MODULI STACKS OF TWO-DIMENSIONAL GALOISREPRESENTATIONSANA CARAIANI, MATTHEW EMERTON, TOBY GEE, AND DAVID SAVITT

Abstract. We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, and relate theirgeometry to the weight part of Serre’s conjecture for GL2 .

Contents
1.
2.
3.
4.
5.

Introduction
1
Integral p-adic Hodge theory with tame descent data
9
Moduli stacks of Breuil–Kisin modules and ϕ-modules with descent data 14
Extensions of rank one Breuil–Kisin modules with descent data
58
Moduli stacks of Galois representations and the geometric Breuil–Mézard
conjecture
107
Appendix A. Formal algebraic stacks
111
Appendix B. Serre weights and tame types
115
Appendix C. The geometric Breuil–Mézard Conjecture for potentially
Barsotti–Tate representations
118
References
121

1. Introduction
1.1. Moduli of Galois representations. Let K/Qp be a finite extension, let K
be an algebraic closure of K, and let r : Gal(K/K) → GLd (Fp ) be a continuous
representation. The theory of deformations of r — that is, liftings of r to continuous
representations r : Gal(K/K) → GLd (A), where A is a complete local ring withresidue field Fp — is extremely important in the Langlands program, and inparticular is crucial for proving automorphy lifting theorems via the Taylor–Wiles
method. Proving such theorems often comes down to studying the moduli spaces of
those deformations which satisfy various p-adic Hodge-theoretic conditions.
From the point of view of algebraic geometry, it seems unnatural to study only
formal deformations of this kind, and Kisin observed about fifteen years ago that
results on the reduction modulo p of two-dimensional crystalline representations
The first author was supported in part by NSF grant DMS-1501064, by a Royal Society
University Research Fellowship, and by ERC Starting Grant 804176. The second author was
supported in part by the NSF grants DMS-1303450 and DMS-1601871. The third author was
supported in part by a Leverhulme Prize, EPSRC grant EP/L025485/1, Marie Curie Career
Integration Grant 303605, ERC Starting Grant 306326, and a Royal Society Wolfson Research
Merit Award. The fourth author was supported in part by NSF CAREER grant DMS-1564367
and NSF grant DMS-1702161.
1

fixme TG: warning: now
contains marginalcomments.

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

suggested that there should be moduli spaces of p-adic representations in which theresidual representations r should be allowed to vary. In particular, the special fibresof these moduli spaces would be would be moduli spaces of mod p representationsof Gal(K̄/K).
In this paper we construct such a space (or rather stack) Z of mod p representations in the case d = 2, and describe its geometry. In particular, we show that theirirreducible components are naturally labelled by Serre weights, and that our spaces
give a geometrisation of the weight part of Serre’s conjecture. More precisely, we
prove the following theorem (see Proposition 3.10.19 and Theorem 5.2.2; we explain
the definition of a Serre weight, and see Section 1.3 below for the notion of a Serreweight associated to a Galois representation).
Theorem 1.1.1. The stack Z is an algebraic stack of finite type over Fp , and isequidimensional of dimension [K : Qp ]. The irreducible components of Z are labelledby the Serre weights σ, in such a way that the Fp -points of the component Z(σ)
labelled by σ are precisely the representations r : GK → GL2 (Fp ) having σ as a
Serre weight.
We also show that generic points of the irreducible components admit a simple
description (they are extensions of characters whose restrictions to inertia are
determined by the corresponding Serre weight).
In the course of proving Theorem 1.1.1, we study a partial resolution of themoduli spaces inspired by a construction of Kisin [Kis09] in the setting of formal
deformations, and show that its irreducible components are also naturally labelled bySerre weights. We use this resolution to show that our moduli spaces are genericallyreduced, and as an illustration of the utility of our constructions, we use this to provethe corresponding result for the special fibres of tamely potentially Barsotti–Tate
deformation rings (see Proposition 5.1.1). It seems hard to prove this result purely
in the setting of formal deformations, and we anticipate that it will have applications
to the theory of mod p Hilbert modular forms.
1.2. The construction. The reason that we restrict to the case of two-dimensional
representations is that in this case one knows that most mod p representations are“tamely potentially finite flat”; that is, after restriction to a finite tamely ramified
extension, they come from the generic fibres of finite flat group schemes. Indeed, theonly representations not of this form are the so-called très ramifiée representations,which are twists of extensions of the trivial character by the mod p cyclotomic
character, and can be described explicitly in terms of Kummer theory. (This is alocal Galois-theoretic analogue of the well-known fact that, up to twist, modular
forms of any weight and level Γ1 (N ), with N prime to p, are congruent modulo p tomodular forms of weight two and level Γ1 (N p); the corresponding modular curves
acquire semistable reduction over a tamely ramified extension of Qp .)These Galois representations, and the corresponding finite flat group schemes, canbe described in terms of semilinear algebra data. Such descriptions also exist for moregeneral p-adic Hodge theoretic conditions (such as being crystalline of prescribed
Hodge–Tate weights), although they are more complicated, and can be used toconstruct analogues, for higher dimensional representations, of the moduli stacks weconstruct here; this construction is the subject of the forthcoming paper [EG19a].
The semilinear algebra data that we use in the present paper are Breuil–Kisin
modules and étale ϕ-modules. A Breuil–Kisin module is a module with Frobenius

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

3

over a power series ring, satisfying a condition on the cokernel of the Frobenius
which depends on a fixed integer, called the height of the Breuil–Kisin module.
Inverting the formal variable in the power series ring gives a functor from thecategory of Breuil–Kisin modules to the category of étale ϕ-modules. By Fontaine’s
theory [Fon90], these étale ϕ-modules correspond to representations of Gal(K/K∞ ),where K∞ is an infinite non-Galois extension of K obtained by extracting p-powerroots of a uniformiser. By work of Breuil and Kisin (in particular [Kis09]), forétale ϕ-modules that arise from a Breuil–Kisin module of height at most 1 thecorresponding representations admit a natural extension to Gal(K/K), and in thisway one obtains precisely the finite flat representations. This is the case that we willconsider throughout this paper, extended slightly to incorporate descent data froma finite tamely ramified extension K 0 /K and thereby allowing us to study tamely
potentially finite flat representations.
Following Pappas and Rapoport [PR09], we then consider the moduli stack C
of rank two projective Breuil–Kisin modules, and the moduli stack R of étale
ϕ-modules, together with the natural map C → R. We deduce from the resultsof [PR09] that the stack C is algebraic (that is, it is an Artin stack); however R is
not algebraic, and indeed is infinite-dimensional. (In fact, we consider versions ofthese stacks with p-adic coefficients, in which case C is a p-adic formal algebraic
stack, but we suppress this for the purpose of this introduction.) The analogous
construction without tame descent data was considered in [EG19b], where it was
shown that one can define a notion of the “scheme-theoretic image” of the morphism
C → R, and that the scheme-theoretic image is algebraic. Using similar arguments,
we define our moduli stack Z of two-dimensional Galois representations to be thescheme-theoretic image of the morphism C → R.
By construction, we know that the closed points of Z are in bijection with the(non-très ramifiée) representations Gal(K/K) → GL2 (Fp ), and by using standard
results on the corresponding formal deformation problems, we know that Z is
equidimensional of dimension [K : Qp ]. The closed points of C correspond topotentially finite flat models of these Galois representations, and we are able todeduce that C is also equidimensional of dimension [K : Qp ] (at least morally, thisis by Tate’s theorem on the uniqueness of prolongations of p-divisible groups).
These constructions are relatively formal. To go further, we combine results fromthe theory of local models of Shimura varieties and Taylor–Wiles patching with an
explicit construction of families of extensions of characters. We begin by describing
the last of these.
Intuitively, a natural source of “families” of representations ρ : Gal(K/K) →GL2 (Fp ) is given by the extensions of two fixed characters. Indeed, given two
×characters χ1 , χ2 : Gal(K/K) → Fp , the Fp -vector space Ext1Gal(K/K) (χ2 , χ1 ) is
usually [K : Qp ]-dimensional, and a back of the envelope calculation suggests that
this should give a ([K : Qp ]−2)-dimensional substack of Z (the difference between an
extension and a representation counts for a −1, as does the Gm of endomorphisms).
Twisting χ1 , χ2 independently by unramified characters gives a candidate for a[K : Qp ]-dimensional family; since Z is equidimensional of dimension [K : Qp ], theclosure of such a family should be an irreducible component of Z.
Since there are only finitely many possibilities for the restrictions of the χito the inertia subgroup I(K/K), this gives a finite list of maximal-dimensionalfamilies. On the other hand, there are up to unramified twist only finitely many

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

irreducible two-dimensional representations of Gal(K/K), which suggests that theirreducible representations should correspond to 0-dimensional substacks. Together
these considerations suggest that the irreducible components of our moduli stackshould be given by the closures of the families of extensions considered in theprevious paragraph, and in particular that the irreducible representations shouldarise as limits of reducible representations. This could not literally be the case forfamilies of Galois representations, rather than families of étale ϕ-modules, and may
seem surprising at first glance, but it is indeed what happens.

1.3. Serre weights. In the body of the paper we make this analysis rigorous, andwe show that the different families that we have constructed exhaust the irreducible
components. We can therefore label the irreducible components of Z as follows. Let kbe the residue field of K; a Serre weight is then an irreducible Fp -representation of
GL2 (k) (or rather an isomorphism class thereof). Such a representation is specified
×
by its highest weight, which can be thought of as a pair of characters k × → Fp ,×which via local class field theory corresponds to a pair of characters I(K/K) → Fp ,and thus to an irreducible component of Z (in fact, we need to make a shift in this
dictionary, corresponding to half the sum of the positive roots of GL2 (k), but we
ignore this for the purposes of this introduction).
This might seem artificial, but in fact it is completely natural, for the following
reason. Following the pioneering work of Serre [Ser87] and Buzzard–Diamond–
Jarvis [BDJ10] (as extended in [Sch08] and [Gee11]), we now know how to associate
a set W (r) of Serre weights to each continuous representation r : GK → GL2 (Fp ),with the property that if F is a totally real field and ρ : GF → GL2 (Fp ) is anirreducible representation coming from a Hilbert modular form, then the possible
weights of Hilbert modular forms giving rise to ρ are precisely determined by thesets W (ρ|GFv ) for places v|p of F (see for example [BLGG13, GK14, GLS15]).
Going back to our labelling of irreducible components above, we have associateda Serre weight σ to each irreducible component of Z. One of our main theorems isthat the representations r on the irreducible component labelled by σ are precisely
the representations with σ ∈ W (r),We emphasise that the existence of such a geometric interpretation of the sets W (r)is far from obvious, and indeed we know of no direct proof using any of the explicit
descriptions of W (r) in the literature; it seems hard to understand in any explicit
way which Galois representations arise as the limits of a family of extensions of given
characters, and the description of the sets W (r) is very complicated (for example,
the description in [BDJ10] relies on certain Ext groups of crystalline characters).
Our proof is indirect, and ultimately makes use of a description of W (r) given
in [GK14], which is in terms of potentially Barsotti–Tate deformation rings of r
and is motivated by the Taylor–Wiles method. We interpret this description in thegeometric language of [EG14], which we in turn interpret as the formal completionof a “geometric Breuil–Mézard conjecture” for our stacks.
We also study the irreducible components of the stack C. This stack admits a
decomposition as a disjoint union of substacks C τ , indexed by the tame inertial
types τ (the substack C τ is the moduli of those Breuil–Kisin modules which havedescent data given by τ ). The inertial local Langlands correspondence assigns afinite set of Serre weights JH(σ(τ )) to τ (the Jordan–Hölder factors of the reduction

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

5

mod p of the representation σ(τ ) of GL2 (OK ) corresponding to τ ), and we showthat the scheme-theoretic image of the morphism C τ → Z is Z τ = ∪σ∈JH(σ(τ )) Z(σ).
The set JH(σ(τ )) can be identified with a subset of the power set S of the set of
embeddings k ,→ Fp . For generic choices of τ , it is equal to S, and in this case weshow that the morphism C τ → Z τ is a generic isomorphism on the source. We areable to show (using the theory of Dieudonné modules) that for any non-scalar type τ ,the irreducible components of C τ can be identified with S, and those irreducible
components not corresponding to elements of JH(σ(τ )) have image in Z τ of positive
codimension. (In the case of scalar types, both C τ and Z τ are irreducible.) It follows
from the results described that Z τ is generically reduced, which is not at all obvious
from its definition.
An important tool in our proofs is that C has rather mild singularities, and inparticular is Cohen–Macaulay and reduced. We show this by relating the singularitiesof the various C τ to the local models at Iwahori level of Shimura varieties of GL2 -type;such a relationship was first found in [Kis09] (in the context of formal deformation
spaces, with no descent data) and [PR09] (in the context of the stacks C, althoughagain without descent data) and developed further by the first author and Levinin [CL18].
1.4. An outline of the paper. In Section 2 we recall the theory of Breuil–Kisin
modules and étale ϕ-modules, and explain how it extends to the setting of tamedescent data. In Section 3 we define the stacks C, R and Z, and prove some of theirbasic properties following [EG19b]. We relate the singularities of C to those of local
models, define the Dieudonné stack, and explain how the morphism from C to the
Dieudonné stack can be thought of in terms of effective Cartier divisors.
In Section 4 we build our families of reducible Galois representations, and showthat they are dense in Z. We begin with a thorough study of spaces of extensionsof Breuil–Kisin modules, before considering their scheme-theoretic images in R.
After some general considerations we specialise to the case of extensions of rank one
Breuil–Kisin modules, where we explicitly calculate the dimensions of the extension
groups. We also show that the Kisin variety corresponding to an irreducible Galois
representation has “small” dimension, by using a base change argument, and proving
an upper bound on the Kisin variety for reducible representations via an explicit
calculation.
In Section 5 we prove our main results, by combining the hands-on study of C
and Z of Section 4 with the results on the weight part of Serre’s conjecture and theBreuil–Mézard conjecture from [GK14].
We finish with several appendices, summarising results that we use earlier in thepaper. Appendix A recalls some properties of formal algebraic stacks from [Eme],and proves a technical result that we use in Section 5. Appendix B recalls somestandard facts about Serre weights and the inertial local Langlands correspondence,and finally Appendix C combines the results of [GK14] and [EG14] to prove ageometric Breuil–Mézard result for tamely potentially Barsotti–Tate deformation
rings, which we use in Section 5.
1.5. Final comments. As explained above, our construction excludes the trèsramifiée representations, which are twists of certain extensions of the trivial characterby the mod p cyclotomic character. From the point of view of the weight part of
Serre’s conjecture, they are precisely the representations which admit a twist of

