × G
b m , and we compose with the morphism
phism Spf O[[IK]] × Gcycb m → R1 given by taking the tensor product with the universal unramified
R1 × G
rank one étale ϕ-module.
By construction (and the usual explicit description of the universal Galois
deformation ring of a character), (7.2.12) is versal at finite type points, as wellas surjective on finite type points. We claim that the canonical morphismabb m ×R Spf O[[I ab ]] × Gbm
Spf O[[IK]] × G
Kcyc
1
cyc


ab
b m ×O Spf O[[I ab ]] × Gbm→ Spf O[[IK]] × G
Kcyc
cyc(7.2.13)

7.2. MODULI STACKS IN THE RANK ONE CASE

185

abb m in the target; given
factors through the diagonal copy of Spf O[[IK]] × G
cycthis, it follows from Lemmas 7.1.15 and 7.2.1 that there is an isomorphism ofgroupoids



∼
ab
ab
b m ×Gb m −→
b m ×R Spf O[[I ab ]]× Gbm
Spf O[[IK]]× G
Spf O[[IK
]]× G
Kcyc
1
cyc
cyc

b m acting trivially on Spf O[[I ab ]]× G
b m . The proposition will then follow
with G
Kcyc
from Lemma 7.1.8 provided we show that (7.2.12) is representable by algebraic
spaces (or, equivalently, by algebraic stacks).
It remains to verify the claimed factorization of (7.2.13), as well as theclaimed representability by algebraic stacks of (7.2.12). We establish each ofthese claims in turn.
ab
To show that (7.2.13) factors through the diagonal diagonal copy of Spf O[[IK]]×cycb m in the target, it suffices, by Lemma 7.1.14, to show that ifGabb m × Spf O[[I ab ]] × G
bm
Spec A → Spf O[[IK]] × G
Kcyc
cyc
is a morphism whose source is a finite type Artinian local O-algebra, then thepull back of (7.2.13) to Spec A factors through the pull-back to Spec A of the
diagonal morphismabb m → Spf O[[I ab ]] × G
b m ×O Spf O[[I ab ]] × G
bm .
Spf O[[IK]] × G
Kcyc
Kcyc
cyc
ab
b m correspond to characters χ :Since morphisms Spec A → Spf O[[IK]] × Gcycab×
GKcyc → A , this amounts to the evident fact that if χ, χ0 are two such characters, then the locus in Spec A over which χ and χ0 become isomorphic coincides
with the locus over which they coincide.
We now turn to proving that (7.2.12) is representable by algebraic stacks.
To see this, it suffices to study the corresponding question modulo some power$a of the uniformizer in O, so we work with the stack RaK,1 from now on.
We next recall some of the constructions we made in Section 3.2. We havethe subfield K basic ⊆ K of Definition 3.2.3, and the natural morphism RaK,1 →RaK basic ,[K:K basic ] given by the forgetful map which regards a rank one AK,A module as an AK basic ,A -module of rank [K : K basic ]. By Lemma 3.2.11, we canwrite RaK,1 = limh RaK,1,K basic ,h , where the algebraic stack RaK,1,K basic ,h is the−→scheme-theoretic image of the base-changed morphisma
C[K:KRK,1 → RK,1 ,basic ],h ×R basic
K,[K:K basic ]+awhere C[K:K
basic ],h is the algebraic stack of ϕ-modules over AK basic ,A of rank [K :K basic ] and T -height at most h. Accordingly, it suffices to show that each basechangeb m ×R Ra
Y a := Spf O[[I ab ]] × Gbasichis an algebraic stack.

Kcyc

K,1

K,1,K

,h

186

CHAPTER 7. THE RANK ONE CASEab
By construction, Yha is a closed subsheaf of Spf(O/$a )[[IK]] × (Gm )O/$a .
cycab
We claim that it is in fact a closed subsheaf of Spec(O/$a )[IK/U
]×(Gm )O/$a ,cycabfor some open subgroup U of IKcyc .
ab
Since Yha and Spec(O/$a )[IK/U ] × (Gm )O/$a are both closed subsheavescycaabof Spf(O/$ )[[IKcyc ]] × (Gm )O/$a , it follows from Lemma 7.1.14 that it isenough to show that if A is a finite local Artinian O/$a -algebra, then given any
abmorphism Spec A → Yha , the composite Spec A → Yha → Spf(O/$a )[[IK
]] ×
cyc
a
ab
(Gm )O/$a factors through Spec(O/$ )[IKcyc /U ]×(Gm )O/$a for the open subgroup U := I a,h of Proposition 7.3.17 below. If A is in fact a field, thenby [EG21, Lem. 3.2.14], the étale ϕ-module MA corresponding to the morphism Spec A → RK,1 attains (K basic , T )-height at most h in the sense of Definition 7.2.14 below after passing to a finite extension of A, and the required
factorisation is immediate from Proposition 7.3.17.
Having established the result in the case that A is a finite type field, in orderto prove the general local Artinian case, it suffices to prove a factorisation on
the level of versal rings. More precisely, let F0 /F be a finite extension, and let
x : Spec F0 → Yha be a finite type point of Yha , corresponding to a representation0ρ : GKcyc → (F0 )× . As above, we write Rρ,O for the corresponding universal
lifting O0 -algebra. Write Spf Rha for the scheme-theoretic image of the morphism
0

0

a
C[K:K
RK,1 ×RK,1 Spf Rρ,O → Spf Rρ,O .
basic ],h ×R basic
K,[K:K basic ]By Lemma A.30, the induced morphism Spf Rha → RaK,1,K basic ,h is a versal
00morphism at x. Write Rρ,O ,U for the quotient of Rρ,O corresponding toliftings which are trivial on U ; then it suffices to show that Spf Rha is a closed
0formal subscheme of Spf Rρ,O ,U .
We now employ Lemma A.32. Exactly as in the proof of Lemma 3.4.8, it
is enough (after possibly increasing F0 ) to show that if A is a finite type local
0
Artinian Rρ,O -algebra with residue field F0 for which the induced morphisma
C[K:KRK,1 ×RK,1 Spec A → Spec A admits a section,
basic ],h ×R basic
K,[K:K basic ]00then the morphism Spec A → Spf Rρ,O factors through Spf Rρ,O ,U . Since theexistence of the section implies (by definition) that the étale ϕ-module MA has(T, K basic )-height at most h (in the sense of Definition 7.2.14), we are done by
Proposition 7.3.17.
In the previous argument, we used the following definition, which will play
an important role in the technicalities of Section 7.3 below. (See in particular
the statements of Propositions 7.3.17 and 7.3.18.)
7.2.14 Definition. Let A be a finite local Artinian O/$a -algebra for some a ≥1, and let M be a rank one projective étale ϕ-module over AK,A . We regard M
as a rank [K : K basic ] étale ϕ-module M 0 over AK basic ,A , and we say that M has(K basic , T )-height at most h if there is a projective ϕ-module M over A+
K basic ,Aof T -height at most h such that M 0 = M[1/T ].

7.2. MODULI STACKS IN THE RANK ONE CASE

187

We next turn to describing RΓ1 disc explicitly. We first note that since IK ∩b
GKcyc = IKcyc , we have an exact sequence 1 → IKcyc → GK → FrobZ
K ×Γ.
We let H ⊆ FrobZ
K ×Γ denote the image of GK ; it is an open subgroup of
b
FrobZ
×Γ.
We
write
Hdisc := H ∩ (FrobZK
K ×Γdisc ); then Hdisc is dense in H,
and is isomorphic to Z × Z. Write O[Hdisc ] for the group ring of Hdisc over O,\∼
and O[Hdisc ] for its $-adic completion. Then Spec O[Hdisc ] = Gm ×O Gm , and
0
ab
\
∼ b
b
Spf O[Hdisc ] = Gm ×O Gm . Let I denote the image of IKcyc in GK ; equivalently,0abab
I is the quotient of IKcyc by the closure of its subgroup of commutators [IK
, Γ].
cyc
b

7.2.15 Proposition. There is an isomorphism
h

i
∼
Γdisc
\
b
Spf O[[I 0 ]] × Spf O[Hdisc ] /Gm −→ R1
b m -action is taken to be(where, in the formation of the quotient stack, the Gtrivial).
Proof. We leave the construction of this isomorphism to the reader, by combining the isomorphism of Proposition 7.2.11 with the definition of RΓ1 disc .
7.2.16 Remark. Note that the case d = 1 is not representative of the general casein so far as the structure of R, and so also of RΓdisc , is concerned. Namely, whend = 1, the stacks R and RΓdisc are formal algebraic stacks. This will not be thecase when d > 1. (They will instead be Ind-algebraic stacks whose underlying
reduced substacks are Ind-algebraic but not algebraic.) The reason for thisis that the 1-dimensional mod p representations of GKcyc may be described as
Frobenius twists of a finite number of characters when d = 1, so that in this case
Rred is indeed an algebraic stack, while the spaces Ext1GKcyc (χ, ψ) are infinite
dimensional, for any two mod p characters χ and ψ of GKcyc , so that already inthe case d = 2, the stack Rred is merely Ind-algebraic.
We now give an explicit description of the stack X1 .
7.2.17 Proposition. There is an isomorphism
h

i

∼
ab
b m /Gb m −→Spf O[[IK]] × GX1
b m -action is taken to be(where, in the formation of the quotient stack, the Gtrivial).
Proof. We prove this in the same way as Proposition 7.2.11; we leave the details
to the reader, indicating only the key differences. We can construct a morphismabb m → X1 in exactly the same way as in the proof of PropoSpf O[[IK]] × G
sition 7.2.11 (by choosing a lift σK ∈ GK of FrobK , splitting the short exact
sequence (7.2.4)), and we need to prove that this morphism is representable byalgebraic spaces. This can be done by arguing exactly as in the proof of Propoaasition 7.2.11, with the isomorphism XK,1
= limh,s XK,1,K
basic ,h,s of Lemma 3.5.5−→
replacing the isomorphism RaK,1 = limh RaK,1,K basic ,h .
−→

188

CHAPTER 7. THE RANK ONE CASEa
Bearing in mind the definition of XK,1,K
basic ,h,s , we find that we have to

h,s,a
ab
show that there is an open subgroup IK
of IK
such that if A is a finite localaArtinian O/$ -algebra, and M is a rank one étale ϕ-module over AK,A withthe property that there is a rank [K : K basic ] ϕ-module M over A+, suchK basic ,Athat:• M[1/T ] = M (where we are regarding M as an étale ϕ-module over AK basic ,A ),• M is of T -height at most h, and• the action of ΓK basic ,disc = ΓK,disc on M[1/T ] extends the canonical actionof ΓKsbasic ,disc given by Corollary 3.5.2,
h,s,athen the action of IK
on TA (M ) is trivial. This is immediate from Proposition 7.3.18.

7.2.18 Remark. Note that Propositions 7.2.15 and 7.2.17 describe X as a certain
formal completion of RΓdisc (in the case d = 1). In particular, the monomorphism X ,→ RΓdisc is not a closed immersion (in the case d = 1, and presumably
not in the general case either). This helps to explain why somewhat elaborate arguments were required in Chapter 3 to deduce properties of X from thecorresponding properties of RΓdisc .
7.2.19 Remark. It follows easily from Proposition 7.2.17 that the stack X1 may
be described as a moduli stack of 1-dimensional continuous representations of
the Weil group WK ; indeed, we see that if A is a p-adically complete O-algebra,ab× Z → A× (ifthen X1 (Spf A) is the groupoid of continuous characters IKb
we identify Gm with the p-adically completed group ring of Z). Mapping thegenerator 1 ∈ Z to σK (a lift of Frobenius, as in the proof of Proposition 7.2.17)
∼
ab
ab
.
× Z −→ WK
yields an isomorphism IK
This phenomenon doesn’t persist in the general case. Indeed, as noted in
Section 1.4, already in the case d = 2, there seems to be no description of X2
as the moduli space of representations of a group that is compatible with thedescription of its closed points in terms of representations of GK .
One can attempt to adapt the proof of Proposition 7.2.17 to the case d > 1,
by constructing a morphism from the moduli stack of continuous d-dimensional
Weil–Deligne representations (suitably understood2 ) to the stack Xd . However,when d > 1, this morphism of stacks will not be representable by algebraic
spaces. The precise point where the proof that we’ve given in the rank 1 casefails to generalise is that once we are not passing to abelianized Galois groups,
the upper numbered ramification groups are not open in the inertia group.
2 Since we are considering representations in rings that are p-power torsion, we cannot use
the usual formulation of the Weil–Deligne group. Rather, we choose a finitely generated dense
subgroup Z n Z[1/p] of the tame Galois group of GK (by choosing a lift of Frobenius and a liftof a topological generator of tame inertia), and form the topological group W DK by taking
its preimage in GK (and equipping Z n Z[1/p] with its discrete topology).

7.3. A RAMIFICATION BOUND

7.3

189

A ramification boundWe end this chapter by proving the bounds on the ramification of a character
valued in a finite Artinian O-algebra that were used in the proofs of Propositions 7.2.11 and 7.2.17. There are well-established techniques for proving sucha bound, going back to [Fon85]. We find it convenient to follow the arguments of [CL11], and indeed we will follow some of their arguments very closely.
The main differences between our setting and that of [CL11] are that we areworking in the cyclotomic setting, rather than the Kummer setting; that weare considering representations of GKcyc of finite height, rather than semistable
representations of GK ; and that to define a representation of finite height, weneed to pass between K and K basic . None of these changes make a fundamental
difference to the argument.
In fact, while [CL11] go to some effort to work modulo an arbitrary powerof p, and to optimise the bounds that they obtain, we are content to give thesimplest proof that we can of the existence of a bound of the kind that we need.
In particular, we are able to reduce to the case of mod p representations, whichsimplifies much of the discussion.
Recall that Kcyc /K is a Galois extension with Galois group ΓK ∼= Zp . For
each s ≥ 0, we write Kcyc,s for the unique subfield of Kcyc which is cyclic over Kof degree ps . We write e = e(K/K0 ).
Suppose that K is basic, and that M is a free ϕ-module over A+
K /p of T height at most h. Then M := AK ⊗A+ M is an étale ϕ-module over AK /p, andK
we have a corresponding GKcyc -representation T (M ) as in Section 3.6. (Notethat while elsewhere in the book we have worked with O/$a -coefficients, it ismore convenient to use Z/pZ-coefficients in most of this section.)
In order to compare directly to the arguments of [CL11], it is more convenient to work with the contragredient GKcyc -representation T (M )∨ , which
can also be computed as T (M ∨ ) = (C[ ⊗AK M ∨ )ϕ=1 = HomAK ,ϕ M, C[ =
HomA+ ,ϕ M, C[ . By [Fon90, A.1.2.7, B.1.8.3], this can also be computed via
Kthe functor[T ∨ (M) := HomA+ ,ϕ M, OC.
K

More precisely, there is a diagram[
T ∨ (M) := HomA+ ,ϕ M, OC
HomA+ ,ϕ M, C[ = T (M ∨ )



K

K

(
v 
HomA+ ,ϕ M, C[
K[(the first arrow being induced by the inclusion of OC
in C[ , and the second by
restriction from M to M), with both arrows in fact being isomorphisms (by op.
cit.).
We will now follow the arguments of [CL11, §4]; note that the ring denoted R[
in [CL11] is OC. For the time being we will assume that we are in the following

190

CHAPTER 7. THE RANK ONE CASE

situation; the rather complicated hypotheses here will allow us to interpret the
canonical actions of Corollary 3.5.2 in terms of the constructions of [CL11, §4].
7.3.1 Hypothesis. Assume that K is basic, and fix some h ≥ 1. Let M be a freeϕ-module over A+
K /p of T -height at most h. Fix the following:• an integer b ≥ eh/(p − 1).
• An integer N > pb/e such that N ≥ N (1, h), where N (1, h) is as inCorollary 3.5.2.
• An integer s such that ps−1 > b/e, and s ≥ s(1, h, N ), where s(1, h, N ) is
as in Corollary 3.5.2.
Note in particular that under Hypothesis 7.3.1 we have ps > ph/(p − 1) > h.
We normalize the valuation v on K so that v(K × ) = Z, and continue towrite v for the unique compatible valuation on K, and for the induced valuation[on OC= lim OK /p. Note that with this convention we have v(T ) = v(p) = e←−[(where we are abusively continuing to denote the image of T in OC
by T ; this0
follows straightforwardly from the definition of T as the trace of T := ([ε] − 1),which shows (after a simple computation) that v(T ) = (p − 1)v(T 0 ), along withthe formula v(ζps −1) = v(p)/ps−1 (p−1), which shows that v(T 0 ) = v(p)/(p−1)).
[
We write a>cfor the set of elements of OC
of valuation greater than c, andO[C

define a≥cin the same way. For any c ≥ 0, we write
O[
C


[
TO∨[ ,c (M) := HomA+ ,ϕ M, OC
/a>c
,
O[
K

C

so that there is a natural map T∨

C(M) → TO∨[ ,c (M). For any c0 ≥ c we writeCTO∨[ ,c0 ,c (M) = Im (TO∨[ ,c0 (M) → TO∨[ ,c (M)),
C

C

C

0

where the morphism is induced by the natural map a>c
→ a>c
.
O[
O[
C

C

7.3.2 Lemma. Assume that we are in the setting of Hypothesis 7.3.1. Then the
induced morphism T ∨ (M) → TO∨[ ,b (M) is injective, with image TO∨[ ,pb,b (M).
CC

Proof. This can be proved by a standard Frobenius amplification argument
exactly as in the proof of [CL11, Prop. 2.3.3]. For example, to see the injectivity,
we note that since M is finitely generated, any element of the kernel corresponds
to a ϕ-equivariant morphism M → a≥cfor some c > b. Since M has T -heightO[C

at most h, and v(T h ) = eh, the ϕ-equivariance implies that the image of thismorphism is contained in a≥pc−eh. Since c > b ≥ eh/(p − 1) by hypothesis, weO[C

see that the sequence c, pc − eh, p(pc − eh) − eh, . . . goes to infinity, and theinjectivity follows.
We leave surjectivity to the reader; it is a simpler version of the proof of
Lemma 7.3.11 below. (See also the proof of Lemma 4.3.2 for an almost identicalargument.)

7.3. A RAMIFICATION BOUND

191

Suppose given c ≥ 0 satisfying ps > c/e. (E.g. b and pb both satisfy this[condition.) The Frobenius ϕ is an automorphism of the perfect k-algebra OC−swhich multiplies valuations by p, and so ϕ induces the first of the following
sequence of GK -equivariant isomorphisms of k-algebras
>c/ps

∼

[
[
OC
/a>c
−→ k ⊗k,ϕs OC
/aC
O[
C

>c/ps

∼

−→ k ⊗k,ϕs OK /aK,(7.3.3)>c/ps

where aK
has the evident meaning, and the second isomorphism is induced[
by projection onto the first factor in the projective limit OC= limϕ OC /p =←−>c/ps
, so that thislimϕ OK /p. (The upper bound on c ensures that p ∈ aK←−
projection does indeed induce the indicated morphism; it is then straightforward
to verify that it is an isomorphism.) This isomorphism in turn induces anisomorphism
∼
>c/ps
).
(7.3.4)
TO∨[ ,c (M) −→ HomA+ ,ϕ (M, OK /aK
KC[
7.3.5 Lemma. If ps > c/e ≥ 0, then the action of GKcyc,s on OC
induces an∨action of GKcyc,s on TO[ ,c (M).
C[
Proof. The action of ϕ on OC
is GK -equivariant,and so the GK -action on[[OCinduces an action on Homϕ M, OC
/a>c
.
The
claim
of the lemma is thatO[C

the restriction of this action to GKcyc,s preserves the subobject TO∨[ ,c (M) :=
C


[
[
HomA+ ,ϕ M, OC/a>cof Homϕ M, OC
/a>c
.
O[
O[
K

C

C

Rather than proving this directly, we use the isomorphisms (7.3.3) and (7.3.4);
taking these into account, it suffices to prove that the action of GKcyc,s on
>c/ps
>c/ps
Homϕ (M, OK /aK) preserves the subobject HomA+ ,ϕ (M, OK /aK). InKother words, we have to check that the GKcyc,s -action preserves the property of
being A+K -equivariant. For this, it suffices to check that GKcyc,s acts trivially>c/ps
under the isomorphism (7.3.3). Since T can beon the image of T in OK /aK0
expressed as a power series in T , it in fact suffices to check this for the image>c/psof T 0 . But this image is equal to ζps+1 − 1 mod aK, which is fixed by GKcyc,s(since ζps+1 ∈ Kcyc,s ).
7.3.6 Corollary. Assume that we are in the setting of Hypothesis 7.3.1. Then[the natural action of GKcyc,s on OC
induces an extension of the action of GKcyc∨
on T (M) to an action of GKcyc,s .
Proof. This follows directly from Lemmas 7.3.2 and 7.3.5, once we note that,by hypothesis, ps > pb/e > b/e.
By Corollary 3.5.2 (and our hypothesized bound on s), we can give M[1/T ]the structure of a (ϕ, ΓKcyc,s )-module; this in particular induces an action of GKcyc,son T (M ). The following corollary expresses the compatibility of this action withthe action constructed in Corollary 7.3.6.

192

CHAPTER 7. THE RANK ONE CASE

7.3.7 Corollary. Assume that we are in the setting of Hypothesis 7.3.1. Thenthe two actions of GKcyc,s on T (M )∨ (from Corollary 7.3.6, and from Corollary 3.5.2) agree.
Proof. By the proof of Corollary 3.5.2, the semilinear action of ΓKcyc,s on M[
extends to a semilinear action of GKcyc,s on OC⊗A+ M, which is uniquely deK
termined by the properties that it commutes with ϕ, and satisfies (g − 1)(M) ⊂[T N (OC⊗A+ M) for all g ∈ GKcyc,s . As explained in Section 3.6.4, we haveK

T (M ) = (C[ ⊗A+ M)ϕ=1 , with the action of GKcyc,s on T (M ) being that inherK[ited from its action on OC⊗A+ M.
K
As noted above, we also have

T (M )∨ = T (M ∨ ) = T ∨ (M)[= HomC[ ,ϕ C[ ⊗A+ M, C[ .
= HomA+ ,ϕ M, OC
K

K

Now evaluation induces a pairingC[ ⊗A+ M × HomC[ C[ ⊗A+ M, C[ → C[ ,K

Kwhich restricts to a pairing(C[ ⊗A+ M)ϕ=1 × HomC[ ,ϕ C[ ⊗A+ M, C[ → (C[ )ϕ=1 = Z/pZ,
K(7.3.8)K

and this latter pairing is the natural pairing of GKcyc -representations
T (M ) × T (M )∨ → Z/pZ.
Therefore, to prove the lemma, we have to show that (7.3.8) is furthermore
GKcyc,s -equivariant (with the action on T (M ) being the action recalled above,coming from Corollary 3.5.2, and the action on T (M )∨ = T ∨ (M) given by
Corollary 7.3.6).
To see this, note firstly that by the same argument that we used in the proof
of Lemma 7.3.2, it follows from Hypothesis 7.3.1 that(C[ ⊗A+ M)ϕ=1 ⊆ (a≥b )−1 ⊗A+ M.
KKIn addition, we have
HomC[ ,ϕ C[ ⊗A+ M, C[ = HomOC[ ,ϕ OC[ ⊗A+ M, OC[
K
K
 ∼

⊆ HomOC[ OC[ ⊗A+ M, OC[ ←− HomOC[ (a≥b )−1 ⊗A+ M, (a≥b )−1
K

K

with the last isomorphism being induced by restricting the domain of a homomorphism from (a≥b )−1 ⊗A+ M to OC[ ⊗A+ M. Bearing in mind this isomorKK
phism, we obtain an evaluation pairing(a≥b )−1 ⊗A+ M × HomOC[ (OC[ ⊗A+ M, OC[ → (a≥b )−1
KK

7.3. A RAMIFICATION BOUND

193(and taking ϕ-invariants again recovers the pairing (7.3.8)).
Fix some α ∈ C[ with v(α) = −b, so(α) = (a≥b )−1 . Let x be an element
Pthat
n
[
ϕ=1
[
of (C ⊗A+ M), and write x = α i=1 λi ⊗ xi where λi ∈ OC
and xi ∈ M.
K

Pn
[
[
Let f ∈ HomO[ ,ϕ OC ⊗A+ M, OC , so that x and f pair to αf ( i=1 λi xi ) ∈CK[(a≥b )−1 ; of course, they in fact pair to an element of Fp ⊂POC
⊂ (a≥b )−1 .
n
Let g ∈ GKcyc ,s be arbitrary. Then gx and gf pair to g(α)(gf )( i=1 g(λi )g(xi )).
To establish the claimed GKcyc ,s -equivariance, we need to show thatg(α) · (gf )

n
X

n
X


g(λi )g(xi ) = g αfλi xi .

i=1

i=1This is an equality of elements of Fp , and it therefore suffices to show that it[
holds modulo mOC when each side is considered as an element of OC. Equiva−1
lently, after multiplication by α it suffices to show that
(gf )

n
X

n
X


λi xi
g(λi )g(xi ) ≡ g f(mod a>b ).

i=1

i=1

[
Recall that (g − 1)(M) ⊂ T N (OC⊗A+ M). By Hypothesis 7.3.1, we haveKv(T N ) = eN > pb, so that in particular we have T N ∈ a>b. Thus g(xi ) ≡ xi
O[
C(mod a>b ), so it suffices to show that
(gf )

n
X

n
X


λi xi
g(λi )xi ≡ g f(mod a>b ),

i=1

i=1

or equivalently thatnXg(λi )(gf )(xi ) ≡

i=1

n
X

g(λi )g(f (xi ))(mod a>b ),i=1

so in turn it is enough to show that if x ∈ M then (gf )(x) ≡ g(f (x)) (mod a>b ).
But this is true by the very definition of gf (which was defined via the actionof GKcyc ,s on TO∨[ ,b (M), which in turn is defined via the action of GKcyc ,s on
C[OC/a>b ), so we are done.

Recall that we write k for the residue field of K, and since K is assumed
basic we have k∞ = k, so that we can and do regard M as a free A+
K /p = k[[T ]]module. Write Ts for tr K(ζps+1 )/Kcyc,s (ζps+1 − 1) ∈ OKcyc,s , and note that wehave a homomorphismA+
K /p = k[[T ]] → OKcyc,s /p(7.3.9)which sends T → Ts , and whose restriction to k is given by ϕ−s .
Let L be any algebraic extension of Kcyc,s inside K. We continue to write v
for the unique valuation on L with v(p) = e. Note that we have v(Tsh ) =194

CHAPTER 7. THE RANK ONE CASE

eh/ps < e = v(p). For any c ≥ 0 we write a>c
L for the set of elements of L ofvaluation greater than c, and define a≥cinthesame way.
L
Choose an (ordered) A+/p-basis
e
,
.
.
.
,
e
of
M, and write ϕ(e1 , . . . , ed ) =1dK(e1 , . . . , ed )A for some A ∈ Md (A+
/p).
By
our
assumptionthat M has T -heightK
at most h, we can write AB = T h Idd for some B ∈ Md (A+K /p). We can and doeechoose matrices A, B ∈ Md (OKcyc,s ) which respectively lift the images of A, B
under the homomorphism A+
K /p → OKcyc,s /p defined above. Since the image
e by anof Tsh in OKcyc,s /p is nonzero, we see that after possibly multiplying B
invertible matrix which is trivial mod p/Tsh , we can and do assume thateB
e = Tsh Idd .
A(7.3.10)We setde
TeL∗ (M) := {(x1 , . . . , xd ) ∈ OL
| (xp1 , . . . , xpd ) = (x1 , . . . , xd )A}.