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTthe Steinberg representation as their only Serre weight. In accordance with thepicture described above, this means that the full moduli stack of 2-dimensional
representations of Gal(K/K) can be obtained from our stack by adding in theirreducible components consisting of the très ramifiée representations. This is carriedout in [EG19a], and the geometrisation of the weight part of Serre’s conjecture
described above is extended to this moduli stack, using the results of this paper as
an input.
We assume that p > 2 in much of the paper; while we expect that our results
should also hold if p = 2, there are several reasons to exclude this case. We
are frequently able to considerably simplify our arguments by assuming that theextension K 0 /K is not just tamely ramified, but in fact of degree prime to p; this isproblematic when p = 2, as the consideration of cuspidal types involves a quadraticunramified extension. We also use results on the Breuil–Mézard conjecture which
ultimately depend on automorphy lifting theorems that are not available in the casep = 2 at present (although it is plausible that the methods of [Tho17] could be used
to prove them).
1.6. Acknowledgements. We would like to thank Ulrich Görtz, Wansu Kim and
Brandon Levin for helpful conversations and correspondence.
1.7. Notation and conventions.
Topological groups. If M is an abelian topological group with a linear topology,
then as in [Sta13, 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 insome other references this would be referred to as being complete and separated. In
particular, any p-adically complete ring A is by definition p-adically separated.
Galois theory and local class field theory. If M is a field, we let GM denote its
absolute Galois group. If M is a global field and v is a place of M , let Mv denotethe completion of M at v. If M is a local field, we write IM for the inertia subgroupof GM .
Let p be a prime number. Fix a finite extension K/Qp , with ring of integers
OK and residue field k. Let e and f be the ramification and inertial degrees ofK, respectively, and write #k = pf for the cardinality of k. Let K 0 /K be a finite
tamely ramified Galois extension. Let k 0 be the residue field of K 0 , and let e0 , f 0 be
the ramification and inertial degrees of K 0 respectively.
Our representations of GK will have coefficients in Qp , a fixed algebraic closure of
Qp whose residue field we denote by Fp . Let E be a finite extension of Qp containedin Qp and containing the image of every embedding of K 0 into Qp . Let O be thering of integers in E, with uniformiser $ and residue field F ⊂ Fp .
Fix an embedding σ0 : k 0 ,→ F, and recursively define σi : k 0 ,→ F for all i ∈ Z
pso that σi+1
= σi ; of course, we have σi+f 0 = σi for all i. We let ei ∈ k 0 ⊗Fp F
denote the idempotent satisfying (x ⊗ 1)ei = (1 ⊗ σi (x))ei for all x ∈ k 0 ; notethat ϕ(ei ) = ei+1 . We also denote by ei the natural lift of ei to an idempotent in
W (k 0 ) ⊗Zp O. If M is an W (k 0 ) ⊗Zp O-module, then we write Mi for ei M .
ab
We write ArtK : K × → WK
for the isomorphism of local class field theory,
normalised so that uniformisers correspond to geometric Frobenius elements.

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

7

×Lemma 1.7.1. Let π be any uniformiser of OK . The composite IK → OK
→ k× ,−1×
where the map IK → OK is induced by the restriction of ArtK , sends an elementffg ∈ IK to the image in k × of g(π 1/(p −1) )/π 1/(p −1) .

Proof. This follows (for example) from the construction in [Yos08, Prop. 4.4(iii),
Prop. 4.7(ii), Cor. 4.9, Def. 4.10].
For each σ ∈ Hom(k, Fp ) we define the fundamental character ωσ to σ to be thecomposite
IK

/ O×

K

/ k×

σ

/ F×
p,

×
where the map IK → OK
is induced by the restriction of Art−1
K . Let ε denotethe p-adic cyclotomic character and ε the mod p cyclotomic character, so thatQ×eσ∈Hom(k,Fp ) ωσ = ε. We will often identify characters IK → Fp with characters×k × → Fp via the Artin map, and similarly for their Teichmüller lifts.
Inertial local Langlands. A two-dimensional tame inertial type is (the isomorphismclass of) a tamely ramified representation τ : IK → GL2 (Zp ) that extends to arepresentation of GK and whose kernel is open. Such a representation is of theform τ ' η ⊕ η 0 , and we say that τ is a tame principal series type if η, η 0 both
extend to characters of GK . Otherwise, η 0 = η q , and η extends to a character of GL ,where L/K is a quadratic unramified extension. In this case we say that τ is a tamecuspidal type.
Henniart’s appendix to [BM02] associates a finite dimensional irreducible Erepresentation σ(τ ) of GL2 (OK ) to each inertial type τ ; we refer to this association
as the inertial local Langlands correspondence. Since we are only working with tameinertial types, this correspondence can be made very explicit as follows.
If τ ' η ⊕ η 0 is a tame principal series type, then we also write η, η 0 : k × → O× forthe multiplicative characters determined by η ◦ ArtK |O× , η 0 ◦ ArtK |O× respectively.
KK
If η = η 0 , then we set σ(τ ) = η ◦ det. Otherwise, we write I for the Iwahori
subgroup of GL2 (OK ) consisting of matrices which are upper triangular modulo a
uniformiser $K of K, and write χ = η 0 ⊗ η : I → O× for the characterab
7→ η 0 (a)η(d).
$K c dGL (O )

Then σ(τ ) := IndI 2 K χ.
If τ = η ⊕ η q is a tame cuspidal type, then as above we write L/K for a quadraticunramified extension, and l for the residue field of OL . We write η : l× → O× forthe multiplicative character determined by η ◦ ArtL |O× ; then σ(τ ) is the inflationL
to GL2 (OK ) of the cuspidal representation of GL2 (k) denoted by Θ(η) in [Dia07].
p-adic Hodge theory. We normalise Hodge–Tate weights so that all Hodge–Tateweights of the cyclotomic character are equal to −1. We say that a potentiallycrystalline representation ρ : GK → GL2 (Qp ) has Hodge type 0, or is potentially
Barsotti–Tate, if for each ς : K ,→ Qp , the Hodge–Tate weights of ρ with respectto ς are 0 and 1. (Note that this is a more restrictive definition of potentially
Barsotti–Tate than is sometimes used; however, we will have no reason to deal withrepresentations with non-regular Hodge-Tate weights, and so we exclude them from

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

consideration. Note also that it is more usual in the literature to say that ρ ispotentially Barsotti–Tate if it is potentially crystalline, and ρ∨ has Hodge type 0.)
We say that a potentially crystalline representation ρ : GK → GL2 (Qp ) has
inertial type τ if the traces of elements of IK acting on τ and on
Dpcris (ρ) = lim (Bcris ⊗Qp Vρ )GK 0
−0→
K /K

are equal (here Vρ is the underlying vector space of Vρ ). A representation r : GK →GL2 (Fp ) has a potentially Barsotti–Tate lift of type τ if and only if r admits a lift
to a representation r : GK → GL2 (Zp ) of Hodge type 0 and inertial type τ .
Serre weights. By definition, a Serre weight is an irreducible F-representation of
GL2 (k). Concretely, such a representation is of the form
f −1
σ~t,~s := ⊗j=0
(dettj Symsj k 2 ) ⊗k,σj F,

where 0 ≤ sj , tj ≤ p − 1 and not all tj are equal to p − 1. We say that a Serre weightis Steinberg if sj = p − 1 for all j, and non-Steinberg otherwise.
A remark on normalisations. Given a continuous representation r : GK → GL2 (Fp ),there is an associated (nonempty) set of Serre weights W (r) whose precise definition
we will recall in Appendix B. There are in fact several different definitions of W (r) in
the literature; as a result of the papers [BLGG13, GK14, GLS15], these definitions
are known to be equivalent up to normalisation.
However, the normalisations of Hodge–Tate weights and of inertial local Langlands
used in [GK14, GLS15, EGS15] are not all the same, and so for clarity we lay out
how they differ, and how they compare to the normalisations of this paper.
Our conventions for Hodge–Tate weights and inertial types agree with thoseof [GK14, EGS15], but our representation σ(τ ) is the representation σ(τ ∨ ) of [GK14,
EGS15] (where τ ∨ = η −1 ⊕ (η 0 )−1 ); to see this, note the dual in the definition of σ(τ )in [GK14, Thm. 2.1.3] and the discussion in §1.9 of [EGS15].1
In all cases one chooses to normalise the set of Serre weights so that the conditionof Lemma B.5(1) holds. Consequently, our set of weights W (r) is the set of dualsof the weights W (r) considered in [GK14]. In turn, the paper [GLS15] has the
opposite convention for the signs of Hodge–Tate weights to our convention (andto the convention of [GK14]), so we find that our set of weights W (r) is the set ofduals of the weights W (r∨ ) considered in [GLS15].
Stacks. We follow the terminology of [Sta13]; in particular, we write “algebraic stack”
rather than “Artin stack”. More precisely, an algebraic stack is a stack in groupoidsin the fppf topology, whose diagonal is representable by algebraic spaces, which
admits a smooth surjection from a scheme. See [Sta13, Tag 026N] for a discussionof how this definition relates to others in the literature, and [Sta13, Tag 04XB] for
key properties of morphisms representable by algebraic spaces.
For a commutative ring A, an fppf stack over A (or fppf A-stack) is a stack fibredin groupoids over the big fppf site of Spec A.
1However, this dual is erroneously omitted when the inertial local Langlands correspondence ismade explicit at the end of [EGS15, §3.1]. See Remark B.1.

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

9Scheme-theoretic images. We briefly remind the reader of some definitions from [EG19b,§3.2]. Let X → F be a proper morphism of stacks over a locally Noetherian basescheme S, where X is an algebraic stack which is locally of finite presentation over S,
and the diagonal of F is representable by algebraic spaces and locally of finite
presentation.
We refer to [EG19b, Defn. 3.2.8] for the definition of the scheme-theoretic image Zof the proper morphism X → F. By definition, it is a full subcategory in groupoidsof F, and in fact by [EG19b, Lem. 3.2.9] it is a Zariski substack of F. By [EG19b,
Lem. 3.2.14], the finite type points of Z are precisely the finite type points of F forwhich the corresponding fibre of X is nonzero.
The results of [EG19b, §3.2] give criteria for Z to be an algebraic stack, andprove a number of associated results (such as universal properties of the morphism
Z → F, and a description of versal deformation rings for Z); rather than recallingthese results in detail here, we will refer to them as needed in the body of the paper.
2. Integral p-adic Hodge theory with tame descent data
In this section we introduce various objects in semilinear algebra which arise in
the study of potentially Barsotti–Tate Galois representations with tame descentdata. Much of this material is standard, and none of it will surprise an expert, but
we do not know of a treatment in the literature in the level of generality that we
require; in particular, we are not aware of a treatment of the theory of tame descentdata for Breuil–Kisin modules. However, the arguments are almost identical tothose for strongly divisible modules and Breuil modules, so we will be brief.
The various equivalences of categories between the objects we consider and finite
flat group schemes or p-divisible groups will not be relevant to our main arguments,
except at a motivational level, so we largely ignore them.
2.1. Breuil–Kisin modules and ϕ-modules with descent data. Recall that
we have a finite tamely ramified Galois extension K 0 /K. Suppose further that there
0
exists a uniformiser π 0 of OK 0 such that π := (π 0 )e(K /K) is an element of K, wheree(K 0 /K) is the ramification index of K 0 /K. Recall that k 0 is the residue field of K 0 ,while e0 , f 0 are the ramification and inertial degrees of K 0 respectively. Let E(u) be
the minimal polynomial of π 0 over W (k 0 )[1/p].
Let ϕ denote the arithmetic Frobenius automorphism of k 0 , which lifts uniquely
to an automorphism of W (k 0 ) that we also denote by ϕ. Define S := W (k 0 )[[u]],and extend ϕ to S byX X
ϕai ui =ϕ(ai )upi .
0

By our assumptions that (π 0 )e(K /K) ∈ K and that K 0 /K is Galois, for eachg ∈ Gal(K 0 /K) we can write g(π 0 )/π 0 = h(g) with h(g) ∈ µe(K 0 /K) (K 0 ) ⊂ W (k 0 ),and we let Gal(K 0 /K) act on S via
X Xgai ui =g(ai )h(g)i ui .
Let A be a p-adically complete Zp -algebra, set SA := (W (k 0 ) ⊗Zp A)[[u]], andextend the actions of ϕ and Gal(K 0 /K) on S to actions on SA in the obvious(A-linear) fashion.
Lemma 2.1.1. An SA -module is projective if and only if it is projective as anA[[u]]-module.

10

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTProof. Suppose that M is an SA -module that is projective as an A[[u]]-module.
Certainly W (k 0 ) ⊗Zp M is projective over SA , and we claim that it has M as an
SA -module direct summand. Indeed, this follows by rewriting M as W (k 0 )⊗W (k0 ) M
and noting that W (k 0 ) is a W (k 0 )-module direct summand of W (k 0 ) ⊗Zp W (k 0 ). 
The actions of ϕ and Gal(K 0 /K) on SA extend to actions on SA [1/u] =(W (k 0 ) ⊗Zp A)((u)) in the obvious way. It will sometimes be necessary to consider the subring S0A := (W (k) ⊗Zp A)[[v]] of SA consisting of power series in0v := ue(K /K) , on which Gal(K 0 /K) acts trivially.
Definition 2.1.2. Fix a p-adically complete Zp -algebra A. A weak Breuil–Kisin
module with A-coefficients and descent data from K 0 to K is a triple (M, ϕM , {ĝ}g∈Gal(K 0 /K) )consisting of a SA -module M and a ϕ-semilinear map ϕM : M → M such that:• the SA -module M is finitely generated and u-torsion free, and• the induced map ΦM = 1 ⊗ ϕM : ϕ∗ M → M is an isomorphism after
inverting E(u) (here as usual we write ϕ∗ M := SA ⊗ϕ,SA M),
together with additive bijections ĝ : M → M, satisfying the further properties thatthe maps ĝ commute with ϕM , satisfy gˆ1 ◦ gˆ2 = g\
1 ◦ g2 , and have ĝ(sm) = g(s)ĝ(m)for all s ∈ SA , m ∈ M. We say that M is has height at most h if the cokernel ofΦM is killed by E(u)h .
If M as above is projective as an SA -module (equivalently, if the condition that
the M is u-torsion free is replaced with the condition that M is projective) then we
say that M is a Breuil–Kisin module with A-coefficients and descent data from K 0
to K, or even simply that M is a Breuil–Kisin module.
The Breuil–Kisin module M is said to be of rank d if the underlying finitely
generated projective SA -module has constant rank d. It is said to be free if theunderlying SA -module is free.
A morphism of (weak) Breuil–Kisin modules with descent data is a morphismof SA -modules that commutes with ϕ and with the ĝ. In the case that K 0 = K
the data of the ĝ is trivial, so it can be forgotten, giving the category of (weak )Breuil–Kisin modules with A-coefficients. In this case it will sometimes be convenient
to elide the difference between a Breuil–Kisin module with trivial descent data, anda Breuil–Kisin module without descent data, in order to avoid making separate
definitions in the case of Breuil–Kisin modules without descent data; the same
convention will apply to the étale ϕ-modules considered below.
Lemma 2.1.3. Suppose either that A is a Z/pa Z-algebra for some a ≥ 1, or that A
is p-adically separated and M is projective. Then in Definition 2.1.2 the condition
that ΦM is an isomorphism after inverting E(u) may equivalently be replaced withthe condition that ΦM is injective and its cokernel is killed by a power of E(u).
Proof. If A is a Z/pa Z-algebra for some a ≥ 0, then E(u)h divides ue(a+h−1) in SA(see [EG19b, Lem. 5.2.6] and its proof), so that M[1/u] is étale in the sense thatthe induced map
ΦM [1/u] : ϕ∗ M[1/u] → M[1/u]is an isomorphism. The injectivity of ΦM now follows because M, and thereforeϕ∗ M, is u-torsion free.
If instead A is p-adically complete, then no Eisenstein polynomial over W (k 0 ) is azero divisor in SA : this is plainly true if p is nilpotent in A, from which one deduces

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

11

the same for p-adically complete A. Assuming that M is projective, it follows thatthe maps M → M[1/E(u)] and ϕ∗ M → (ϕ∗ M)[1/E(u)] are injective, and we aredone.

Remark 2.1.4. We refer the reader to [EG19b, §5.1] for a discussion of foundational
results concerning finitely generated modules over the power series ring A[[u]]. In
particular (using Lemma 2.1.1) we note the following.
(1) An SA -module M is finitely generated and projective if and only if it isu-torsion free and u-adically complete, and M/uM is a finitely generated
projective A-module ([EG19b, Prop. 5.1.8]).
(2) If the SA -module M is projective of rank d, then it is Zariski locally free ofrank d in the sense that there is a cover of Spec A by affine opens Spec Bi
such that each of the base-changed modules M ⊗SA SBi is free of rank d([EG19b, Prop. 5.1.9]).
(3) If A is coherent (so in particular, if A is Noetherian), then A[[u]] is faithfully
flat over A, and so SA is faithfully flat over A, but this need not hold if A
is not coherent.
Definition 2.1.5. If Q is any (not necessarily finitely generated) A-module, and M
b A Q denote the u-adic completion of M ⊗A Q.
is an A[[u]]-module, then we let M ⊗Lemma 2.1.6. If M is a Breuil–Kisin module and B is an A-algebra, then theb A B is a Breuil–Kisin module.
base change M ⊗b AB ∼
Proof. We claim that M ⊗= M⊗A[[u]] B[[u]] for any finitely generated projective
A[[u]]-module; the lemma then follows immediately from Definition 2.1.2.
To check the claim, we must see that the finitely generated B[[u]]-module M⊗A[[u]]B[[u]] is u-adically complete. But M is a direct summand of a free A[[u]]-module offinite rank, in which case M⊗A[[u]] B[[u]] is a direct summand of a free B[[u]]-moduleof finite rank and hence is u-adically complete.

Remark 2.1.7. If I ⊂ A is a finitely generated ideal then A[[u]] ⊗A A/I ∼= (A/I)[[u]],b A A/I; so in this case M ⊗A A/I itself
and M ⊗A A/I ∼= M ⊗A[[u]] (A/I)[[u]] ∼= M⊗
is a Breuil–Kisin module.
Note that the base change (in the sense of Definition 2.1.5) of a weak Breuil–Kisin
module may not be a weak Breuil–Kisin module, because the property of being
u-torsion free is not always preserved by base change.
We make the following two further remarks concerning base change.
Remark 2.1.8. (1) If A is Noetherian, if Q is finitely generated over A, and if N
is finitely generated over A[[u]], then N ⊗A Q is finitely generated over A[[u]], andhence (by the Artin–Rees lemma) is automatically u-adically complete. Thus inb A Q is an isomorphism.
this case the natural morphism N ⊗A Q → N ⊗b(2) Note that A[[u]] ⊗A Q = Q[[u]] (the A[[u]]-module consisting of power series
with coefficients in the A-module Q), and so if N is Zariski locally free on Spec A,
b A Q is Zariski locally isomorphic to a direct sum of copies of Q[[u]], andthen N ⊗hence is u-torsion free (as well as being u-adically complete). In particular, by
Remark 2.1.4(2), this holds if N is projective.
Definition 2.1.9. Let A be a Z/pa Z-algebra for some a ≥ 1. A weak étale ϕmodule with A-coefficients and descent data from K 0 to K is a triple (M, ϕM , {ĝ})consisting of:

12

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT• a finitely generated SA [1/u]-module M ;• a ϕ-semilinear map ϕM : M → M with the property that the induced map
ΦM = 1 ⊗ ϕM : ϕ∗ M := SA [1/u] ⊗ϕ,SA [1/u] M → M
is an isomorphism,
together with additive bijections ĝ : M → M for g ∈ Gal(K 0 /K), satisfying thefurther properties that the maps ĝ commute with ϕM , satisfy gˆ1 ◦ gˆ2 = g\
1 ◦ g2 , andhave ĝ(sm) = g(s)ĝ(m) for all s ∈ SA [1/u], m ∈ M .
If M as above is projective as an SA [1/u]-module then we say simply that M isan étale ϕ-module. The étale ϕ-module M is said to be of rank d if the underlying
finitely generated projective SA [1/u]-module has constant rank d.
Remark 2.1.10. We could also consider étale ϕ-modules for general p-adicallycomplete Zp -algebras A, but we would need to replace SA [1/u] by its p-adic
completion. As we will not need to consider these modules in this paper, we do not
do so here, but we refer the interested reader to [EG19a].
A morphism of weak étale ϕ-modules with A-coefficients and descent data from
K 0 to K is a morphism of SA [1/u]-modules that commutes with ϕ and with theĝ. Again, in the case K 0 = K the descent data is trivial, and we obtain the usual
category of étale ϕ-modules with A-coefficients.
Note that if A is a Z/pa Z-algebra, and M is a Breuil–Kisin module (resp., weak
Breuil–Kisin module) with descent data, then M[1/u] naturally has the structure ofan étale ϕ-module (resp., weak étale ϕ-module) with descent data.
Suppose that A is an O-algebra (where O is as in Section 1.7). In making
calculations, it is often convenient to use the idempotents ei (again as in Section 1.7).
In particular if M is a Breuil–Kisin module, then writing as usual Mi := ei M, wewrite ΦM,i : ϕ∗ (Mi−1 ) → Mi for the morphism induced by ΦM . Similarly if M isan étale ϕ-module then we write Mi := ei M , and we write ΦM,i : ϕ∗ (Mi−1 ) → Mi
for the morphism induced by ΦM .
2.2. Dieudonné modules. Let A be a Zp -algebra. We define a Dieudonné moduleof rank d with A-coefficients and descent data from K 0 to K to be a finitely generated
projective W (k 0 ) ⊗Zp A-module D of constant rank d on Spec A, together with:• A-linear endomorphisms F, V satisfying F V = V F = p such that F isϕ-semilinear and V is ϕ−1 -semilinear for the action of W (k 0 ), and• a W (k 0 ) ⊗Zp A-semilinear action of Gal(K 0 /K) which commutes with F
and V .
Definition 2.2.1. If M is a Breuil–Kisin module of height at most 1 and rank dwith descent data, then there is a corresponding Dieudonné module D = D(M)of rank d defined as follows. We set D := M/uM with the induced action ofGal(K 0 /K), and F given by the induced action of ϕ. The endomorphism V isdetermined as follows. Write E(0) = cp, so that we have p ≡ c−1 E(u) (mod u).
The condition that the cokernel of ϕ∗ M → M is killed by E(u) allows us to factor
the multiplication-by-E(u) map on M uniquely as V ◦ ϕ, and V is defined to be
c−1 V modulo u.
2.3. Galois representations. The theory of fields of norms [FW79] was usedin [Fon90] to relate étale ϕ-modules with descent data to representations of a certain
absolute Galois group; not the group GK , but rather the group GK∞ , where K∞ isMODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

13

a certain infinite extension of K (whose definition is recalled below). Breuil–Kisin
modules of height h ≤ 1 are closely related to finite flat group schemes (definedover OK 0 , but with descent data to K on their generic fibre). Passage from a
Breuil–Kisin module to its associated étale ϕ-module can morally be interpreted as
the passage from a finite flat group scheme (with descent data) to its corresponding
Galois representation (restricted to GK∞ ). Since the generic fibre of a finite flatgroup scheme over OK 0 , when equipped with descent data to K, in fact gives rise
to a representation of GK , in the case h = 1 we may relate Breuil–Kisin modules
with descent data (or, more precisely, their associated étale ϕ-modules), not only torepresentations of GK∞ , but to representations of GK .
In this subsection, we recall some results coming from this connection, and draw
some conclusions for Galois deformation rings.
2.3.1. From étale ϕ-modules to GK∞ -representations. We begin by recalling from
[Kis09] some constructions arising in p-adic Hodge theory and the theory of fields ofnorms, which go back to [Fon90]. Following Fontaine, we write R := limx7→xp OK̄ /p.
←−√n
Fix a compatible system ( p π )n≥0 of pn th roots of π in K̄ (compatible in the
√ p
√
√
n+1
n
n
0
obvious sense that p
π = p π ), and let K∞ := ∪n K( p π), and also K∞
:=
√
√
0 pn
0
pn
∪n K ( π). Since (e(K /K),
p)
=
1,
the
compatible
system(π)
determinesn≥0√√ e(K 0 /K) a
pn
pn
n
0
0)
unique compatible system
(
π
of
p
th
roots
of
π
such
that
(
π0 )
=
n≥0
√
√
n
n
p
0
0
p
π. Write π = ( π 0 )n≥0 ∈ R, and [π ] ∈ W (R) for its Teichmüller representative.
We have a Frobenius-equivariant inclusion S ,→ W (R) by sending u 7→ [π 0 ]. We can0naturally identify Gal(K∞/K∞ ) with Gal(K 0 /K), and doing this we see that theaction of g ∈ GK∞ on u is via g(u) = h(g)u.
We let OE denote the p-adic completion of S[1/u], and let E be the field of fractionsof OE . The inclusion S ,→ W (R) extends to an inclusion E ,→ W (Frac(R))[1/p].
Let E nr be the maximal unramified extension of E in W (Frac(R))[1/p], and let
OE nr ⊂ W (Frac(R)) denote its ring of integers. Let OEd
nr be the p-adic completionof OE nr . Note that OEd
isstableundertheactionofG
K∞ .
nr
Definition 2.3.2. Suppose that A is a Z/pa Z-algebra for some a ≥ 1. If M
is a weak étale ϕ-module with A-coefficients and descent data, set TA (M ) :=
ϕ=1
OEd
, an A-module with a GK∞ -action (via the diagonal action on
nr ⊗S[1/u] M
OEd
nr and M , the latter given by the ĝ). If M is a weak Breuil–Kisin module with
A-coefficients and descent data, set TA (M) := TA (M[1/u]).
Lemma 2.3.3. Suppose that A is a local Zp -algebra and that |A| < ∞. Then TA
induces an equivalence of categories from the category of weak étale ϕ-modules with
A-coefficients and descent data to the category of continuous representations of
GK∞ on finite A-modules. If A → A0 is finite, then there is a natural isomorphism∼
TA (M ) ⊗A A0 −→ TA0 (M ⊗A A0 ). A weak étale ϕ-module with A-coefficients anddescent data M is free of rank d if and only if TA (M ) is a free A-module of rank d.
Proof. This is due to Fontaine [Fon90], and can be proved in exactly the same way
as [Kis09, Lem. 1.2.7].

We will frequently simply write T for TA . Note that if we let M 0 be the étale
ϕ-module obtained from M by forgetting the descent data, then by definition we
have T (M 0 ) = T (M )|GK 0 .
∞14

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

2.3.4. Relationships between GK -representations and GK∞ -representations. We will
later need to study deformation rings for representations of GK in terms of thedeformation rings for the restrictions of these representations to GK∞ . Note thatthe representations of GK∞ coming from Breuil–Kisin modules of height at most 1
admit canonical extensions to GK by [Kis09, Prop. 1.1.13].
Lemma 2.3.5. If r, r0 : GK → GL2 (Fp ) are continuous representations, both ofwhich arise as the reduction mod p of potentially Barsotti–Tate representations oftame inertial type, and there is an isomorphism r|GK∞ ∼= r0 |GK∞ , then r ∼= r0 .
Proof. The extension K∞ /K is totally wildly ramified. Since the irreducible
Fp -representations of GK are induced from tamely ramified characters, we seethat r|GK∞ is irreducible if and only if r is irreducible, and if r or r0 is irreducible
then we are done. In the reducible case, we see that r and r0 are extensions of
the same characters, and the result then follows from [GLS15, Lem. 5.4.2] andLemma B.5 (2).
Let r : GK → GL2 (F) be a continuous representation, let Rr denote the universal[0,1]framed deformation O-algebra for r, and let Rr be the quotient with the property
that if A is an Artinian local O-algebra with residue field F, then a local O-morphism[0,1]
Rr → A factors through Rr if and only if the corresponding GK -module (ignoring
the A-action) admits a GK -equivariant surjection from a potentially crystalline
O-representation all of whose Hodge–Tate weights are equal to 0 or 1, and whose
restriction to GK 0 is crystalline. (The existence of this quotient follows as in [Kim11,§2.1].)Let Rr|GK be the universal framed deformation O-algebra for r|GK∞ , and let
≤1
Rr|
G

∞

denote the quotient with the property that if A is an Artinian local O-

K∞

algebra with residue field F, then a morphism Rr|GK∞≤1→ A factors through Rr|
GK∞

if and only if the corresponding GK∞ -module is isomorphic to T (M) for someweak Breuil–Kisin module M of height at most one with A-coefficients and descentdata from K 0 to K. (The existence of this quotient follows exactly as for [Kim11,
Thm. 1.3].)
Proposition 2.3.6. The natural map induced by restriction from GK to GK∞[0,1]≤1induces an isomorphism Spec Rr → Spec Rr|.
GK∞

Proof. This can be proved in exactly the same way as [Kim11, Cor. 2.2.1] (which isthe case that E = Qp and K 0 = K).