Note that since x 7→ xp is not a ring homomorphism on OL , this is just a set
with a GKcyc,s -action, rather than an OL -module.
For any c ∈ [0, eps ), we write
>c/ps

∗
TL,c
(M) := HomA+ /p,ϕ (M, (OL /p)/aL),

K

where the A+
K -algebra structure on OL /p is that induced by the composite(7.3.9)s
A+
K /p −→ OKs /p → OL /p. (The use of c/p rather than c in the definition∗
of TL,c comes from the definition of the homomorphism (7.3.9); see also (7.3.3).)
For any c0 ≥ c we write
∗
∗
∗
TL,c
0 ,c (M) = Im (TL,c0 (M) → TL,c (M)),>c0 /ps>c/ps

where the morphism is induced by the natural map aL→ aL.
Evaluating on the members of the basis e1 , . . . , ed of M yields an identification∼>c/ps d∨TL,c(M) −→ {(x1 , . . . , xd ) ∈ ((OL /p)/aL) | (xp1 , . . . , xpd ) = (x1 , . . . , xd )A}>c/ps)), so that there is a(where A denotes the image of A in Md ((OL /p)/aL∨
natural map TeL∨ (M) → TL,c
(M).
The following lemma is the analogue of [CL11, Lem. 4.1.4] in our setting, andthe proof is essentially identical (and in fact simpler, since we are only working
modulo p).
7.3.11 Lemma. Assume that we are in the setting of Hypothesis 7.3.1. Then∨∨the morphism TeL∨ (M) → TL,b(M) is injective, and has image TL,pb,b(M).
Proof. We begin with injectivity. Suppose that we have two distinct tuples (x1 , . . . , xd )∨and (y1 , . . . , yd ) ∈ TeL∗ (M) whose images in TL,b
(M) coincide; then if we write zi :=syi − xi , we have v(zi ) > b/p for each i. By (7.3.10), we havee = (z1 , . . . , zd )Tsh .
((x1 + z1 )p − xp1 , . . . , (xd + zd )p − xpd )B

7.3. A RAMIFICATION BOUND

195

Writing (xi + zi )p − xpi = zip + pzi (xip−1 + . . . ), we see that if z := mini v(zi ),then we have eh/ps + z ≥ min(pz, e + z). Since ps > h, we see that necessarilye + z > eh/ps + z. Hence it must be that eh/ps + z ≥ pz, and thus thateh/ps (p − 1) ≥ z. (This final deduction is where we use our assumption thatthe tuples (x1 , . . . , xd ) and (y1 , . . . , yd ) are distinct; this assumption ensures thatz < ∞, so that it is legitimate to subtract if from the two sides of an inequality.)
However this contradicts our assumption that z > b/ps .
We now turn to determining the image. By definition, the morphism TeL∨ (M) →∨∨∨(M). It
TL,b (M) factors through TL,pb,b(M). Choose some (x1 , . . . , xd ) ∈ TL,pbs∨esuffices to construct (ex1 , . . . , xed ) ∈ TL (M) with x
ei lifting the xi modulo a>b/p
by successive approximation.
Begin by choosing arbitrary lifts xi of xi to OL .
>b/ps−1>(pb−eh)/ps>b/ps
We have Ts−h aL
= aL
⊆ aL
, so by our assumption that∨(M), we have
(x1 , . . . , xd ) ∈ TL,pbe − (x1 , . . . , xd ) ∈ (Ts−h a>b/p
Ts−h (xp1 , . . . , xpd )BL

s−1>b/ps d)d ⊆ (aL) ,so in fact we have0se − (x1 , . . . , xd ) ∈ (a≥b /p )dTs−h (xp1 , . . . , xpd )BL(j)≥b0 /psfor some b0 > b. We will construct Cauchy sequences (zi ) ∈ aL
limits zi are such that taking x
ei := xi + zi gives the required lift.
(0)
To this end, we set zi = 0, and define
(j+1)

(z1

(j+1)

, . . . , zd

(j+1)

To see that zi
(j+1)

(z1

(j)

whose

(j)

e − (x1 , . . . , xd ).
) := Ts−h ((x1 + z1 )p , . . . , (xd + zd )p )B≥b0 /ps

∈ aL
(j+1)

, . . . , zd

, we write

(j)
(j)
e
) := Ts−h ((x1 + z1 )p − xp1 , . . . , (xd + zd )p − xpd )Be − (x1 , . . . , xd )).
+(Ts−h (xp , . . . , xp )B1d≥b0 /ps(j). Since (xi +so we need only check that each (xi + zi )p − xpi ∈ Tsh aL
(j) p
(j)
(j) p
(j)
≥b0 /pspzi ) − xi ∈ (pzi , (zi ) ), and zi ∈ aL
, this holds.
(j)
It remains to show that each (zi ) is Cauchy. Set  = min(e − eh/ps , ((p −(j+1)1)b0 −eh)/ps ), so that in particular  > 0. It is enough to show that mini v(zi
−
(j)
(j+1)
(j)
zi ) ≥ mini v(zi− zi ) + . To this end, we have
(j+1)

(z1

(j+1)

(j)

− z1 , . . . , zd
(j)

(j)

− zd )(j−1) p= Ts−h (x1 + z1 )p − (x1 + z1

(j)(j−1) p  e) , . . . , (xd + zd )p − (xd + zd
(j)

) B,

(j−1) pso that it is enough to show that for each i, we have (xi +zi )p −(xi +zi
(j)
(j−1)
(j)
(j−1) ≥eh/ps +
(zi − zi
)Tsh a≥)aL.
L = (zi − zi

) ∈

196

CHAPTER 7. THE RANK ONE CASE
For each i, we write(j)(j−1) p(xi + zi )p − (xi + zi
) =

p  
X
p p−k (j) k(j−1) k((zi ) − (zi
) ),
x
k i

k=1


so it is enough to note that for 1 ≤ k ≤ p − 1 we have v( kp ) = e ≥ eh/ps + ,
while for k = p, we write(j)(j−1) p(zi )p − (zi

(j)

(j−1)

− zi

) = (zi(j)(j)(j−1) p−1)((zi )p−1 + · · · + (zi)),≥b0 /ps(j−1)

and note that since zi , zi
∈ aL
, the second term here is contained in≥(p−1)b0 /ps
0saL, and we have (p − 1)b /p ≥ eh/ps + .
The following corollary is the analogue of [CL11, Thm. 4.1.1] in our setting.
7.3.12 Corollary. Assume that we are in the setting of Hypothesis 7.3.1, andlet L be an algebraic extension of Kcyc,s inside K, so that GL acts natu∨
rally on T ∨ (M) by Corollary 7.3.6. Then the natural injection TL,pb,b(M) ⊆∨TK,pb,b (M) is an isomorphism if and only if the action of GL on T ∨ (M) is
trivial.
∨(M) ⊆Proof. By Lemma 7.3.11 (applied to each of L and K), the inclusion TL,pb,b∨
TK,pb,b (M) is an isomorphism if and only if the same is true of the inclusionTe∨ (M) ⊆ Te∨ (M). Directly from the definition of these latter sets, we see that
L

K∨
TeL∨ (M) = (TeK(M))GL , so this containment holds if and only if GL acts trivially∨on Te (M), or equivalently, if and only if GL acts trivially on T ∨(M).
KK,pb,b∼∨
Now Lemmas 7.3.2 and 7.3.5 show that T ∨ (M) −→ TK,pb,b(M), and by its∨
construction, the GKcyc,s -action on T (M) arises, via this isomorphism, from∨the GKcyc,s -action on TK,pb,b(M). The lemma follows.

7.3.13 Corollary. Assume that we are in the setting of Hypothesis 7.3.1, andlet ρ denote the representation T (M )∨ = T ∨ (M) of GKcyc,s given by Corollary 7.3.6. Let L be an algebraic extension of Kcyc,s inside K. If there exists an>b/ps−1

OKcyc,s -algebra homomorphism OK ker ρ → OL /aL, then K

ker ρ

⊆ L.

Proof. Suppose that we have an OKcyc,s -algebra homomorphism η : OK ker ρ →s−1

OL /a>b/p. We claim that η induces an injective homomorphism>b/ps−1

OK ker ρ /a

K

ker ρ

>b/ps−1→ OL /aL.

To begin with, let F/Kcyc,s be the maximal unramified subextension of
K

ker ρ

/Kcyc,s , and let kF denote its residue field (which is then also the residue

field of K

ker ρ). If kL denotes the residue field of L, then the morphism η induces

7.3. A RAMIFICATION BOUND

197

an embedding kF ,→ kL , which in turn lifts to an embedding W (kF ) ,→ OL .
This embedding then induces an embedding
OF = OKcyc,s ⊗W (k) W (kL ) ,→ OL .

(7.3.14)

In particular, we find that OL contains OF , or, equivalently, that L contains F ,although the embedding (7.3.14) may not be the inclusion, but rather the composite of this inclusion with an element of Gal(F/Kcyc,s ). If we regard OK ρ
(resp. OL ) as a OF -algebra via the inclusion (resp. via the embedding (7.3.14)),then we see (from the very construction of (7.3.14)) that η is a morphism of OF algebras.
>b/ps−1
Now, let π be a uniformizer of OK ker ρ , with image η(π) ∈ OL /aL; weclaim that η(π) is non-zero, so that its valuation is well-defined, and that thisvaluation is equal to that of π. It follows immediately from this that the kernel>b/ps−1of η is generated by a ker ρ , as required. To verify the claim, let Eπ be theK(Eisenstein) minimal polynomial of π over OF . Then η(π) is a root of η(Eπ ), andthe non-leading coefficients of η(Eπ ) all have valuation at least v(Ts ) = e/ps ,with equality holding for the constant coefficient. Since e/ps ≤ eh/ps ≤ (p −1)b/ps < pb/ps , it follows that v(η(π)) = v(π).
We then have a composite of injections∨∨∨
TK(M) ⊆ TL,pb,b(M) ⊆ TK,pb,b
(M)
ker ρ
,pb,b

(the second inclusion being induced by the natural map OL → OK ). By Corollary 7.3.12 the composite is an isomorphism (being an injection of isomorphic finite sets), so all of these inclusions are isomorphisms. Applying Corollary 7.3.12
ker ρ
again, we conclude that K⊆ L, as required.
Recall that for each finite extension L of Qp and each real number v, we havethe upper numbered ramification subgroups GvL , as defined in [Ser79, Chapter(v+1)
IV]; these are the groups denoted GL
in [Fon85] (see [Fon85, Rem. 1.2]). We
can now prove our first bound on the ramification of T (M ), from which our
subsequent bounds will be deduced.
7.3.15 Corollary. Assume that we are in the setting of Hypothesis 7.3.1. If v >pb − 1, then GvKcyc,s acts trivially on T (M ).
Proof. Let ρ denote T (M )∨ . Recall that for a real number m, the extension
K ker ρ /Kcyc,s satisfies (Pm ) if, for any subfield L of K containing Kcyc,s , thesexistence of an OKcyc,s -algebra homomorphism OK ker ρ → OL /a≥m/p implies
that K ker ρ ⊆ L. (This property was introduced by Fontaine, and is discussed
extensively in [Yos10], to which we refer the reader for further explanations andsreferences. Note that the denominator ps appears in a≥m/p because we always
compute valuations with respect to K, and Kcyc,s is a totally ramified extensionof K of degree ps .)Corollary 7.3.13 shows that inf{m | (Pm ) holds for K ker ρ /Kcyc,s } ≤ pb. The
present corollary then follows from this, together with [Yos10, Thm. 1.1].

198

CHAPTER 7. THE RANK ONE CASE

We now return to the setting of a general K (i.e. we do not assume that K
is basic). Let A be a finite local Artinian O/$a -algebra for some a ≥ 1, andlet M be a rank one projective étale ϕ-module over AK,A , which we assume
to be of (K basic , T )-height at most h in the sense of Definition 7.2.14; that is,
we regard M as a rank [K : K basic ] étale ϕ-module M 0 over AK basic ,A , and weassume there is a projective ϕ-module M over A+of T -height at most h
K basic ,Asuch that M 0 = M[1/T ].
0
We have the GKcyc
basic -representation TA (M ), which we may compute as
TA (M 0 ) = (W (C[ ) ⊗AK basic M 0 )ϕ=1 = (W (C[ ) ⊗AK basic M )ϕ=1 .
It follows that there is a GKcyc -equivariant surjection
TA (M 0 ) = (W (C[ )⊗AK basic M )ϕ=1  (W (C[ )⊗AK M )ϕ=1 = TA (M ). (7.3.16)
GK basic

In fact (2.7.4) shows that TA (M 0 ) can be identified with IndGKcyc
TA (M ), andcyc
the morphism (7.3.16) then becomes the natural GKcyc -equivariant surjection.
7.3.17 Proposition. Suppose that A is an Artinian O/$a -algebra for some a ≥1, and M is a rank 1 projective étale ϕ-module over AK,A of (K basic , T )-heightat most h.
abThen there is an open subgroup I a,h of IK
depending only on K, a and hcyc(and not on A or M ) such that the action of I a,h on TA (M ) is trivial.
Proof. We choose integers b, s, N satisfying the following conditions:• b ≥ eh/(p − 1).
• N > pb/e is such that N ≥ N (i, h) for each i ∈ [1, a], where the N (i, h)
are as in Corollary 3.5.2, with K there being replaced by K basic .
• s is such that ps−1 > b/e, and such that s ≥ s(i, h, N ) for each i ∈ [1, a],where the s(i, h, N ) are as in Corollary 3.5.2, with K there being replaced
by K basic .
Note in particular that Hypothesis 7.3.1 holds, with K in the hypothesis replaced by K basic . Note also that our assumptions imply that the hypotheses of Corollary 3.5.2 hold (with the field K there taken to be K basic ), so0that the action of GKcyc
basic on TA (M ) has a canonical extension to an action
0of GKcyc,sbasic . This canonical action on TA (M ) induces the canonical action oni0i0$ TA (M ) = TA ($ M ) for each i ∈ [0, a], and hence on each of the quotients$i−1 TA (M 0 )/$i TA (M 0 ) = TA/$ ($i−1 M 0 /$i M 0 ), for i ∈ [1, a].
To ease notation, write Mi := $i−1 M/$i M , for 1 ≤ i ≤ a, and let Mi0
denote Mi regarded as an étale ϕ-module over AK basic ,A/$ . By Corollary 7.3.15,
if v is sufficiently large (depending only on K, a and h, and our subsequent
choices of b, N and s, and in particular not on A or M ), then the canonicalaction of GvK basic on each TA/$ (Mi0 ) is trivial. It follows that there is some mcyc,s

7.3. A RAMIFICATION BOUND

199

depending only on a (and the degree [K : K basic ]) such that for any g ∈ GvK basic ,the action of g p

m

cyc,s

on TA (M 0 ) is trivial.
ψK

cyc,s /K basic(v)

It follows from the definitions that GvK basic ∩ GKcyc,s = GKcyc,s cyc,s ,cyc,swhere ψKcyc,s /Kcyc,sbasic (v),basic is as in [Ser79, Chapter IV]), so if we take any w ≥ ψK
cyc,s /Kcyc,s
w
pm
pm
w
then (GKcyc,s )(:= {g | g ∈ GKcyc,s }; note that this is just a subset, not a
subgroup, of GKcyc,s ) acts trivially on TA (M 0 ).
pm
The product GKcyc (Gwis a priori a subset of GK , but we claim that
Kcyc,s )it is in fact an open subgroup of GK , and thus (since it contains GKcyc ) is equal
to GKcyc,s0 for some s0 ≥ s which depends only on s and m (and so ultimately
only on K, a and h).
To prove the claim, we briefly recall the relationship between upper numbered ramification groups and the filtration of the unit group: If L is a finite×
extension of Qp , then the Artin map identifies OL
with ILab , and for each in×v
teger w ≥ 1, identifies the subgroup 1 + (mL ) of OLwith the image of GvLabin GL (see [Ser79, Chapter XV, §2, Thm. 2]). It then identifies the subgroup
m
m
is an
(1 + (mL )v )p with the image of (GvL )p in Gab
L . This former subgroupm×
open subgroup of OL, and so we find that for each v, the image of (GvL )p in ILabis open.
Now the extension Kcyc /Kcyc,s is an infinitely ramified ΓKcyc,s ∼= Zp -extension,and so the inertia group I(Kcyc /K) is finite index (equivalently, open) in ΓKcyc,s .
ab, the discussion of the preceding paragraph
Since I(Kcyc /K) is a quotient of IK
mw
then shows that the image of (GKcyc,s )p in ΓKcyc,s is also an open subgroup.
Since ΓKcyc,s is itself open in GK , this implies the claim.
pm
Since (7.3.16) is GKcyc -equivariant, and since (Gwacts trivially on TA (M 0 ),Kcyc,s )we see that the kernel of (7.3.16) is in fact GKcyc,s0 -invariant, and thus that theGKcyc -action on TA (M ) extends to a GKcyc,s0 -action, which is abelian (since thepmGKcyc -action is abelian, as TA (M ) has rank one over A, while (Gwacts
Kcyc,s )trivially).
Now another application of [Ser79, Chapter IV]) shows that Gw
Kcyc,s ∩ GKcyc,s0 =00GwKcyc,s0 for some w depending only on w, and our preceding discussion of0mp
abelian ramification theory shows that (Gwhas open image in Gab
Kcyc,s0 )Kcyc,s0 .
0
If we let I denote the preimage of this open image (i.e.
0mp
I 0 = [GKcyc,s0 , GKcyc,s0 ](Gw,Kcyc,s0 )where the overline denotes closure in GKcyc,s0 ), then I 0 is an open subgroup of
GKcyc,s0 which acts trivially on TA (M ). Thus I 00 := I 0 ∩ GKcyc also acts triviallyabon TA (M ). Finally, we let I a,h denote the image of I 00 in IK. (As the notationa,hindicates, I
depends only on a, h, and K, but not on A or M , since this istrue of each of the groups I 0 and I 00 from which it is constructed.)
The same argument allows us to prove the following variant of Proposition 7.3.17.

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CHAPTER 7. THE RANK ONE CASE

7.3.18 Proposition. Suppose that A is an Artinian O/$a -algebra for some a ≥1, and M is a rank 1 projective étale (ϕ, Γ)-module over AK,A . Assume furthermore that M has (K basic , T )-height at most h, and that s is some sufficiently
large integer with the property that the action of ΓK basic ,disc = ΓK,disc on M extends the canonical action of ΓKsbasic ,disc given by Corollary 3.5.2. (In particular,
we are assuming that s has been chosen to be large enough that this canonicalaction is defined.)a,h,sab
Then there is an open subgroup IK
of IK
depending only on K, a, h and sa,h,s(and not on A or M ) such that the action of IK
on TA (M ) is trivial.
Proof. The proof is very similar to that of Proposition 7.3.17, and we content
ourselves with indicating the key modifications. The main difference between thesetting of that proposition and the present one is that, in our present setting, themorphism (7.3.16) is a GK -equivariant morphism from a GK basic -representation
to a GK -representation. Furthermore, by assumption, the GKcyc,sbasic -action on its
source is the canonical action.
We now follow the argument of Proposition 7.3.17, and find that, after possibly enlarging s in a manner depending only on a, h, and K, some ramification
0
group Gw
Kcyc,s acts trivially on TA (M ) (the index w also depending only on a,h, and s). In the present argument, there is no need to pass to an auxiliary
subgroup GKcyc,s0 , since the morphism (7.3.16) is in particular already a GKcyc,s equivariant morphism of GKcyc,s -representations; and the action of GKcyc,s on
T (M ) is already abelian. The argument then proceeds in the same manner, toproduce an open subgroup I 0 of the inertia subgroup of GKcyc,s (and thus of IK )which acts trivially on TA (M ). (And, by construction, the group I 0 depends
a,h,s
only on a, h, and s, but not on A or M .) We then let IK
denote the image
0abof I in IK .

Chapter 8

A geometric Breuil–Mézard
conjecture
In this chapter we explain a (for the most part conjectural) relationship between
crys,λ,τthe geometry of our potentially semistable and crystalline moduli stacks Xd
ss,λ,τ
and Xd, and the representation theory of GLn (k). Throughout the chapter
we fix a sufficiently large coefficient field E with ring of integers O and residue
field F, and we largely omit it from our notation.
The starting point for this proposed relationship is the Breuil–Mézard conjecture [BM02], which is a conjectural formula for the Hilbert–Samuel multipliccrys,λ,τ
ss,λ,τ
ities of the special fibres of the lifting rings Rρ
and Rρ
, and which has
important applications to proving modularity lifting theorems via the Taylor–Wiles method [Kis09a]. The conjecture was geometrized in [BM14] and [EG14],
by refining the conjectural formula for the Hilbert–Samuel multiplicity to a
conjectural formula for the underlying cycle of the special fibres of the liftingcrys,λ,τ
ss,λ,τ
rings Rρand Rρ
, considered as (equidimensional) closed subschemes ofthe special fibre of the universal lifting ring Rρ . We refer to this generalization
as the “refined Breuil–Mézard conjecture”, and to the original conjecture (orrather, its generalizations to GLn ) as the “numerical Breuil–Mézard conjecture”.
Our aim in this chapter, then, is to “globalize” the conjectures of [EG14]
crys,λ,τcrys,λ,τby formulating versions of them for the stacks Xd
and Xd; the conjectures of [EG14] can be recovered from these conjectures by passing to versal
rings at finite type points. (We caution the reader that there is another kindof “globalization” that could be considered, namely realizing local Galois representations as the restrictions to decomposition groups of global representations,
as used in the proofs of some of the results of [EG14] and [GK14]. Other than in
the proof of Theorem 8.6.1 below, we don’t consider this kind of globalizationin the present work.) This generalization, which we will call the “geometric
Breuil–Mézard conjecture”, seems to us to be the natural setting in which toconsider the Breuil–Mézard conjecture, and we will use our description of the
201

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CHAPTER 8. A GEOMETRIC BREUIL–MÉZARD CONJECTURE

irreducible components of Xd,red to deduce new results about both the refinedand numerical versions of the Breuil–Mézard conjecture.

8.1

The qualitative geometric Breuil–Mézard conjecture

If λ is a regular Hodge type, and τ is any inertial type, then by Theorems 4.8.12
ss,λ,τ
crys,λ,τ
and Xdare finite type p-adic formal algeand 4.8.14, the stacks Xd
braic stacks over O, which are O-flat and equidimensional of dimension 1 + [K :crys,λ,τ
ss,λ,τ
Qp ]d(d − 1)/2. It follows that their special fibres X dand X dare algebraic stacks over F which are equidimensional of dimension [K : Qp ]d(d − 1)/2.
crys,λ,τcrys,λ,τ

ss,λ,τ

ss,λ,τ

and X dand Xdare closed substacks of Xd , X d
Since Xd
are closed substacks of the special fibre X d , and their irreducible components(with the induced reduced substack structure) are therefore closed substacksof the algebraic stack X d,red (see [Sta, Tag 0DR4] for the theory of irreducible
components of algebraic stacks and their multiplicities). Since X d,red is equidimensional of dimension [K : Qp ]d(d − 1)/2 by Theorem 6.5.1, it follows thatcrys,λ,τ

ss,λ,τ

the irreducible components of X d
and X dare irreducible componentskof X d,red , and are therefore of the form X d,red for some Serre weight k (again
by Theorem 6.5.1).
crys,λ,τ
ss,λ,τ
k
For each k, we write µk (X d
) and µk (X d) for the multiplicity of X d,red
crys,λ,τas a component of X dss,λ,τ

and X dcrys,λ,τ. We write Zcrys,λ,τ = Z(X d)ss,λ,τ
and Zss,λ,τ = Z(X d) for the corresponding cycles, i.e. for the formal sums

Zcrys,λ,τ =X

crys,λ,τµk (X dk

) · X d,(8.1.1)k

Zss,λ,τ =X

ss,λ,τ

µk (X dk

) · X d,(8.1.2)kwhich we regard as elements of the finitely generated free abelian group Z[Xd,red ]kwhose generators are the irreducible components X d .
Now fix some representation ρ : GK → GLd (F), corresponding to a pointx : Spec F → Xd . (More generally, we could consider representations valued
in GLd (F0 ) for some finite extension F0 /F, but in keeping with our attempt
to keep the notation in this chapter as uncluttered as possible by omitting O,
it is convenient to suppose that F has been chosen sufficiently large.) For
each regular Hodge type λ and inertial type τ , we have effective versal morcrys,λ,τ
ss,λ,τ
crys,λ,τ
ss,λ,τ
phisms Spec Rρ
/$ → Xand Spec Rρ
/$ → X(see Corollary 4.8.13), as well as a (non-effective; see Remark 6.5.3) versal morphism
Spf Rρ → Xd (see Proposition 3.6.3).
k

For each k, we may consider the fibre product Spf Rρ ×Xd Xd . This is apriori a closed formal subscheme of Spf Rρ , but since Rρ is a complete local

8.1. THE QUALITATIVE GEOMETRIC BM CONJECTURE

203

ring, it may equally well be regarded as a closed subscheme of Spec Rρ (seeLemma A.3).
k

8.1.3 Lemma. The fibre product Spf Rρ ×Xd Xd , when we regard it as a closed
subscheme of Spec Rρ , is equidimensional of dimension d2 + [K : Qp ]d(d − 1)/2.
Proof. As in the proof of Theorem 6.5.1, we may find a regular Hodge type λcrys,λkksuch that Xd is an irreducible component of (X)red . Thus Spf Rρ ×Xd Xd
is versal to a union of irreducible components of the spectrum of the versal
crys,λcrys,λ)red (those that correspond to the various formal/$)red to (Xring (Rρkbranches of Xd passing through x, in the terminology of [Sta, Tag 0DRA]) andcrys,λcrys,λhence is of the stated dimension, since Rρ
/$, and so also (Rρ/$)red , isequidimensional of this dimension.
We let Ck (ρ) denote the d2 + [K : Qp ]d(d − 1)/2-dimensional cycle inSpec Rρ /$ underlying the fibre product of Lemma 8.1.3.
The following theorem gives a qualitative version of the refined Breuil–Mézard conjecture [EG14, Conj. 4.2.1]. While its statement is purely local,
we do not know how to prove it without making use of the stack Xd .
8.1.4 Theorem. Let ρ : GK → GLd (F) be a continuous representation. Then
there are finitely many cycles of dimension d2 +[K : Qp ]d(d−1)/2 in Spec Rρ /$such that for any regular Hodge type λ and any inertial type τ , each of the special
crys,λ,τ
ss,λ,τ
fibres Spec Rρ
/$ and Spec Rρ
/$ is set-theoretically supported on someunion of these cycles.
crys,λ,τProof. We have Spf Rρ

/$ = Spf Rρ ×Xd X

crys,λ,τ

ss,λ,τ

and Spf Rρ/$ =crys,λ,τ. It follows from (8.1.1) and (8.1.2), together with the
Spf Rρ ×Xd X
definition of Ck (ρ), that we may write the underlying cycles as
crys,λ,τZ(Spec Rρ

/$) =

X

crys,λ,τµk (X d) · Ck (ρ),(8.1.5)kss,λ,τ

Z(Spec Rρ

/$) =

X

ss,λ,τ

µk (X d) · Ck (ρ).

(8.1.6)k(Note that by [Sta, Tag 0DRD], the multiplicities do not change when passing
to versal rings.) The theorem follows immediately (taking our finite set of cycles
to be the Ck (ρ)).
We can regard this theorem as isolating the “refined” part of [EG14, Conj.
4.2.1]; that is, we have taken the original numerical Breuil–Mézard conjecture,formulated a geometric refinement of it, and then removed the numerical partof the conjecture. The numerical part of the conjecture (in the optic of thiscrys,λ,τ
ss,λ,τ
chapter) consists of relating the multiplicities µk (X d
) and µk (X d) tothe representation theory of GLn (k), as we recall in the next section.

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CHAPTER 8. A GEOMETRIC BREUIL–MÉZARD CONJECTURE

8.2

Semistable and crystalline inertial types

We now briefly recall the “inertial local Langlands correspondence” for GLd .
Let recp denote the local Langlands correspondence for Qp -representations of
GLd (K), normalized as in [CEG+ 16, §1.8]; this is a bijection between the isomorphism classes of irreducible smooth Qp -representations of GLd (K) and the isomorphism classes of d-dimensional semisimple Weil–Deligne Qp -representationsof the Weil group WK . We have the following result, which is essentially due toSchneider–Zink [SZ99].
8.2.1 Theorem. Let τ : IK → GLd (Qp ) be an inertial type. Then there
are finite-dimensional smooth irreducible Qp -representations σ crys (τ ) and σ ss (τ )of GLd (OK ) with the properties that if π is an irreducible smooth Qp -representationof GLd (K), then the Qp -vector space HomGLd (OK ) (σ crys (τ ), π) (resp. the Qp vector space HomGLd (OK ) (σ ss (τ ), π)) has dimension at most 1, and is nonzero
precisely if recp (π)|IF ∼= τ , and N = 0 on recp (π) (resp. if recp (π)|IF ∼= τ,and π is generic).
Proof. See [CEG+ 16, Thm. 3.7] for σ crys (τ ), and [Sho18, Thm. 3.7] together
with [Pyv20, Thm. 2.1, Lem. 2.2] for σ ss (τ ).
Note that we do not claim that the representations σ crys (τ ) and σ ss (τ ) are
unique; the possible non-uniqueness of these representations is of no importancefor us.
For each regular Hodge type λ we let Lλ be the corresponding representationof GLd (OK ), defined as follows: For each σ : K ,→ Qp , we write ξσ,i = λσ,i −(d−i), so that ξσ,1 ≥ · · · ≥ ξσ,d . We view each ξσ := (ξσ,1 , . . . , ξσ,d ) as a dominant
weight of the algebraic group GLd (with respect to the upper triangular Borel
subgroup), and we write Mξσ for the algebraic OK -representation of GLd (OK )of highest weight ξσ . Then we define Lλ := ⊗σ Mξσ ⊗OK ,σ O.
For each τ we let σ crys,◦ (τ ), σ ss,◦ (τ ) denote choices of GLd (OK )-stable Olattices in σ crys (τ ), σ ss (τ ) respectively (the precise choices being unimportant).
Then we write σ crys (λ, τ ), (resp. σ ss (λ, τ )) for the semisimplification of the Frepresentation of GLd (k) given by Lλ ⊗O σ crys,◦ (τ )⊗O F (resp. Lλ ⊗O σ ss,◦ (τ )⊗O
F). For each Serre weight k, we write Fk for the corresponding irreducible Frepresentation of GLd (k) (see for example the appendix to [Her09]). Then theress
are unique integers ncrysk (λ, τ ) and nk (λ, τ ) such that⊕nssk (λ,τ )σ crys (λ, τ ) ∼= ⊕k Fk,ss⊕nk (λ,τ )σ ss (λ, τ ) ∼.
= ⊕ k FkOur geometric Breuil–Mézard conjecture is as follows.
8.2.2 Conjecture. There are cycles Zk with the property that
P for each regular
Hodge type λ and each inertial type τ , we have Zcrys,λ,τ = k ncrysk (λ, τ ) · Zk ,P
Zss,λ,τ = k nss
(λ,
τ
)
·
Z
.
k
k

8.3. THE RELATIONSHIP BETWEEN THE BM CONJECTURES

205

For some motivation for the conjecture (coming from the Taylor–Wiles patching method), see for example [EG14, Thm. 5.5.2]. Some evidence for the conjecture is given in the following sections.
The expressions (8.1.1) and (8.1.2) describe Zcrys,λ,τ and Zss,λ,τ as linear
k0

combinations of (the cycles underlying) the irreducible components X d , andthus each cycle Zk (assuming that such cycles exist) will itself be a linear comk0

bination of the various X d . We expect that the cycles Zk are effective, i.e. thatk0each of them is a linear combination of the X d with non-negative (integer)
coefficients. Note that the finitely many (conjectural) cycles Zk are completely
determined by the infinitely many equations in Conjecture 8.2.2 (see [EG14,
Lem. 4.1.1, Rem. 4.1.7(1)]).
While the original motivation for Conjecture 8.2.2 was to understand potentially semistable deformation rings in terms of the representation theory of GLd ,it can also be thought of as giving a geometric interpretation of the multiplicities ncrysk (λ, τ ).

8.3

The relationship between the numerical, refined and geometric Breuil–Mézard conjectures

We now explain the relationship between Conjecture 8.2.2 and the conjecturesof [EG14]. Suppose firstly that Conjecture 8.2.2 holds, and fix some ρ : GK →GLd (F) corresponding to a point x : Spec F → Xd . For each k, we set
Zk (ρ) := Spf Rρ ×Xd Zk ,which we regard as a d2 + [K : Qp ]d(d − 1)/2-dimensional cycle in Spec Rρ /$.
(As noted above, the Zk will be linear combinations of the cycles underlying the
k0

various X d , and so the the Zk (ρ) will be linear combinations of the various cycles
Ck0 (ρ); thus Lemma 8.1.3 shows that they are indeed d2 + [K : Qp ]d(d − 1)/2dimensional cycles.)
8.3.1 Remark. Equivalently, we can define Zk (ρ) to be the image of Zk under
the natural map from Z[Xd,red ] to the group Zd2 +[K:Qp ]d(d−1)/2 (Spec Rρ /$) of
d2 +[K : Qp ]d(d−1)/2-dimensional cycles in Spec Rρ /$ which is defined as follows: We let Rρalg be the quotient of Rρ /$ which is a versal ring to Xd,red at x,
so that by [Sta, Tag 0DRB,Tag 0DRD] we have a multiplicity-preserving surjection from the set of irreducible components of Spec Rρalg to the set of irreducible
components of Xd,red containing x; we then send any irreducible componentof Xd,red not containing x to zero, and send each irreducible component containing x to the sum of the corresponding irreducible components of Spec Rρalg
in its preimage.