3. Moduli stacks of Breuil–Kisin modules and ϕ-modules with descentdata
In this section we define moduli stacks of Breuil–Kisin modules with tame descentdata, following [PR09, EG19b] (which consider the case without descent data). In
particular, we define various stacks Z in Section 3.9, as scheme-theoretic images ofmorphisms from moduli stacks of Breuil–Kisin modules to moduli stacks of étale
ϕ-modules; these stacks are the main objects of interest in the rest of the paper. In
the rest of the section, we use the theories of local models of Shimura varieties andDieudonné modules to begin our study of the geometry of these stacks.

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

15

3.1. Moduli stacks of Breuil–Kisin modules. We begin by defining the moduli
stacks of Breuil–Kisin modules, with and without descent data. We will make useof the notion of a $-adic formal algebraic stack, which is recalled in Appendix A.
dd,aaDefinition 3.1.1. For each integer a ≥ 1, we let Cd,h,K
0 be the fppf stack over O/$
dd,awhich associates to any O/$a -algebra A the groupoid Cd,h,K
0 (A) of rank d Breuil–Kisin modules of height at most h with A-coefficients and descent data from K 0 toK.
dd,aBy [Sta13, Tag 04WV], we may also regard each of the stacks Cd,h,K
0 as an fppf
dd,add
stack over O, and we then write Cd,h,K
0 := lim Cd,h,K 0 ; this is again an fppf stack−→aover O.
We will frequently omit any (or all) of the subscripts d, h, K 0 from this notation
when doing so will not cause confusion. In the case that K = K 0 , we write C a for
C dd,a and C for C dd .

The natural morphism C dd → Spec O factors through Spf O, and by construction,∼
there is an isomorphism C dd,a −→ C dd ×Spf O Spec O/$a , for each a ≥ 1; in particular,
each of the morphisms C dd,a → C dd,a+1 is a thickening (in the sense that its pullback
under any test morphism Spec A → C dd,a+1 becomes a thickening of schemes, as
defined in [Sta13, Tag 04EX]2). In Corollary 3.1.7 below we show that for each
integer a ≥ 1, C dd,a is in fact an algebraic stack of finite type over Spec O/$a , andthat C dd (which is then a priori an Ind-algebraic stack, endowed with a morphism to
Spf O which is representable by algebraic stacks) is in fact a $-adic formal algebraic
stack, in the sense of Definition A.2.
Our approach will be to deduce the statements in the case with descent datafrom the corresponding statements in the case with no descent data, which follow
from the methods of Pappas and Rapoport [PR09]. More precisely, in that reference
it is proved that each C a is an algebraic stack over O/$a [PR09, Thm. 0.1 (i)], andthus that C := lima C a is a $-adic Ind-algebraic stack (in the sense that it is an−→
Ind-algebraic stack with a morphism to Spf O that is representable by algebraic
stacks). (In [PR09] the stack C is described as being a p-adic formal algebraic stack.
However, in that reference, this term is used synonomously with our notion of ap-adic Ind-algebraic stack; the question of the existence of a smooth cover of C by ap-adic formal algebraic space is not discussed. As we will see, though, the existenceof such a cover is easily deduced from the set-up of [PR09].)
We thank Brandon Levin for pointing out the following result to us. The proof isessentially (but somewhat implicitly) contained in the proof of [CL18, Thm. 3.5],
but we take the opportunity to make it explicit. Note that it could also be directlydeduced from the results of [PR09] using Lemma A.3, but the proof that we givehas the advantage of giving an explicit cover by a formal algebraic space.
Proposition 3.1.2. For any choice of d, h, C is a $-adic formal algebraic stack offinite type over Spf O with affine diagonal.
Proof. We begin by recalling some results from [PR09, §3.b] (which is where theproof that each C a is an algebraic stack of finite type over O/$a is given). If A is
2Note that for morphisms of algebraic stacks — and we will see below that C dd,a and C dd,a+1
are algebraic stacks — this notion of thickening coincides with the notion defined in [Sta13, Tag
0BPP], by [Sta13, Tag 0CJ7].

16

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTan O/$a -algebra for some a ≥ 1, then we set
L+ G(A) := GLd (SA ),0

LGh,K (A) := {X ∈ Md (SA ) | X −1 ∈ E(u)−h Md (SA )},0and let g ∈ L+ G(A) act on the right on LGh,K (A) by ϕ-conjugation as g −1 ·X ·ϕ(g).
Then we may write0
C = [LGh,K /φ L+ G].
For each n ≥ 1 we have the principal congruence subgroup Un of L+ G given by
Un (A) = I + un · Md (SA ). As in [PR09, §3.b.2], for any integer n(a) > eah/(p − 1)
we have a natural identification
0∼ [LGh,K 0 /Un(a) ]O/$a(3.1.3)[LGh,K /φ Un(a) ]O/$a =where the Un(a) -action on the right hand side is by left translation by the inverse;
0h,Kmoreover this quotient stack is represented by a finite type scheme (Xn(a))O/$a ,
and we find that
h,K 0
Ca ∼
= [(Xn(a) )O/$a /φ (Gn(a) )O/$a ],where (Gn(a) )O/$a = (L+ G/Un(a) )O/$a is a smooth finite type group scheme over
O/$a .
h,K 0
Now define Ya := [(Xn(a))O/$a /φ (Un(1) )O/$a ]. If a ≥ b, then there is a natural
isomorphism (Ya )O/$b ∼Y= b . Thus we may form the $-adic Ind-algebraic stackh,K 0
Y := lima Ya . Since Y1 := (Xn(1))F is a scheme, each Ya is in fact a scheme [Sta13,−→
Tag 0BPW], and thus Y is a $-adic formal scheme. (In fact, it is easy to checkh,K 0
directly that Un(1) acts freely on Xn(a), and thus to see that Ya is an algebraic
space.) The natural morphism Y → C is then representable by algebraic spaces;
indeed, any morphism from an affine scheme to C factors through some C a , andrepresentability by algebraic spaces then follows from the representability by algebraic
spaces of Ya → C a , and the Cartesianness of the diagram
Ya

/Y


Ca


/C

Similarly, the morphism Y → C is smooth and surjective, and so witnesses the claimthat C is a $-adic formal algebraic stack.
To check that C has affine diagonal, it suffices to check that each C a has affinediagonal, which follows from the fact that (Gn(a) )O/$a is in fact an affine groupscheme over O/$a (indeed, as in [PR09, §2.b.1], it is a Weil restriction of GLd ). 
We next introduce the moduli stack of étale ϕ-modules, again both with andwithout descent data.
aDefinition 3.1.4. For each integer a ≥ 1, we let Rdd,ad,K 0 be the fppf O/$ -stack

which associates to any O/$a -algebra A the groupoid Rdd,ad,K 0 (A) of rank d étale
ϕ-modules with A-coefficients and descent data from K 0 to K.
By [Sta13, Tag 04WV], we may also regard each of the stacks Rdd,ad,K 0 as an fppf
dd
dd,aO-stack, and we then write R := lima R, which is again an fppf O-stack.
−→MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

17

We will omit d, K 0 from the notation wherever doing so will not cause confusion,
and when K 0 = K, we write R for Rdd .
Just as in the case of C dd , the morphism Rdd → Spec O factors through Spf O,∼
and for each a ≥ 1, there is a natural isomorphism Rdd,a −→ Rdd ×Spf O Spec O/$a .
Thus each of the morphisms Rdd,a → Rdd,a+1 is a thickening.
dd
dd
There is a natural morphism Cd,h,K
0 → Rd , defined via(M, ϕ, {ĝ}g∈Gal(K 0 /K) ) 7→ (M[1/u], ϕ, {ĝ}g∈Gal(K 0 /K) ),and natural morphisms C dd → C and Rdd → R given by forgetting the descent data.
In the optic of Section 2.3, the stack Rdd
d may morally be thought of as a moduli
dddd
of GK∞ -representations, and the morphisms Cd,h,K
correspond to passage
0 → Rdfrom a Breuil–Kisin module to its underlying Galois representation.
Proposition 3.1.5. For each a ≥ 1, the natural morphism Rdd,a → Ra is representable by algebraic spaces, affine, and of finite presentation.
Proof. To see this, consider the pullback along some morphism Spec A → Ra (where
A is a O/$a -algebra); we must show that given an étale ϕ-module M of rank dwithout descent data, the data of giving additive bijections ĝ : M → M , satisfying
the further property that:• the maps ĝ commute with ϕ, satisfy gˆ1 ◦ gˆ2 = g\
1 ◦ g2 , and we have ĝ(sm) =g(s)ĝ(m) for all s ∈ SA [1/u], m ∈ M
is represented by an affine algebraic space (i.e. an affine scheme!) of finite presentation
over A.
To see this, note first that such maps ĝ are by definition S0A [1/v]-linear. The
data of giving an S0A [1/v]-linear automorphism of M which commutes with ϕ isrepresentable by an affine scheme of finite presentation over A by [EG19b, Prop.
5.4.8] and so the data of a finite collection of automorphisms is also representable bya finitely presented affine scheme over A. The further commutation and composition
conditions on the ĝ cut out a closed subscheme, as does the condition of SA [1/u]semi-linearity, so the result follows.

Corollary 3.1.6. The diagonal of Rdd is representable by algebraic spaces, affine,and of finite presentation.
∼

Proof. Since Rdd = lima Rdd,a −→ lima Rdd ×Spf O Spec O/$a , and since the−→−→transition morphisms are closed immersions (and hence monomorphisms), we havea Cartesian diagram
Rdd,a/ Rdd,a ×O/$a Rdd,a


Rdd


/ Rdd ×O Rdd

for each a ≥ 1, and the diagonal morphism of Rdd is the inductive limit of the
diagonal morphisms of the various Rdd,a . Any morphism from an affine scheme T
to Rdd ×O Rdd thus factors through one of the Rdd,a ×O/$a Rdd,a , and the fibre
product Rdd ×Rdd ×Rdd T may be identified with Rdd,a ×Rdd,a ×Rdd,a T . It is thus
equivalent to prove that each of the diagonal morphisms Rdd,a → Rdd,a ×O/$a Rdd,ais representable by algebraic spaces, affine, and of finite presentation.

18

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTThe diagonal of Rdd,a may be obtained by composing the pullback over Rdd,a ×O
Rof the diagonal Ra → Ra ×O Ra with the relative diagonal of the morphism
dd,aR→ Ra . The first of these morphisms is representable by algebraic spaces,
affine, and of finite presentation, by [EG19b, Thm. 5.4.11 (2)], and the second is also
representable by algebraic spaces, affine, and of finite presentation, since it is therelative diagonal of a morphism which has these properties, by Proposition 3.1.5. dd,aCorollary 3.1.7.
(1) For each a ≥ 1, C dd,a is an algebraic stack of finite
presentation over Spec O/$a , with affine diagonal.
(2) The Ind-algebraic stack C dd := lima C dd,a is furthermore a $-adic formal−→
algebraic stack.
(3) The morphism Chdd → Rdd is representable by algebraic spaces and proper.
Proof. By Proposition 3.1.2, C a is an algebraic stack of finite type over Spec O/$awith affine diagonal. In particular it has quasi-compact diagonal, and so is quasiseparated. Since O/$a is Noetherian, it follows from [Sta13, Tag 0DQJ] that C a isin fact of finite presentation over Spec O/$a .
By Proposition 3.1.5, the morphism Rdd,a ×Ra C a → C a is representable byalgebraic spaces and of finite presentation, so it follows from [Sta13, Tag 05UM]
that Rdd,a ×Ra C a is an algebraic stack of finite presentation over Spec O/$a . In
order to show that C dd,a is an algebraic stack of finite presentation over Spec O/$a ,it therefore suffices to show that the natural monomorphism(3.1.8)C dd,a → Rdd,a ×Ra C a

is representable by algebraic spaces and of finite presentation. We will in fact showthat it is a closed immersion (in the sense that its pull-back under any morphismfrom a scheme to its target becomes a closed immersion of schemes); since the target
is locally Noetherian, and closed immersions are automatically of finite type andquasi-separated, it follows from [Sta13, Tag 0DQJ] that this closed immersion is offinite presentation, as required.
By [Sta13, Tag 0420], the property of being a closed immersion can be checked
after pulling back to an affine scheme, and then working fpqc-locally. The claim thenfollows easily from the proof of [EG19b, Prop. 5.4.8], as fpqc-locally the condition
that a lattice in an étale ϕ-module of rank d with descent data is preserved by theaction of the ĝ is determined by the vanishing of the coefficients of negative powersof u in a matrix.
To complete the proof of (1), it suffices to show that the diagonal of C dd,a isaffine. Since (as we have shown) the morphism (3.1.8) is a closed immersion, andthus a monomorphism, it is equivalent to show that the diagonal of Rdd,a ×Ra C a isaffine. To ease notation, we denote this fibre product by Y. We may then factor
the diagonal of Y as the composite of the pull-back over Y ×O/$a Y of the diagonal
morphism C a → C a ×O/$a C a and the relative diagonal Y → Y ×C a Y. The former
morphism is affine, by [EG19b, Thm. 5.4.9 (1)], and the latter morphism is alsoaffine, since it is the pullback via C a → Ra of the relative diagonal morphism
Rdd,a → Rdd,a ×Ra Rdd,a , which is affine (as already observed in the proof ofCorollary 3.1.6).
To prove (2), consider the morphism C dd → C. This is a morphism of $-adic
Ind-algebraic stacks, and by what we have already proved, it is representable byalgebraic spaces. Since the target is a $-adic formal algebraic stack, it follows

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

19

from [Eme, Lem. 7.9] that the source is also a $-adic formal algebraic stack, as
required.
To prove (3), since each of C dd,a and Rdd,a is obtained from C dd and Rdd via
pull-back over O/$a , it suffices to prove that each of the morphisms C dd,a → Rdd,ais representable by algebraic spaces and proper. Each of these morphisms factors as(3.1.8)proj.

C dd,a −→ Rdd,a ×Ra C a −→ Rdd,a .
We have already shown that the first of these morphisms is a closed immersion,and hence representable by algebraic spaces and proper. The second morphism isalso representable by algebraic spaces and proper, since it is a base-change of themorphism C a → Ra , which has these properties by [EG19b, Thm. 5.4.11 (1)]. The next lemma gives a concrete interpretation of the points of C dd over $adically complete O-algebras, extending the tautological interpretation of the pointsof each C dd,a prescribed by Definition 3.1.1.
Lemma 3.1.9. If A is a $-adically complete O-algebra then the Spf(A)-points of
C dd are the Breuil–Kisin modules of rank d and height h with A-coefficients anddescent data.
Proof. Let M be a Breuil–Kisin module of rank d and height h with A-coefficientsand descent data. Then the sequence {M/$a M}a≥1 defines a Spf(A)-point of C dd
(cf. Remark 2.1.7), and since M is $-adically complete it is recoverable from thesequence {M/$a M}a≥1 .
In the other direction, suppose that {Ma } is a Spf(A)-point of C dd , so that
Ma ∈ C dd,a (A/$a ). Define M = lima Ma , and similarly define ϕM and {ĝ} as←−inverse limits. Observe that ϕ∗ M = lima ϕ∗ Ma (since ϕ : SA → SA makes SA←−
into a free SA -module). Since each ΦMa is injective with cokernel killed by E(u)hthe same holds for ΦM .
Since the required properties of the descent data are immediate, to complete
the proof it remains to check that M is a projective SA -module (necessarily ofrank d, since its rank will equal that of M1 ), which is a consequence of [GD71, Prop.
0.7.2.10(ii)].

We now temporarily reintroduce h to the notation.
Definition 3.1.10. For each h ≥ 0, write Rah for the scheme-theoretic imageof Cha → Ra in the sense of [EG19b, Defn. 3.2.8]; then by [EG19b, Thms. 5.4.19,
5.4.20], Rah is an algebraic stack of finite presentation over Spec O/$a , the morphism
Cha → Ra factors through Rah , and we may write Ra ∼Ra as an inductive limit
= lim
−→h h
of closed substacks, the natural transition morphisms being closed immersions.
We similarly write Rdd,afor the scheme-theoretic image of the morphism Chdd,a →hRdd,a in the sense of [EG19b, Defn. 3.2.8].
Theorem 3.1.11. For each a ≥ 1, Rdd,a is an Ind-algebraic stack. Indeed, we canwrite Rdd,a = limh Xh as an inductive limit of algebraic stacks of finite presentation−→
over Spec O/$a , the transition morphisms being closed immersions.
Proof. As we have just recalled, by [EG19b, Thm. 5.4.20] we can write Ra = limh Rah ,−→so that if we set Xhdd,a := Rdd,a ×Ra Rah , then Rdd,a = limh Xhdd,a , and the transition−→
morphisms are closed immersions. Since Rah is of finite presentation over Spec O/$a ,20

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

and a composite of morphisms of finite presentation is of finite presentation, it
follows from Proposition 3.1.5 and [Sta13, Tag 05UM] that Xhdd,a is an algebraic
stack of finite presentation over Spec O/$a , as required.

Theorem 3.1.12. Rdd,ais an algebraic stack of finite presentation over Spec O/$a .
h
It is a closed substack of Rdd,a , and the morphism Chdd,a → Rdd,a factors througha morphism Chdd,a → Rdd,awhich is representable by algebraic spaces, schemeh
theoretically dominant, and proper.
Proof. As in the proof of Theorem 3.1.11, if we set Xhdd,a := Rdd,a ×Ra Rah ,then Xhdd,a is an algebraic stack of finite presentation over Spec O/$a , and the natural morphism Xhdd,a → Rdd,a is a closed immersion. The morphism Chdd,a → Rdd,afactors through Xhdd,a (because the morphism Cha → Ra factors through its schemetheoretic image Rah ), so by [EG19b, Prop. 3.2.31], Rdd,ais the scheme-theoretichimage of the morphism of algebraic stacks Chdd,a → Xhdd,a . The required properties
now follow from [EG19b, Lem. 3.2.29] (using representability by algebraic spaces
and properness of the morphism Chdd,a → Rdd,a , as proved in Corollary 3.1.7 (3), to
see that the induced morphism Chdd,a → Rdd,ais representable by algebraic spaceshand proper, along with [Sta13, Tag 0DQJ], and the fact that Xhdd,a is of finite
presentation over Spec O/$a , to see that Rdd,ais of finite presentation).
h3.2. Representations of tame groups. Let G be a finite group.
Definition 3.2.1. We let Repd (G) denote the algebraic stack classifying d-dimensional
representations of G over O: if X is any O-scheme, then Repd (G)(X) is the groupoid
consisting of locally free sheaves of rank d over X endowed with an OX -linear action of G (rank d locally free G-sheaves, for short); morphisms are G-equivariant
isomorphisms of vector bundles.
We now suppose that G is tame, i.e. that it has prime-to-p order. In this case(taking into account the fact that F has characteristic p, and that O is Henselian),
the isomorphism classes of d-dimensional G-representations of G over E and over F
are in natural bijection. Indeed, any finite-dimensional representation τ of G over
E contains a G-invariant O-lattice τ ◦ , and the associated representation of G overF is given by forming τ := F ⊗O τ ◦ .
Lemma 3.2.2. Suppose that G is tame, and that E is chosen large enough so thateach irreducible representation of G over E is absolutely irreducible (or, equivalently,
so that each irreducible representation of G over F is absolutely irreducible), and sothat each irreducible representation of G over Qp is defined over E (equivalently, sothat each irreducible representation of G over Fp is defined over F).
(1) Repd (G) is the disjoint union of irreducible components Repd (G)τ , where τranges over the finite set of isomorphism classes of d-dimensional representations of G over E.
(2) A morphism X → Repd (G) factors through Repd (G)τ if and only if theassociated locally free G-sheaf on X is Zariski locally isomorphic to τ ◦ ⊗O OX .
(3) If we write Gτ := AutO[G] (τ ◦ ), then Gτ is a smooth (indeed reductive)
group scheme over O, and Repd (G)τ is isomorphic to the classifying space[Spec O/Gτ ].

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

21

Proof. Since G has order prime to p, the representation P := ⊕σ σ ◦ is a projective
generator of the category of O[G]-modules, where σ runs over a set of representatives
for the isomorphism classes of irreducible E-representations of G. (Indeed, each σ ◦ isprojective, because the fact that G has order prime to p means that all of the Ext1 s
against σ ◦ vanish. To see that ⊕σ σ ◦ is a generator, we need to show that every
O[G]-module admits a non-zero map from some σ ◦ . We can reduce to the case ofa finitely generated module M , and it is then enough (by projectivity) to provethat M ⊗O F admits such a map, which is clear.) Our assumption that each σ isabsolutely irreducible
furthermore shows that EndG (σ ◦ ) = O for each σ, so thatQ
EndG (P ) = σ O.
Standard Morita theory then shows that the functor M 7→ HomG (P, M ) inducesQ
an equivalence betweenQthe category of O[G]-modules and the category of σ Omodules. Of course, a σ O-module is just given by a tuple (Nσ )σ of O-modules,◦and in this optic, the functor HomG (P, –) can be written
L as◦ M 7→ HomG (σ , M ) σ ,with a quasi-inverse functor being given by (Nσ ) 7→ σ σ ⊗O Nσ . It is easily seen
(just using the fact that HomG (P, –) induces an equivalence of categories) that M isa finitely generated projective A-module, for some O-algebra A, if and only if each
HomG (σ ◦ , M ) is a finitely generated projective A-module.
The preceding discussion shows that giving a rank d representation of G overan O-algebra A amounts
to giving a tuple (Nσ )σ of projective A-modules, ofP
ranks nσ , such that σ nσ dim σ = d. For each such tuple of ranks (nσ ), we obtaina corresponding moduli stack Rep(nσ ) (G) classifying rank d representations of G
which decompose in this manner, and Repd (G) is isomorphic to the disjoint unionof the various stacks Rep(nσ ) (G).
If we write τ = ⊕σ σ nσ , then we may relabel Rep(nσ ) (G) as Repτ (G); statements (1) and (2) are then proved. By construction, there is an isomorphism∼

Repτ (G) = Rep(nσ ) (G) −→Y[Spec O/ GLnσ ].

σ

Noting that Gτ := Aut(τ ) =
well.

Q

σ GLnσ /O , we find that statement (3) follows asFor each τ , it follows from the identification of Repd (G)τ with [Spec O/Gτ ]that there is a natural map Repd (G)τ → Spec O. We let π0 (Repd (G)) denote thedisjoint union of copies of Spec O, one for each isomorphism class τ ; then there
is a natural map Repd (G) → π0 (Repd (G)). While we do not want to develop ageneral theory of the étale π0 groups of algebraic stacks, we note that it is natural
to regard π0 (Repd (G)) as the étale π0 of Repd (G).
3.3. Tame inertial types. Write I(K 0 /K) for the inertia subgroup of Gal(K 0 /K).
Since we are assuming that E is large enough that it contains the image of every
embedding K 0 ,→ Qp , it follows in particular that every Qp -character of I(K 0 /K)
is defined over E.
Recall from Subsection 1.7 that if A is an O-algebra, and M is a Breuil–Kisin
module with A-coefficients, then we write Mi for ei M. Since I(K 0 /K) acts triviallyon W (k 0 ), the ĝ for g ∈ I(K 0 /K) stabilise each Mi , inducing an action of I(K 0 /K)
on Mi /uMi .

22

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

Write
Repd,I(K 0 /K) :=

0
fY
−1


Repd I(K 0 /K) ,i=0

the fibre product being taken over O. If {τi } is an f 0 -tuple of isomorphism classesof d-dimensional representations of I(K 0 /K), we write
Repd,I(K 0 /K),{τi } :=

0
fY
−1


Repd I(K 0 /K) τi .

i=0

Lemma 3.2.2 shows that we may write
Repd,I(K 0 /K) =a

Repd,I(K 0 /K),τi .

{τi }Note that since K 0 /K is tamely ramified, I(K 0 /K) is abelian of prime-to-p order, and each τi is just a sum of characters. If all of the τi are equal to somefixed τ , then we write Repd,I(K 0 /K),τ for Repd,I(K 0 /K),{τi } . We have corresponding
stacks π0 (Repd,I(K 0 /K) ), π0 (Repd,I(K 0 /K),{τi } ) and π0 (Repd,I(K 0 /K),τ ), defined in
the obvious way.
If M is a Breuil–Kisin module of rank d with descent data and A-coefficients,
then Mi /uMi is projective A-module of rank d, endowed with
 an A-linear actionof I(K 0 /K), and so is an A-valued point of Repd I(K 0 /K) . Thus we obtain amorphism
Cddd → Repd,I(K 0 /K) ,(3.3.1)

defined via M 7→ (M0 /uM0 , . . . , Mf 0 −1 /uMf 0 −1 ).
Definition 3.3.2. Let A be an O-algebra, and let M be a Breuil–Kisin module ofrank d with A-coefficients. We say that M has mixed type (τi )i if the composite
Spec A → Cddd → Repd,I(K 0 /K) (the first arrow being the morphism that classifies M,
and the second arrow being (3.3.1)) factors through Repd,I(K 0 /K),{τi } . Concretely,
this is equivalent to requiring that, Zariski locally on Spec A, there is an I(K 0 /K)equivariant isomorphism Mi /uMi ∼= A ⊗O τi◦ for each i.
If each τi = τ for some fixed τ , then we say that the type of M is unmixed, or
simply that M has type τ .
Remark 3.3.3. If A = O then a Breuil–Kisin module necessarily has some (unmixed) type τ , since after inverting E(u) and reducing modulo u the map ΦM,i∼gives an I(K 0 /K)-equivariant E-vector space isomorphism ϕ∗ (Mi−1 /uMi−1 )[ p1 ] →(Mi /uMi )[ p1 ]. However if A = F there are Breuil–Kisin modules which have agenuinely mixed type; indeed, it is easy to write down examples of free Breuil–Kisin
modules of rank one of any mixed type (see also [CL18, Rem. 3.7]), which necessarily cannot lift to characteristic zero. This shows that Cddd is not flat over Zp . In
the following sections, when d = 2 and h = 1 we define a closed substack Cddd,BTof C dd which is flat over Zp , and can be thought of as taking the Zariski closure of
Qp -valued Galois representations that become Barsotti–Tate over K 0 and such thatall pairs of labeled Hodge–Tate weights are {0, 1} (see Remark 3.5.8 below).
(τ )Definition 3.3.4. Let Cd i be the étale substack of Cddd which associates to each(τ )O-algebra A the subgroupoid Cd i (A) of Cddd (A) consisting of those Breuil–Kisin

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

23

modules which are of mixed type (τi ). If each τi = τ for some fixed τ , we write Cdτ(τ )for Cd i .
(τ )Proposition 3.3.5. Each Cd i is an open and closed substack of Cddd , and Cddd is(τ )the disjoint union of its substacks Cd i .
Proof. By Lemma 3.2.2, Repd,I(K 0 /K) is the disjoint union of its open and closed(τ )substacks Repd,I(K 0 /K),{τi } . By definition Cd i is the preimage of Repd,I(K 0 /K),{τi }under the morphism (3.3.1); the lemma follows.