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CHAPTER 8. A GEOMETRIC BREUIL–MÉZARD CONJECTURE

Exactly as in the proof of Theorem 8.1.4, it follows that for each regulartype λ and inertial type τ , we havecrys,λ,τZ(Spec Rρ

/$) =

X

ncrys
k (λ, τ ) · Zk (ρ),(8.3.2)nss
k (λ, τ ) · Zk (ρ).

(8.3.3)k

ss,λ,τ

Z(Spec Rρ

/$) =

X
k

The first of these statements is [EG14, Conj. 4.2.1] (with the cycles Ca there
being our cycles Zk (ρ)), and the second is the corresponding statement forpotentially semistable lifting rings.
We now relate this conjecture to the numerical version of the Breuil–Mézard
conjecture. We have a homomorphism Zd2 +[K:Qp ]d(d−1)/2 (Spec Rρ /$) → Z
defined by sending each cycle to its Hilbert–Samuel multiplicity in the senseof [EG14, §2.1]. Let µk (ρ) denote the Hilbert–Samuel multiplicity of the cycle
crys,λ,τ
ss,λ,τ
/$), e(Spec Rρ
/$) for the Hilbert–SamuelZk (ρ), and write e(Spec Rρ
multiplicities of the indicated rings. Then it follows from (8.3.2) and (8.3.3)that we have
X cryscrys,λ,τe(Spec Rρ/$) =nk (λ, τ )µk (ρ),(8.3.4)k

ss,λ,τ

e(Spec Rρ

/$) =

X

nss
k (λ, τ )µk (ρ).

(8.3.5)kThen (8.3.4) is [EG14, Conj. 4.1.6], and (8.3.5) is the corresponding semistableversion.
Suppose now that for each ρ : GK → GLd (F), all regular Hodge types λ andall inertial types τ we have (8.3.4) and (8.3.5) for some integers µk (ρ), but do notassume Conjecture 8.2.2 (so in particular we do not presuppose any geometricinterpretation for the integers µk (ρ)).
For each k we choose a point xk : Spec F → X d,red which is contained in Xkk0

and not in any X for k 0 6= k. We furthermore demand that xk is a smooth
point of X d,red . (Since X d,red is reduced and of finite type over F, there is akdense set of points of X satisfying these conditions.) Write ρk : GK → GLd (F)for the representation corresponding to Xk , and set
Zk :=

X

k0

µk (ρk0 ) · X .

(8.3.6)k0

Then for each regular Hodge type λ and inertial type τ , it follows from (8.3.4)

8.4. THE WEIGHT PART OF SERRE’S CONJECTURE

207

that
X

ncrys
k (λ, τ ) · Zk =

X

=

X

k

crys,λ,τe(Rρk/$) · Xkkcrys,λ,τµk (X d)·Xkk

= Zcrys,λ,τ ,where we used that xk is a smooth point of X d,red and is only contained in X
crys,λ,τkcrys,λ,τto conclude that µk (X d
) = e(Rρk/$). Similarly we have Zss,λ,τ =P ss
k nk (λ, τ ) · Zk , and we conclude that the geometric Breuil–Mézard conjecture(Conjecture 8.2.2) is equivalent to the numerical conjecture.
8.3.7 Remark. The argument that we just made shows that the geometric conjecture follows from knowing the numerical conjecture for sufficiently generic ρ(indeed, it is enough to check it for a single sufficiently generic ρ for each irreducible component of Xd,red ), while in turn the geometric conjecture implies
the numerical conjecture for all ρ.

8.4

The weight part of Serre’s conjecture

We now briefly explain the relationship between Conjecture 8.2.2 and the weightpart of Serre’s conjecture. For more details, see [GHS18] (particularly Section 6).
We expect that the cycles Zk will be effective, in the sense that they arekcombinations of the Xd with non-negative coefficients. This expectation is borneout in all known examples (see the following sections), and in any case would be
a consequence of standard conjectures about the Taylor–Wiles method. Indeed,
the local cycles Zk (ρ) of Section 8.3 are conjecturally the supports of certain“patched modules”, and in particular are effective; see [EG14, Thm. 5.5.2]. The
effectivity of the Zk (ρ) would immediately imply the effectivity of the Zk .
The “weight part of Serre’s conjecture” is perhaps more of a conjectural conjecture than an actual conjecture: it should assign to each ρ : GK → GLd (F)a set W (ρ) of Serre weights, with the property that if ρ is the restriction to a
decomposition group of a suitable global representation (for example, an irreducible representation coming from an automorphic form on a unitary group),
then k ∈ W (ρ) if and only if k is a weight for the global representation (forexample, in the sense that the global representation corresponds to some mod p
cohomology class for a coefficient system corresponding to Fk ).
Many conjectural definitions of the sets W (ρ) have been proposed. Following [GK14], one definition is to assume the Breuil–Mézard conjecture, for example in the form (8.3.2), and define W (ρ) to be the set of k for which µk (ρ) > 0.
While this is less explicit than other definitions, it has the merit that it would
follow from standard conjectures about modularity lifting theorems that it gives
the correct set of weights; see [GHS18, §3, 4].

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CHAPTER 8. A GEOMETRIC BREUIL–MÉZARD CONJECTURE

Assume Conjecture 8.2.2, and assume that the cycles Zk are effective. As
explained in Section 8.3, it follows that the numerical Breuil–Mézard holds forevery ρ, with µk (ρ) being the Hilbert–Samuel multiplicity of the cycle Zk (ρ),
which (since Zk (ρ) is effective) is positive if and only if Zk (ρ) is nonzero, i.e.
if and only if Zk is supported at ρ. Thus we can rephrase the Breuil–Mézard
version of the weight part of Serre’s conjecture as saying that W (ρ) is the setof k such that Zk is supported at ρ.
Alternatively, we can rephrase this conjecture in the following way: to each
irreducible component of Xd,red , we assign the set of weights k with the property
that Zk is supported on this component. Then W (ρ) is simply the union of thesets of weights for the irreducible components of Xd,red which contain ρ. As
we explain in Section 8.6, if d = 2 then this description agrees with the otherdefinitions of W (ρ) in the literature, and therefore gives a geometrization of theweight part of Serre’s conjecture.

8.5

The case of GL2 (Qp )The numerical Breuil–Mézard conjecture for K = Qp and d = 2 is completely
known, thanks in large part to Kisin’s paper [Kis09a] (which gave a proof inmany cases by a mixture of local and global techniques), and Paškūnas’ paper [Paš15] which reproved these results by purely local means (the p-adic local
Langlands correspondence), relaxed the hypotheses on ρ, and also proved therefined version of the correspondence. The remaining cases not handled bythese papers are proved in the papers [HT15, San14, Tun21a], culminating in
the proof of the final cases for p = 2 by Tung in [Tun21b].
Consequently, by the discussion of Section 8.3, Conjecture 8.2.2 holds for K =Qp and d = 2. It follows easily from the explicit description of µk (ρ) in thepapers cited above (or alternatively from the description for GL2 (K) in Secktion 8.6 below) that Zk = X2 unless k = (a + p − 1, a) for some a, in which case(a+p−1,a)(a,a)
Z(a+p−1,a) = X2
+ X2
.

8.6

GL2 (K): potentially Barsotti–Tate types

In this section we assume that p is odd, and explain some consequences of theresults of [GK14] (which proved the numerical Breuil–Mézard conjecture for
2-dimensional potentially Barsotti–Tate representations) and [CEGS19] (which
studied moduli stacks of rank 2 Breuil–Kisin modules with tame descent data).
We will take advantage of Remark 8.3.7.
We begin with the following slight extension of one of the main resultsof [GK14]. Let BT denote the minimal regular Hodge type, i.e. we have BTσ,1 =1, BTσ,2 = 0 for all σ : K ,→ Qp .
8.6.1 Theorem. Let K/Qp be a finite extension with p > 2, and let ρ : GK →GL2 (Qp ) be arbitrary. Then the numerical Breuil–Mézard conjecture holds for8.6. GL2 (K): POTENTIALLY BARSOTTI–TATE TYPES

209

potentially crystalline and potentially semistable lifts of ρ of Hodge type BT andarbitrary inertial type τ .
More precisely, there are unique non-negative integers µk (ρ) such that (8.3.4)
and (8.3.5) both hold for λ = BT and τ arbitrary.
Proof. If we remove the potentially semistable case and consider only potentially crystalline representations, the theorem is [GK14, Thm. A], which isproved as [GK14, Cor. 4.5.6]. We now briefly explain how to modify the proofs
in [GK14] to prove the more general result; as writing out a full argument would
be a lengthy exercise, and the arguments are completely unrelated to those ofthis book, we only explain the key points. Examining the proof of [GK14, Cor.
4.5.6], we see that we just need to verify that the assertion of the first sentenceof the proof of [GK14, Thm. 4.5.5] holds in this setting, i.e. that the equivalentconditions of [GK14, Lem. 4.3.9] hold. Exactly as in the proof of [GK14, Cor.
4.4.3], it is enough to show that every irreducible component of a product of
local deformation rings is witnessed by an automorphic representation.
By the usual Khare–Wintenberger argument, it suffices to prove this after
making a solvable base change, and in particular we can suppose that all of theresidual local Galois representations are trivial, the mod p cyclotomic characteris trivial, the trivial mod p representation admits a non-ordinary crystalline lift,
and the inertial types are all trivial. Now, any non-crystalline semistable representation of Hodge type BT is necessarily ordinary (indeed, it follows from a
direct computation of the possible weakly admissible modules that all such representations are unramified twists of an extension of the inverse of the cyclotomic
character by the trivial character), and by results of Kisin and Gee (see [Sno18,
Prop. 4.3.1], [Kis09b, Cor. 2.5.16], and [Gee06, Prop. 2.3]), as p > 2 and ρ istrivial, the semistable ordinary deformation ring in question is a domain, as
are the crystalline ordinary deformation ring, and the crystalline non-ordinary
deformation ring (all in Hodge type BT). It therefore suffices to show that in
the situation of the proof of [GK14, Cor.
`4.4.3], given` any decomposition of theset S of places of F1+ lying over p as Sss Scrys-ord Snon-ord , we can arrange tohave a congruence to an automorphic representation π 00 , having the` properties
that π 00 is unramified at the places lying over a place in Scrys-ord Snon-ord , isan unramified twist of the Steinberg representation at the places lying over aplace in Sss , and is furthermore ordinary (resp. not ordinary) at the places lying
over a place in Scrys-ord (resp. Snon-ord ).
The existence of such a representation if Sss = S follows from the construction of the global representation r used in the proof of numerical Breuil–Mézard
conjecture in [GK14]; more precisely, it follows from [Sno09, Prop. 8.2.1], which
is applied in the proof of [GK14, Thm. A.2] (with the type function in the senseof [Sno09] being C at all places above p). Thus, to establish the general case,it suffices to establish the existence of suitable “level lowering” congruences.
This is easily done by switching to a group which is ramified at the places lying
over Sss , and then making congruences to automorphic representations which
are either unramified and ordinary (in the case of places lying over Scrys-ord ) or
have cuspidal type (in the case of places lying over Snon-ord ), and then applying

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CHAPTER 8. A GEOMETRIC BREUIL–MÉZARD CONJECTURE

the Khare–Wintenberger argument again at the places lying over Snon-ord . (Seee.g. [Kis09b, Lem. 3.5.3] for a similar argument for places away from p.)
We say that a Serre weight k for GL2 is Steinberg if for each σ we havekσ,1 − kσ,2 = p − 1. If k is Steinberg then we define k̃ by k̃σ,1 = k̃σ,2 = kσ,2 .
8.6.2 Theorem. Continue to assume that d = 2 and p > 2. Then Conjecture 8.2.2 holds for λ = BT and τ arbitrary, with the cycles Zk being as follows:
k

k

k̃

if k is not Steinberg, then Zk = X2 , while if k is Steinberg, Zk = X2 + X2 .
Proof. We use the notation of Section 8.3. By Theorem 8.6.1 and the discussionof Section 8.3, we need only show that the cycles Zk in the statement of thetheorem are those determined by (8.3.6). Suppose firstly that k is not Steinberg.
kThen by [CEGS19, Thm. 5.2.2 (2)], X2 has a dense set of finite type points withthe property that their only non-Steinberg Serre weight is k. It follows fromthis and [CEGS19, Thm. 5.2.2 (3)] that if neither k nor k 0 is Steinberg, thenµk (ρk0 ) = δk,k0 . By construction, if k 0 is Steinberg, then ρk0 is a twist of a trèsramifiée extension of the trivial character by the mod p cyclotomic character.
By [CEGS19, Lem. B.5], this implies that µk (ρk0 ) = 0. The cycles Zk are
therefore as claimed if k is non-Steinberg.
It remains to determine the values of µk (ρk0 ) in the case that k is Steinberg. By twisting, we can and do assume that kσ,2 = 0 for all σ. If we apply
Theorem 8.6.1 with τ being the trivial type, and recall that LBT is the trivial
representation and that σ ss (τ ) is the Steinberg type, the reduction of which isprecisely the representation Fk , we find that
Zss,BT,τ = Zk .

(8.6.3)Thus µk (ρk0 ) 6= 0 if and only if ρk0 admits a semistable lift of Hodge type BT.
If this lift is in fact crystalline, then ρk0 has k̃ as a Serre weight, so by anotherapplication of [CEGS19, Thm. 5.2.2 (2)], we see that either k 0 = k̃ or else thatk 0 is also Steinberg. In the former case, ρk0 is (by Remark 5.5.16) an unramified
twist of an extension of inverse of the mod p cyclotomic character by a nontrivial unramified character, so it does not admit a semistable non-crystalline
lift of Hodge type BT (as all such lifts are unramified twists of an extensionof the inverse of the cyclotomic character by the trivial character); so we haveµk (ρk̃ ) = µk̃ (ρk̃ ) = 1.
Finally, we are left with the task of computing µk (ρk0 ) when both k and k 0
are Steinberg. Recall that by twisting, we are assuming that kσ,2 = 0 for all σ.
The weight k 0 is then a twist of k. We have to show that in this case we again
have µk (ρk0 ) = δk,k0 . To see this, first note that since ρk0 is très ramifiée, it does
not admit a crystalline lift of Hodge type BT, so all of its semistable lifts ofHodge type BT are given by unramified twists of extensions of the inverse of
the cyclotomic character by the trivial character. Furthermore, the reductionof any lattice in such an extension is an unramified twists of an extension of theinverse of the mod $ cyclotomic character by the trivial character, and hence8.6. GL2 (K): POTENTIALLY BARSOTTI–TATE TYPES

211

cannot equal ρk0 unless k 0 equals k (rather than being a non-trivial twist of it).
If we take into account (8.6.3), we find that indeed µk (ρk0 ) = 0 when k 0 6= k.
Finally, any lift of ρk which is an unramified twist of an extension of theinverse of the cyclotomic character by the trivial character is automaticallyss,BT,τ
is precisely the ordinary (framed)semistable of Hodge type BT, so that Rρkdeformation ring parameterizing such lifts of ρk . A standard Galois cohomology
calculation (that we leave to the reader) shows that this ring is formally smooth,ss,BT,τ
/$ is also formally smooth. It follows that µk (ρk ) = 1,
and thus that Rρk
as claimed.
The following lemma makes precise which Galois representations occur on
Steinberg components. Note that the twist in the statement of the lemma isdetermined by the values of kσ,2 .
k

8.6.4 Lemma. If k is Steinberg, then the Fp -points of X2 are twists of an extension of the inverse of the mod p cyclotomic character by the trivial character.
Proof. Twisting, we may assume that kσ,2 = 0 for all σ. Let ρ : GK → GL2 (F)kcorrespond to a closed point of X2 . We claim that ρ is an unramified twist
BT,Stof an extension of  by 1. To see this, let Rρ
denote the Zp -flat quotient
BT,ss

of Rρ

determined by the irreducible components of the generic fibre which
BT,crysare not components of Rρ

.
BT,St

It follows from Theorem 8.6.2 that the cycle of Spec Rρ
/$ is nonzero,so that in particular ρ must admit a semistable non-crystalline lift of Hodge
type BT. As in the proof of Theorem 8.6.1, any such representation is anunramified twist of an extension of −1 by the trivial character, so we are done.
(We could presumably also phrase this argument on the level of the crystalline
and semistable moduli stacks, but have chosen to present it in the more familiar
setting of deformation rings.)
8.6.5 Remark. The paper [CEGS19] constructs and studies moduli stacks of
two-dimensional representations of GKπ[ ,∞ (for a fixed choice of π [ ) which aretamely potentially of height at most 1. Since restriction from Barsotti–Taterepresentations of GK to representations of GKπ[ ,∞ of height at most 1 is anequivalence (by the results of [Kis09b]), these stacks can be interpreted as stacks
of GK -representations (and indeed are interpreted as such in [CEGS19]). It is
presumably straightforward (using that all the stacks under consideration are offinite type, and are Zp -flat and reduced, in order to reduce to a comparison ofpoints over finite extensions of Zp ) to identify them with the stacks consideredin this book (for λ = BT and τ a tame inertial type). (We were able to apply the
results of [CEGS19] in the proof of Theorem 8.6.2 without doing this because
we only needed to use them on the level of versal rings, which are given by
universal Galois lifting rings for both our stacks and those of [CEGS19].) It may
well be the case that the stacks of Kisin modules considered in [CEGS19] canbe identified with certain of the moduli stacks of Breuil–Kisin–Fargues modules

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CHAPTER 8. A GEOMETRIC BREUIL–MÉZARD CONJECTURE

that we defined in Section 4.5. Since we do not need to know this, we leave it
as an exercise for the interested reader.

8.6.6

A lower boundThe patching arguments of [GK14] easily imply a general inequality, as we now
record.
8.6.7 Proposition. Assume that p > 2. Let d = 2, and let the Zk be as
in the statement of Theorem 8.6.2. Then forregular Hodge type λ andP eacheach inertial type τ , we have Zcrys,λ,τ ≥ncrys(λ, τ ) · Zk , and Zss,λ,τ ≥kk
P ss
k nk (λ, τ ) · Zk .
8.6.8 Remark. The meaning of the inequalities in the statement of Proposition 8.6.7 is the obvious one: each side is a linear combination of irreducible
components, and the assertion is that for each irreducible component, the multiplicity on the left hand side is at least the multiplicity on the right hand side.
Proof of Proposition 8.6.7. As in Section 8.3, it is enough to show that foreach ρ, we have
X cryscrys,λ,τnk (λ, τ )µk (ρ),
e(Spec Rρ/$) ≥kss,λ,τ

e(Spec Rρ

/$) ≥

X

nss
k (λ, τ )µk (ρ),

kwhere the µk (ρ) are the uniquely determined integers from [GK14]. (In fact,
it is enough to show these inequalities when ρ = ρk for some k, but assuming
this does not simplify our arguments.) Since [GK14] considers only potentiallycrys,λ,τcrystalline representations, we only give the proof for Rρ; the proof in thepotentially semistable case is essentially identical, and we leave it to the reader.
We now examine the proof of [GK14, Thm. 4.5.5], setting r there to be
our ρ. As we’ve done throughout this book, we write k rather than σ for Serreweights. If we ignore the assumption that every potentially crystalline lift ofHodge type λ and inertial type τ is potentially diagonalizable, then we do notknow that the equivalent conditions of [GK14, Lem. 4.3.9] hold, but examining
the proof of [GK14, Lem. 4.3.9], we do know that the statement of part (4) ofloc. cit. can be replaced with an inequality (with equality holding if and only
if M∞ is a faithful R∞ -module).
Returning to the proof of [GK14, Thm. 4.5.5], we note that the definitionof µk (ρ) is such that the quantity µ0σgl (r) is simply the product of the corresponding µk (ρ). In particular, if we take λv = λ and τv = τ for each v, then(bearing in mind the discussion of the previous paragraph), the main displayed
equality becomes an inequality
NX
crys,λ,τ ,e(Spec Rρ
/$)N ≥ ncrys
k (λ, τ )µk (ρ)
k8.7. BRIEF REMARKS ON GLd , d > 2

213

where there are N places of F + lying over p. Since each side is non-negative,
the result follows.
8.6.9 Remark. The argument used to prove Proposition 8.6.7 is well known
to the experts, and goes back to [Kis09a, Lem. 2.2.11]. Despite the relatively
formal nature of the argument, it does not seem to be easy to prove an analogous
statement for GLd with d > 2; the difficulty is in the step where we used [GK14,
Thm. 4.5.5] to replace a global multiplicity with a product of local multiplicities.
Without this argument (which crucially relies on the modularity lifting theoremsof [Kis09b]) one only obtains inequalities involving cycles in products of copiesof Xd .

8.7

Brief remarks on GLd , d > 2

We expect the situation for GLd , d > 2, to be considerably more complicatedthan that for GL2 . Experience to date suggests that the weight part of Serre’s
conjecture in high dimension is consistently more complicated than is anticipated, and so it seems unwise to engage in much speculation. In particular, theresults of [LHLM20] show that even for generic weights k, we should not expect
the cycles Zk to have as simple a form as those for GL2 .
k
In the light of Lemma 5.5.4, it seems reasonable to expect Xd to contributeto Zk , and it also seems reasonable to expect contributions from the “shifted”
weights, as in Definition 5.5.2 (see also [GHS18, §7.4]). The comparative weakness of the automorphy lifting theorems available to us in dimension greaterthan 2 prevents us from saying much more than this, although we refer thereader to [GHS18, §§3,4,6] for a discussion of the conjectural general relationship between the numerical Breuil–Mézard conjecture, the weight part of Serre’s
conjecture, automorphy lifting theorems, and the stacks Xd .

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CHAPTER 8. A GEOMETRIC BREUIL–MÉZARD CONJECTURE

Appendix A

Formal algebraic stacks
The theory of formal algebraic stacks is developed in [Eme]. In this appendix webriefly summarise the parts of this theory that are used in the body of the book,and also introduce some additional terminology and establish some additional
results which we will require.

Algebraic stacks
We follow the terminology of [Sta]; in particular, we write “algebraic stack”
rather than “Artin stack”. More precisely, an algebraic stack is a stack ingroupoids in the fppf topology, whose diagonal is representable by algebraic
spaces, which admits a smooth surjection from a scheme. See [Sta, Tag 026N]
for a discussion of how this definition relates to others in the literature. If S isa scheme, then by “a stack over S” we mean a stack fibred in groupoids overthe big fppf site of S.
We say that a morphism X → Y of stacks over S is representable by algebraic
stacks if for any morphism of stacks Z → Y whose source is an algebraic stack,the fibre product X ×Y Z is again an algebraic stack. (In the Stacks Project, theterminology algebraic is used instead [Sta, Tag 06CF].) Note that a morphismfrom a sheaf to a stack is representable by algebraic stacks if and only if it
is representable by algebraic spaces (this is easily verified, or see e.g. [Eme,Lem. 3.5]).
Following [Sta, Tag 03YK, Tag 04XB], we can define properties of morphisms
representable by algebraic stacks in the following way.
A.1 Definition. If P is a property of morphisms of algebraic stacks which isfppf local on the target, and preserved by arbitrary base-change, then we saythat a morphism f : X → Y of stacks which is representable by algebraic stacks
has property P if and only if for every algebraic stack Z and morphism Z → Y,
the base-changed morphism of algebraic stacks Z ×Y X → Z has property P .
When applying Definition A.1, it suffices to consider the case when Z is
actually a scheme, or even an affine scheme.
215

216

APPENDIX A. FORMAL ALGEBRAIC STACKS

Some properties P to which we can apply Definition A.1 are being locallyof finite type, locally of finite presentation, and smooth. The following lemma
provides alternative descriptions of the latter two properties.
A.2 Lemma. Let f : X → Y be a morphism of stacks which is representableby algebraic stacks.
1. The morphism f is locally of finite presentation if and only if it is limit
preserving on objects, in the sense of [Sta, Tag 06CT].
2. The morphism f is smooth if and only if it is locally of finite presentationand formally smooth, in the sense that it satisfies the usual infinitesimal
lifting property: for every affine Y-scheme T , and every closed subscheme
T0 ,→ T defined by a nilpotent ideal sheaf, the functor HomY (T, X ) →HomY (T0 , X ) is essentially surjective.
Proof. We first recall that a morphism of algebraic stacks is locally of finite
presentation if and only if it is limit preserving on objects [EG21, Lem. 2.3.16].
Suppose now that f is locally of finite presentation, that T is an affine∼
scheme, written as a projective limit of affine schemes T −→ lim Ti , and suppose←−
given a commutative diagram
T

/ Ti


X


/Y

for some value of i. We must show that we can factor the left-hand verticalarrow through Ti0 , for some i0 ≥ i, so that the evident resulting diagram again
commutes. The original diagram induces a morphism T → Ti ×Y X , and obtaining a factorization of the desired type amounts to obtaining a factorization ofthis latter morphism through some Ti0 . In other words, we must show that themorphism Ti ×Y X → Ti is limit preserving on objects. But this is a morphismof algebraic stacks which is locally of finite presentation, and so it is indeed limit
preserving on objects.
Conversely, suppose that f is limit preserving on objects. We must showthat for any morphism T → Y, the base-changed morphism T ×Y X → T is
locally of finite presentation. However, this base-change is again limit preserving
on objects [Sta, Tag 06CV], and since it is a morphism of algebraic stacks, it isindeed locally of finite presentation.
We now turn to (2). By Definition A.1 and [Ryd11, Cor. B.9] (that is, by
the result at hand in the case that Y is itself an algebraic stack), it is enoughto show that f satisfies the infinitesimal lifting property if and only if the basechanged morphism Z ×Y X → Z satisfies the infinitesimal lifting property for
all morphisms Z → Y whose source is a scheme.
Assume first that f satisfies the infinitesimal lifting property, and let T0 → T
be a nilpotent closed immersion whose target is an affine Z-scheme. Givena morphism T0 → Z ×Y X , the infinitesimal lifting property lets us lift the217
composite T0 → Z ×Y X → X to a morphism T → X , and since we also have amorphism T → Z, we obtain the required morphism T → Z ×Y X .
Conversely, assumed that the base changed morphism Z ×Y X → Z satisfies
the infinitesimal lifting property for all Z → Y with Z a scheme, and let T0 → T
be as in the statement of the proposition. Taking Z = T , we can lift the induced
morphism T0 → T ×Y X to a morphism T → T ×Y X , and the composite of thismorphism with the second projection gives us the required lifting.

Formal algebraic spaces
Following [Sta, Tag 0AHW], an affine formal algebraic space over a base scheme S
is a sheaf X on the fppf site of S which admits a description as an Ind-scheme∼
X −→ limi Xi , where the Xi are affine schemes and the transition morphisms−→
are thickenings (in the sense of [Sta, Tag 04EX]). Here we allow the indexing
set in the inductive limit to be arbitrary; if it can be chosen to be the natural
numbers, then we say that X is countably indexed. A countably indexed affine
formal algebraic space can be written in the form X ∼Spec A/In , where
= lim
−→n
A is a complete topological ring equipped with a decreasing sequence {In }nof open ideals which are weak ideals of definition, i.e. consist of topologicallynilpotent elements ([Sta, Tag 0AMV]), and which form a fundamental basisof 0 in A (see the discussion of [Sta, Tag 0AIH]). We then write X := Spf A(following [Sta, Tag 0AIF]).
A particular example of a countably indexed affine formal algebraic space isan adic affine formal algebraic space. By definition, this is of the form Spf A,
where A is adic: that is, A is a topological ring which is complete (and separated,by our convention throughout this book), and which admits an ideal of definition(that is, an ideal I whose powers form a basis of open neighbourhoods of zero).
We say that Spf A is Noetherian if A is adic and Noetherian. We say that A(or Spf A) is adic* if I can be taken to be finitely generated.
More generally, we say that a complete topological ring A is weakly admissible if A contains an open ideal I consisting of topologically nilpotent elements,[Sta, Tag 0AMV]).
A formal algebraic` space over S is a sheaf X on the fppf site of S which receives a morphism Ui → X which is representable by schemes, étale, and surjective, and whose source is a disjoint union of affine formal algebraic spaces Ui .
We say that X is locally countably indexed if the Ui can be chosen to be countably indexed. We say that X is locally Noetherian if the Ui can be taken to be
Noetherian.
We will find the following lemmas useful.
A.3 Lemma. Let X be an affine formal algebraic space over S, which is either
countably indexed (e.g. an adic or adic* affine formal algebraic space), and henceof the form Spf A for some weakly admissible topological ring A, by [Sta, Tag
0AIK], or else is of the form Spf A, for a pro-Artinian ring A. Then if Y → X
is a closed immersion of formal algebraic spaces over S, we have that Y is of

218

APPENDIX A. FORMAL ALGEBRAIC STACKS

the form Spf B for some quotient B of A by a closed ideal, endowed with its
quotient topology.
Proof. In the countably indexed case, this is immediate from [Sta, Tag 0AIK,Tag
0ANQ,Tag 0APT]. If A is pro-Artinian, say A = lim Ai with the Ai Artinian,←−then Spec Ai ×Spf A Y is a closed subscheme of Spec Ai for each index i, andhence is of the form Spec Bi for some quotient Bi of B. Thus∼∼∼

Y −→ lim Spec Ai ×Spf A Y −→ lim Spec Bi −→ Spf B,−→−→
iwhere B = limi Bi . The general theory of pro-Artinian rings shows that B is←−a quotient of A, since each Bi is a quotient of the corresponding Ai [EG21,
Rem. 2.2.7].
A.4 Lemma. ([Eme, Lem. 8.18]) A morphism of Noetherian affine formal algebraic spaces Spf B → Spf A which is representable by algebraic spaces is (faithfully) flat if and only if the corresponding morphism A → B is (faithfully) flat.