3.4. Local models: generalities. Throughout this section we allow d to be arbitrary; in Section 3.5 we specialise to the case d = 2, where we relate the local
models considered in this section to the local models considered in the theory of
Shimura varieties. We will usually omit d from our notation, writing for example C
for Cd , without any further comment. We begin with the following lemma, for which
we allow h to be arbitrary.
Lemma 3.4.1. Let M be a rank d Breuil–Kisin module of height h with descentdata over an O-algebra A. Assume further either that A is pn -torsion for some n,or that A is Noetherian and p-adically complete. Then im ΦM /E(u)h M is a finite
projective A-module, and is a direct summand of M/E(u)h M as an A-module.
Proof. We follow the proof of [Kis09, Lem. 1.2.2]. We have a short exact sequence
0 → im ΦM /E(u)h M → M/E(u)h M → M/ im ΦM → 0
in which the second term is a finite projective A-module (since it is a finite projective
OK 0 ⊗Zp A-module), so it is enough to show that the third term is a projective
A-module. It is therefore enough to show that the finitely generated A-module
M/ im ΦM is finitely presented and flat.
To see that it is finitely presented, note that we have the equality
M/ im ΦM = (M/E(u)h )/(im ΦM /E(u)h ),and the right hand side admits a presentation by finitely generated projective
A-modulesϕ∗ (M/E(u)h ) → M/E(u)h → (M/E(u)h )/(im ΦM /E(u)h ) → 0.
To see that it is flat, it is enough to show that for every finitely generated ideal I
of A, the map
I ⊗A M/ im ΦM → M/ im ΦM
is injective. It follows easily (for example, from the snake lemma) that it is enoughto check that the complex
0 → ϕ∗ M → M → M/ im ΦM → 0,
which is exact by Lemma 2.1.3, remains exact after tensoring with A/I. Since I isb A A/I by Remark 2.1.7, and the desired
finitely generated we have M ⊗A A/I ∼= M⊗
exactness amounts to the injectivity of ΦM ⊗b A A/I for the Breuil–Kisin module
b A A/I. This follows immediately from Lemma 2.1.3 in the case that A is killedM⊗by pn , and otherwise follows from the same lemma once we check that A/I is
p-adically complete, which follows from the Artin–Rees lemma (as A is assumed
Noetherian and p-adically complete).
24

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

We assume from now on that h = 1, but we continue to allow arbitrary d. We
allow K 0 /K to be any Galois extension such that [K 0 : K] is prime to p (so inparticular K 0 /K is tame).
K 0 /K,aDefinition 3.4.2. We let Mloc

be the algebraic stack of finite presentation
K 0 /Kaover Spec O/$ defined as follows: if A is an O/$a -algebra, then Mloc (A) is thegroupoid of tuples (L, L+ ), where:• L is a rank d projective OK 0 ⊗Zp A-module, with a Gal(K 0 /K)-semilinear,
A-linear action of Gal(K 0 /K);
• L+ is an OK 0 ⊗Zp A-submodule of L, which is locally on Spec A a direct
summand of L as an A-module (or equivalently, for which L/L+ is projective
as an A-module), and is preserved by Gal(K 0 /K).
K 0 /K

K 0 /K,a

K 0 /KWe set Mloc := lima Mloc
, so that Mloc is a $-adic formal algebraic stack,−→of finite presentation over Spf O (indeed, it is easily seen to be the $-adic completionof an algebraic stack of finite presentation over Spec O).
dd
Definition 3.4.3. By Lemma 3.4.1, we have a natural morphism Ψ : C1,K
0 →
K 0 /K

Mloc
pair, which takes a Breuil–Kisin module with descent data M of height 1 to the(M/E(u)M, im ΦM /E(u)M).
K 0 /K,aRemark 3.4.4. The definition of the stack Mloc
does not include any condition
that mirrors the commutativity between the Frobenius and the descent data on a
K 0 /Kdd
Breuil–Kisin module, and so in general the morphism ΨA : C1,K
(A)
0 (A) → Mloc
cannot be essentially surjective.
It will be convenient to consider the twisted group rings SA [Gal(K 0 /K)] and (OK 0 ⊗ZpA)[Gal(K 0 /K)], in which the elements g ∈ Gal(K 0 /K) obey the following commutation relation with elements s ∈ SA (resp. s ∈ OK 0 ):g · s = g(s) · g.
(In the literature these twisted group rings are more often written as SA ∗Gal(K 0 /K),
(OK 0 ⊗Zp A) ∗ Gal(K 0 /K), in order to distinguish them from the usual (untwisted)group rings, but as we will only use the twisted versions in this paper, we prefer to
use this notation for them.)
By definition, endowing a finitely generated SA -module P with a semilinear
Gal(K 0 /K)-action is equivalent to giving it the structure of a left SA [Gal(K 0 /K)]module. If P is projective as an SA -module, then it is also projective as an
0
SA [Gal(K 0 /K)]-module. Indeed, SA is a direct summand of SA [Gal(KP /K)] as a
1
0
SA [Gal(K /K)]-module, given by the central idempotent | Gal(K 0 /K)| g∈Gal(K 0 /K) g,
so P is a direct summand of the projective module SA [Gal(K 0 /K)] ⊗SA P . Similar
remarks apply to the case of (OK 0 ⊗Zp A)[Gal(K 0 /K)]-modules.
K 0 /K

dd
Theorem 3.4.5. The morphism Ψ : C1,K
0 → Mloc
spaces and smooth.

is representable by algebraic

Proof. We first show that the morphism Ψ is formally smooth, in the sense that it
satisfies the infinitesimal lifting criterion for nilpotent thickenings of affine test objects[EG19b, Defn. 2.4.2]. For this, we follow the proof of [Kis09, Prop. 2.2.11] (see also
the proof of [CL18, Thm. 4.9]). Let A be an O/$a -algebra and I ⊂ A be a nilpotent

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

25
K 0 /K+dd
ideal. Suppose that we are given MA/I ∈ C1,K
0 (A/I) and (LA , LA ) ∈ Mloc
together with an isomorphism

(A)

∼

+
Ψ(MA/I ) −→ (LA , L+A ) ⊗A A/I =: (LA/I , LA/I ).
dd
We must show that there exists MA ∈ C1,K
0 (A) together with an isomorphism∼+Ψ(MA ) −→ (LA , LA ) lifting the given isomorphism.
As explained above, we can and do think of MA/I as a finite projective SA/I [Gal(K 0 /K)]module, and LA as a finite projective (OK 0 ⊗Zp A)[Gal(K 0 /K)]-module. Since theclosed 2-sided ideal of SA [Gal(K 0 /K)] generated by I consists of nilpotent elements,
we may lift MA/I to a finite projective SA [Gal(K 0 /K)]-module MA . (This ispresumably standard, but for lack of a reference we indicate a proof. In fact theproof of [Sta13, Tag 0D47] goes over unchanged to our setting. Writing MA/I as adirect summand of a finite free SA/I [Gal(K 0 /K)]-module F , it is enough to lift thecorresponding idempotent in EndSA/I [Gal(K 0 /K)] (F ), which is possible by [Sta13,
Tag 05BU]; see also [Lam91, Thm. 21.28] for another proof of the existence of lifts
of idempotents in this generality.) Note that since MA/I is of rank d as a projective
SA/I -module, MA is of rank d as a projective SA -module.
Since MA /E(u)MA is a projective (OK 0 ⊗Zp A)[Gal(K 0 /K)]-module, we may lift
the composite∼
MA /E(u)MA  MA/I /E(u)MA/I −→ LA/I
to a morphism θ : MA /E(u)MA → LA . Since the composite of θ with LA  LA/I
is surjective, it follows by Nakayama’s lemma that θ is surjective. But a surjective
map of projective modules of the same rank is an isomorphism, and so θ is an∼
isomorphism lifting the given isomorphism MA/I /E(u)MA/I −→ LA/I .
−1
We let M+(L+
A denote the preimage in MA of θA ). The image of the induced+∼b A A/I = MA/I is precisely im ΦMA/I , since the same
map f : MA ⊂ MA  MA ⊗is true modulo E(u) and because M+
A , im ΦMA/I contain E(u)MA , E(u)MA/I
respectively. Observing that
MA /M+ ∼= (MA /E(u)MA )/(M+ /E(u)MA ) ∼= LA /L+AA
A++
we deduce that MA /MA is projective as an A-module, whence MA is an A-module

direct summand of MA . By the same argument im ΦMA/I is an A-module directbsummand of MA/I , and we conclude that the map M+
A ⊗A A/I → im ΦMA/I induced
by f is an isomorphism.
Finally, we have the diagram
ϕ∗ MA

M+
A


ϕ∗ MA/I/ im ΦM

A/I

where the horizontal arrow is given by ΦMA/I , and the right hand vertical arrow is f .
Since ϕ∗ MA is a projective SA [Gal(K 0 /K)]-module, we may find a morphism of
SA [Gal(K 0 /K)]-modules ϕ∗ MA → M+
A which fills in the commutative square. Since∼ + b A A/I is surjective, it follows
the composite ϕ∗ MA → M+→
im
Φ
MA/I = MA ⊗A
by Nakayama’s lemma that ϕ∗ MA → M+
A is also surjective, and the composite
ϕ ∗ MA → M+
⊂
M
gives
a
map
Φ
.
Since
ΦMA [1/E(u)] is a surjective map ofA
MAA
projective modules of the same rank, it is an isomorphism, and we see that MA

26

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

together with ΦMA is our required lifting to a Breuil–Kisin module of rank d withdescent data.
Since the source and target of Ψ are of finite presentation over Spf O, and $-adic,
we see that Ψ is representable by algebraic spaces (by [Eme, Lem. 7.10]) and locallyof finite presentation (by [EG19b, Cor. 2.1.8] and [Sta13, Tag 06CX]). Thus Ψ is infact smooth (being formally smooth and locally of finite presentation).

We now show that the inertial type of a Breuil–Kisin module is visible on thelocal model.
K 0 /K

Lemma 3.4.6. There is a natural morphism Mloc

→ π0 (RepI(K 0 /K) ).

0

K /K

Proof. The morphism Mloc → π0 (RepI(K 0 /K) ) is defined by sending (L, L+ ) 7→
L/π 0 . More precisely, L/π 0 is a rank d projective k 0 ⊗Zp A-module with a linear action

Qf 0 −1of I(K 0 /K), so determines an A/p-point of π0 (RepI(K 0 /K) ) = i=0 π0 (Repd I(K 0 /K) ).
Since the target is a disjoint union of copies of Spec O, the morphism Spec A/p →π0 (RepI(K 0 /K) ) lifts uniquely to a morphism Spec A → π0 (RepI(K 0 /K) ), as required.
Definition 3.4.7. We let(τ )K 0 /K

Mloci := Mloc

⊗π0 (RepI(K 0 /K) ) π0 (RepI(K 0 /K),{τi } ).
(τ )K 0 /KIf each τi = τ for some fixed τ , we write Mτloc for Mloci . By Lemma 3.2.2, Mloc(τ )is the disjoint union of the open and closed substacks Mloci .
(τ )Lemma 3.4.8. We have C (τi ) = C dd ×MK 0 /K Mloci .
loc

Proof. This is immediate from the definitions.



In particular, C (τi ) is a closed substack of C dd .
3.5. Local models: determinant conditions. Write N = K · W (k 0 )[1/p], sothat K 0 /N is totally ramified. Since I(K 0 /K) is cyclic of order prime to p and acts
trivially on ON , we may write(3.5.1)(L, L+ ) = ⊕ξ (Lξ , L+ξ )where the sum is over all characters ξ : I(K 0 /K) → O× , and Lξ (resp. L+ξ ) is theON ⊗ A-submodule of L (resp. of L+ ) on which I(K 0 /K) acts through ξ.
K 0 /K

Definition 3.5.2. We say that an object (L, L+ ) of Mloc (A) satisfies the strong
determinant condition if Zariski locally on Spec A the following condition holds: forall a ∈ ON and all ξ, we haveY
(3.5.3)detA (a|L+
)
=
ψ(a)
ξ
ψ:N ,→E

as polynomial functions on ON in the sense of [Kot92, §5].
Remark 3.5.4. An explicit version of this determinant condition is stated, in thisgenerality, in [Kis09, §2.2], specifically in the proof of [Kis09, Prop. 2.2.5]. We recall
this here, with our notation. We have a direct sum decompositionM∼
ON ⊗Zp A →ON ⊗W (k0 ),σi A
σi :k0 ,→F

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

27

Recall that ei ∈ ON ⊗Zp O denotes the idempotent that identifies ei · ON ⊗Zp A withthe summand ON ⊗W (k0 ),σi A. For j = 0, 1, . . . , e − 1, let Xj,σi be an indeterminate.
Then the strong determinant condition on (L, L+ ) is that for all ξ, we have
(3.5.5)detA 

X=ei π j Xj,σi | L+
ξ

j,σi

YX


ψ(ei π j )Xj,σi ,ψ j,σiwhere j runs over 0, 1, . . . , e − 1, σi over embeddings k 0 ,→ F, and ψ over embeddings
ON ,→ O. Note that ψ(ei ) = 1 if ψ|W (k0 ) lifts σi and is equal to 0 otherwise.
K 0 /K,BT

K 0 /K

Definition 3.5.6. We write Mloc
for the substack of Mloc
given bythose (L, L+ ) which satisfy the strong determinant condition. For each (possi(τ ),BT(τ )K 0 /K,BT
bly mixed) type (τi ), we write Mloci:= Mloci ×MK 0 /K Mloc.
loc

Suppose for the remainder of this section that d = 2 and h = 1, so that C dd
consists of Breuil–Kisin modules of rank two and height at most 1. We thenK 0 /K,BT(τ ),BTset C dd,BT := C dd ×MK 0 /K Mloc
, and C (τi ),BT := C (τi ) ×M(τi ) Mloci
.
loc

loc

A Breuil–Kisin module M ∈ C dd (A) is said to satisfy the strong determinant
K 0 /K
condition if and only if its image Ψ(M) ∈ Mloc (A) does, i.e. if and only if it lies
in C dd,BT .
Proposition 3.5.7. C (τi ),BT (resp. C dd,BT ) is a closed substack of C (τi ) (resp. C dd );in particular, it is a $-adic formal algebraic stack of finite presentation over O.
Proof. This is immediate from Corollary 3.1.7 and the definition of the strong
determinant condition as an equality of polynomial functions.

Remark 3.5.8. The motivation for imposing the strong determinant condition is asfollows. One can take the flat part (in the sense of [Eme, Ex. 9.11]) of the $-adic
formal stack C dd , and on this flat part, one can impose the condition that thecorresponding Galois representations have all pairs of labelled Hodge–Tate weights
equal to {0, 1}; that is, we can consider the substack of C dd corresponding to the
Zariski closure of the these Galois representations.
We will soon see that C dd,BT is flat (Corollary 3.8.3). By Lemma 3.5.16 below,
it follows that the substack of the previous paragraph is equal to C dd,BT ; so we
may think of the strong determinant condition as being precisely the condition
which imposes this condition on the labelled Hodge–Tate weights, and results in
a formal stack which is flat over Spf O. Since the inertial types of p-adic Galois
representations are unmixed, it is natural from this perspective to expect that C dd,BTshould be the disjoint union of the stacks C τ,BT for unmixed types, and indeed thiswill be proved shortly at Corollary 3.5.13.
To compare the strong determinant condition to the condition that the type of a
Breuil–Kisin module is unmixed, we make some observations about these conditionsin the case of finite field coefficients.
K 0 /K

Lemma 3.5.9. Let F0 /F be a finite extension, and let (L, L+ ) be an object of Mloc (F0 ).
Then (L, L+ ) satisfies the strong determinant condition if and only if the following
property holds: for each i and for each ξ : I(K 0 /K) → O× we have dimF0 (L+i )ξ = e.

28

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTProof. This is proved in a similar way to [Kis09, Lemma 2.5.1], using the explicit
formulation of the strong determinant condition from Remark 3.5.4. In the notationof that remark, we see that the strong determinant condition holds at ξ if and only
if for each embedding σi : k 0 ,→ F we have


X
YX

=
(3.5.10)detA 
π j Xj,σi | (L+ψ(π)j Xj,σi ,
i )ξ
j

ψ

j

where the product runs over the embeddings ψ : ON ,→ O with the property
that ψ|W (k0 ) lifts σi . Since π induces a nilpotent endomorphism of (L+i )ξ the leftdim 0 (L+ )hand side of (3.5.10) evaluates to X0,σi F i ξ while the right-hand side, which canebe viewed as a norm from ON ⊗Zp F0 down to W (k 0 ) ⊗Zp F0 , is equal to X0,σ. iLemma 3.5.11. Let F0 /F be a finite extension, and let M be a Breuil–Kisin moduleof rank 2 and height at most one with F0 -coefficients and descent data.
(1) M satisfies the strong determinant condition if and only if the following property holds: for each i and for each ξ : I(K 0 /K) → O× we havedimF0 (im ΦM,i /E(u)Mi )ξ = e.
(2) If M satisfies the strong determinant condition, then the determinant of ΦM,iwith respect to some choice of basis has u-adic valuation e0 .
Proof. The first part is immediate from Lemma 3.5.9. For the second part, letΦM,i,ξ be the restriction of ΦM,i to ϕ∗ (Mi−1 )ξ . We think of Mi and ϕ∗ (Mi−1 ) as0
free F0 [[v]]-modules of rank 2e(K 0 /K), where v = ue(K /K) . We havee(K 0 /K)
detF0 [[v]] (ΦM,i ) = detF0 [[u]] (ΦM,i ).
Since ΦM,i commutes with the descent datum, we also haveY
detF0 [[v]] (ΦM,i ) =detF0 [[v]] (ΦM,i,ξ ),ξ
0where ξ runs over the e(K /K) characters I(K 0 /K) → O× .
The proof of the second part of [Kis09, Lemma 2.5.1] implies that, for each ξ,0
detF0 [[v]] (ΦM,i,ξ ) is v e = ue times a unit. Indeed, each Mi,ξ is a free F0 [[v]]-module ofrank 2. It admits a basis {e1,ξ , e2,ξ } such that im ΦM,i,ξ = hv i e1,ξ , v j e2,ξ i for some
non-negative integers i, j. The strong determinant condition on (im ΦM,ı,ξ /v e Mi,ξ )implies that i + j = 2e − e = e, and this is precisely the v-adic valuation ofe(K 0 /K)
detF0 [[v]] (ΦM,i,ξ ). We deduce that the u-adic valuation of detF0 [[u]] (ΦM,i )ise(K 0 /K) · e0 , which implies the second part of the lemma.

By contrast, we have the following criterion for the type of a Breuil–Kisin module
to be unmixed.
Proposition 3.5.12. Let F0 /F be a finite extension, and let M be a Breuil–Kisin
module of rank 2 and height at most one with F0 -coefficients and descent data. Thenthe type of M is unmixed if and only if dimF0 (im ΦM,i /E(u)Mi )ξ is independent ofξ for each fixed i. In particular, if M satisfies the strong determinant condition,then the type of M is unmixed.
Proof. We begin the proof of the first part with the following observation. Let Λ
be a rank two free F0 [[u]]-module with an action of I(K 0 /K) that is F0 -linear andMODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

29

u-semilinear with respect to a character χ (i.e., such that g(uλ) = χ(g)ug(λ) forλ ∈ Λ). In particular I(K 0 /K) acts on Λ/uΛ through a pair of characters which wecall η and η 0 . Let Λ0 ⊂ Λ be a rank two I(K 0 /K)-sublattice. We claim that there
are integers m, m0 ≥ 0 such that the multiset of characters of I(K 0 /K) occurring inΛ/Λ0 has the shape{ηχi : 0 ≤ i < m} ∪ {η 0 χi : 0 ≤ j < m0 }0and the multiset of characters occurring in Λ0 /uΛ0 is {ηχm , η 0 χm }.
To check the claim we proceed by induction on dimF0 Λ/Λ0 , the case Λ = Λ0 being
trivial. Suppose dimF0 Λ/Λ0 = 1, so that Λ0 lies between Λ and uΛ. Consider the
chain of containments Λ ⊃ Λ0 ⊃ uΛ ⊃ uΛ0 . If without loss of generality I(K 0 /K)
acts on Λ/Λ0 via η, then it acts on Λ0 /uΛ by η 0 and uΛ/uΛ0 by χη, proving theclaim with m = 1 and m0 = 0. The general case follows by iterated application of
the case dimk Λ/Λ0 = 1, noting that since I(K 0 /K) is abelian the quotient Λ/Λ0
has a filtration by I(K/K 0 )-submodules whose graded pieces have dimension 1.
Now return to the statement of the proposition. Let (τi ) be the mixed type of
M and write τi−1 = η ⊕ η 0 . We apply the preceding observation with Λ = im ΦM,i0and Λ0 = E(u)Mi = ue Mi . Note that χ is a generator of the cyclic group I(K 0 /K)of order e0 /e. Since ΦM commutes with descent data, the group I(K 0 /K) acts on
Λ/uΛ via η and η 0 . Then the the multiset{ηχi : 0 ≤ i < m} ∪ {η 0 χi : 0 ≤ j < m0 }contains each character of I(K 0 /K) with equal multiplicity if and only if one of η, η 0is the successor to ηχm−1 in the list η, ηχ, ηχ2 , . . ., and the other is the successor to00η 0 χm −1 in the list η 0 , η 0 χ, η 0 χ2 , . . ., i.e., if and only if {ηχm , η 0 χm } = {η, η 0 }. Since00
Mi /uMi ∼= ue Mi /ue +1 Mi = Λ0 /uΛ0 , this occurs if and only if that τi = τi−1 .
Finally, the last part of the proposition follows immediately from the first partand Lemma 3.5.11.
Corollary 3.5.13. C dd,BT is the disjoint union of its closed substacks C τ,BT .
Proof. This follows from Propositions 3.3.5 and 3.5.12. Indeed, from Proposition 3.3.5, it suffices to show that if (τi ) is a mixed type, and C (τi ),BT is nonzero,then (τi ) is in fact an unmixed type. Indeed, note that if C (τi ),BT is nonzero, thenit contains a dense set of finite type points, so in particular contains an F0 -point
for some finite extension F0 /F. It follows from Proposition 3.5.12 that the type isunmixed, as required.

Remark 3.5.14. Since our primary interest is in Breuil–Kisin modules, we will have(τ ),BTno further need to consider the stacks Mloci
or C (τi ),BT for types that are notunmixed.
Let τ be a tame type; since I(K 0 /K) is cyclic, we can write τ = η ⊕ η 0 for(possibly equal) characters η, η 0 : I(K 0 /K) → O× . Let (L, L+ ) be an object ofMτloc (A). Suppose that ξ 6= η, η 0 . Then elements of Lξ are divisible by π 0 in L, andso multiplication by π 0 induces an isomorphism of projective ei (ON ⊗ A)-modulesof equal rank∼pi,ξ : ei Lξχ−1 −→ ei Lξ
iwhere χi : I(K 0 /K) → O× is the character sending g 7→ σi (h(g)). The induced map++
p+i,ξ : ei Lξχ−1 −→ ei Lξ
i30

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

is in particular an injection. The following lemma will be useful for checking thestrong determinant condition when comparing various different stacks of local model
stacks.
Lemma 3.5.15. Let (L, L+ ) be an object of Mτloc (A). Then (L, L+ ) is an objectof Mτ,BTloc (A) if and only if both
(1) the condition (3.5.3) holds for ξ = η, η 0 , and
π0

+
+
0
(2) the injections p+i,ξ : ei Lξχ−1 −→ ei Lξ are isomorphisms for all ξ 6= η, η andifor all i.
The second condition is equivalent to(2 0 ) we have (L+ /π 0 L+ )ξ = 0 for all ξ 6= η, η 0 .

Proof. The equivalence between (2) and (20 ) is straightforward. Suppose now thatξ 6= η, η 0 . Locally on Spec A the module ei L+
is by definition a direct summandξχ−1i

of ei Lξχ−1 . Since pi,ξ is an isomorphism, the image of p+i,ξ is locally on Spec A ai−1
direct summand of ei L+
ξ . Under the assumption that (3.5.10) holds for i and ξχi ,the condition (3.5.10) for i and ξ is therefore equivalent to the surjectivity of p+i,ξ .
The lemma follows upon noting that χi is a generator of the group of characters
I(K 0 /K) → O× .


To conclude this section we describe the OE 0 -points of C dd,BT , for E 0 /E a finite
extension; recall that our convention is that a two-dimensional Galois representationis Barsotti–Tate if all its labelled pairs of Hodge–Tate weights are equal to {0, 1}(and not just that all of the labelled Hodge–Tate weights are equal to 0 or 1).
Lemma 3.5.16. Let E 0 /E be a finite extension. Then the Spf(OE 0 )-points of
C dd,BT correspond precisely to the potentially Barsotti–Tate Galois representations
GK → GL2 (OE 0 ) which become Barsotti–Tate over K 0 ; and the Spf(OE 0 )-points ofC τ,BT correspond to those representations which are potentially Barsotti–Tate oftype τ .
Proof. In light of Lemma 3.1.9 and the first sentence of Remark 3.3.3, we are reduced
to checking that a Breuil–Kisin module of rank 2 and height 1 with OE 0 -coefficientsand descent data corresponds to a potentially Barsotti–Tate representation if andonly if it satisfies the strong determinant condition, as well as checking that thedescent data on the Breuil–Kisin module matches the type of the corresponding
Galois representation.
Let MOE0 ∈ C dd,BT (Spf(OE 0 )). Plainly MOE0 satisfies the strong determinantcondition if and only if M := MOE0 [1/p] satisfies the strong determinant condition(with the latter having the obvious meaning). Consider the filtration
Fil1 (ϕ∗ (Mi )) := {m ∈ ϕ∗ (Mi ) | ΦM,i+1 (m) ∈ E(u)Mi+1 } ⊂ ϕ∗ Mi
inducing
Fil1i ⊂ ϕ∗ (Mi )/E(u)ϕ∗ (Mi ).
Note that ϕ∗ (Mi )/E(u)ϕ∗ (Mi ) is isomorphic to a free K 0 ⊗W (k0 ),σi E 0 -module ofrank 2. Then M corresponds to a Barsotti–Tate Galois representation
GK 0 → GL2 (E 0 )if and only, if for every i, Fil1i is isomorphic to K 0 ⊗W (k0 ),σi E 0 as a K 0 ⊗W (k0 ),σi E 0 submodule of ϕ∗ Mi /E(u)ϕ∗ Mi . This follows, for example, by specialising theMODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

31

proof of [Kis08, Cor. 2.6.2] to the Barsotti–Tate case (taking care to note thatthe conventions of loc. cit. for Hodge–Tate weights and for Galois representations
associated to Breuil–Kisin modules are both dual to ours).
Let ξ : I(K 0 /K) → O× be a character. Consider the filtration
Fil1i,ξ ⊂ ϕ∗ (Mi )ξ /E(u)ϕ∗ (Mi )ξ ' N 2 ⊗W (k0 ),σi E 0
induced by Fil1i . The strong determinant condition on (im ΦM,i+1 /E(u)Mi+1 )ξ
holds if and only if Fil1i,ξ is isomorphic to N ⊗W (k0 ),σi E 0 . By [CL18, Lemma 5.10],
we have an isomorphism of K 0 ⊗W (k0 ),σi E 0 -modules
Fil1i ' K 0 ⊗N Fil1i,ξ .
This, together with the previous paragraph, allows us to conclude. Note that, sinceu acts invertibly when working with E 0 -coefficients and after quotienting by E(u),the argument is independent of the choice of character ξ.
For the statement about types, let SK00 be Breuil’s period ring (see e.g. [Bre00,§5.1]) endowed with the evident action of Gal(K 0 /K) compatible with the embedding
S ,→ SK00 . Here K00 is the maximal unramified extension in K 0 . Recall that by[Liu08, Cor. 3.2.3] there is a canonical (ϕ, N )-module isomorphism(3.5.17)SK00 ⊗ϕ,S M ∼= SK00 ⊗K00 Dpcris (T (M)).

One sees from its construction that the isomorphism (3.5.17) is in fact equivariant
for the action of I(K 0 /K), and the claim follows by reducing modulo u and all its
divided powers.