Ind-algebraic and formal algebraic stacks
A.5 Definition. ([Eme, Defn. 4.2]) An Ind-algebraic stack over a scheme S is a
X , where we are taking thestack X over S which can be written as X ∼= lim−→i∈I i2-colimit in the 2-category of stacks of a 2-directed system {Xi }i∈I of algebraic
stacks over S.
By [Eme, Rem. 4.9], if X → Y is representable by algebraic spaces, and Y
is an Ind-algebraic stack, then X is also an Ind-algebraic stack.
A.6 Definition. ([Eme, Defn. 5.3]) A formal algebraic stack over S is a stack X
which admits a morphism U → X which is representable by algebraic spaces,smooth, and surjective, and whose source is a formal algebraic space. If thesource can be chosen to be locally countably indexed, then we say that X is
locally countably indexed.
A.7 Definition. ([Eme, Defn. 7.6]) A p-adic formal algebraic stack is a formal
algebraic stack X over Spec Zp which admits a morphism X → Spf Zp whichis representable by algebraic stacks; so in particular, we may write X as an
Ind-algebraic stack by writing X ∼X ×Spf Zp Spec Z/pn Z.
= lim
−→n
The relationship between formal algebraic stacks and Ind-algebraic stacks
is discussed in detail in [Eme, §6]. Before explaining some of this material, we
need to recall some preliminaries on finiteness properties and underlying reduced
substacks.

Underlying reduced substacks
If X /S is any stack, then we let (Xred )0 be the full subcategory of X whose setof objects consists of those T → X for which T is a reduced S-scheme.

219
A.8 Definition. ([Eme, Lem. 3.27]) The underlying reduced substack Xred of X
is the intersection of all of the substacks of X which contain (Xred )0 .
A.9 Lemma. ([Eme, Lem. 4.16] ) If X is an Ind-algebraic stack, and we write∼
limi Xi −→ X , for some 2-directed system {Xi }i∈I of algebraic stacks, then the−→
induced morphism limi (Xi )red → Xred is an isomorphism, and so in particular−→
Xred is again an Ind-algebraic stack.
As the following lemma records, in the case that X is a formal algebraic
stack, then Xred is algebraic, and (just as in the case of formal schemes, orformal algebraic spaces), X is a thickening of Xred .
A.10 Lemma. ([Eme, Lem. 5.26] ) If X is is a formal algebraic stack overS, then Xred is a closed and reduced algebraic substack of X , and the inclusion
Xred ,→ X is a thickening (in the sense that its base-change over any algebraic
space induces a thickening of algebraic spaces). Furthermore, any morphism
Y → X with Y a reduced algebraic stack factors through Xred .

Finiteness properties
Let X be a formal algebraic stack X . We say that X is quasi-compact if thealgebraic stack Xred is quasi-compact. We say that X is quasi-separated if
the diagonal morphism of X (which is automatically representable by algebraic
spaces) is quasi-compact and quasi-separated.
We say that X is locally Noetherian if and only if it admits a morphism
U → X which is representable by algebraic spaces, smooth, and surjective,
whose source is a locally Noetherian formal algebraic space. Finally, we saythat X is Noetherian if it is locally Noetherian, and it is quasi-compact andquasi-separated.

The relationship between Ind-algebraic and formal algebraic stacks
In one direction, we have the following lemma.
A.11 Lemma. ([Eme, Lem. 6.2] ) If X is a quasi-compact and quasi-separated
formal algebraic stack, then X ∼X for a 2-directed system {Xi }i∈I of quasi= lim−→i icompact and quasi-separated algebraic stacks in which the transition morphisms
are thickenings.
A partial converse to Lemma A.11 is proved as [Eme, Lem. 6.3]. In particular, we have the following useful criteria for being a formal algebraic stack.
A.12 Proposition. ([Eme, Cor. 6.6] ) Suppose that X is an Ind-algebraic stack∼that can be written as the 2-colimit X −→ lim Xn of a directed sequence (Xn )n≥1−→
in which the Xn are algebraic stacks, and the transition morphisms are closed
immersions. If Xred is a quasi-compact algebraic stack, then X is a locallycountably indexed formal algebraic stack.

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APPENDIX A. FORMAL ALGEBRAIC STACKS

A.13 Proposition. Suppose that {Xn }n≥1 is an inductive system of algebraic
stacks over Spec Zp , such that each Xn in fact lies over Spec Z/pn Z, and eachof the induced morphisms Xn → Xn+1 ×Spec Z/pn+1 Z Z/pn Z is an isomorphism.
Then the Ind-algebraic stack X := limn Xn is a p-adic formal algebraic stack.
−→
Proof. This is a special case of [Eme, Lem. 6.3, Ex. 7.8].

Scheme-theoretic images
The notion of the scheme-theoretic image of a (quasi-compact) morphism ofalgebraic stacks is developed in [Sta, Tag 0CMH]; equivalent alternative presentations are given in [EG21, §3.1] and in [Eme, Ex. 9.9]. In [Eme, §6] thisis extended to a definition of the scheme-theoretic image between formal algebraic stacks that are quasi-compact and quasi-separated. In fact, the definition
there uses Lemma A.11 to write the formal stacks in question as Ind-algebraic
stacks with transition morphisms being closed immersions, and this is the levelof generality that is appropriate to us here.
A robust theory of scheme-theoretic images requires a quasi-compactness assumption on the morphism whose scheme-theoretic image is being formed, andwe begin by introducing the notion of quasi-compactness which seems appropriate to our context.
A.14 Definition. Let X be an Ind-algebraic stack which can be written as a 2colimit of quasi-compact algebraic stacks for which the transition morphisms areclosed immersions, say X ∼X . We say that a morphism of stacks X → Y
= lim
−→ λ
is Ind-representable by algebraic stacks if each of the induced morphisms
Xλ → Y(A.15)

is representable by algebraic stacks. We say that such a morphism is furthermore
Ind-quasi-compact if each of the morphisms Xλ → Y (representable by algebraic
stacks by assumption) is quasi-compact.
Using the fact that any two descriptions of X as an Ind-algebraic stack asabove (i.e. as the 2-colimit of quasi-compact algebraic stacks with respect totransition morphisms that are closed immersions) are mutually cofinal, one seesthat the property of a morphism being Ind-representable by algebraic stacks(resp. Ind-representable by algebraic stacks and Ind-quasi-compact) is independent of the choice of such a description of X .
Suppose now, in the context of the Definition A.14, that Y is also an
Ind-algebraic stack, which admits a description as the 2-colimit of algebraic
stacks with respect to transition morphisms that are closed immersions, say
Y∼
Y . Then the induced morphism (A.15) factors through Yµ for some µ
= lim
−→ µ
(since the Xλ are quasi-compact and the transition morphisms between the Yµ
are monomorphisms). Since any morphism of algebraic stacks is representableby algebraic stacks, and since closed immersions are representable by algebraic
stacks, we find that (A.15) is necessarily representable by algebraic stacks, and221
thus the morphism X → Y is necessarily Ind-representable by algebraic stacks.
Furthermore, the morphism (A.15) is quasi-compact if and only if the induced
morphism Xλ → Yµ is quasi-compact for some (or equivalently any) allowablechoice of µ. For example, if the Yµ are all quasi-separated, then these induced
morphisms are necessarily quasi-compact (via the usual graph argument, sinceeach Xλ is quasi-compact), and so in this case any morphism X → Y is necessarily Ind-quasi-compact.
Suppose now that the morphism X → Y is Ind-quasi-compact, or equivalently (as we have just explained), that the various induced morphisms Xλ → Yµ
are quasi-compact. Then each of these induced morphisms has a schemetheoretic image Zλ . One easily checks that Zλ , thought of as a closed substackof Y, is independent of the particular choice of the index µ used in its definition. Evidently Zλ is a closed substack of Zλ0 if λ ≤ λ0 . In particular, we may
form the 2-colimit lim Zλ , which is an Ind-algebraic stack. There is a natural−→morphism lim Zλ → Y.
−→
A.16 Definition. In the preceding context, we define the scheme-theoretic
image of the Ind-quasi-compact morphism X → Y to be Z := lim Zλ .
−→It is easily verified that the scheme-theoretic image, so defined, is independent of the chosen descriptions of X and Y. There is a canonical monomorphism
Z ,→ Y. (We don’t claim that this monomorphism is necessarily a closed immersion in this level of generality.)
A very special case of this definition is the case that X → Y is representableby algebraic spaces and quasi-compact, and Y = Spf A for a pro-Artinian ring A.
In this case the definition of the scheme-theoretic image coincides with [EG21,
Defn. 3.2.15] (i.e. the scheme-theoretic image can be computed via the schemetheoretic images of the pull-backs of X to the discrete Artinian quotients of A).
In this particular case the scheme-theoretic image is a closed formal subspaceof Y, i.e. of the form Spf B for some topological quotient B of A.
A.17 Definition. Let X and Y be Ind-algebraic stacks which satisfy the hypotheses introduced above, i.e. which may each be written as the 2-colimit withrespect to closed immersions of algebraic stacks, which are furthermore quasicompact in the case of X . Then we say that an Ind-quasi-compact morphism
X → Y is scheme-theoretically dominant if the induced map Z → Y is an
isomorphism, where Z is the scheme-theoretic image of X → Y.
A.18 Remark. If X is a quasi-compact and quasi-separated formal algebraic
stack, then Lemma A.11 shows that X ∼X with the Xλ being quasi= lim−→ λcompact and quasi-separated algebraic stacks, and the morphisms being thickenings, and so in particular closed immersions. The preceding discussion thus
applies to morphisms X → Y of quasi-compact and quasi-separated formal algebraic stacks, and shows that such morphisms are necessarily Ind-representableby algebraic stacks and Ind-quasi-compact. In particular, Definition A.17 applies to such morphisms, and in this case recovers the definition of schemetheoretic dominance given in [Eme, Def. 6.13].

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APPENDIX A. FORMAL ALGEBRAIC STACKSA closely related context in which the preceding definition applies if thefollowing: if A is an adic* topological ring, with finitely generated ideal ofdefinition I, and X , Y → Spf A are morphisms of quasi-compact formal algebraic stacks that are representable by algebraic stacks, then we may write
Y , where Xn := X ×Spf A Spec A/I n and Yn :=X and Y ∼
X ∼
= lim
= lim
−→n n
−→n n
n
Y ×Spf A Spec A/I are quasi-compact algebraic stacks. A morphism X → Yof stacks over Spf A is then necessarily representable by algebraic stacks [Eme,Lem. 7.10], and so is Ind-quasi-compact if and only if it quasi-compact in theusual sense. We may then define its scheme-theoretic image, following Definition A.16, or speak of such a morphism being scheme-theoretically dominant.
We now show (in Proposition A.21) that under certain hypotheses the schemetheoretic image of a p-adic formal algebraic stack is also a p-adic formal algebraic
stack. The deduction of this result from those of [Eme] will involve the following
definition.
A.19 Definition. ([Eme, Defn. 8.26, Rem. 8.39]) We say that a formal algebraic
stack X over a scheme S is Ind-locally of finite type over S if there exists an∼
isomorphism X −→ lim Xi , where each Xi is an algebraic stack locally of finite−→
type over S, and the transition morphisms are thickenings.
A.20 Remark. If X is a quasi-compact and quasi-separated formal algebraic
stack, then in order to verify that X is Ind-locally of finite type, it suffices to∼exhibit an isomorphism X −→ lim Xi , where each Xi is an algebraic stack locally−→of finite type over S, and the transition morphisms are closed immersions (notnecessarily thickenings) [Eme, Lem. 8.29].
A.21 Proposition. Suppose that we have a commutative diagram
X

/Y
" 
Spf Zpin which the diagonal arrow makes X into a p-adic formal algebraic stack offinite presentation, and where Y is an Ind-algebraic stack which can be written as
the inductive limit of algebraic stacks, each of finite presentation over Spec Z/pa
for some a ≥ 1, with the transition maps being closed immersions.
Suppose also that the horizontal morphism X → Y (which, by the usual graphargument, is seen to be representable by algebraic stacks) is proper. Then thescheme-theoretic image Z of this morphism is a p-adic formal algebraic stack offinite type. If X is flat over Spf Zp , then so is Z.
Proof. We deduce this from [Eme, Prop. 10.5]. Note firstly that X is quasicompact and quasi-separated, since it is of finite presentation over Spf Zp . Examining the hypotheses of [Eme, Prop. 10.5], we need to show that Z is formal
algebraic, that it is quasi-compact and quasi-separated, that it is Ind-locally offinite type over Spec Zp , and that the induced morphism X → Z (which will be
representable by algebraic stacks, by the usual graph argument) is proper.

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By hypothesis, we can write Y ∼Y , where each Yµ is an algebraic
= lim
−→µ µ
a
stack of finite presentation over some Z/p (with a depending on µ), and theX , where Xa :=transition maps are closed immersions. If we write X ∼
= lim
−→a a
Spec Z/pa ⊗Spf Zp X , then by assumption each Xa is an algebraic stack. By
definition, then, we have that
Z := lim Za ,−→a

(A.22)

where Za is the scheme-theoretic image of Xa → Yµ , for µ sufficiently large. It
follows from Lemma A.23 below that each of the closed immersions Za ,→ Za+1
is a finite order thickening. We conclude from [Eme, Lem. 6.3] that Z is in fact
a formal algebraic stack.
Since Yµ is of finite presentation over Spec Zp , each closed substack Za isof finite presentation over Z/pa , and in particular quasi-compact and quasiseparated; it follows that Z is quasi-compact and quasi-separated. The description (A.22) of Z furthermore exhibits Z as being Ind-locally of finite type overSpec Zp .
It remains to show that X → Z is proper. If T → Z is any morphism whose
source is a scheme, then since Z → Y is a monomorphism, we find that there
is an isomorphism T ×Z X → T ×Y X . Since by assumption the target of thisisomorphism is an algebraic stack, proper over T , the same is true of the source.
The claim regarding flatness follows from Lemma A.24 below.
We used the following lemmas in the proof of the Proposition A.21.
A.23 Lemma. Consider a commutative diagram of morphisms of algebraic
stacks
/Z
X

X0


/ Z0

with Z 0 being quasi-compact, in which the left-hand vertical arrow is an nthorder thickening for some n ≥ 1, the right-hand vertical arrow is a closed
immersion, and the lower horizontal arrow is quasi-compact, surjective, andscheme-theoretically dominant. Then the right-hand vertical arrow is then also
an nth order thickening.
Proof. Since X → X 0 and X 0 → Z 0 are surjective, by assumption, the same istrue of their composite X → Z 0 , and hence of the closed immersion Z → Z 0 ;thus this closed immersion is a thickening. To see that it is of order n, we
first note that since Z 0 is quasi-compact, we find a smooth surjection U 0 → Z 0
whose source is an affine scheme; since the property of a thickening being of finite
order may be checked fppf locally, and since the property of being quasi-compact
and scheme-theoretically dominant is also preserved by flat base-change, after
pulling back our diagram over U 0 , we may assume that Z 0 = U 0 is an affine
scheme, and that Z = U is a closed subscheme. Let I be the ideal sheaf on

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APPENDIX A. FORMAL ALGEBRAIC STACKSU 0 cutting out U , and let a be a section of I. If a0 denotes the pull-back of
a to X 0 , then a0|X = 0, and so (a0 )n = 0, by assumption. Since X 0 → U 0 is
scheme-theoretically dominant, we find that an = 0. Thus U ,→ U 0 is indeed
an nth order thickening.
A.24 Lemma. If X → Y is a quasi-compact scheme-theoretically dominantmorphism of p-adic formal algebraic stacks which are locally of finite type, and
if X is flat over Zp , then the same is true of Y.
Proof. Let V → Y be a morphism which is representable by algebraic spaces
and smooth, whose source is an affine formal algebraic space; so V = Spf B forsome p-adically complete Zp -algebra B that is topologically of finite type. It
suffices to show that B is flat over Zp . Since V → Y is in particular flat (being
smooth), the base-change morphism X ×Y V → V is again scheme-theoreticallydominant. Since it is also quasi-compact, and since V is formally affine (and soquasi-compact), we may find a formal algebraic space U = Spf A endowed with amorphism U → X ×Y V which is representable by algebraic spaces, smooth, andsurjective; and thus also scheme-theoretically dominant. Thus we may replace
our original situation with the composite morphism Spf A → Spf B. But in thiscontext, scheme-theoretic dominance amounts to the morphism B → A being
injective; thus B is Zp -flat if A is.

Versality and versal rings
We will sometimes find it useful to study scheme-theoretic images in terms ofversal rings, and so we will recall some notation and results from [EG21, §2.2]
related to this topic.
If Λ is a Noetherian ring, equipped with a finite ring map Λ → k whosetarget is a field, then we let CΛ denote the category whose objects are Artinian∼
local Λ-algebras A equipped with an isomorphism of Λ-algebras A/mA −→ k.
We let pro -CΛ be the corresponding category of formal pro-objects, which (viapassage to projective limits) we identify with the category of topological pro(discrete Artinian) local Λ-algebras A equipped with a Λ-algebra isomorphism∼
A/mA −→ k. By [EG21, Rem. 2.27], any morphism A → B in pro -CΛ has closed
image, and induces a topological quotient map from its source to its image, sothat in particular A → B is surjective if and only if it is induced by a compatible
system of surjective morphisms in CΛ .
Fix a locally Noetherian base scheme S, and let k be a finite type OS -field,
i.e. k is a field equipped with a morphism Spec k → S of finite type. Choose, as
we may, an affine open subscheme Spec Λ ⊆ S for which Spec k → S factors asSpec k → Spec Λ → S, with Λ → k being finite. (In what follows we fix such a
choice of Λ, although the notions that we define in terms of it are independentof this choice.) We now define a category fibred in groupoids Fbx in the following
way; this category is an example of a deformation category in the sense of [Sta,Tag 06J9]. In the notation of [Sta, Tag 07T2], our category Fbx is denoted FF ,k,x(with a slightly unfortunate clash of notation in the two instances of F).

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A.25 Definition. If F is a category fibred in groupoids over S, and if x :Spec k → F is a morphism, then we define a category Fbx , cofibred in groupoids
over CΛ , as follows: For any object A of CΛ , the objects of Fbx (A) consist of∼morphisms y : Spec A → F, together with an isomorphism α : x −→ y compatible with the given identification of A/m with k; here y denotes the induced
morphism Spec A/m → F. The set of morphisms between two objects (y, α)and (y 0 , α0 ) of Fbx (A) consists of the subset of morphisms β : y → y 0 in F(A) for
which α0 ◦ β = α; here β denotes the morphism y → y 0 induced by β. If A → B
is a morphism in CΛ , then the corresponding pushforward Fbx (A) → Fbx (B) is
defined by pulling back morphisms to F along the corresponding morphism ofschemes Spec B → Spec A.
If F is a category fibred in groupoids over the locally Noetherian scheme
S, and if x : Spec k → F is a k-valued point of F, for some finite type OS field, then the notion of a versal ring to F at x is defined in [EG21, Def. 2.2.9].
Rather than recalling that definition here, we will give a definition of versality
in a greater level of generality that is convenient for us. We will then explain
how the notion of versal ring is obtained as a particular case.
A.26 Definition. Let f : F → G be a morphism of categories fibred ingroupoids over the locally Noetherian scheme S, let k be a finite type OS -field,
let x : Spec k → F be a k-valued point of F, and let y : Spec k → G be a k-valued∼
point of G equipped with an isomorphism of k-valued points α : f ◦x −→ y. The
morphism f induces in an evident way a morphism fbx : Fbx → Gby of categories
cofibred in groupoids. We say that f is versal at x if the the morphism fbx issmooth in the sense of [Sta, Tag 06HG]; that is, given a commutative diagram/ Spec A

Spec B
 z
b
Fx

fbx


/ Gbx

in which the upper arrow is the closed immersion corresponding to a surjection
A → B in CΛ , we can fill in the dotted arrow so that the diagram remains
commutative. (Clearly this notion is independent of the particular choice of yand α; more precisely it holds for any such choice if it holds for one such choice,such as y = f ◦ x and α = id.)A.27 Example. Let S be a locally Noetherian scheme, let k be a finite type OS field k, and let A be an object of pro -CΛ ; recall in particular then that A comes∼∼equipped with a chosen isomorphism Spec A/m −→ k. We let x0 : Spec k −→Spec A/m ,→ Spf A denote the induced k-valued point of Spf A.
Now let F be a category fibred in groupoids over S, let x : Spec k → F
be a k-valued point of F, and suppose that f : Spf A → F is a morphism, for∼which we can find an isomorphism f ◦ x0 −→ x of k-valued points of F. Thenthe morphism f is versal at the point x0 if and only if A is a versal ring to thek-valued point x of F in the sense of [EG21, Def. 2.2.9]

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APPENDIX A. FORMAL ALGEBRAIC STACKS

A.28 Example. Let S be a locally Noetherian scheme, let U be a locally finite
type S-scheme, and let f : U → F be a morphism to a category fibred ingroupoids over S. If u ∈ U is a finite type point, giving rise to a morphism
Spec κ(u) → U , then f is versal at this κ(u)-valued point of U if and only
if f is versal at the point u in the sense of [EG21, Def. 2.4.4 (1)]; cf. [EG21,
Rem. 2.4.5].
A.29 Remark. If X is an algebraic stack which is locally of finite presentation
over a locally Noetherian scheme S, then it admits (effective, Noetherian) versal
rings at all finite type points [Sta, Tag 0DR1]. If X is an Ind-locally finite type
algebraic space over S, then it admits (canonical) versal rings at all finite typepoints, by [EG21, Lem. 4.2.14]. We don’t prove a general statement about
the existence of versal rings for Ind-algebraic stacks, since in all the cases weconsider in the body of the book we are able to construct them “explicitly” (forexample, in terms of Galois lifting rings, or lifting rings for étale ϕ-modules).
The following lemma and its proof are essentially [EG21, Lem. 3.2.16], butsince the setup there is different, we give the details here. Before stating thelemma, we introduce the setup. Let S be a locally Noetherian scheme. We
suppose given a morphism X → Y of stacks over S which is representable byalgebraic stacks and proper, and that Y is an Ind-algebraic stack which may
be written as the 2-colimit of algebraic stacks which are of finite presentation
over S (and so in particular quasi-compact and quasi-separated) with respect
to transition morphisms that are closed immersions, say Y ∼Y . Then= lim−→ λXλ := X ×Y Yλ is an algebraic stack, and the projection Xλ → Yλ is proper,
so that Xλ is finite type over S (and in particular quasi-compact, and also
quasi-separated, although we won’t use this latter fact). Furthermore, we have
X , and so X → Y is a morphism of Indan induced isomorphism X ∼
= lim
−→λ λ
algebraic stacks whose scheme-theoretic image Z may be defined. Of course,
in this context, since the Ind-structures on X and Y are compatible, if we let
Zλ denote the scheme-theoretic image of Xλ in Yλ , then this coincides with thescheme-theoretic image of Xλ in Yλ0 for any λ0 ≥ λ, and we may write Z ∼=limλ Zλ . Note that Zλ is also of finite presentation over S, by Lemma 4.5.10.
−→
A.30 Lemma. Suppose that we are in the preceding situation, so that X → Y
is a morphism of stacks over a locally Noetherian base S which is representableby algebraic stacks and proper, where Y is an Ind-algebraic stack which may
be written as the 2-colimit of algebraic stacks which are of finite presentation
over S, with respect to transition morphisms that are closed immersions, andwe write Z for the scheme-theoretic image of X → Y.
Suppose that x : Spec k → Z is a finite type point, and that Spf Ax → Y isa versal morphism for the composite x : Spec k → Z ,→ Y. Let Spf Bx be thescheme-theoretic image of XSpf Ax → Spf Ax . Then the morphism Spf Bx → Y
factors through a versal morphism Spf Bx → Z.
Proof. By definition, we may write Ax = lim Ai , Bx = lim Bi , where the Ai are←−←−objects of CΛ , and Spec Bi is the scheme-theoretic image of XAi → Spec Ai .
As explained immediately above, we write X ∼X , Y ∼
Y , and
= lim
= lim
−→ λ
−→ λ

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Z ∼
Z , where the transition morphisms are closed immersions of alge= lim−→ λbraic stacks, and Zλ is the scheme-theoretic image of the morphism Xλ → Yλ ;in particular, the morphism Xλ → Zλ is proper and scheme-theoretically dominant.
If follows that for each i, the composite Spec Bi → Spec Ai → Y factors
through Z. Indeed, for λ sufficiently large the morphism (Xλ )Bi → Spec Bi isscheme-theoretically surjective, and the morphism (Xλ )Bi → Y factors throughZλ , so the morphism Spec Bi → Y also factors through Zλ .
We now show that a morphism Spec A → Spf Ax , with A an object of CΛ ,factors through Spf Bx if and only if the composite Spec A → Spf Ax → Y
factors through Z. In one direction, if Spec A → Y factors through Spf Bx ,then it factors through Spec Bi for some i, and hence through Z, as we saw
above.
Conversely, if the composite Spec A → Spf Ax → Y factors through Z,
then we claim that there exists factorisation Spec A → Spec B → Y, where
B is an object of CΛ , the morphism Spec A → Spec B is a closed immersion,and XB → Spec B is scheme-theoretically dominant. To see this, note firstly
that the composite Spec A → Y factors through Zλ for some λ. Since Zλ isan algebraic stack which is locally of finite presentation over a locally Noetherian base, it admits effective Noetherian versal rings by Remark A.29, so thatSpec A → Zλ factors through a versal morphism Spec Cx → Zλ at the finite
type point of Zλ induced by x. By [EG21, Lem. 1.6.3], we can find a factorisation Spec A → Spec Rx → Spec Cx , where Rx is complete local Noetherian,Spec Rx → Spec Cx is faithfully flat, and Spec A → Spec Rx is a closed immersion. Since Spec Rx → Spec Cx is faithfully flat, and Spec Cx → Zλ is flatby [Sta, Tag 0DR2], we see that the base changed morphism (Xλ )Rx → Spec Rx
is scheme-theoretically dominant. By [EG21, Lem. 3.2.4] (applied to the propermorphism of algebraic stacks Xλ → Zλ ), Rx admits a cofinal collection of Artinian quotients Ri for which (Xλ )Ri → Spec Ri is scheme-theoretically dominant. The closed immersion Spec A → Spec Rx factors through Spec Ri forsome Ri , so we may take B = Ri .
Now, by the versality of Spf Ax → Y, we may lift the morphism Spec B →Y to a morphism Spec B → Spf Ax , which furthermore we may factor as
Spec B → Spec Ai → Spf Ax , for some value of i. Since XB → Spec B isscheme-theoretically dominant, the morphism Spec B → Spec Ai then factors
through Spec Bi , and thus through Spf Bx , as required.
It follows in particular that the composite Spf Bx → Spf Ax → Y factors
through a morphism Spf Bx → Z. It remains to check that this morphism isversal. This is formal. Suppose we are given a commutative diagram
Spec A0/ Spf Bx

/ Spf AxSpec A/Z

/Y

where the left hand vertical arrow is a closed immersion, and A0 , A are objects

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APPENDIX A. FORMAL ALGEBRAIC STACKS

of CΛ . By the versality of Spf Ax → Y, we may lift the composite Spec A → Y
to a morphism Spec A → Spf Ax . Since the composite Spec A → Spf Ax → Y
factors through Z, the morphism Spec A → Spf Ax then factors through Spf Bx ,as required.
The following lemma records a useful property of versal rings in the Noetherian context.
A.31 Lemma. Let X is a locally Noetherian formal algebraic stack, let R be
a complete Noetherian local ring with residue field k, and let f : Spf R → X bea morphism for which the induced morphism x : Spec k → Xred is a finite typepoint, and which is versal to X at x. Then the morphism f is flat, in the senseof [Eme, Def. 8.42].
Proof. Since X is locally Noetherian, by assumption, we may find a morphism
Spf A → X whose source is a Noetherian affine formal algebraic space, whichis representable by algebraic spaces and smooth, and whose image contains
the point x. By definition, we have to show that the base-changed morphism
Spf A ×X Spf R → Spf A is flat.
The fibre product Spf A×X Spf R is a formal algebraic space (which is locally
Noetherian,since Spf R is so [Sta, Tag 0AQ7]), and so admits a morphism`
Spf Bi → Spf A ×X Spf R whose source is the disjoint union of Noetherianaffine formal algebraic spaces, and which is representable by algebraic spaces,étale, and surjective. Again, it suffices to show that each of the morphisms
Spf Bi → Spf A is flat, which by definition is equivalent to each of the induced
morphisms A → Bi being flat. If we let I denote an ideal of definition of the
topology on A, then it suffices to show that each morphism A/I n → Bi /I n isflat (see e.g. the proof of [Eme, Lem. 8.18]).
For this, it suffices in turn to show, for each maximal ideal n of Bi , thatthe induced morphism A/I n → (Bi /I n )ˆis flat (where (–)ˆdenotes n-adic completion; note that if m denotes the maximal ideal of R, then the topology on
Bi is the m-adic topology, so that any maximal ideal n of Bi contains m, andbi → Spf A is versal to thealso contains IB). Now the induced morphism Spf B
induced morphism Spec Bi /n → Spf A (as one sees by chasing through the constructions, beginning from the versality of Spf R → X , and taking into accountthe infinitesimal lifting property for étale morphisms), and thus the induced
morphism Spf(Bi /I n )ˆ → Spec A/I n is also versal (to the induced morphism
Spec Bi /n → Spec A/I n ). Thus this morphism is flat, e.g. by [Sta, Tag 0DR2],
and the lemma is proved.
We will frequently find the following lemma useful.
A.32 Lemma. Let R → S be a continuous surjection of objects in pro -CΛ , and
let X → Spf R be a finite type morphism of formal algebraic spaces.
Make the following assumption: if A is any finite-type Artinian local Ralgebra for which the canonical morphism R → A factors through a discrete
quotient of R, and for which the canonical morphism XA → Spec A admits a
section, then the canonical morphism R → A furthermore factors through S.

229
Then the scheme-theoretic image of X → Spf R is a closed formal subschemeof Spf S.
Proof. Writing Spf T for the scheme-theoretic image of X → Spf R, we need toshow that the surjection R → T factors through S. By definition, we can write T
as an inverse limit of discrete Artinian quotients B for which the canonical morphism XB → Spec B is scheme-theoretically dominant. It follows from [EG21,
Lem. 5.4.15] that for any such B, the surjection R → B factors through S; so
the surjection R → T factors through S, as required.

Immersed substacks
The following lemma is often useful for studying morphisms from a stack interms of morphisms from a cover of the stack.
A.33 Lemma. [Eme, Lem. 3.18, 3.19] Suppose that X is a stack over a basescheme S, and that U → X is a morphism over S which is representable byalgebraic spaces and whose source is a sheaf. Write R := U ×X U .
1. Suppose that the projections R := U ×X U ⇒ U (which are again representable by algebraic spaces, being the base-change of morphisms which
are so representable) are flat and locally of finite presentation. Then themorphism U → X induces a monomorphism [U/R] → X .
2. Suppose that U → X is flat, surjective, and locally of finite presentation.
∼
Then the morphism U → X induces an isomorphism [U/R] −→ X .
In particular, if X is a formal algebraic stack (over some base scheme S), thenby definition there exists a morphism U → X which is representable by algebraic
spaces, smooth, and surjective, and whose source U is a formal algebraic space.
In this case R := U ×X U is a formal algebraic space which is endowed in anatural way with the structure of a groupoid in formal algebraic spaces over U ,and (since smooth morphisms are in particular flat), by Lemma A.33 there is∼
an isomorphism of stacks [U/R] −→ X .
Suppose now that Z ,→ X is an immersion (in the sense that it is representable by algebraic spaces, and pulls back to an immersion over any test
morphism T → X whose source is a scheme). The induced morphism W :=U ×X Z ,→ U is then an immersion of formal algebraic spaces, and W is Rinvariant, in (an evident generalization of) the sense of [Sta, Tag 044F]. Conversely, if W is an R-invariant locally closed formal algebraic subspace of U , andif we write RW := R ×U W = W ×U R for the restriction of R to W , then the∼
induced morphism [W/RW ] → [U/R] −→ X is an immersion. (In the context
of algebraic stacks, this is [Sta, Tag 04YN].)

Flat parts
Let O be the ring of integers in a finite extension of Qp , and let X → Spf O bea p-adic formal algebraic stack which is locally of finite type over Spf O. Then

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APPENDIX A. FORMAL ALGEBRAIC STACKS

by [Eme, Ex. 9.11], there is a closed substack Xfl of X , the flat part of X , whichis the maximal substack of X which flat over Spf O.
More precisely, Xfl → Spf O is flat, and if Y → X is a morphism of locally
Noetherian formal algebraic stacks for which the composite Y → Spf O is flat,then Y → X factors through Xfl .

Obstruction theory
Let X be a limit preserving Ind-algebraic stack over a locally Noetherian scheme
S. If x : Spec A → X is a morphism for which the composite morphism
Spec A → S factors through an affine open subscheme of S, then in [Sta, Tag
07Y9] there is defined a functor Tx from the category of A-modules to itself,
whose formation is also functorial in the pair (x, A) [Sta, Tag 07YA], and suchthat for any finitely generated A-module M , there is a natural identification ofTx (M ) with the set of lifts of x to morphisms x0 : Spec A[M ] → X (where A[M ]denotes the square zero extension of A by M ). (These definitions apply to X
by [Eme, Lem. 4.22].)
We make the following definition, which is a special case of [Eme, Defn. 11.6];
that definition incorporates an auxiliary module in its definition for technicalreasons, but in our applications this module is zero, so we have suppressed it
here. We have also incorporated [Eme, Rem. 11.9], which allows us to restrict
to the case of finitely generated A-modules M .
A.34 Definition. We say that X admits a nice obstruction theory if, for eachfinite type S-algebra A which lies over an affine open subscheme of S, equipped•with a morphism x : Spec A → X , there exists a complex of A-modules K(x,A)
such that the following conditions are satisfied:•
1. The complex K(x,A)
is bounded above and has finitely generated cohomology modules (or, equivalently, is isomorphic in the derived category to abounded above complex of finitely generated A-modules).
•2. The formation of K(x,A)
is compatible (in the derived category) with pullback. More precisely, if f : Spec B → Spec A, inducing the morphismy : Spec B → Spec A → X , then there is a natural isomorphism in the∼••derived category f ∗ K(x,A)−→ K(y,B).

3. For any pair (x, A) as above, and for any finitely generated A-module M ,∼•we have an isomorphism Tx (M ) −→ H 1 (K(x,A)⊗L M ) whose formation isfunctorial in M and in the pair (x, A).
4. For any pair (x, A) as above, and for any finitely generated A-module•
M , the cohomology module H 2 (K(x,A)⊗L M ) serves as an obstruction
module. In particular, for any square zero extension
0 → I → A0 → A → 0

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for which A0 is of finite type over S (or, equivalently, for which I isa finite A-module), we have a functorial obstruction element ox (A0 ) ∈•H 2 (K(x,A)⊗L I), which vanishes if and only if x can be lifted to Spec A0 .
The following result generalises the familiar fact that a pro-representing
object for a formal deformation ring whose tangent space is finite dimensional
is necessarily a complete Noetherian local ring.
A.35 Theorem. ([Eme, Thm. 11.13] ) If X is a locally countably indexed andInd-locally of finite type formal algebraic stack over a locally Noetherian scheme
S, and if X admits a nice obstruction theory, then X is locally Noetherian.
(When comparing this result with the statement of [Eme, Thm. 11.13], thereader should bear in mind that an Ind-locally of finite type formal algebraic
stack over S is in particular the 2-colimit of algebraic stacks that are locallyof finite type (or equivalently, locally of finite presentation) over S, and hence
is limit preserving. Also, [Eme, Lem. 8.38] ensures that an Ind-locally of finite
type formal algebraic stack over S is also locally of Ind-locally finite type —which is the condition that appears in [Eme, Thm. 11.13].)

232

APPENDIX A. FORMAL ALGEBRAIC STACKSAppendix B

Graded modules and rigid
analysis
In this appendix we establish some (mostly straightforward) results that combine graded ring techniques with various results related to completions that areof a rigid analytic flavour.

B.1

Associated graded algebras and modules

Let R be an Artinian local ring, with maximal ideal I and residue field k.
B.2 Definition. If M is an R-module, then we let Gr• M denote the graded kvector space associated to the I-adic filtration on M (so Gri M := I i M/I i+1 M ).
The formation of Gr• M is functorial in M . If M and N are two R-modules,
then there is a natural surjection (of graded k-vector spaces)
Gr• M ⊗k Gr• N → Gr• (M ⊗R N ).

(B.3)

In particular, it follows that if A is an R-algebra, then Gr• A is naturally a gradedk-algebra, and if M is an A-module, then Gr• M is naturally a Gr• A-module.
We recall the following basic lemmas.
B.4 Lemma. If A is an R-algebra, then A is Noetherian if and only if Gr• A
is Noetherian.
Proof. This is easy and standard; see e.g. [ST03, Prop. 1.1] for a statement of
the “if” direction of this result (which is the less obvious of the two directions)in a significantly more general setting.
B.5 Lemma. If A is an R-algebra and if M is an A-module, then the naturalmorphism Gr• A ⊗Gr0 A Gr0 M → Gr• M is surjective. Furthermore, M is nonzero if and only if Gr0 M is non-zero, if and only if Gr• M is non-zero.
233

234

APPENDIX B. GRADED MODULES AND RIGID ANALYSIS

Proof. The first claim is immediate. This implies in turn that if Gr0 M is zerothen Gr• M is zero (the converse being evident), and thus that M is zero (since
Gr• M is the associated graded of M with respect to a finite length filtration
beginning at M and ending at 0). (Essentially equivalently, one can observe
directly that since I is nilpotent, if M/IM = 0 then M = 0.)
B.6 Lemma. If A is an R-algebra and M is an A-module, then M is finitely
generated over A if and only if Gr0 M is finitely generated over Gr0 A, if andonly if Gr• M is finitely generated over Gr• A.
Proof. Clearly if M is finitely generated over A, then Gr0 M := M/IM is finitely
generated over A/I; the converse assertion follows from Nakayama’s lemma withrespect to the nilpotent ideal I. If Gr0 M is finitely generated over Gr0 A, thenLemma B.5 shows that Gr• M is finitely generated over Gr• A; the converse assertion follows from the fact that (again by Lemma B.5) Gr0 M may be obtained
as the quotient of Gr• M by the ideal Gr•>0 A of Gr• A.
B.7 Lemma. Let f : M → N be a morphism of A-modules. Then f is an
isomorphism if and only if the induced morphism Gr• f : Gr• M → Gr• N is an
isomorphism.
Proof. If f is an isomorphism then certainly Gr• f is an isomorphism. Conversely, if Gr• f is an isomorphism, then in particular M/IM → N/IN is surjective, so f is surjective by Nakayama’s lemma with respect to the nilpotent
ideal I. In addition, for each i the morphism I i M/I i+1 M → I i N/I i+1 N is injective, so by an easy induction on i we see that any element of ker f is containedin I i M for all i, and is thus zero, as required.
B.8 Lemma. If A is an R-algebra, then an A-module M is (faithfully) flat over
A if and only if Gr• M is (faithfully) flat over Gr• A. Furthermore, if any of these
conditions holds, then the natural morphism Gr• A ⊗Gr0 A Gr0 M → Gr• M is an
isomorphism.
Proof. This is [EG21, Lem. 5.5.37].
B.9 Lemma. If A is an R-algebra, and if M and N are A-modules, then thenatural surjection (B.3) induces a natural surjection
Gr• M ⊗Gr• A Gr• N → Gr• (M ⊗A N ).
Proof. The natural surjection M ⊗R N → M ⊗A N induces a surjection Gr• (M ⊗R
N ) → Gr• (M ⊗A N ). It is immediate that its composite with the surjection (B.3)
factors through Gr• M ⊗Gr• A Gr• N.
B.10 Lemma. If A is an R-algebra, and if M and N are A-modules with either
M or N flat over A, then the surjection of Lemma B.9 is a natural isomorphism∼

Gr• M ⊗Gr• A Gr• N −→ Gr• (M ⊗A N ).

B.11. A GENERAL SETTING

235

Proof. Suppose (without loss of generality) that N is A-flat. Taking into accountLemma B.8, we see that we have to show that natural morphism Gr• M ⊗Gr0 A
Gr0 N → Gr• (M ⊗A N ) is an isomorphism, i.e. that for each i ≥ 0, the naturalmorphism
I i M/I i+1 M ⊗A/I N/I → I i (M ⊗ N )/I i+1 (M ⊗ N )is an isomorphism. This follows easily by tensoring N over A with the various
short exact sequences
0 → I n M → I m M → I m M/I n M → 0,
taking into account the flatness of N over A.

B.11

A general setting

We fix an Artinian local ring R with maximal ideal I and residue field k, as wellan R-algebra C + , and an element u ∈ C + , satisfying the following properties:(A) u is a regular element (i.e. a non-zero divisor) of C + .
(B) C + /u is a flat R-algebra.
(C) C + /I is a rank one complete valuation ring, and the image of u in C + /I
(which is necessarily non-zero, by (A)) is of positive valuation (i.e. lies in
the maximal ideal of C + /I).
B.12 Remark. In the preceding context, if v ∈ C + has non-zero image in C + /I,
then v n divides um in C + , for some m, n > 0. (Indeed, since C + /I is a rank
one valuation ring in which the image of u has positive valuation, we may write
ua = vw + x, for some a > 0, some w ∈ C + , and some x ∈ IC + . Raisingboth sides of this equation to a sufficiently large power, remembering that I isnilpotent, gives the claim.)
In particular, if v ∈ C + is another element satisfying conditions (A), (B),and (C) above, then the powers of u and of v are mutually cofinal with respect to divisibility (i.e. the u-adic and v-adic topologies on C + coincide), andconsequently C + [1/u] = C + [1/v].
B.13 Remark. Recall that since R is an Artinian local ring, an R-module is flat
if and only if it is free [Sta, Tag 051G]. (We will use this fact in the proof of
Lemma B.25 below. For a proof, note first that if M is any R-module, thenany basis of M/IM lifts to a generating set of M . If M is furthermore flat,then lifting a basis of M/IM , we obtain a surjection F → M whose source isfree, with the additional property that, if N denotes its kernel, then N/IN = 0.
∼
Thus N = 0, and so F −→ M , as required.)
We begin by establishing some simple consequences of our assumptions on
C + and u.
B.14 Lemma.

236

APPENDIX B. GRADED MODULES AND RIGID ANALYSIS(1) For each n ≥ 1, the quotient C + /un is flat over R.
(2) C + itself is flat over R.
(3) The ring C + is u-adically complete.
Proof. Claim (1) follows by an evident induction from assumptions (A) and (B)together with a consideration of the short exact sequencesu·

0 → C + /un −→ C + /un+1 → C + /u → 0.
Now choose an increasing filtration (Im ) of R by ideals so that I0 = 0,
and such that each quotient Im+1 /Im is of length one (i.e. is isomorphic to k).
Taking into account (1), the short exact sequences
0 → Im → Im+1 → Im+1 /Im (∼= k = R/I) → 0
give rise to short exact sequences
0 → Im ⊗R (C + /un ) → Im+1 ⊗R (C + /un ) → C + /(I, un ) → 0.
Suppose that for some m we know that the natural morphism
Im ⊗R C + → lim Im ⊗R (C + /un )←−nis an isomorphism. Then passing to the inverse limit over n, and taking into
account both assumption (C) and the fact that the transition maps Im ⊗R(C + /un+1 ) → Im ⊗R (C + /un ) are surjective, so that the relevant lim1 vanishes,←−
we obtain a short exact sequence
0 → Im ⊗R C + → lim Im+1 ⊗R (C + /un ) → C + /I → 0.
←−n
The five lemma shows that the natural morphism to this sequence from theexact sequence
Im ⊗R C + → Im+1 ⊗R C + → C + /I → 0
is then an isomorphism. Proceeding by induction (the case m = 0 being trivial),
we find (once we reach the top of our filtration, so that Im = R) that C + is
u-adically complete, and we also find that each of the morphisms
I m ⊗R C + → C +is injective. Since any ideal J of R can be placed in such a filtration (Im ), we
find that C + is flat over R. Thus (2) and (3) are proved.
We write C := C + [1/u]. Since C is a localisation of C + , which is R-flatby Lemma B.14 (2), it is flat over R. Note also that Gr0 C ∼= (C + /I)[1/u] isa field that is complete with respect to a non-archimedean absolute value. In
particular, we can do rigid geometry over Gr0 C; this is a key point, which wewill exploit below.

B.11. A GENERAL SETTING

237

b R C + to denote theB.15 Definition. If M is an R-module, then we write M ⊗+b R C :=
u-adic completion of the tensor product M ⊗R C . We also write M ⊗b R C + )[1/u].
(M ⊗b R C + and M ⊗bR C
B.16 Remark. Remark B.12 shows that the formation of M ⊗is independent of the choice of u satisfying (A), (B), and (C) above.
B.17 Remark. In applications, we will apply this construction primarily in theb R C + and A ⊗b R C arecase when M = A is an R-algebra, in which case A ⊗again R-algebras.
b R C+ =B.18 Example. If we take C + := R[[u]] (for an indeterminate u), then A ⊗b R C = A((u)).
A[[u]], and A ⊗We next state and prove some additional properties of u-adic completions
that we will need.
B.19 Lemma. If M is an R-module, then multiplication by u is injective on
b A C + . Consequently (indeed, equivalently), the natural map M ⊗b R C+ →M⊗
b R C is injective.
M⊗
b R C + → M ⊗R (C + /um )Furthermore, for each m ≥ 0, the natural map M ⊗is surjective, and induces an isomorphism∼b R C + )/um (M ⊗b R C + ) −→ M ⊗R (C + /um ).
(M ⊗Proof. Note that the claim of the second paragraph is a general property ofcompletion with respect to finitely generated ideals (see e.g. the statement andproof of [Sta, Tag 05GG]). We will also deduce it as a byproduct of the proofof the claims in the first paragraph, to which we now turn.
b R C as the localization M ⊗b R C + [1/u], the second
Given the definition of M ⊗claim of the first paragraph is clearly equivalent to the first. As for this firstclaim, we note that for each m, n ≥ 1, Lemma B.14 (1) shows that the terms inthe short exact sequenceum ·0 → C + /un −→ C + /un+m → C + /um → 0
are flat R-modules. Thus, tensoring this short exact sequence with M over R
yields a short exact sequence
0 → M ⊗R (C + /un ) → M ⊗R (C + /un+1 ) → M ⊗R (C + /u) → 0.
Passing to the inverse limit over n, we obtain the exact sequenceum ·b R C + −→ M ⊗b R C + → M ⊗R C + /um C +0→M⊗from which the claims of the first paragraph follow. If we use the surjectivity ofthe transition maps in the inverse limit to infer that the relevant lim1 vanishes,−→
then we see that this sequence is even exact on the right, so that we also obtaina confirmation of the claim of the second paragraph.
B.20 Lemma.

238

APPENDIX B. GRADED MODULES AND RIGID ANALYSISb R C + and M 7→ M ⊗b R C are exact.
(1) The functors M 7→ M ⊗(2) If A is an R-algebra, if J is a finitely generated ideal in A, and if M isan A-module, then the natural map
b R C + ) → JM ⊗b R C+J(M ⊗is an isomorphism, and consequently there is a short exact sequenceb R C +) → M ⊗b R C + → (M/JM ) ⊗b R C + → 0.
0 → J(M ⊗(3) If J1 and J2 are ideals in an R-algebra A, with J1 ⊇ J2 , and if M is an
A-module, then there is a natural isomorphism∼b R C + )/J2 (M ⊗b R C + ) −→ (J1 M/J2 M ) ⊗b R C +.
J1 (M ⊗
Proof. If 0 → M1 → M2 → M3 → 0 is a short exact sequence of R-modules,
then Lemma B.14 (1) implies that
0 → M1 ⊗R (C + /un ) → M2 ⊗R (C + /un ) → M3 ⊗R (C + /un ) → 0
is exact for each n. If we pass to the inverse limit over n, and note that thetransition morphisms are evidently surjective, so that the relevant lim1 vanishes,←−
we obtain a short exact sequenceb R C + → M2 ⊗R C + → M3 ⊗R C + → 0,
0 → M1 ⊗proving the first exactness claim of (1). The second exactness claim follows fromthe first, together with the fact that the localization C := C + [1/u] is flat over∼b R C −→b R C + ) ⊗C + C.
C, and that there is an isomorphism M ⊗(M ⊗In order to prove (2), we apply (1) to the short exact sequence 0 → JM →M → M/JM → 0, obtaining a short exact sequenceb R C+ → M ⊗b R C + → (M/JM ) ⊗b R C + → 0.
0 → (JM ) ⊗(B.21)b R C + as a submodule of M ⊗b R C + . ThereThus we may (and do) regard (JM ) ⊗is a consequent inclusionb R C + ) ⊆ (JM ) ⊗b R C +,J(M ⊗(B.22)

whose image is u-adically dense in the target. Since J is a finitely generatedb R C + ) is furthermore u-adically complete, and thus
ideal, we see that J(M ⊗that (B.22) is actually an equality, establishing the first claim of (2).
b R C + by J(M ⊗b R C + ) in the short exact sequence (B.21)
Replacing (JM ) ⊗yields the short exact sequence whose existence is asserted in the second claimof (2).
Claim (3) follows directly from (2), and the exactness result of (1).
The following lemma will allow us to use grading techniques to study u-adic
completions.

B.24. LOCI OF VANISHING AND OF EQUIVARIANCE

239

B.23 Lemma. If M is an R-module, then there are natural
isomorphisms
∼(1) Gr• M ⊗k Gr0 (C + /un ) −→ Gr• M ⊗R (C + /un ) (for any n ≥ 1),∼
b R C + ) −→b k Gr0 C + , and(2) Gr• (M ⊗Gr• M ⊗∼••b R C) −→ Gr M ⊗b k Gr0 C.
(3) Gr (M ⊗Proof. Lemma B.14 (1) shows that C + /un is flat over R, and (1) follows from
this, together with Lemmas B.8 and B.10.
Lemma B.20 (3) (taking the inclusion J2 ⊆ J1 to be the various inclu∼
b R C + ) −→b R C +.
sions I i+1 ⊆ I i in turn) shows that each Gri (M ⊗Gri M ⊗i
Since I annihilates Gr (M ), the target of this isomorphism can be rewritten asb k Gr0 C+ . This proves (2).
Gri M ⊗∼The R-algebra C may be described as the direct limit C −→ limn u−n C +−→(the transition maps being the obvious inclusions). Since the formation of direct
limits is compatible with tensor products, we find that∼

Gr0 C −→ lim Gr0 u−n C + .
−→n
The isomorphism of (3) is thus obtained from the isomorphisms∼

b R u−n C + ) −→ Gr• M ⊗b k Gr0 u−n C +Gr• (M ⊗of (2) by passing to the direct limit over n.

B.24

Loci of vanishing and of equivariance

We maintain the notation of Section B.11, and prove some slightly technicalresults about the “locus of vanishing” and “locus of equivariance” of morphisms
b R C-modules.
between finitely generated projective A ⊗B.25 Lemma. Let A be an R-algebra, and let M be a finitely generated prob R C. If m ∈ M , then the “locus of vanishing” of m
jective module over A ⊗is a Zariski closed subset of Spec A. More precisely, there is an ideal J of
A such that for any morphism φ : A → B of R-algebras, the image of m inb R C) ⊗A ⊗MB := (B ⊗b R C M vanishes if and only if J is contained in the kernelof φ.
b R C, we may find anProof. Since M is finitely generated projective over A ⊗bother finitely generated projective A ⊗R C-module N such that M ⊕ N is freeb R C. Replacing m ∈ M by m ⊕ 0 ∈ M ⊕ N , we reof finite rank over A ⊗b R C ⊕r . Writing
duce to the case when M is free of finite rank, say M = A ⊗bm = (x1 , . . . , xr ) with xi ∈ A ⊗R C, we see that
P it suffices to construct a corresponding ideal Ji for each xi ; the ideal J := i Ji will then satisfy the claimof the lemma. Thus we reduce to the proving the lemma in the case of an elebR C = A ⊗b R C + [1/u]. Recall from Lemma B.19 that A ⊗b R C + isment x ∈ A ⊗b R C. Thus, since u is a unit in A ⊗R C, we may replace x bya subring of A ⊗b R C +.
un x for some sufficiently large value of n, and assume that x ∈ A ⊗240

APPENDIX B. GRADED MODULES AND RIGID ANALYSISb R C + ,→ A ⊗b R C with B ⊗b R C + overNote that tensoring the inclusion A ⊗+b R C induces the inclusionA⊗b R C.
b R C + ,→ B ⊗bR C = B ⊗b R C + [1/u] = (B ⊗b R C + ) ⊗A ⊗B⊗
b R C+ A ⊗b R C+
Thus it suffices to construct an ideal J in A such that the image of x in B ⊗vanishes if and only if the morphism φ : A → B factors through A/J. Let xn
denote the image of x in A ⊗R (C + /un ), so thatb R C + = lim A ⊗R (C + /un ).
x = (xn ) ∈ A ⊗←−n

If we let yn denote the image of xn in B ⊗R (C + /un ), then the image y of x inb R C is equal toB⊗b R C + = lim B ⊗R (C + /un ).
(yn ) ∈ B ⊗←−nThus it suffices to construct an ideal Jn in A such that yn vanishes if andonly if
Pthe morphism A → B factors through A/Jn ; indeed, we may then take
J := n Jn .
Lemma B.14 shows that C + /un is flat over R, and thus free over R. (SeeRemark B.13.) If we choose an isomorphism C + /un ∼= R⊕S , then we may write⊕S
xn = (xPn,s ) ∈ A . If we let Jn,s denote the ideal in A generated by xn,s , thenJn := s∈S Jn,s is the required ideal.
B.26 Corollary. Let A be an R-algebra, and let M and N be finitely generatedb R C. If f : M → N is an A ⊗b R C-module homomorprojective modules over A ⊗phism, then the “locus of vanishing” of f is a Zariski closed subset of Spec A.
More precisely, there is an ideal J of A such that for any morphism φ : A → B
of R-algebras, the base-changeb R C) ⊗A ⊗b R C) ⊗A ⊗fB : MB := (B ⊗b R C M → NB := (B ⊗bR C N
vanishes if and only if J is contained in the kernel of φ.
Proof. We may regard f ∈ HomA ⊗b R C (M, N ) as an element of the moduleM ∨ ⊗A ⊗b R C N.
The corollary then follows from Lemma B.25 applied to f so regarded.
B.27 Corollary. Let σ be a ring automorphism of C, let A be an R-algebra,b R C eachand let M and N be finitely generated projective modules over A ⊗endowed with a σ-semi-linear automorphism, denoted σM and σN respectively.
b R C-module homomorphism, then the “locus of σIf f : M → N is an A ⊗equivariance” of f is a Zariski closed subset of Spec A. More precisely, there
is an ideal J of A such that for any morphism φ : A → B of R-algebras, thebase-changeb R C) ⊗A ⊗b R C) ⊗A ⊗fB : MB := (B ⊗b R C M → NB := (B ⊗bR C Nsatisfies fB ◦ σM = σN ◦ fB if and only if J is contained in the kernel of φ.

B.24. LOCI OF VANISHING AND OF EQUIVARIANCE

241

Proof. This follows from Corollary B.26, applied to the morphism f − σN ◦ f ◦−1
σM.
b R C + -modules.
We also have the following variants on these results for A ⊗B.28 Lemma. Let A be an R-algebra, and let M and N be finitely generatedb R C + . Let f : M → N [1/u] be an A ⊗b R C + -module
projective modules over A ⊗homomorphism. Then the locus where f (M ) ⊆ N is a Zariski closed subsetof Spec A. More precisely, there is an ideal J of A such that for any morphism
φ : A → B of R-algebras, the base-changeb R C) ⊗A ⊗b R C) ⊗A ⊗fB : MB := (B ⊗b R C M → NB [1/u] := (B ⊗b R C N [1/u]satisfies fB (MB ) ⊆ NB if and only if J is contained in the kernel of φ.
Proof. We argue as in the proof of Lemma B.25. Since M, N are finitely generb R C + , we may find finitely generated projective modated projective over A ⊗b R C + such that M ⊕ P and N ⊕ Q are free. Replacing M
ules P, Q over A ⊗by M ⊕ P , N by N ⊕ Q, and f by (f, 0), we reduce to the case that M , N
b R C + -modules. After choosing bases we may suppose
are both finite free A ⊗b R C + )⊕r , N = (A ⊗b R C + )⊕s , and considering the matrix reprethat M = (A ⊗b R C, there issenting f , we reduce to the showing that for any element x ∈ A ⊗b R C + if and only if J is contained in
an ideal J of A such that (φ ⊗ 1)(x) ∈ B ⊗the kernel of φ. This is a special case of Lemma B.29 below.
B.29 Lemma. Let A be an R-algebra, and let M be a finitely generated prob R C + -module. Let x be an element of M [1/u]. Then the locus where
jective A ⊗x ∈ M is a Zariski closed subset of Spec A. More precisely, there is an ideal
J of A such that for any morphism φ : A → B of R-algebras, the image of x
in MB [1/u] lies in MB if and only if J is contained in the kernel of φ.
Proof. We begin by arguing as in the proof of Lemma B.28. Since M is finitely
b R C + , we may find a finitely generated projective
generated projective over A ⊗+bmodule P, Q over A ⊗R C such that M ⊕ P is free. Replacing M by M ⊕ P
b R C + -module.
and x by (x, 0), we reduce to the case that M is a finite free A ⊗+ ⊕rb
After choosing a basis we may suppose that M = (A ⊗R C ) , so that we canb R C +.
reduce to the case that M = A ⊗b R C + . Then (φ ⊗ 1)(x) ∈Choose n sufficiently large so that un x ∈ A ⊗+b
B ⊗R C if and only if the image of (φ ⊗ 1)(x) inb R C + )/(B ⊗b R C + ) = B ⊗R (u−n C + )/C + )u−n (B ⊗vanishes. As in the proof of Lemma B.25, u−n C + /C + is a free R-module,
so we can and do choose an isomorphism u−n C + /C + ∼= R⊕S . Then if we−n ++write x = (xs ) ∈ A⊕S ∼
A
⊗
(u
C
/C
),
we
can
take
J to be the ideal=R
generated by the xs .

242

B.30

APPENDIX B. GRADED MODULES AND RIGID ANALYSIS

Extending actions

We continue to remain in the context of Section B.11. We will show that various
sorts of actions on C + or C can be extended to the completed tensor products
b R C + and A ⊗b R C.
A⊗
Throughout our discussion, we endow C + with the u-adic topology, while
we endow C with the unique topology which makes it a topological group, andin which C + is an open subgroup. More generally, if A is any R-algebra, we
b R C + with the u-adic topology, while we endow A ⊗b R C withagain endow A ⊗b R C+
the unique topology which makes it a topological group, and in which A ⊗(endowed with its u-adic topology) is an open subgroup.
B.31 Lemma. Suppose that C + (resp. C) is equipped with a continuous Rlinear endomorphism ϕ. Then there is a unique extension of ϕ to a continuousb R C + (resp. A ⊗b R C).
A-linear endomorphism of A ⊗b R C + coincides with its
Proof. Lemma B.19 shows that u-adic topology on A ⊗+b R C := lim A ⊗R (C + /un ), andinverse limit topology, if we recall that A ⊗←−n
we endow each of the objects appearing in the inverse limit with its discrete
topology. Given this, it is immediate that a u-adically continuous R-linear
endomorphism ϕ of C + extends uniquely to a u-adically continuous A-linearb R C +.
endomorphism of A ⊗The case when ϕ is a continuous R-linear endomorphism of C is only slightly
more involved. Namely, to say that ϕ : C → C is continuous is to say that,for some m ≥ 0, we have an inclusion ϕ(um C + ) ⊆ C + , and that the induced
morphism ϕ : um C + → C + is continuous, when each of the source and thetarget are endowed with their u-adic topologies. Again taking into accountLemma B.19, we then obtain a unique A-linear and continuous morphismb R C = lim A ⊗R (um C + /um+n ) [1/u]
A⊗
←−
n

b R C,−→ lim A ⊗R (C + /un ) [1/u] = A ⊗←−n
as required.
If G is a topological group acting continuously on C + (i.e. acting in sucha way that the action map G × C + → C + is continuous), then for any g ∈ G
and m ≥ 0, we may find n ≥ 0, and an open neighbourhood U of g, such thatU × un C + ⊆ um C + . (This just expresses the continuity of the action map at
the point (g, 0) ∈ G × C + .) If G is furthermore compact, then G is covered byfinitely many such open sets U , and thus for any given choice of m, we may infact find n such that G · un C + ⊆ um C + . An identical remark applies in thecase of a compact topological group acting on C.
We will need to introduce an additional condition on G-actions on C, whichis related to, but doesn’t seem to follow from, the previous considerations.
B.32 Definition. If G is a group, we say that a G-action on C is bounded if,
for any M ≥ 0, there exists N ≥ 0 such that G · u−M C + ⊆ u−N C + .

B.35. A RIGID ANALYTIC PERSPECTIVE

243

B.33 Remark. It follows from Remark B.12 that the notion of a group action
being bounded is independent of the choice of u ∈ C + satisfying conditions (A),(B), and (C) of Section B.11.
We then have the following lemma.
B.34 Lemma. Suppose that G is a compact topological group, and that C +(resp. C) is equipped with a continuous (resp. continuous and bounded) R-linear
G-action. Then there is a unique extension of this action to a continuous Ab R C + (resp. A ⊗b R C).
linear action of G on A ⊗Proof. The proof of this lemma is very similar to that of Lemma B.31. Indeed,
applying that lemma (and taking into account its uniqueness statement) we
obtain the desired action of G. It remains to confirm that this action is again
continuous.
Consider first the case of a G-action on C + . For each m ≥ 0, we saw abovethat G · un C + ⊆ um C + if n is sufficiently large, so that we obtain a wellb R C + = G × lim A ⊗R C + /un → A ⊗R C + /um which
defined map G × A ⊗←−nis continuous (since it is just obtained from the continuous G-action on C + by
tensoring with A). Passing to the projective limit in m then yields the desired
continuity.
In the case of a G-action on C, we argue similarly: namely, the assumptionsof continuity and boundedness of the G-action yield continuous morphisms
G × u−M C + /un C + → u−N C + /um C + ;Passing to the projective limit in n, then in m, and then to the inductive limit
in N , and then in M , we deduce the desired continuity of the G-action on
b R C.
A⊗

B.35

A rigid analytic perspective

We keep ourselves in the setting of Section B.11. We are interested in studyb R C, especially their descenting finitely generated projective modules over A ⊗properties. In the present general context, we don’t know whether an analogue of Drinfeld’s descent result for Tate modules (see [Dri06, Thm. 3.3, 3.11]
and [EG21, Thm. 5.1.18]) holds for arbitrary R-algebras A. However, we will
see that such a descent result is valid in the more restricted setting of finite
type R-algebras. In the case of finite type k-algebras, we will prove this using
results from rigid analysis. For finite type R-algebras that are not necessarily annihilated by I, we will use grading techniques to reduce to the case of
k-algebras.
b k Gr0 C + ) (where the Spf is taken
If A is a finite type k-algebra, then Spf(A ⊗b k Gr0 C + ) is a formal scheme of finite
with respect to the u-adic topology on A ⊗0 +b k Gr0 C) is
type over Spf Gr C , whose rigid analytic generic fibre Max Spec(A ⊗an affinoid rigid analytic space over the complete non-archimedean field Gr0 C.
As an application of this rigid analytic point-of-view (together with gradedtechniques), we first establish the following proposition.

244

APPENDIX B. GRADED MODULES AND RIGID ANALYSIS

B.36 Proposition.
b R C is Noetherian.
(1) If A is a finite type R-algebra, then A ⊗(2) If A → B is a (faithfully) flat morphism of finite type R-algebras, thenb R C+ → B ⊗b R C + and A ⊗bR C → B ⊗b R C arethe induced morphisms A ⊗again (faithfully) flat.
Proof. In the case when A is in fact a k-algebra, it follows from the precedb R C is an affinoid algebra over Gr0 C, and is therefore
ing discussion that A ⊗Noetherian by [BGR84, Thm. 1, §5.2.6]. In the general case, Lemma B.23 (3)∼b R C) −→b k Gr0 C. Since Gr• A is a finite
gives an isomorphism Gr• (A ⊗Gr• A ⊗
type k-algebra, it follows from the case already proved that the target of thisb R C is itself
isomorphism is Noetherian, and thus so is its source. Thus A ⊗Noetherian, by Lemma B.4.
We explain how our rigid analytic perspective proves a part of (2). If we
suppose first that A → B is a flat morphism of k-algebras, then so is themorphism A ⊗R C → B ⊗R C. Now the preceding discussion shows that theb R C) → Max Spec(A⊗R C) and Max Spec(B ⊗b R C) →morphisms Max Spec(A ⊗Max Spec(B ⊗R C) are open immersions of rigid analytic spaces over Gr0 C, sob R C) → Max Spec(B ⊗b R C) is alsothat the induced morphism Max Spec(A ⊗flat. This proves (the k-algebra case of) the second flatness claim of (2). If
A → B is a flat morphism of R-algebras that are not necessarily k-algebras,then we deduce from Lemma B.8 that Gr• A → Gr• B is flat, and thus, fromwhat we’ve already shown, thatb k Gr0 C → Gr• B ⊗b k Gr0 C
Gr• A ⊗
is flat. Lemma B.23 (3) then shows that the morphismb R C) → Gr• (B ⊗b R C)Gr• (A ⊗is flat, and one more application of Lemma B.8 completes the proof of the second
flatness claim of (2).
It is not clear to us whether one can deduce the second faithful flatness claimof (2) by this style of argument. In order to prove this result, as well as to obtainthe first set of claims of (2), we appeal to the results of [FGK11]. Indeed, it isclear that the second set of claims in (2) follows immediately from the first, andit the first set of claims that we will now prove.
We first note that if the morphism A → B is flat (resp. faithfully flat), thennso are each of the morphisms A ⊗R C + /un A → B ⊗+R C/u . In the terminology++bbof [FGK11, §5.2], the morphism A ⊗R C → B ⊗R C is adically flat (resp. adically faithfully flat); here we regard the source and target as being endowed with
their u-adic topologies. The discussion at the beginning of [FGK11, §5.2] (seeb R C+ → B ⊗b R C + is flat. In thealso our Proposition 2.2.2) then shows that A ⊗adically faithfully flat case, we deduce in addition from [FGK11, Prop. 5.2.1 (2)]b R C+ → B ⊗b R C + is faithfully flat;(again, see also Proposition 2.2.2) that A ⊗b R C + and B ⊗b R C + are Noetherian outside u by (1) of the present
note that A ⊗proposition.

b R C-MODULES
B.37. DESCENT FOR A ⊗

B.37

245

b R C-modules
Descent for A ⊗We now establish the descent result alluded to above; it is analogous to Drinfeld’s[EG21, Thm. 5.1.18 (1)] but is restricted to the context of finite type R-algebras.
We use grading techniques to reduce to the case of finite type k-algebras, where
we can then apply known results from rigid analysis.
B.38 Theorem. Let A → B be a faithfully flat morphism of finite type Rb R C) induces an equivalence
algebras. Then the functor M 7→ M ⊗A ⊗b R C (B ⊗bof categories between the category of finite type A ⊗R C-modules and the categoryb R C-modules equipped with descent data.
of finite type B ⊗B.39 Remark. Recall (for example from [BLR90, §6.1]) that if S → S 0 is a ringhomomorphism, then a descent datum for an S 0 -module M 0 is an isomorphism∼of S 0 ⊗S S 0 -modules S 0 ⊗S M 0 −→ M 0 ⊗S S 0 , satisfying a certain cocycle condition.
Given a descent datum, we have a pair of morphisms of S-modules M 0 →M 0 ⊗S S 0 , namely the obvious morphism and the composite of the obvious
morphism M 0 → S 0 ⊗S M 0 with the descent datum isomorphism. We let K
denote the S-module given by the kernel of the difference of these two maps.
We refer to the functor M 0 7→ K, from the category of S 0 -modules with descentdata to the category of S-modules, as the kernel functor. (If S → S 0 is faithfully
flat, then the theory of faithfully flat descent shows that M 0 is the extension ofscalars from S to S 0 of K.)
Proof. We first treat the case when A and B are k-algebras. In this case, theb R C-modules (resp. finite type B ⊗b R C-modules) is
category of finite type A ⊗equivalent to the category of coherent sheaves on the affinoid rigid analytic
b R C) (resp. Max Spec(B ⊗b R C)) over the field Gr0 C, andspace Max Spec(A ⊗the statement follows from faithfully flat descent for rigid analytic coherent
sheaves [BG98, Thm. 3.1].
We reduce the general case of R-algebras to the case of k-algebras via theusual graded arguments. To ease notation, we let F : C → D denote the baseb R C) from the category C of finite typechange functor M 7→ M ⊗A ⊗b R C (B ⊗b R C-modules to the category D of finite type B ⊗b R C-modules equipped withA⊗
descent data, and let G : D → C denote the kernel functor. There are evident
natural transformations
G ◦ F → idC

andF ◦ G → idD ,(B.40)

which we claim are natural isomorphisms (so that G provides a quasi-inverse to
F , proving in particular that F induces an equivalence of categories).
b R C)-modules, and let D
We let C denote the category of finite type Gr0 (A ⊗0bdenote the category of finite type Gr (B ⊗R C)-modules equipped with descentb R C). We let F : C → D denote the base-change functor
data to Gr0 (A ⊗0b R C), and let G : D → C denote the kernel functor. The– ⊗Gr0 (A ⊗Gr(B ⊗b R C)
case of k-algebras that we have already treated shows that there are natural
isomorphisms∼∼
G ◦ F −→ idC and F ◦ G −→ idD .
(B.41)

246

APPENDIX B. GRADED MODULES AND RIGID ANALYSIS

We think of passage to the associated graded Gr• as inducing functors C →C and also D → D. It follows from Lemmas B.8 and B.10, together with
Proposition B.36, that there is a natural isomorphism∼Gr• ◦ F −→ F ◦ Gr• ;there is also an evident natural isomorphism∼Gr• ◦ G −→ G ◦ Gr• ;furthermore, these natural isomorphisms are compatible with the natural transformations (B.40) and (B.41). Since the natural transformations (B.41) are isomorphisms, the same is true of the natural transformations (B.40), by Lemma B.7.
This completes the proof that F is an equivalence.

Appendix C

Topological groups andmodulesIn this appendix we recall some more-or-less well-known facts regarding topological groups and modules, for which we haven’t located a convenient reference.
We found the note [Kay96] to be a useful reference for the basic facts regarding
Polish topological groups.
Recall the following definition.
C.1 Definition. A topological space is called Polish if it is separable (i.e.
contains a countable dense subset) and completely metrizable. A topologicalgroup is called Polish if its underlying topological space is Polish. Similarly, a
topological ring is called Polish if its underlying topological space is Polish.
Our interest in Polish groups is due to the following lemma and corollary;see also [Sta, Tag 0CQW] for a more algebraic proof of a closely related result.
C.2 Lemma. If φ : G → H is a continuous homomorphism between two Polishtopological groups, then the following are equivalent:
1. φ is open.
2. φ(G) is not meagre in H.
Proof. If (1) holds, then φ(G) is a non-empty open subset of the Polish, andhence Baire, space H, and so is not meagre. Thus (1) implies (2). For theconverse, see e.g. [Kay96, Thm. 18].
C.3 Corollary. A continuous surjective homomorphism of Polish topologicalgroups is necessarily open.
Proof. A completely metrizable space is Baire, and hence is not a meagre subset
of itself. The corollary thus follows from the implication “(2) =⇒ (1)” of
Lemma C.2.
247

248

APPENDIX C. TOPOLOGICAL GROUPS AND MODULES

Any metrizable space is first countable (i.e. each point contains a countableneighbourhood basis). Recall that, conversely, if G is a topological group, then
G is first countable if and only if the identity element 1 admits a countableneighbourhood basis, and in this case, if (the underlying topological space of) G
is Hausdorff then it is in fact metrizable (this is the Birkhoff–Kakutani theorem),and even admits a (left or right) translation invariant metric. In particular, (theunderlying topological space of) a Hausdorff topological group is separable andmetrizable if and only if it is second countable (i.e. admits a countable basis for
its topology).
Recall also that a topological group G admits a canonical uniform structure,so that it makes sense to speak of G being complete. For the sake of completeness, we note that the underlying topological space of G being Polish forces G
to be complete.
C.4 Lemma. If G is a Polish topological group, then G is complete.
Proof. Since G is metrizable, it is Hausdorff, and so we may consider the canonb of G into its completion. Since G
ical embedding of topological groups G ,→ Gbbb is a complete
is separable and dense in G, we see that G is separable. Also, G
metric space (it may be identified with the metric space completion of G withb is Polish.
respect to any invariant metric inducing the topology on G). Thus G
Any Polish subset of a Polish space is Gδ (i.e. a countable intersection of openb Since G
b is Polish, and thus Baire,sets), and so in particular G is Gδ inside G.
b It follows from Lemma C.2 that the inclusion
we find that G is not meagre in G.
b is open, and thus that G is an open subgroup of its completion G.
b Since
G ,→ Gan open subgroup of a topological group is also closed, we find that G closed,b and thus that G = G,b which is to say, G is complete, as
as well as dense, in G,
claimed.
C.5 Remark. It follows from Lemma C.4 and the discussion preceding it that a
topological group is Polish if and only if it is complete and second countable.
We next recall a result about Noetherian and Polish topological rings, whichis inspired by a result of Grauert and Remmert in the theory of classical Banach
algebras [GR71, App. to §I.5]. (See [BGR84, Prop. 3, §3.7.3] for the analogous
result in the context of non-archimedean Banach algebras.)
C.6 Proposition. Let A be a Polish (or, equivalently, a (Hausdorff ) complete
and second countable) topological ring which is Noetherian (as an abstract ring),
and which contains an open additive subgroup that is closed under multiplication,and consists of topologically nilpotent elements. Then any finitely generated Amodule M has a unique completely metrizable topology with respect to which it
becomes a topological A-module. Furthermore, M is complete with respect to this
topology, any submodule of M is closed, and any morphism of finitely generated
A-modules is automatically continuous, has closed image, and induces an open
mapping from its domain onto its image.
Proof. To begin with, suppose that M is a finitely generated A-module which
is furthermore endowed with a completely metrizable topology which makes it249
a topological A-module. We claim first that M is in fact complete, and thatany surjection of A-modules An → M (for some n ≥ 1) is continuous and open.
It then follows that M is endowed with the quotient topology via this map,and thus the completely metrizable topology on M is uniquely determined (ifit exists).
To see the claim, note that since M is a finitely generated A-module, we may
choose a surjective homomorphism of A-modules An → M for some n ≥ 1.
Furthermore, since M is a topological A-module, any such surjection is continuous. Since A is separable as a topological space by assumption, we see that M
is as well. (The image of any countable dense subset of An is a countable dense
subset of M .) Thus M is Polish, as is An , and it follows from Lemma C.4 that
M is complete (as a topological module), while it follows from Corollary C.3
that the given surjection An → M is open, as claimed.
We next claim that any A-submodule of M is closed. To see this, let N
be a submodule of M , and let N denote its closure. Since A is Noetherian
by assumption, so is its finitely generated module M , and thus N is finitely
generated, say by the elements x1 , . . . , xn . Applying the results proved above
for M to its closed (and hence completely metrizable topological)PsubmodulenN , we find that the surjection An → N given by (a1 , . . . , an ) 7→ i=1 ai xi is
an open mapping. Consequently, if we let I be an open additive subgroup of
A which satisfies I 2 ⊆ I, and which consists of topologically nilpotent elements(such an I exists by assumption), then Ix1 + · · · + Ixn is an open subset of
N . Since N is dense in N , we conclude that N + Ix1 + · · · + Ixn = N , andconsequently we may write
xi =

n
X

aij xj + yi

i=1

for each 1 ≤ i ≤ n, for some yi ∈ N and aij ∈ I. Rearranging, we find that   
1 − a11
−a12
···
−a1n
x1
y1
 −a21





1
−
a
·
·
·
−a
x
22
2n   2  y2 
= . 
 ..



..
.
.
.
..
..
 .
  ..   .. .
−an1−a2n · · · 1 − annxnyn
1 − a11
 −a21
The determinant of the matrix  .
 ..

−a12
1 − a22
..
.

···
···
..
.


−a1n
−a2n 
 lies in 1 + I,
..

.

−an1
−a2n · · · 1 − ann
and thus is a unit (since A is complete and I consists of topologically nilpotent
elements). Thus we find that x1 , . . . , xn lies in the A-span of y1 , . . . , yn , so thatin fact N ⊆ N . Thus N is indeed closed, as claimed.
Applying the preceding result to An , we find that any A-submodule of An
is closed, and hence that any finitely generated A-module M is isomorphic to aquotient An /N , where N is a closed submodule of An . The quotient topology

250

APPENDIX C. TOPOLOGICAL GROUPS AND MODULES

on An /N makes it into a complete and metrizable topological A-module, andthus M does indeed admit a complete and metrizable topological A-module
structure. We have already seen that this topology is unique, and that any Asubmodule of M is closed. Finally, we see that homomorphisms between finitely
generated A-modules are necessarily continuous and that their images (being
A-submodules of their targets) are necessarily closed, and a final application ofCorollary C.3 shows that they induce open mappings onto their images.

Appendix D

Tate modules andcontinuity
In this appendix we study continuity conditions for group actions on modules
over Laurent series rings. We begin with some results on the topology of such
modules, before introducing group actions and related notions.

D.1

Topologies and lattices

We fix a finite extension E/Qp with ring of integers O, uniformizer $, andresidue field F, and we also fix a finite extension k/Fp . If A is a p-adicallycomplete O-algebra, we write A+
A := (W (k) ⊗Zp A)[[T ]], and we let AA be thep-adic completion of A+
[1/T
].
A
D.2 Remark. Any finitely generated projective AA -module M has a natural
topology. Indeed, we may write M as a direct summand of AnA for some n ≥ 1.
We then endow this latter module with its product topology, and endow M withthe subspace topology. More intrinsically, since A is p-adically complete (byassumption), we may write M = lima M/pa M . Each of the quotients M/pa M←−
is then a projective (A/pa A)((T ))-module, and so has natural topology, making
it a Tate A/pa A-module in the sense of [Dri06]. The topology on M is then the
projective limit of these Tate module topologies. We note that multiplication
by T is topologically nilpotent on M .
Similarly, any Zariski locally finite free A+
A -module M has a natural topology.
We may describe this topology in an analogous manner to the case consideredin the preceding paragraph. Namely, such a module is a finitely generated+ n
and projective A+
A -module, and thus is a direct summand of (AA ) for somen ≥ 0. If we endow this latter module with its product topology then the natural
topology on M is its corresponding subspace topology. In this case, though, this
topology admits a more intrinsic and succinct description: it is the (p, T )-adic
topology on M.
251

252

APPENDIX D. TATE MODULES AND CONTINUITY

We remind the reader that a topological group G is said to be Polish if its
underlying topological space is Polish, i.e. is separable and completely metrizable. As explained in Remark C.5, this is equivalent to G being complete (as atopological group) and second countable (as a topological space).
D.3 Lemma. If A is a p-adically complete O-algebra for which A/p is countable, then AA is Polish, and consequently any finitely generated projective AA module is Polish, when endowed with its canonical topology.
Proof. We will use the fact that a countable product of Polish spaces is Polish,
as is a closed subspace of a Polish space. It follows from these facts, and fromthe description of the natural topology on a finitely generated projective AA module given in Remark D.2, that the canonical topology on any such module
is Polish, provided that that AA itself is Polish.
We may write AA = lima AA/pa A , so that AA is a closed subset of the←−(countable) product of the various spaces AA/pa A . It suffices, then, to showthat each of these spaces is Polish; in other words, we reduce to the case when
A is a countable Z/pa -algebra for some a ≥ 1.
n
Since A is countable, so is each of the quotients A+
a subset
A /T . ChooseS++nXn ⊆ AA which maps bijectively onto AA /T , and write X := m,n T −m Xn .
Then X is a countable dense subset of AA , and thus AA is separable.
The topology on A+
A is the T -adic topology, and thus is metrizable (as isany I-adic topology on a ring). Since A+
A is T -adically complete, it is in factcompletely metrizable. The same is then evidently true for[
AA := A+T −m A+A [1/T ] =A.
m

D.4 Remark. Clearly T A+
A is an open subgroup of AA that is closed undermultiplication and consists of topologically nilpotent elements. Thus if A/p iscountable, then AA satisfies the conditions of Proposition C.6, and thus the
canonical topology on finitely generated projective AA -modules constructed in
Remark D.2 is a particular case of the canonical topology constructed on any
finitely generated AA -module in Proposition C.6.
We now present some additional facts related to the preceding concepts
which will be needed in the sequel.
D.5 Lemma. A finitely generated A+
A -module is projective of rank d if and only
if for each a ≥ 1, the quotient M/pa M is projective of rank d as an A+
A/pa A module. Similarly, a finitely generated AA -module M is projective of rank d if
and only if for each a ≥ 1, the quotient M/pa M is projective of rank d as an
AA/pa A -module.
Proof. This is immediate from [GD71, Prop. 0.7.2.10(ii)], applied to the padically complete rings A+
A and AA .
D.6 Definition. If M is a finitely generated AA -module, then a lattice in M
is a finitely generated A+
A -submodule M ⊆ M whose AA -span is M .

D.1. TOPOLOGIES AND LATTICES

253

Note that any finitely generated AA -module contains a lattice. (If f : ArA →M is a surjection from a finitely generated free AA -module onto the finitelyrgenerated AA -module M , then the image of the restriction of f to (A+A ) is alattice in M .)We record some additional lemmas and a remark which apply in the casewhen A is an O/$a -algebra for some a ≥ 1.
D.7 Lemma. Let A be an O/$a -module for some a ≥ 1, and let M be a finitely
generated AA -module.
1. If M and N are two lattices contained in a finitely generated AA -module M ,then there exists n ≥ 0 such that T n M ⊆ N ⊆ T −n M.
2. If in addition either A is Noetherian or M is projective, then any latticein M is T -adically complete.
3. If A is Noetherian, if M is a lattice in M , and if N is an A+
A -submoduleof M such that T n M ⊆ N ⊆ T −n M for some n ≥ 0, then N is a latticein M .
Proof. To prove (1), it suffices to prove one of the inclusions; the reverse inclusion may then be obtained (possibly after increasing n) by switchingroles
S∞ the−nof M and N. Since AA = A+
[1/T
],
we
find
that
N
⊆
M[1/T
]
=
T
M.
A
n=0
−n
Since N is finitely generated as an A+-module,weobtain
that
N
⊆
T
M
for
A
some sufficiently large value of n, as required.
To prove (2), we note that if A is Noetherian, than A+
A is also Noetherian,and thus any finitely generated A+-moduleisT-adically
complete, since A+AA
itself is. If M is projective, then we write M as a direct summand of a finitely
generated free AA -module, so that M may then be embedded into a finitely
generated free A+
A -module. Thus M is T -adically separated, and hence the+ rrkernel of any surjection (A+A ) → M is T -adically closed. Since (AA ) is T adically complete, we conclude that the same is true of M.
Suppose now that A is Noetherian, and that we are in the situation of (3).
The inclusion T n M ⊆ N shows that M = M[1/T ] ⊆ N[1/T ], so that N generates M as an AA -module. We must show that N is furthermore finitely
generated over A+
A . For this, we first note that, since M is T -adically complete(by (2)), the inclusion T n M ⊆ N ⊆ T −n M shows that N is also T -adicallycomplete. It also shows that N/T N is a subquotient of the finitely generated
A-module T −n M/T n+1 M, and thus is a finitely generated A-module (as A is
Noetherian). Since N is T -adically complete, an application of the topological
Nakayama lemma shows that N is finitely generated over A+
A , as required.
D.8 Remark. By [EG21, Thm. 5.1.14] (a theorem of Drinfeld), if A is an O/$a algebra for some a ≥ 1, then we may think of a finitely generated projective AA -module as a Tate A-module M together with a topologically nilpotent
automorphism T . In this optic a lattice in our sense is precisely a lattice in the
Tate module M which is also an A+
A -submodule (by definition, a lattice L ⊆ M

254

APPENDIX D. TATE MODULES AND CONTINUITY

is an open submodule with the property that for every open submodule U ⊆ L,
the A-module L/U is finitely generated), as the following lemma shows.
D.9 Lemma. If A is an O/$a -algebra for some a ≥ 1, and if M is an A+
Asubmodule of a finitely generated projective AA -module M , then the following
conditions on M are equivalent:
1. M is a lattice in M (in the sense of Definition D.6)
2. M is open in M , and for every A+
A -submodule U of M which is open in M ,the quotient M/U is finitely generated over A.
3. M is open in M , and for every A-submodule U of M which is open in M ,the quotient M/U is finitely generated over A (i.e. M is a lattice in M ,when M is thought of as a Tate module over A in the sense of Remark D.8).
Proof. Suppose that M is a lattice in M , in the sense of Definition D.6. If werchoose a surjection (A+A ) → M for some n ≥ 0, then the induced surjectionrr
AA → M is a continuous and open map, and so M is open in M , since (A+
A)
+ r
r
n
is open in AA . Furthermore, since the submodules T (AA ) (n ≥ 0) formrna neighbourhood basis of 0 in (A+A ) , we see that their images T M form aneighbourhood basis of 0 in M. Thus if U is any open A-submodule of M, thenM/U is a quotient of M/T n M for some n ≥ 0, and thus is finitely generated
over A. Thus (1) implies (3).
Clearly (3) implies (2), and so we suppose that (2) holds. Let N be anA+of M that is a lattice inSthe sense of Definition D.6, so that
A -submodule S
M = N[1/T ] = n≥0 T −n N. Then M = n≥0 M ∩ T −n N, and thus
M/(M ∩ N) =[(M ∩ T −n N)/(M ∩ N).

n≥0

But M/(M ∩ N) is finitely generated over A by assumption, and thus
M/M ∩ N = (M ∩ T −n N)/(M ∩ N),
for some sufficiently large value of n, implying that M ⊆ T −n N. In particular
M is T -adically complete, being open (and hence closed) in the lattice T −n N,
which is T -adically complete by Lemma D.7 (2). Since T is an automorphismof M , we find that T M is an open submodule of M, so that M/T M is finitely
generated over A. As M is T -adically complete, the topological Nakayama
lemma implies that M is finitely generated over A+A.
Since T is topologically nilpotent and N is finitely generated over A+
A , wealso find that T n N ⊆ M for some sufficiently large value of n. Thus M[1/T ] =N[1/T ] = M. This completes the proof that M is a lattice in M in the sense of
Definition D.6, showing that (2) implies (1).
D.10 Remark. It follows from Lemmas D.7 and D.9 that if A is a O/$a -algebra,
and M is a finitely generated projective AA -module, then the lattices in M forma neighbourhood basis of the origin in M .

D.12. GROUP ACTIONS

255

We also note the following technical lemma.
D.11 Lemma. If A ⊆ B is an inclusion of O/$a -algebras for some a ≥ 1,with A Noetherian, if M is a projective AA -module M with extension of scalars
MB := AB ⊗AA M to B, and if MB is a lattice in MB , then M := M ∩MB (theintersection takes place in MB ) is a lattice in M , having the additional property
that M ∩ T n MB = T n M for any integer n.
Proof. Choose a projective complement to M , i.e. a finitely generated projective
AA -module N such that M ⊕ N is free over AA , and let NB be a lattice in NB .
Then it suffices to prove the lemma with M replaced by M ⊕ N and with MB
replaced by MB ⊕ NB ; thus we may and do suppose that M is free over AA .
+00
Choose a lattice M0 in M which is free over A+
A , and note that MB := AB Mn0n0
is a lattice in MB , and that M ∩ T MB = T M for any integer n.
Lemma D.7 (1) shows that T n M0B ⊆ MB ⊆ T −n M0B for some sufficiently
large value of n. Intersecting with M , we find that T n M0 ⊆ M ⊆ T −n M0 , sothat M is a lattice in M , by Lemma D.7 (3). Finally, by the definition of M,
we find that M ∩ T n MB = T n M for any integer n.

D.12Group actions

We now prove some lemmas which allow us to check whether the action of a
topological group on a Tate module is continuous, and to extend such an action
from a group to its completion.
D.13 Lemma. Let G be a topological group acting on a Tate module M , andassume that G admits a neighbourhood basis of the identity consisting of open
subgroups. Then the following are equivalent:
1. The action G × M → M is continuous.
2. (a) For each m ∈ M , the map G → M , g 7→ gm is continuous at theidentity of G,(b) for each g ∈ G, the map M → M , m 7→ gm is continuous, and(c) for any lattice L in M , there is an open subgroup H of G that preserves L.
Furthermore, if these equivalent conditions hold, then the following stronger
form of condition (2)(a) holds:(2)(a’) For each m ∈ M , the map G → M , g 7→ gm is equicontinuous at theidentity of G.
D.14 Remark. The assumption in Lemma D.13 that G admits a neighbourhoodbasis of open subgroups holds in particular if G is a profinite group, or if G isa subgroup of a profinite group with the subspace topology. In particular, itholds for the groups Zp and for Z ⊂ Zp .

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Proof of Lemma D.13. Suppose first that (1) holds; then clearly conditions (2)(a)and (2)(b) hold. Let L be a lattice in M . Since the action morphism G×M → M
is continuous, we may find an open subgroup H of G and an open submodule
U of M , such that HU ⊆ L. Replacing G by H, we may thus suppose thatGU ⊆ L. Now consider the morphism G × L/U → M/L induced by the action
morphism. Since L is a lattice, the A-module L/U is finitely generated. Let{mi } be a finite generating set. Since M/L is discrete, for each generator mi ,there is an open subgroup Hi of G such
T that Hi mi is constant, and thus equal
to zero, modulo L. If we write H := i Hi , then H is an open subgroup of G
such Hmi ⊆ L for every i. Consequently, HL ⊆ L, verifying that (2)(c) holds
as well.
Suppose conversely that (2) holds. We first note that we may strengthen (2)(a)as follows: if m ∈ M , then the orbit map g 7→ gm is a continuous map G → M .
Indeed, if g0 ∈ G, then we may write this map as the composite of the continuous automorphism g 7→ gg0−1 of G and the orbit map g 7→ gg0 m. The latterorbit map is continuous at the identity of G, by (2)(a), and so the orbit map ofm is continuous at g0 .
We now wish to prove that the action map G × M → M is continuous.
Let (g, m) ∈ G × M. Any neighbourhood of the image gm ∈ M contains aneighbourhood of the form gm + L, where L is a lattice in M . Given a latticeL, then, we must find a neighbourhood of (g, m) whose image lies in gm + L.
Since g is a continuous automorphism of M , by (2)(a), we may find a lattice
L0 such that gL0 ⊆ L. By (2)(c), we may find an open subgroup H 0 ⊆ G that
preserves L0 . Then H 0 acts on M/L0 , and since this quotient is discrete, andsince the orbit maps for the action are continuous (by what we proved above),it follows that the action of H 0 on M/L0 is smooth, and so we may find an
open subgroup H of H 0 that fixes the image of m in M/L0 . Then we find that(gH)(m + L0 ) ⊆ gm + L, and since gH × (m + L0 ) is an open neighbourhood of(g, m) ∈ G × M , we have proved the required continuity.
Finally, suppose that conditions (1) and (2) hold; we must show that thecontinuity condition of (2)(a) can be upgraded to the equicontinuity conditionof (2)(a’). To do this, we need to show that for any two lattices L, L0 ⊆ M , there
is an open subgroup H of G such that for each m ∈ L, we have Hm ⊆ m + L0 .
Replacing L0 by L ∩ L0 , we can assume that L0 ⊆ L. By (2)(c), we canchoose H such that HL0 ⊆ L0 and HL ⊆ L, so that we have an induced
continuous map H × (L/L0 ) → L/L0 . Since L/L0 is a finitely generated Amodule and is discrete, after replacing H by an open subgroup we can assume
that H acts trivially on L/L0 , as required.
D.15 Lemma. Let G be a Hausdorff topological group, which admits a neighbourhood basis of the identity consisting of open subgroups, and suppose furthermore that for any open subgroup H of G, the quotient G/H is finite; equivalently,
b of G is profinite.
suppose that the completion G
If M is a Tate module, endowed with a continuous action G × M → M , thenb × M → M.
the action G × M → M extends to a continuous action G

D.16. T -QUASI-LINEAR ENDOMORPHISMS

257

Proof. We have to show that the continuous action G × M → M extends (necb to a continuous map G×Mbessarily uniquely, since G is dense in G)→ M . (Thebfact that this map will induce a G-action on M follows from the correspondingb Since M is the union of its lattices, it
fact for G, and the density of G in G.)
suffices to show that the induced map G × L → M extends to a continuous map
b × L → M , for each lattice L ⊆ M . For this, it suffices to show that the map
G
G × L → M is uniformly continuous, for each lattice L. That is, for any lattice
L0 ⊆ M , we have to find an open subgroup H ⊆ G and a sublattice L00 ⊆ Lsuch that gHm + HL00 ⊆ gm + L0 for all g ∈ G and all m ∈ L.
By Lemma D.13, the conditions (2)(a)-(c) of that lemma hold. We begin byshowing that we may find a sublattice L00 of L such GL00 ⊆ L0 . For this, we firstnote that, by (2)(c), we may find an open subgroup H ⊆ G such that HL0 = L0 .
If let {gi } denote a (finite!) set of coset representatives for H\G, then sinceeach gi induces a continuous automorphism of M , we may find a lattice Li such0that gi Li ⊆TL . Taking into account Remark D.10, we may then find a lattice
L00 ⊆ L0 ∩ i Li , and by construction GL00 ⊆ L0 .
Condition (2)(a’) allows us to choose an open subgroup H ⊆ G such thatHm ⊆ m + L00 for all m ∈ L. We then find thatgHm + HL00 ⊆ gm + GL00 ⊆ gm + L0
for all g ∈ G, as required.

D.16

T -quasi-linear endomorphisms

In order to apply the preceding results to the case of interest to us (the semilinear
action of Γ on (ϕ, Γ)-modules), we now introduce and study the notion of a T quasi-linear endomorphism of a finite projective AA -module. The relevance of
this notion to (ϕ, Γ)-modules is explained in Lemma D.27 below.
D.17 Definition. If M is a finite projective AA -module, then a T -quasi-linear
endomorphism of M is a morphism f : M → M which is W (k)⊗Zp A-linear, andwhich furthermore satisfies the following (T -quasi-linearity) condition: there+×
exist power series a(T ) ∈ (A+A ) and b(T ) ∈ (p, T )AA such thatf (T m) = a(T )T f (m) + b(T )T m
for every m ∈ M .
D.18 Lemma. If f is T -quasi-linear, then for all n ∈ Z, we may writef (T n m) = a(T )n T n f (m) + bn (T )T n m+
×for all m ∈ M , where a(T ) ∈ (A+A ) and bn (T ) ∈ (p, T )AA .

Proof. Note that the case n = 0 is trivial (taking b0 (T ) = 0), while if the claim
holds for some n ≥ 1, then we may writef (T −n m) = a(T )−n T −n f (m) − a(T )−n bn (T )T −n m.

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APPENDIX D. TATE MODULES AND CONTINUITY

It therefore suffices to prove the result for n ≥ 1. This may be proved byinduction on n, the case n = 1 being the definition of T -quasi-linearity. Indeed,
if the claim holds for n, then we havef (T n+1 m) = f (T n (T m)) = a(T )n T n f (T m) + bn (T )T n (T m)= a(T )n T n (a(T )T f (m) + b(T )T m) + bn (T )T n+1 m= a(T )n+1 T n+1 f (m) + (bn (T ) + b(T )a(T )n )T n+1 m,
as required.
D.19 Lemma. Let A be an O/$a -algebra for some a ≥ 1, and let M be a
finite projective AA -module. Let f be a T -quasi-linear endomorphism of M ,and let M be a lattice in M . Then there is an integer m ≥ 0 such that foreach s ∈ Z and n ≥ 0 we have f n (T s M) ⊆ T s−mn M.
Proof. Choose a finite set {mi } of generators for M as an A+
A -module. Choose m
sufficiently large that we have T m f (mi ) ∈ M for each i. Then by Lemma D.18
and the W (k)⊗Zp A-linearity of f , we see that for each s ∈ Z we have f (T s+m M) ⊆T s M, which gives the result in the case n = 1. The general case follows by induction on n.
As noted in Remark D.2, any finite projective AA -module M has a natural
topology, so it makes sense to speak of an endomorphism of M being continuous,or topologically nilpotent.
D.20 Lemma. If A is an O/$a -algebra for some a ≥ 1, and M is a finite projective AA -module, then any T -quasi-linear endomorphism of M is necessarily
continuous.
Proof. This is immediate from Lemma D.19.
D.21 Lemma. If A is an O/$a -algebra for some a ≥ 1, and M is a finite
projective AA -module, and if f is a T -quasi-linear endomorphism of M , thenthe following are equivalent:
1. f is topologically nilpotent.
2. There exists a lattice M in M and some n ≥ 1 such that f n (M) ⊆ T M.
3. There exists a lattice M in M and some n ≥ 1 such that f n (M) ⊆ (p, T )M.
4. There exists a lattice M in M such that for any m ≥ 1, there exists n0such that for any s ∈ Z and any n ≥ n0 , we have f n (T s M) ⊆ T s+m M.
5. For any lattice M in M and any m ≥ 1, there exists n0 such that forany s ∈ Z and any n ≥ n0 , we have f n (T s M) ⊆ T s+m M.

D.16. T -QUASI-LINEAR ENDOMORPHISMS

259

Proof. By Lemma D.7 (1), we see that (4) =⇒ (5) =⇒ (1) =⇒ (2) =⇒ (3),so we only need to that (3) =⇒ (4). To this end, note firstly that it follows
from Lemma D.18 and the Zp -linearity of f that for each i, j ≥ 0 and s ∈ Z wehavef ((p, T )j T s f i (M)) ⊆ (p, T )j T s f i+1 (M) + (p, T )j+1 T s f i (M).
It follows (by induction on m) that for each m ≥ 0 and s ∈ Z we havef m (T s M) ⊆m
X

(p, T )m−i T s f i (M).

i=0

In particular, if we take m = n and recall that f n (M) ⊆ (p, T )M by hypothesis,
we find thatf n (T s M) ⊆ (p, T )T s M +n−1
X(p, T )n−i T s f i (M).

(D.22)

i=1

The same argument shows that if N is an A+
A -submodule of M with theproperty thatn−1X
N⊆(p, T )ai T s f i (M)i=0

for non-negative integers a0 , . . . , an−1 , thenf (N) ⊆n−1
X

(p, T )bi T s f i (M)i=0

where b0 = min(a0 + 1, an−1 + 1), and bi = min(ai + 1, ai−1 ) if i > 0. It follows
by an easy induction on N (with the base case being given by (D.22)) that for
all N ≥ n, if we write N + 1 = (q + 1)n + r with 0 ≤ r < n, then we havef N (T s M) ⊆n−1
X(p, T )ci T s f i (M)

i=0

where
(cn−1 , . . . , c0 ) = (q, q, . . . , q) + (r, r + 1, . . . , n − 1, 1, 2, . . . , r).
In particular we see that for all N ≥ n we havef N (T s M) ⊆ (p, T )b(N +1)/nc−1 T sn−1X

f i (M)

i=0

Now, by Lemma D.19, for any sufficiently large t we have f i (M) ⊆ T −t M
for 0 ≤ i ≤ n − 1, so it follows that for N ≥ n we havef N (T s M) ⊆ (p, T )b(N +1)/nc−1 T s−t M.

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APPENDIX D. TATE MODULES AND CONTINUITY

Since pa = 0 in A, we also have (p, T )n+a−1 ⊆ (T n ) for all n ≥ 0, so that if
N ≥ an − 1 then we havef N (T s M) ⊆ T s+b(N +1)/nc−a−t M.
Since b(N + 1)/nc − a − t → ∞ as N → ∞, we have (4), as required.
D.23 Corollary. Suppose that A is an O/$a -algebra for some a ≥ 1. If M
is a finite projective AA -module, and if f is a T -quasi-linear endomorphism of
M , then the following are equivalent:
1. f is topologically nilpotent.
2. The action of f on M ⊗O/$a F is topologically nilpotent.
Proof. Obviously (1) =⇒ (2). Conversely, if (2) holds, then by the equivalenceof conditions (1) and (2) of Lemma D.21 for the action of f on M ⊗O/$a F,
we see that condition (3) of Lemma D.21 holds (for the action of f on M ), andtherefore condition (1) of Lemma D.21 holds, as required.
The following lemmas provide the key examples of T -quasi-linear endomorphisms, and explain our interest in the concept.
We suppose that AA is endowed with a continuous action of Zp by A-algebra
automorphisms which preserve A+
A , and suppose further that, for some topological generator γ of Zp , thatγ(T ) − T ∈ (p, T )T A+
A.

(D.24)

D.25 Lemma. If (D.24) holds, then γ preserves the ideals (T ) and (p, T )of A+
A . Furthermore for each integer n ≥ 1 we haveγ n (T ) − T ∈ (p, T )T A+
A.

(D.26)

Proof. The first claim follows immediately from (D.24), which shows that γ(T )is a unit multiple of T , together with the fact that γ preserves A+
A . Then (D.26)
follows by induction on n (the case n = 1 being (D.24)).
D.27 Lemma. If M is a finite projective AA -module which is endowed with anaction of the subgroup hγi of Zp which is semi-linear with respect to the given
action of this group on AA (obtained by restricting the Zp -action), then for any
integer n ≥ 1, f := γ n − 1 is a T -quasi-linear endomorphism of M .
Proof. We have f (T m) = γ n (T )f (m) + (γ n (T ) − T )m, so it follows from D.26
that f is T -quasi-linear.
D.28 Lemma. Suppose that A is an O/$a -algebra for some a ≥ 1, and that AA
is endowed with an action of Zp satisfying (D.24). Let M be a finite projective
AA -module, equipped with a semi-linear action of hγi ⊂ Zp . Then the following
are equivalent:D.16. T -QUASI-LINEAR ENDOMORPHISMS

261

1. The action of hγi extends to a continuous action of Zp .
2. The action of hγi on M ⊗O/$a F extends to a continuous action of Zp .
3. For any lattice M ⊆ M , and any n ≥ 1, there exists s ≥ 0 such thats(γ p − 1)i (M) ⊆ (p, T )n M for all i ≥ 1.
s

4. For any lattice M ⊆ M , there exists s ≥ 0 such that (γ p − 1)(M) ⊆ T M.
s

5. For some lattice M ⊆ M and some s ≥ 0, we have (γ p −1)(M) ⊆ (p, T )M.
6. The action of γ − 1 on M ⊗O/$a F is topologically nilpotent.
7. The action of γ − 1 on M is topologically nilpotent.
Proof. Noting that if n ≥ a then (p, T )n ⊆ (T ), we see that (3) =⇒ (4), andby Lemma D.21 and Corollary D.23 we have (4) =⇒ (5) =⇒ (6) =⇒ (7). (Forss(5) =⇒ (6), we also use that (γ − 1)p ≡ (γ p − 1) (mod $).)
We next show that (7) =⇒ (3). Suppose that (7) holds. Recalling againssthat for each s ≥ 0 we have (γ − 1)p ≡ (γ p − 1) (mod $), it follows froms
Corollary D.23 and Lemma D.27 that the action of (γ p −1) on M is topologicallynilpotent for each s ≥ 0. We now argue by induction on a, noting that for a = 1,
the implication (7) =⇒ (3) is immediate from Lemma D.21. We may therefore
assume thats(γ p − 1)i (M) ⊆ (p, T )n M + $a−1 Msfor all i ≥ 1. It follows in particular that p(γ p −1)i (M) ⊆ (p, T )n M for all i ≥ 1,
so that for any t ≥ 0 and any i ≥ 1, we have
((γ p

s+t

s

t

− 1)i − (γ p − 1)ip )(M) ⊆ (p, T )n M.

s+t

s

t

(To see this, write (γ p− 1) = ((γ p − 1) + 1)p − 1 and use the binomialstheorem.) It therefore suffices to show that there is some t ≥ 1 for which (γ p −ts1)ip (M) ⊆ (p, T )n M for all i ≥ 1; but we have already seen that (γ p − 1) acts
topologically nilpotently on M , so by Lemma D.21 we can even arrange thatst(γ p − 1)ip (M) ⊆ T n M for all i ≥ 1.
We have shown the equivalence of conditions (3)–(7). Suppose now that (1)sholds. Then Lemma D.13 shows that, for each lattice M ⊆ M , we have γ p (M) ⊆M for all sufficiently large s ≥ 0, and that furthermore, for each m ∈ M, there istsome t(m) such that if t ≥ t(m), then γ p (m) ∈ m+(p, T )M. Letting m1 , . . . , mnbe generators for M as an A+
A -module, we see that if s is sufficiently large, thens(γ p − 1)(mi ) ∈ (p, T )M for each i. Then if λi ∈ A+
A , we have
X
X
X s
s
s
s
(γ p − 1)(
λi mi ) =γ p (λi )(γ p − 1)(mi ) +(γ p − 1)(λi )mi .
i

i

i
s

ps
It follows from (D.26) that (γ p − 1)(λi ) ∈ T A+
A ; hence (γ − 1)(M) ⊆ (p, T )M,and so (5) holds.
Suppose now that the equivalent conditions (3)–(7) hold. We will showthat (1) holds. By Lemma D.15 (taking the group G there to be hγi ∼
= Z

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APPENDIX D. TATE MODULES AND CONTINUITY

b ∼endowed with its p-adic topology, so that G= Zp ), it is enough to show thatthe conditions of Lemma D.13 (2) hold.
We begin with Lemma D.13 (2)(b), the condition that any g ∈ hγi acts
continuously on M . It is enough to show that if M ⊆ M is a lattice, thenthere is a lattice N with g(N) ⊆ M. This is in fact a general property ofsemilinear automorphisms of AA which preserve A+
A . Indeed, let m1 , . . . , mnbe generators of M as an A+-module,
and
let
N
be
the A+A
A -module generated−1−1
by g (m1 ), . . . , g (mn ). This is a lattice, because g is an automorphismof M , and it follows easily from the semi-linearity of the action of g on M thatg(N) ⊆ M, as required.
We now check Lemma D.13 (2)(c). Let M ⊆ M be some lattice, fix achoice of n ≥ 1, and then choose s as in (3). It suffices to show that thesssubgroup H = hγ p i of hγi preserves M. Since (γ p − 1)(M) ⊆ (p, T )n M ⊆ M,s
ps
we certainly have γ (M) ⊆ M, so it suffices to show that γ −p (M) ⊆ M.
For this, note that since (as recalled above) it follows from the congruencesss(γ − 1)p ≡ (γ p − 1) (mod $) that γ p − 1 acts topologically nilpotently on M ,and preserves M, we see that for any m ∈ M we have
s

s

s

s

γ −p (m) = (1 − (1 − γ p ))−1 (m) = m + (1 − γ p )(m) + (1 − γ p )2 (m) + · · · ∈ M,
as required.
To complete the verification of the conditions of Lemma D.13 (2), we needto show that for any lattice M ⊆ M , the orbit maps hγi → M , for the variousm ∈ M, are (equi)continuous at the identity of hγi. It is enough to show that forseach n ≥ 1, we can find s sufficiently large such that H = hγ p i satisfies Hm ⊆m + (p, T )n M, for each m ∈ M, or equivalently, that (h − 1)(M) ⊆ (p, T )n M,
for all h ∈ H. We accomplish this by choosing s as in (3).
s
It then suffices to prove the stronger claim that (γ rp − 1)(M) ⊂ (p, T )n Msfor all r ∈ Z. We have already seen that γ rp (M) ⊆ M. If r ≥ 1 we may
s
s
s
s
s
write (γ rp − 1) = (γ p − 1)(1 + γ p + · · · + γ (r−1)p ), and since (1 + γ p +sss· · · + γ (r−1)p )(M) ⊂ M, we have (γ rp − 1)(M) ⊆ (γ p − 1)(M) ⊂ (p, T )n M,ss
as required. If r ≤ 0, then the result follows by writing (γ −rp − 1) = −(γ rp −s1)(γ −p )r .
Finally, applying the equivalence of (1) and (6) with M replaced by M ⊗O/$a
F, we see that (2) and (6) are equivalent, as required.
The following lemmas will allow us to reduce the problem of investigating
the topological nilpotency of a T -quasi-linear endomorphism from the projective
case to the free case.
D.29 Lemma. If M1 and M2 are Tate modules over a ring A, endowed with
continuous endomorphisms f1 and f2 respectively, then f1 and f2 are both topologically nilpotent if and only if the direct sum f := f1 ⊕ f2 is a topologicallynilpotent endomorphism.
Proof. This is immediate from the definitions.

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263

D.30 Lemma. If M is a finite projective AA -module endowed with a T -quasilinear endomorphism f , then we may find a finite projective AA -module N ,endowed with a T -quasi-linear endomorphism g which is furthermore topologically nilpotent, such that M ⊕ N is a free AA -module.
Proof. By definition, there is a AA -module N such that M ⊕ N is free. We
may then take g to be the zero endomorphism of N ; this is evidently both
T -quasi-linear and topologically nilpotent.
If A is an O/$a -algebra for some a ≥ 1, and M is a finite projective AA module, equipped with a T -quasi-linear endomorphism f , and if A → B is amorphism of O/$a -algebras, then we have a base-changed W (k) ⊗Zp B-linear
endomorphism of B ⊗A M . This base-changed endomorphism is T -adically continuous (by Lemma D.19) and it therefore induces an endomorphism fB of thebase-changed projective AB -module MB (which by definition is the completionof B ⊗A M for the T -adic topology). The endomorphism fB is evidently also
T -quasi-linear. (Note that since f is not necessarily A[[T ]]-linear, we do not
define fB by viewing MB as B[[T ]] ⊗A[[T ]] M .) If f is furthermore topologicallynilpotent, then so is fB (for example, by Lemma D.21).
D.31 Lemma. Let {Ai }i∈I be a directed system of O/$a -algebras, with A =limi∈I Ai , and suppose that I admits a least element i0 . Let M be a finite−→
projective AAi0 -module, let f be a T -quasi-linear endomorphism of M , andsuppose that the base-changed endomorphism fA is topologically nilpotent. Thenfor some i ∈ I, the base-changed endomorphism fAi is topologically nilpotent.
Proof. By Lemma D.30, we may choose a finite projective AAi0 -module N ,endowed with a topologically nilpotent T -quasi-linear endomorphism g, suchthat M ⊕N is free. By Lemma D.29, the base-changed endomorphism fA ⊕gA =(f ⊕ g)A of MA ⊕ NA is topologically nilpotent, and it suffices to show thatfAi ⊕ gAi = (f ⊕ g)Ai is topologically nilpotent for some i ∈ I. Thus wemay reduce to the case when M is free, in which case we may also choose afree lattice M contained in M . Taking into account Lemma D.21, for somesufficiently large n we have fAn (MA ) ⊆ T MA , and it suffices to show thatfAni (MAi ) ⊆ T MAi(D.32)

for some i ∈ I. Lemma D.19 shows that f n (T r M) ⊆ T M for some r ≥ 0, andthat f n (M) ⊆ T −s M for some s ≥ 0. Thus f n induces a morphism M/T r M →T −s M/T M of finite rank free Ai0 -modules, which, by assumption, vanishes
after base-change to A. Thus this morphism in fact vanishes after base-change
to some Ai , and consequently (D.32) does indeed hold for this choice of Ai .

264

APPENDIX D. TATE MODULES AND CONTINUITY

Appendix E

Points, residual gerbes, andisotrivial families
Recall that if X is an algebraic stack, then the underlying set |X | of points of
X is defined as the set of equivalence classes of morphisms Spec K → X , with
K being a field; two such morphisms are regarded as equivalent if they may be
dominated by a common morphism Spec L → X .
If X is a scheme (regarded as an algebraic stack), then the set of underlying
points |X| is naturally identified with the underlying set of points of X in theusual sense, and the equivalence class of morphisms representing a given pointx ∈ |X| has a canonical representative, namely the morphism Spec k(x) →X, where k(x) is the residue field of x. More abstractly, this morphism isa monomorphism, and this property characterizes it uniquely, up to unique
isomorphism, among the morphisms in the equivalence class corresponding to x.
If X is an algebraic space, then it is not the case that every equivalenceclass in |X| admits a representative which is a monomorphism (see e.g. [Sta,Tag 02Z7]). However, under mild assumptions on X, the points of |X| do admit
such representatives (which are then unique up to unique isomorphism); thisin particular is the case if X is quasi-separated. See e.g. the discussion at the
beginning of [Sta, Tag 03I7].
If X is genuinely an algebraic stack, then it is not reasonable to expect theequivalence classes in |X | to admit monomorphism representatives in general(even if X is quasi-separated), since points of X typically admit non-trivial
stabilizers. The aim of the theory of residual gerbes is to provide a replacement
for such representatives.
We recall the relevant definition [Sta, Tag 06MU]):
E.1 Definition. Let X be an algebraic stack. We say that the residual gerbe
at a point x ∈ |X | exists if we may find a monomorphism Zx ,→ X such that
Zx is reduced and locally Noetherian, |Zx | is a singleton, and the image of |Zx |in |X | is equal to x. If such a monomorphism exists, then Zx is unique up tounique isomorphism, and we refer to it as the residual gerbe at x.
265

266

APPENDIX E. POINTS AND RESIDUAL GERBESIn the case that X is an algebraic space, the residual gerbe Zx exists atevery point x ∈ |X| [Sta, Tag 06QZ], and is itself an algebraic space, called theresidual space of X at x [Sta, Tag 06R0]. However, as already noted, in thiscase, if X is furthermore quasi-separated, then the residual space at a point is
simply the spectrum of a field.
If X is an algebraic stack with quasi-compact diagonal (e.g. if X is quasiseparated), then the residual gerbe exists at every point of |X | [Sta, Tag 06RD]).
If X is a scheme, then a finite type point of X is a point x ∈ X that is locallyclosed. Equivalently, these are the points for which the morphism Spec k(x) →X is a finite type morphism, and may be characterized more abstractly as thoseequivalence classes of morphisms Spec K → X which admit a representative
which is locally of finite type. This latter notion makes sense for an arbitrary
algebraic stack, and allows us to define the notion of a finite type point of analgebraic stack (see e.g. [EG21, 1.5.3]).
We then have the following results, which provide analogues of the topological characterization of the finite type points of a scheme as being those points
that are locally closed.
E.2 Lemma. Suppose that X is an algebraic space, and that x ∈ |X| is a finite
type point with the property that for any étale morphism U → X whose sourceis an affine scheme, the fibre over x is finite. Then, if Zx denotes the residual
space at x in X, the canonical monomorphism Zx ,→ X is an immersion.
Proof. Consider the construction of Zx given in [Sta, Tag 06QZ]: we choose asurjective étale morphism U → X, form the uniona
U0 =Spec k(u),u∈U lying over x

and then realize Zx ,→ X as a descent of the monomorphism U 0 ,→ U . In
making this construction, we may replace U by any open subscheme containing
x in its image, and thus we may assume that U is affine. By assumption, there
are then only finitely many points u in U lying over x, these points are all finite
type points of U , and furthermore, none of these points are specializations of
any of the others [Sta, Tag 03IM]. Thus U 0 ,→ U is in fact an immersion, andthus the same is true of Zx ,→ X.
E.3 Lemma. If X is an algebraic stack whose diagonal is quasi-compact, if
x ∈ |X | is a finite type point, and if Zx is the residual gerbe of X at x, then the
canonical monomorphism Zx ,→ X is in fact an immersion.
Proof. We prove this by examining the construction of Zx carried out in the
proof of [Sta, Tag 06RD]. The first step of the proof is to replace X by theclosure of x in |X |, regarded as a closed substack of X with its induced reduced
structure. The assumption that X has quasi-compact diagonal implies thatthe morphism IX → X is quasi-compact, and thus that we may find a dense
open substack U of X over which this morphism is flat and locally of finite
presentation. Since x is dense in |X |, we find that x ∈ |U |, and so replacing X

267
by U, we may assume that IX → X is flat and locally of finite presentation.
By [Sta, Tag 06QJ], this implies that X is a gerbe over some algebraic space X.
If we let Zx denote the residual space at (the image in |X| of) x in |X|, thenZx is obtained as the base-change of Zx over the morphism X → X.
Note that, in the various reduction steps undertaken in the preceding argument, we replaced X by an open substack of a closed substack; it thus suffices toverify that Zx → X is an immersion after making these reductions. Note thatsince immersions are monomorphisms, the diagonal of the stack obtained after
these reductions are made is the base-change of the diagonal of the original stack
X , and thus continues to be quasi-compact. Consequently, we may assume that
we are in the case where X is a gerbe over X. Since Zx ,→ X is then obtained
as the base-change of Zx ,→ X, it suffices to show that this latter morphism is
an immersion. For this, if we take into account Lemma E.2, it suffices to showthat X can be chosen to be quasi-separated (as every étale morphism from anaffine scheme to a quasi-separated algebraic space has finite fibres over every
point of |X |; see the discussion at the beginning of [Sta, Tag 03I7]).
If we examine the proof of [Sta, Tag 06QJ], we see that X is constructed
as follows: We choose a smooth surjective morphism U → X whose source isa scheme, and write R = U ×X U, so that X = [U/R]. We then factor themorphism R → U ×Spec Z U through a morphism R0 → U ×Spec Z U , whereR0 → U ×Spec Z U is a flat and locally of finite presentation equivalence relation,and set X = U/R0 . We note one additional aspect of the situation, namely thatthe morphism R → R0 is surjective. The assumption that X has quasi-compact
diagonal then implies that R → U ×Spec Z U is quasi-compact, and since R
surjects onto R0 , we find that R0 → U ×Spec Z U is again quasi-compact. Thus
X = U/R0 is quasi-separated, as required.
E.4 Example. We note that quasi-separatedness (or some such hypothesis) is
necessary for the truth of Lemma E.3. To illustrate this, let G be an algebraic group of positive dimension over an algebraically closed field k, and let
X := G/G(k). Then X contains a unique finite type point x — namely, theequivalence class of the monomorphism Spec k = G(k)/G(k) ,→ G/G(k) — andthis monomorphism realizes Spec k as the residual space Zx . This monomorphism is not an immersion, since its pull-back to G induces the monomorphism
G(k) ,→ G, which is not an immersion.
Traditionally, in the theory of moduli problems, a family of some objects
parameterized by a base scheme T is called isotrivial if the isomorphism class ofthe members of the family is constant over T . From the view-point of morphisms
to a moduli stack X , this corresponds to the image of T in |X | being a singleton,say x. We would like to conclude that the morphism T → X factors through theresidual gerbe Zx . In practice, we often verify the “constancy” of the morphism
T → X only at finite type points. The following result gives sufficient conditions,
under such a constancy hypothesis on the finite type points, for a morphism tofactor through the residual gerbe.

268

APPENDIX E. POINTS AND RESIDUAL GERBES

E.5 Lemma. Let X be an algebraic stack, and let x ∈ |X | be a point for
which the residual gerbe Zx exists, and for which the canonical monomorphism
Zx ,→ X is an immersion. If f : T → X is a morphism whose domain is areduced scheme, and for which all the finite type points of T map to the given
point x, then f factors through Zx .
Proof. Consider the fibre product T ×X Zx . By assumption, the projection
from this fibre product to T is an immersion whose image contains every finite
type point of T . A locally closed subset of T that contains every finite typepoint is necessarily equal to T , and thus, since T is reduced, this immersion isin fact an isomorphism. Consequently the morphism f factors through Zx , asclaimed.
We end this discussion by explaining how the preceding discussion generalizes
to certain Ind-algebraic stacks. Let {Xi }i∈I be a 2-directed system of algebraic
stacks, and assume that the transition morphisms are monomorphisms. Let
X := limi Xi be the Ind-algebraic stack obtained as the 2-direct limit of the−→
Xi . We may define the underlying set of points |X | as equivalence classes ofmorphisms from spectra of fields in the usual way, and since any such morphism
factors through some Xi , we find that |X | = limi |Xi |.
−→Now suppose that X has quasi-compact diagonal, or, equivalently (since
the transition morphisms are monomorphisms) that each Xi has quasi-compact
diagonal. If x ∈ |X |, then x ∈ |Xi | for some i, and the residual gerbe Zx at xin Xi exists. The composite monomorphism Zx ,→ Xi ,→ Xi0 realizes Zx as theresidual gerbe at x in Xi0 , for any i0 ≥ i, and so we may regard Zx as being
the residual gerbe at x in X . Lemmas E.3 and E.5 immediately extend to thecontext of such Ind-algebraic stacks X .

Appendix F

Breuil–Kisin–Fargues
modules and potentiallysemistable representations(by Toby Gee and Tong
Liu)
In this appendix we briefly discuss the relationship between Breuil–Kisin–Fargues
modules with semilinear Galois actions, and potentially semistable Galois representations. As explained in Remark F.12 below, we do not expect our results
to be optimal, but they suffice for our applications in the body of the book.
The results of this appendix were originally inspired by [Car13]; the recent paper [Gao19] corrects a mistake in [Car13] and independently proves related (andin some cases stronger) versions of some of our results, by different methods.
We do not make any use of the arguments of either [Car13, Gao19], but instead
combine [Liu18] with a result of Fargues ([BMS18, Thm. 4.28]).
We use the notation introduced in the body of the book, in particular in
Section 2.1. The kernel of the usual ring homomorphism θ : Ainf → OC is aprincipal ideal (ξ); one possible choice of ξ is µ/ϕ−1 (µ), where µ = [ε] − 1 (for[some compatible choice of roots of unity ε = (1, ζp , ζp2 , . . . ) ∈ OC). Recall that
1
B+
is
the
ker(θ)-adic
completion
of
A
[
]
and
that
t
:=
log[ε]
∈ B+
inf p
dR
dR is a+generator of ker(θ) in BdR .
As usual, we let K be a finite extension of Qp with residue field k. Recall[
that for each choice of uniformizer π of K, and each choice π [ ∈ OCof p-powerroots of π, we write Sπ[ for S = W (k)[[u]], regarded as a subring of Ainf via
u 7→ [π [ ]. Write Eπ (u) for the Eisenstein polynomial for π, and Eπ[ for its image
in Ainf ; then Eπ[ ∈ (ξ), because θ(Eπ[ ) = Eπ (π) = 0. Indeed (Eπ[ ) = (ξ) (this
269

270

APPENDIX F. BKF MODULES AND GK -REPRESENTATIONS

follows for example from the criterion given in [BMS18, Rem. 3.11] and thedefinition of an Eisenstein polynomial).
In contrast to the body of the book, we do not use coefficients in most of
this appendix (we briefly consider O-coefficients at the end). Accordingly, we
have the following definitions.
F.1 Definition. An étale (ϕ, GK )-module is a finite free W (C[ )-module M
equipped with a ϕ-semilinear map ϕ : M → M which induces an isomorphism∼ϕ∗ M −→ M , together with a continuous semilinear action of GK which commutes with ϕ.
There is an equivalence of categories between the category of étale (ϕ, GK )modules M of rank d and the category of free Zp -modules T of rank d which
are equipped with a continuous action of GK . The Galois representation corresponding to M is given by T (M ) = M ϕ=1 . (See Section 3.6.4.)
F.2 Definition. Fix a choice of π [ . We define a Breuil–Kisin module of heightat most h to be a finite free Sπ[ -module M equipped with a ϕ-semi-linear
morphism ϕ : M → M, with the property that the corresponding morphism
ΦM : ϕ∗ M → M is injective, with cokernel killed by Eπh[ .
F.3 Definition. A Breuil–Kisin–Fargues module of height at most h is a finite
free Ainf -module Minf equipped with a ϕ-semi-linear morphism ϕ : Minf →Minf , with the property that the corresponding morphism ΦMinf : ϕ∗ Minf →Minf is injective, with cokernel killed by ξ h .
F.4 Remark. If M is a Breuil–Kisin module we write Minf for the Breuil–Kisin–
Fargues module Ainf ⊗Sπ[ M (this is indeed a Breuil–Kisin–Fargues module,
because Sπ[ → Ainf is faithfully flat, and (Eπ[ ) = (ξ)). As noted in Remark 4.1.1, we are not twisting the embedding S → Ainf by ϕ, and accordingly,
in Definition F.3, we demand that the cokernel of ϕ is killed by a power of ξ,
rather than a power of ϕ(ξ) as in [BMS18].
Leaving this difference aside, our definition of a Breuil–Kisin–Fargues module is less general than that of [BMS18], in that we require ϕ to take M to itself;
this corresponds to only considering Galois representations with non-negative
Hodge–Tate weights. This definition is convenient for us, as it allows us to makedirect reference to the literature on Breuil–Kisin modules. The restriction to
non-negative Hodge–Tate weights is harmless in our main results, as we canreduce to this case by twisting by a large enough power of the cyclotomic character (the interpretation of which on Breuil–Kisin–Fargues modules is explainedin [BMS18, Ex. 4.24]). (We are also only considering free Breuil–Kisin–Fargues
modules, rather than the more general possibilities considered in [BMS18].)
F.5 Definition. A Breuil–Kisin–Fargues GK -module of height at most h is
a Breuil–Kisin–Fargues module of height at most h which is equipped with asemilinear GK -action which commutes with ϕ.
F.6 Remark. Note that if Minf is a Breuil–Kisin–Fargues GK -module, thenW (C[ ) ⊗Ainf Minf is naturally an étale (ϕ, GK )-module in the sense of Definition F.1.

271
sRecall that for each choice of π [ and each s ≥ 0 we write Kπ[ ,s for K(π 1/p ),and Kπ[ ,∞ for ∪s Kπ[ ,s .
F.7 Definition. Let Minf be a Breuil–Kisin–Fargues GK -module of height at
most h with a semilinear GK -action. Then we say that Minf admits all descentsif the following conditions hold.
1. For every choice of π and π [ , there is a Breuil–Kisin module Mπ[ of heightGKat most h with Mπ[ ⊂ (Minf ) π[ ,∞ for which the induced morphism
Ainf ⊗Sπ[ Mπ[ → Minf is an isomorphism.
2. The W (k)-submodule Mπ[ /[π [ ]Mπ[ of W (k) ⊗Ainf Minf is independent of
the choice of π and π [ .
3. The OK -submodule ϕ∗ Mπ[ /Eπ[ ϕ∗ Mπ[ of OC ⊗θ,Ainf ϕ∗ Minf is independent of the choice of π and π [ .
F.8 Remark. In fact condition (2) in Definition F.7 is redundant; see Remark F.20 below. However, we include the condition as it is useful when considering versions of the theory with coefficients and descent data.
F.9 Definition. Let Minf be a Breuil–Kisin–Fargues GK -module which admits
all descents. We say that Minf is furthermore crystalline if for each choice of π
and π [ , and each g ∈ GK , we have(g − 1)(Mπ[ ) ⊂ ϕ−1 (µ)[π [ ]Minf .

(F.10)

Definitions F.7 and F.9 are motivated by the following result, whose proof
occupies most of the rest of this appendix.
F.11 Theorem. Let M be an étale (ϕ, GK )-module. Then V (M ) is semistablewith Hodge–Tate weights in [0, h] if and only if there is a (necessarily unique)
Breuil–Kisin–Fargues GK -module Minf which is of height at most h, which admits all descents, and which satisfies M = W (C[ ) ⊗Ainf Minf .
Furthermore, V (M ) is crystalline if and only if Minf is crystalline.
F.12 Remark. As already noted in Remark F.8, condition (2) of Definition F.7
is redundant, and it is plausible that condition (3) is redundant as well (i.e. thatboth condition condition (2) and condition (3) in Definition F.7 are consequencesof condition (1)); but this does not seem to be obvious. Note though that it is
not sufficient to demand the existence of a descent for a single choice of π, asthere are representations of finite height which are potentially semistable but
not semistable (see for example [Liu10, Ex. 4.2.1]).
We begin with some preliminary results. The following proposition and its
proof are due to Heng Du, and we thank him for allowing us to include themhere.
F.13 Proposition. Let Minf be a Breuil–Kisin–Fargues GK -module with theproperty that C⊗θ,Ainf ϕ∗ Minf has a basis consisting of GK -fixed vectors. Let M =W (C[ ) ⊗Ainf Minf ; then the GK -representation V (M ) is de Rham.

272

APPENDIX F. BKF MODULES AND GK -REPRESENTATIONS

F.14 Remark. Using the equivalence of categories of [BMS18, Thm. 4.28] (a theorem of Fargues), one can easily check that Proposition F.13 admits a converse:
namely that if M is an étale (ϕ, GK )-module with the property that T (M ) isa Zp -lattice in a de Rham representation of GK with non-negative Hodge–Tateweights, then there is a Breuil–Kisin–Fargues module Minf with the properties
in the statement of Proposition F.13.
Proof of Proposition F.13. By [BMS18, Thm. 4.28] (and our assumption thatϕ(Minf ) ⊆ Minf ), we have injections
∗
inf
inf
B+
,→ B+
,→ BdR ⊗Zp T (M ).
dR ⊗Ainf ϕ M
dR ⊗Ainf M

To show that V (M ) is de Rham, we need to show that the BdR -vector space
BdR ⊗Zp T (M ) has a basis consisting of GK -fixed vectors, so it suffices to show+∗
infthat the B+
has a basis consisting of GK -fixed vectors.
dR -lattice BdR ⊗Ainf ϕ M+
Since BdR is a complete discrete valuation ring with maximal ideal (ξ), it isn∗
infenough to show that there are compatible bases of B+
fordR /(ξ ) ⊗Ainf ϕ M
all n ≥ 1, consisting of GK -fixed vectors.
In the case n = 1, since ξ generates the kernel of θ, we have such a basis(n)by hypothesis. Suppose that {ei }i=1,...,d is a basis of GK -fixed vectors for
(n+1)
n
∗
inf
n+1
B+
ei
}i=1,...,d be any basis of B+)⊗AinfdR /(ξ
dR /(ξ )⊗Ainf ϕ M , and let {e
(n)
+
∗
inf
ϕ M lifting {ei }. Recall that t = log[ε] ∈ BdR is a generator of (ξ), and foreach g ∈ GK , write(n+1)

g · (ee1(n+1), . . . , eed(n+1)) = (ee1(n+1), . . . , eed)(1d + tn Bg(n+1) )

(n+1)

+
as an element of Md (BdR/(ξ)) = Md (C).
where we can view Bg(n+1)
is a continuous 1-cocycle valued
A simple calculation shows that g 7→ Bg
in Md (C(n)). Since n ≥ 1, the corresponding cohomology group H 1 (GK , Md (C(n)))
vanishes, because H 1 (GK , C(n)) = 0 by a theorem of Tate–Sen. There is therefore some A ∈ Md (C) such that for all g we haveBg(n+1) = χn (g)g(A) − A,
where χ is the p-adic cyclotomic character. Then
(n+1)

(e1

(n+1)

, . . . , ed(n+1)

) = (ee1(n+1), . . . , eed)(1d − tn A)

n+1
is the required basis of B+) ⊗Ainf ϕ∗ Minf consisting of GK -fixed vectorsdR /(ξ
(n)lifting {ei }.

F.15 Lemma. There is no proper closed subgroup of GK containing all of thesubgroups GKπ[ ,∞ .
Proof. Let s be the greatest integer with the property that K contains a primitive ps th root of unity; equivalently, it is the greatest integer with the property
that Kπ[ ,s /K is Galois over K (for one, or equivalently every, choice of π and π [ ).

273Then Kπ[ ,s0 depends only on π if s0 ≤ s, and so we write Kπ,s0 = Kπ[ ,s0 forsuch s0 .
The subgroups GKπ[ ,∞ for any fixed choice of π topologically generate GKπ,s .
It therefore suffices to show that as π varies, the various normal subgroups GKπ,sof GK collectively generate GK . Let H be the normal subgroup that theygenerate; then for every uniformizer π, GK /H is a quotient of GK /GKπ,s , acyclic group of order ps . If H were a proper subgroup of GK , then necessarilys ≥ 1, and the subgroups GKπ,1 would coincide for every π (since they would
coincide with the unique index p subgroup of H). Thus the extensions Kπ,1 /K
would have to all coincide. By Kummer theory, this would imply that the ratioof any two uniformizers of K is a pth power in K, which is nonsense (for example,
consider the uniformizers π and π + π 2 ).
Proof of Theorem F.11. We begin with the semistable case. If V (M ) is semistablethen the existence of Minf is a straightforward consequence of the results of [Liu18].
In particular, the existence of a unique Minf satisfying condition (1) of Definition F.7 follows from [Liu18, Thm. 2.2.1].
The proofs that conditions (2) and (3) hold are implicit in the proof of [Liu18,
Prop. 4.2.1], as we now explain. Write Mπ[ = Mπ[ /[π [ ]Mπ[ . Since W (k)⊗W (k)Mπ[ = W (k) ⊗Ainf Minf is independent of π [ , it suffices to show that the K0 vector space Dπ[ := Mπ[ [ p1 ] is independent of π [ . Let Sπ[ be the p-adic compleEition of Sπ[ [ i!π[ , i ≥ 1]. By [Bre97, Prop. 6.2.1.1] and [Liu12, §2], Dπ[ admits a
section sπ[ : Dπ[ → Sπ[ [ p1 ] ⊗ϕ,Sπ[ Mπ[ so that1
1
Sπ[ [ ] ⊗K0 sπ[ (Dπ[ ) = Sπ[ [ ] ⊗ϕ,Sπ[ Mπ[ .
pp(F.16)

In particular, we may regard sπ[ (Dπ[ ) as a submodule of B+
cris ⊗ϕ,Sπ[ Mπ [ ⊆
B+
⊗
T
(M
).
Z
p
cris
+
+
Write u = log([π [ ]) ∈ B+
st = Bcris [u]. Note that the set Bst does not[[depend on the choice of π , because, if $ is another choice and we write+0[[
u0 = log([$[ ]), then B+
cris [u ] = Bcris [u]. Indeed, we have log([$ ]) = log([π ])+λ[+GK]) ∈ B+iswith λ = log([ $cris . In particular, Dst (T (M )) := (Bst ⊗Zp T (M ))π[[independent of π . Furthermore, [Liu12, Prop. 2.6] shows that we have a commutative diagram (where iπ[ is a K0 -linear isomorphism)
Dst (T (M )) 




/ B+
st ⊗K Dst (T (M ))
0

o iπ[



sπ[ (Dπ[ ) 



mod u

/ B+ ⊗ϕ,S Mπ[
cris
π[



/ B+
st ⊗Z T (M )
p



mod u

/ B+ ⊗Z T (M )pcris

To show (2), it therefore suffices to show that the image of the composite
Dst (T (M )) → B+st ⊗K0 Dst (T (M ))mod u/

infB+
cris ⊗ϕ,Sπ[ Mπ [ → W (k̄)[1/p] ⊗ϕ,Ainf M

274

APPENDIX F. BKF MODULES AND GK -REPRESENTATIONS

1
is independent of π [ . Here the last map is induced by ν : B+
cris → W (k)[ p ]which extends the natural projection Ainf → W (k). It suffices in turn to show
mod u/ +ν /B
W (k)[ 1 ] is independent of π [ . Thisthat the composite B+
st
cris

p

[
follows from the fact that λ = log([ $]) is in ker(ν) (see the proof of [Liu12,π[Lem. 2.10]).
To prove (3), it again suffices to prove the statement after inverting p. For+11
any subring A ⊂ B+
dR , set F A = A ∩ ξBdR . It is easy to see that F Sπ [ =1Eπ[ Sπ[ and also that Sπ[ /F Sπ[ = OK . (Note that the inclusion Eπ[ Sπ[ ⊆F 1 Sπ[ is strict.) Now we again use the isomorphism (F.16). By reducing modulo
F 1 Sπ[ on the both sides of this identification, we conclude thatK ⊗K0 Dπ[ = K ⊗K0 (sπ[ (Dπ[ )1
mod [π [ ]) = ϕ∗ Mπ[ /Eπ[ ϕ∗ Mπ[ [ ].
pOn the other hand, by tensoring Ainf via Sπ[ to (F.16), we obtain the isomor+
∗
inf
1 +
phism B+cris ⊗K0 sπ [ (Dπ [ ) = Bcris ⊗Ainf ϕ M . Modulo F Bcris on both sides, asimilar argument to the above shows that C⊗K0 Dπ[ = (ϕ∗ Minf /ξϕ∗ Minf )[ p1 ] =C ⊗W (k) ϕ∗ Minf , where we write ϕ∗ Minf = W (k) ⊗Ainf ϕ∗ Minf . In summary
we see that(ϕ∗ Mπ[ /Eπ[ ϕ∗ Mπ[ )[ p1 ] = K ⊗K0 Dπ[ ⊆ C ⊗W (k̄) ϕ∗ Minf = (ϕ∗ Minf /ξϕ∗ Minf )[ p1 ].

Since we have shown that Dπ[ ⊆ ϕ∗ Minf is independent of π [ , it follows thatϕ∗ Mπ[ /Eπ[ ϕ∗ Mπ[ is independent of π [ , as required.
Suppose conversely that we are given Minf as in Definition F.7. Write M
0
for the W (k)-module of Definition F.7 (2), and M for the OK -module of Definition F.7 (3). Since for each π [ the action of GKπ[ ,∞ on Mπ[ is trivial by0

assumption, it follows from Lemma F.15 that GK acts trivially on M and M .
0Since C⊗Ainf ,θ ϕ∗ Minf = C⊗OK ϕ∗ M , we see that the hypotheses of Proposition F.13 are satisfied, so that V (M ) is de Rham, and consequently potentiallysemistable. To show that V (M ) is semistable, we may replace K with an (infinite) unramified extension and assume that K is a complete discretely valued
field with residue field k = k. In particular, we now have M = W (k) ⊗Ainf Minf ,and we claim that M[ p1 ] with its GK -action is isomorphic to Dpst (V (M )). If theclaim holds, then since GK acts trivially on M, GK acts trivially on Dpst (V (M )),so that V (M ) is semistable, as required.
We now prove the claim. Let L/K be a finite Galois extension so that
V (M )|GL is semi-stable, and let ML be the Breuil–Kisin module attached to[
T (M )|GL for some choice of πL. By [Liu12, (2.12)], Ainf ⊗Sπ[ ϕ∗ ML injects into
L∗
Ainf ⊗Zp T (M ). Furthermore, by [Liu12, Lem. 2.9], ϕ∗ Minf
L := Ainf ⊗Sπ[ ϕ ML
L

is stable under the GK -action on Ainf ⊗Zp T (M ), and by [Liu12, Cor. 2.12],
W (k)⊗Sπ[ ϕ∗ ML = W (k)⊗Ainf ϕ∗ Minf
L (recall that k = k) together with its GK L

GL
action is isomorphic to a W (k)-lattice inside Dst,L (T (M )) = (B+.
st ⊗Zp T (M ))275
In summary, W (k) ⊗Ainf ϕ∗ Minf
L [1/p] is isomorphic to Dst,L (T (M )) as GK modules.
∗
inf
It remains to show that ϕ∗ Minf
L = ϕ M . By [BMS18, Thm. 4.28], it
+
+
∗
inf
suffices to show that BdR ⊗Ainf ϕ ML = BdR ⊗Ainf ϕ∗ Minf . By the proof of
Proposition F.13, we see that
∗
inf
B+
= B+
dR ⊗Ainf ϕ M
dR ⊗K DdR (T (M )).

(F.17)

Since T (M ) is semi-stable over L, it is well-known that+∗B+
st ⊗Sπ[ ϕ ML = Bst ⊗W (kL ) Dst (T (M )|GL ).

(F.18)

L(This can be easily seen from the proof of [Kis06, Cor.1.3.15], or see [Liu12, §2]
for a more detailed discussion; the key point, in terms of the diagram above, isthat [Liu12, §2] shows that K0 [u] ⊗K0 Dst (T (M )) = K0 [u] ⊗K0 sπ[ (Dπ[ ).)From (F.18) we obtain+∗
B+
dR ⊗Sπ[ ϕ ML = BdR ⊗L DdR (T (M )|GL ).

(F.19)

L

+
∗
inf
∗
inf
Comparing (F.17) and (F.19) we have B+
dR ⊗Ainf ϕ ML = BdR ⊗Ainf ϕ M ,as required.
Finally, we turn to the crystalline case. Given the above, the result isa consequence of [Oze18, Thm. 3.8], as we now explain. In our case, f (u)in [Oze18] is equal to up , so the assumptions of [Oze18, Thm. 3.8] are automatically satisfied (as noted at the beginning of [Oze18, §3.2]). More preb be the Galois closure of Kπ[ . There is a subringcisely, fix some π [ , and let Kb ⊆ Ainf (constructed in [Oze18]) such that Sπ[ ⊂ Rb and GK acts on R
b
R
n
1/p
b
through G := Gal(∪n≥1 K(ζpn , π)/K). When T (M ) is crystalline, [Oze18,bThm. 3.8] shows that T (M ) admits a (ϕ, G)-module,
which by definition consistsof the following data:• The Breuil-Kisin module Mπ[ attached to T (M )|Gπ[ ,∞ .
c := Rbb ⊗S ϕ∗ Mπ[ which commutes with the action of ϕ
• A G-actionon M
π[and satisfies (F.10).
c ϕ=1 = T (M ) as GK -modules.
• (W (C[ ) ⊗R
b M)
(In fact [Oze18] uses contravariant functors, which can be easily translated tothe covariant functors used here.) This proves that if T (M ) is crystalline then
Minf satisfies (F.10). Conversely, if Minf satisfies (F.10) for one fixed π [ , then∗GK
[Oze18, Lem. 3.15] shows that sπ (Dπ[ ) ⊂ (B+. Hence T (M )cris ⊗Sπ[ ϕ Mπ [ )is crystalline, as required.
F.20 Remark. Note that condition (2) in Definition F.7 is redundant for proving
Theorem F.11. In fact, if we remove (2) from Definition F.7 then the necessity
part of Theorem F.11 of course still holds. For the sufficiency, note that in the276

APPENDIX F. BKF MODULES AND GK -REPRESENTATIONS

above proof, we extended K so that the residue field is k̄; so we only use that
GK (the inertia subgroup in this situation) acts trivially on W (k̄) ⊗Ainf Minf ,and we do not need that Mπ[ /[π [ ]Mπ[ ⊆ W (k̄) ⊗Ainf Minf is independent of π [ .
However, it is not immediately clear that the analogous condition is redundant in the version of the theory with coefficients that we consider in the bodyof the book (see Definition 4.2.4), and this condition is used in defining ourmoduli stacks of potentially semistable representations of given inertial type, so
we include it here.

F.21

Potentially semistable representations

It will be convenient for us to have a slight refinement of these results, allowing
us to discuss potentially semistable (and potentially crystalline) representations.
To this end, fix a finite Galois extension L/K.
F.22 Definition. Let Minf be a Breuil–Kisin–Fargues GK -module of height at
most h. Then we say that Minf admits all descents over L if the corresponding Breuil–Kisin–Fargues GL -module (obtained by restricting the GK -action
on Minf to GL ) admits all descents (in the sense of Definition F.7).
F.23 Corollary. Let M be an étale (ϕ, GK )-module. Then V (M )|GL is semistablewith Hodge–Tate weights in [0, h] if and only if there is a (necessarily unique)
Breuil–Kisin–Fargues GK -module Minf which is of height at most h, which admits all descents over L, and which satisfies M = W (C[ ) ⊗Ainf M. Furthermore T (M )|GL is crystalline if and only if Minf is crystalline as a Breuil–Kisin–
Fargues GL -module.
Proof. The sufficiency of the condition is immediate from Theorem F.11. For
the necessity, by Theorem F.11 there is a Breuil–Kisin–Fargues GL -module Minf
L
which admits all descents and satisfies M = W (C[ ) ⊗Ainf ML , so we need only
show that Minf
L is GK -stable. This follows from [Liu12, Lem. 2.9], or one canargue as follows: note that the proof of Theorem F.11 shows that the Breuil–Kisin–Fargues module Minf
L := Ainf ⊗Sπ[ ML corresponds via [BMS18, Thm.
L4.28] to the pair (T (M ), B+
dR ⊗L DdR (T (M )|GL )). This pair has a GK -action,
because V (M ) is de Rham, hence Minf
L is GK -stable by [BMS18, Thm. 4.28].

F.24

Hodge and inertial types

We finally recall how to interpret Hodge–Tate weights and inertial types in termsof Breuil–Kisin modules. Fix a finite extension E/Qp with ring of integers O,
which is sufficiently large that E contains the image of every embedding σ :K ,→ Qp . The definitions above admit obvious extensions to the case of Breuil–Kisin–Fargues modules with O-coefficients (see Definition 4.2.4), and since O isa finite free Zp -module, the proofs of Theorem F.11 and Corollary F.23 go overunchanged in this setting.

F.24. HODGE AND INERTIAL TYPES

277

Suppose that Minf is as in the statement of Corollary F.23; so it is a Breuil–Kisin–Fargues GK -module of height at most h, which admits all descents over L.
Write l for the residue field of L, write M for the W (l)-module Mπ[ /[π [ ]Mπ[0of Definition F.7 (2), and M for the OL -module ϕ∗ Mπ[ /Eπ[ ϕ∗ Mπ[ of Definition F.7 (2). These modules are independent of the choice of π (which now
denotes a uniformizer of L) and of π [ , by definition.
The semilinear GK -action on Minf induces semilinear actions on M and0
on M , and as noted in the proof of Theorem F.11, the action of GL on
both modules is trivial. Furthermore, by [Liu12, Cor. 2.12], the inertial type
Dpst (V (M ))|IK is given by M[1/p] with its action of IL/K . More precisely,[Liu12, §2.3] shows that W (k)[1/p] ⊗Ainf ϕ∗ Minf is isomorphic to Dpst (T (M )) 'W (k) ⊗W (l) Dst (T (M )|GL ) as W (k)[1/p][GK ]-modules. Furthermore, this isomorphism is compatible with the isomorphism (from the proof of Theorem F.11)
ιπ[ : M[ p1 ] = Dπ[ ' sπ[ (Dπ[ ) oiπ [∼Dst (T (M )|GL ) .

Hence M[ p1 ] ⊂ W (k)[1/p] ⊗Ainf ϕ∗ Minf is endowed with an action of IL/K andis isomorphic to Dpst (V (M )).
We now turn to the Hodge–Tate weights. We have a filtration on the
L ⊗Qp E-module DL = M0 [1/p], which is defined as follows. For each π [ wewrite ΦMπ[ : ϕ∗ Mπ[ → Mπ[ and fπ : ϕ∗ Mπ[ [ p1 ]  DL . For each i ≥ 0 weii ∗idefine Fili ϕ∗ Mπ[ = Φ−1Mπ[ (Eπ [ Mπ [ ) and Fil DL = fπ (Fil ϕ Mπ [ ). Considering the isomorphism ιπ[ : Dst (V (M )|GL ) ' Dπ[ , we obtain an isomorphism
DdR (V (M )|GL ) ' L ⊗W (l)[ p1 ] Dst (V (M )|GL ) ' DL . By [Liu08b, Cor. 3.2.3,
Thm. 3.4.1], this isomorphism respects the filtrations on each side. In particular, Fili DL is independent of the choice of π [ .
By Hilbert 90, DL and its filtration descend to a filtration on a K ⊗Qp Emodule DK ' DdR (V (M )). Then for each σ : K ,→ E, and each i ∈ [0, h], themultiplicity of i in the multiset HTσ (V (M )) is the dimension of the i-th gradedpiece of eσ DK , where eσ ∈ (K ⊗Qp E) is the idempotent corresponding to σ.
We summarise the preceding discussion in the following corollary.
F.25 Corollary. In the setting of Corollary F.23, the inertial type and Hodge
type of V (M ) are determined as follows: the inertial type Dpst (V (M ))|IK isgiven by M[1/p] with its action of IL/K , while the Hodge type of V (M ) is givenby the jumps in the filtration on DK described above.

278

APPENDIX F. BKF MODULES AND GK -REPRESENTATIONS

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Index(Gm )dk , 153
DdR (Minf
A◦ ), 106
Kcyc , 20
K∞ , 20
crys,λ,τ,ORρ
, 17ss,λ,τ,ORρ

, 17Rρ,O , 16[

W (C ), 19
W (ρ), 207
AK , 22A+
K , 23Ainf , 19C, 19C[ , 19
K basic , 58
b cyc , 20
K
WD(MinfA◦ ), 102Cπa[ ,s,d,h , 93
Cd,crys,h , 91
L/K
Cd,crys,h , 97
Cd,ss,h , 91
L/K
Cd,ss,h , 97
OC , 19[
OC, 19
OE , 24
RK,d , 58
RBK,d , 78
Wd,h,s , 65
Wd,h , 65aXd,h,s, 68
XK,d , 57
crys,λ,τXK,d, 115
ss,λ,τ
XK,d
, 115
S, 24
λ, 16

p-adic formal algebraic stack, 218
flat part, 230étale (ϕ, GK )-module, 51
étale (ϕ, GK∞ )-module, 52
étale (ϕ, GKcyc )-module, 51
étale (ϕ, ΓK )-module, 51
étale ϕ-module, 45
k, 16
adic*, 217
algebraic stack, 215
almost Galois extension, 34
almost module, 34
basic (field K), 23
Breuil–Kisin module, 84
Breuil–Kisin–Fargues module, 84
GK -module, 84
admitting all descents, 84
Breuil–Mézard conjecture, 204
canonical GK -action, 91
countably indexed, 217
eigenvalue morphism, 153
essentially twistable, 137
field of norms, 20
finite height ϕ-module, 56
finite type point, 172
formal étale (ϕ, Γ)-module, 74
formal algebraic space, 217
generically maximallyniveau 1, 138
good complex, 17
Herr complex, 122287

nonsplit

of

288

INDEX

Hodge type, 16, 107
bounded by h, 111
effective, 111

versal, 225
versal at finite type points, 177
virtual partial semi-simplification, 172

Ind-algebraic stack, 218
Ind-locally of finite type, 222
inertial type, 16, 102
iso-projective, 37
isotrivial, 267

weak topology, 19
weak Wach module, 65
weakly admissible, 217

labeled Hodge–Tate weights, 16
lattice, 68
level (of weak Wach module), 65
locally countably indexed, 217
maximally nonsplit of niveau 1, 138
maximally nonsplit of niveau 1of weight k, 150
nice obstruction theory, 230
order, 107
ordinary, 168partial semi-simplification, 172
perfect complex, 17
perfection (of étale ϕ-module), 47
Polish group, 247
potentially diagonalizable, 168
representable by algebraic stacks, 215
residual gerbe, 265
residual space, 266
scheme-theoretic image, 221
scheme-theoretically dominant, 221
Serre weight, 16
Steinberg, 210
shift (of a Serre weight), 150
strictly APF, 20
surjective on finite type points, 179
Tate module, 251, 252très ramifiée, 155
twistable, 137
underlying reduced substack, 219
unramified twisting, 137