3.6. Change of extensions. We now discuss the behaviour of various of ourconstructions under change of K 0 . Let L0 /K 0 be a tame extension such that L0 /K is00
Galois. We suppose that we have fixed a uniformiser π 00 of L0 such that (π 00 )e(L /K ) =π 0 . Let S0A := (W (l0 )⊗Zp A)[[u]], where l0 is the residue field of L0 , and let Gal(L0 /K)
and ϕ act on S0A via the prescription of Section 2.1 (with π 00 in place of π 0 ).
There is a natural injective ring homomorphism OK 0 ⊗Zp A → OL0 ⊗Zp A,
which is equivariant for the action of Gal(L0 /K) (acting on the source via thenatural surjection Gal(L0 /K) → Gal(K 0 /K)). There is also an obvious injective00ring homomorphism SA → S0A sending u 7→ ue(L /K ) , which is equivariant for00
the actions of ϕ and Gal(L0 /K); we have (S0A )Gal(L /K ) = SA . If τ is an inertial
00
type for I(K /K), we write τ for the corresponding type for I(L0 /K), obtained byinflation.
K 0 /K
L0 /K
For any (L, L+ ) ∈ Mloc , we define (L0 , (L0 )+ ) ∈ Mloc by(L0 , (L0 )+ ) := OL0 ⊗OK 0 (L, L+ ),with the diagonal action of Gal(L0 /K). Similarly, for any M ∈ C dd (A), we let M0 :=S0A ⊗SA M, with ϕ and Gal(L0 /K) again acting diagonally.
Proposition 3.6.1.
(1) The assignments (L, L+ ) → (L0 , (L0 )+ ) and M 7→ M0 induce compatible
K 0 /K
L0 /K
dd
dd
monomorphisms Mloc → Mloc and CK
0 → CL0 , i.e., as functors they
are fully faithful.
0
dd
dd
τ
→ C τ , as well(2) The monomorphism CK
0 → CL0 induces an isomorphism C
0dd,BT
as a monomorphism CK→ CLdd,BTand an isomorphism C τ,BT → C τ ,BT .
0
0

32

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTProof. (1) One checks easily that the assignments (L, L+ ) → (L0 , (L0 )+ ) and M 7→
M0 are compatible. For the rest of the claim, we consider the case of the functor
M 7→ M0 ; the arguments for the local models case are similar but slightly easier,and we leave them to the reader. Let A be a $-adically complete O-algebra. If N isa rank d Breuil–Kisin module with descent data from L0 to K, consider the Galois
0000invariants NGal(L /K ) . Since (S0A )Gal(L /K ) = SA , these invariants are naturally
a SA -module, and moreover they naturally carry a Frobenius and descent datasatisfying the conditions required of a Breuil–Kisin module of height at most h. In
general the invariants need not be projective of rank d, and so need not be rank d
Breuil–Kisin module with descent data from K 0 to K. However, in the case N = M0
we have
0

0

0

0

0

0

(M0 )Gal(L /K ) = (S0A ⊗SA M)Gal(L /K ) = (S0A )Gal(L /K ) ⊗SA M = M.
Here the second equality holds e.g. because Gal(L0 /K 0 ) has order prime to p, sothat taking Gal(L0 /K 0 )-invariants is exact, and indeed is given by multiplication
by an idempotent i ∈ S0A (use the decomposition S0A = iS0A ⊕ (1 − i)S0A and notethat i kills the latter summand). It follows immediately that the functor M → M0
dd,ais fully faithful, so CK→ CLdd,a
is a monomorphism.
00(2) Suppose now that N has type τ 0 . In view of what we have proven so far,0in order to prove that C τ → C τ is an isomorphism, it is enough to show that
Gal(L0 /K 0 )N
is a rank d Breuil–Kisin module of type τ , and that the natural map of
S0A -modules(3.6.2)0

0

S0A ⊗SA NGal(L /K ) → N

is an isomorphism. For the remainder of this proof, for clarity we write uK 0 , uL0
instead of u for the variables of SA and S0A respectively. Since the type τ 0 of N
is inflated from τ , the action of Gal(L0 /K 0 ) on N/uL0 N factors through Gal(l0 /k 0 );noting that W (l0 ) has a normal basis for Gal(l0 /k 0 ) over W (k 0 ), we obtain an
isomorphism(3.6.3)00∼

W (l0 ) ⊗W (k0 ) (N/uL0 N)Gal(L /K ) → N/uL0 N.
00

In particular the W (k 0 ) ⊗Zp A-module (N/uL0 N)Gal(L /K ) is projective of rank d.
00
00
Observe however that (N/uK 0 N)Gal(L /K ) = (N/uL0 N)Gal(L /K ) . To see this,00
by the exactness of taking Gal(L /K ) invariants it suffices to check that uiL0 N/ui+1
L0
has trivial Gal(L0 /K 0 )-invariants for 0 < i < e(L0 /K 0 ). Multiplication by uiL0 gives
an isomorphism N/uL0 N ∼= uiL0 N/ui+1
L0 , so that for i in the above range, the inertia
000group I(L /K ) acts linearly on uiL0 N/ui+1
L0 through a twist of τ by a nontrivial
00character; so there are no I(L /K )-invariants, and thus no Gal(L0 /K 0 )-invariants
either.
It follows that the isomorphism (3.6.3) is the map (3.6.2) modulo uL0 . By
Nakayama’s lemma it follows that (3.6.2) is surjective. Since N is projective, thesurjection (3.6.2) is split, and is therefore an isomorphism, since it is an isomorphism
00modulo uL0 . This isomorphism exhibits NGal(L /K ) as a direct summand (as an SA module) of the projective module N, so it is also projective; and it is projective ofrank d, since this holds modulo uK 0 .
Finally, we need to check the compatibility of these maps with the strong
determinant condition. By Corollary 3.5.13, it is enough to prove this for the case0of the morphism C τ → C τ for some τ ; by the compatibility in part (1), this comesMODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

33
0down to the same for the corresponding map of local model stacks Mτloc → Mτloc . If
(L, L+ ) ∈ Mτloc , it therefore suffices to show to show that conditions (1) and (20 ) of
Lemma 3.5.15 for (L, L+ ) and (L0 , (L0 )+ ) := OL0 ⊗OK 0 (L, L+ ) are equivalent. Thisis immediate for condition (20 ), since we have (L0 )+ /π 00 (L0 )+ ∼= l0 ⊗k0 (L+ /π 0 L+ ) as
0
I(L /K)-representations.
Writing τ = η ⊕ η 0 , it remains to relate the strong determinant conditions on
the η, η 0 -parts over both K 0 and L0 . Unwinding the definitions using Remark 3.5.4,
one finds that the condition over L0 is a product of [l0 : k 0 ] copies (with different sets
of variables) of the condition over K 0 . Thus the strong determinant condition overK 0 implies the condition over L0 , while the condition over L0 implies the condition
over K 0 up to an [l0 : k 0 ]th root of unity. Comparing the terms involving only copiesof X0,σi ’s shows that this root of unity must be 1.
0

Remark 3.6.4. The morphism of local model stacks Mτloc → Mτloc is not an isomorphism (provided that the extension L0 /K 0 is nontrivial). The issue is that,
as we observed in the preceding proof, local models (L0 , (L0 )+ ) in the image of
τ0
6 0 only for charactersthe morphism Mτloc → Mloc
can have ((L0 )+ /π 00 (L0 )+ )ξ =0×
ξ : I(L /K) → O that are inflated from I(K 0 /K). However, one does obtain an
isomorphism from the substack of Mτloc of pairs (L, L+ ) satisfying condition (2)0of Lemma 3.5.15 to the analogous substack of Mτloc ; therefore the induced map
0Mτ,BT → Mτ ,BT will also be an isomorphism. Analogous remarks will apply tothe maps of local model stacks in §3.7.
3.7. Explicit local models. We now explain the connection between the moduli
stacks C τ and local models for Shimura varieties at hyperspecial and Iwahori level.
This idea has been developed in [CL18] for Breuil–Kisin modules of arbitrary rankwith tame principal series descent data, inspired by [Kis09], which analyses the casewithout descent data.
The results of [CL18] relate the moduli stacks C τ (in the case that τ is a principal
series type) via a local model diagram to a mixed-characteristic deformation of theaffine flag variety, introduced in this generality by Pappas and Zhu [PZ13]. The
local models in [PZ13, §6] are defined in terms of Schubert cells in the genericfibre of this mixed-characteristic deformation, by taking the Zariski closure of these
Schubert cells. The disadvantage of this approach is that it does not give a directmoduli-theoretic interpretation of the special fibre of the local model. Therefore, it ishard to check directly that our stack C τ,BT , which has a moduli-theoretic definition,corresponds to the desired local model under the diagram introduced in [CL18, Cor.
4.11] 3.
In our rank 2 setting, the local models admit a much more explicit condition, using
lattice chains and Kottwitz’s determinant condition, and in the cases of nonscalar
types, we will relate our local models to the naive local model at Iwahori level for
the Weil restriction of GL2 , in the sense of [PRS13, §2.4].
We begin with the simpler case of scalar inertial types. Suppose that K 0 /K
is totally ramified, and that τ is a scalar inertial type, say τ = η ⊕ η. In thiscase we define the local model stack Mloc,hyp (“hyp” for “hyperspecial”) to be
the fppf stack over Spf O (in fact, a p-adic formal algebraic stack), which to each
3One should be able to check this by adapting the ideas in [HN02, §2.1] and [PZ13, Prop. 6.2]
to ResK/Qp GLn where K/Qp can be ramified.

34

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

p-adically complete O-algebra A associates the groupoid Mloc,hyp (A) consisting ofpairs (L, L+ ), where• L is a rank 2 projective OK ⊗Zp A-module, and• L+ is an OK ⊗Zp A-submodule of L, which is locally on Spec A a direct
summand of L as an A-module (or equivalently, for which the quotient L/L+
is projective as an A-module).
+
We let MBT
loc,hyp be the substack of pairs (L, L ) with the property that for alla ∈ OK , we haveY
(3.7.1)
detA (a|L+ ) =ψ(a)ψ:K,→Eas polynomial functions on OK .
τ
Lemma 3.7.2. The functor (L0 , (L0 )+ ) 7→ ((L0 )η , (L0 )+η ) defines a morphism Mloc →τ,BTMloc,hyp which induces an isomorphism Mloc → MBT
loc,hyp . (We remind the reader
that K 0 /K is assumed totally ramified, and that τ is assumed to be a scalar inertial
type associated to the character η.)

Proof. If (L0 , (L0 )+ ) is an object of Mτloc , the proof that ((L0 )η , (L0 )+η ) is indeed an
object of Mloc,hyp (A) is very similar to the proof of Proposition 3.6.1, and is left tothe reader. Similarly, the reader may verify that the functor(L, L+ ) 7→ (L0 , (L0 )+ ) := OK 0 ⊗OK (L, L+ ),where the action of Gal(K 0 /K) is given by the tensor product of the natural action
on OK 0 with the action on (L, L+ ) given by the character η, defines a morphism
Mloc,hyp → Mτloc .
The composition Mloc,hyp → Mτloc → Mloc,hyp is evidently the identity. The
composition in the other order is not, in general, naturally equivalent to the identity
morphism, because for (L, L+ ) ∈ Mτloc (A) one cannot necessarily recover L+ fromthe projection to its η-isotypic part. However, this will hold if L+ satisfies condition(2) of Lemma 3.5.15 (and so in particular will hold after imposing the strong
determinant condition).
Indeed, suppose (L, L+ ) ∈ Mτloc (A). Then there is a natural Gal(K 0 /K)equivariant map of projective OK 0 ⊗Zp A-modules(3.7.3)

OK 0 ⊗OK Lη → Lof the same rank (in which Gal(K 0 /K) acts by η on Lη ). This map is surjective
because it is surjective on η-parts and the maps pi,ξ are surjective for all ξ 6= η;therefore it is an isomorphism. One further has a a natural Gal(K 0 /K)-equivariant
map of OK 0 ⊗Zp A-modules(3.7.4)+
OK 0 ⊗OK L+
η →Lthat is injective because locally on Spec(A) it is a direct summand of the isomorphism(3.7.3). If one further assumes that L+ satisfies condition (2) of Lemma 3.5.15 then
(3.7.4) is an isomorphism, as claimed.
It remains to check the compatibility of these maps with the strong determinantcondition. If (L, L+ ) ∈ Mloc,hyp then certainly condition (2) of Lemma 3.5.15
holds for (L0 , (L0 )+ ) := OK 0 ⊗OK (L, L+ ) ∈ Mτloc . By Lemma 3.5.15 the strong
determinant condition holds for (L0 , (L0 )+ ) if and only if (3.5.3) holds for L0 with ξ =η; but this is exactly the condition (3.7.1) for L, as required.
MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

35

Next, we consider the case of principal series types. We suppose that K 0 /K
is totally ramified. We begin by defining a p-adic formal algebraic stack Mloc,Iw
over Spf O (“Iw” for Iwahori). For each complete O-algebra A, we let Mloc,Iw (A)+be the groupoid of tuples (L1 , L+
1 , L2 , L2 , f1 , f2 ), where• L1 , L2 are rank 2 projective OK ⊗Zp A-modules,• f1 : L1 → L2 , f2 : L2 → L1 are morphisms of OK ⊗Zp A-modules, satisfying f1 ◦ f2 = f2 ◦ f1 = π,• both coker f1 and coker f2 are rank one projective k ⊗Zp A-modules,+• L+
1 , L2 are OK ⊗Zp A-submodules of L1 , L2 , which are locally on Spec A
direct summands as A-modules (or equivalently, for which the quotients
Li /L+i (i = 1, 2) are projective A-modules), and moreover for which the+++morphisms f1 , f2 restrict to morphisms f1 : L+
1 → L2 , f2 : L2 → L1 .
We let MBT
loc,Iw be the substack of tuples with the property that for all a ∈ OK andi = 1, 2, we haveY
(3.7.5)detA (a|L+
ψ(a)
i )=
ψ:K,→Eas polynomial functions on OK .
Write τ = η ⊕ η 0 with η 6= η 0 . Recall that the character h : Gal(K 0 /K) =I(K 0 /K) → W (k)× is given by h(g) = g(π 0 )/π 0 . Since we are assuming that η 6= η 0 ,for each embedding σ : k ,→ F (which we also think of as σ : W (k) ,→ O) there areintegers 0 < aσ , bσ < e(K 0 /K) with the properties that η 0 /η = σ◦haσ , η/η 0 = σ◦hbσ ;in particular, aσ + bσ = e(K 0 /K). Recalling that eσ ∈ W (k) ⊗Zp O ⊂ OK is thePP
idempotent corresponding to σ, we set π1 = σ (π 0 )aσ eσ , π2 = σ (π 0 )bσ eσ ; so we0000
have π1 π2 = π, and π1 ∈ (OK 0 ⊗Zp O)I(K /K)=η /η , π2 ∈ (OK 0 ⊗Zp O)I(K /K)=η/η .
We define a morphism Mτloc → Mloc,Iw as follows. Given a pair (L, L+ ) ∈+++Mτloc (A), we set (L1 , L+1 ) = (Lη , Lη ), (L2 , L2 ) = (Lη 0 , Lη 0 ), and we let f1 , f2 begiven by multiplication by π1 , π2 respectively. The only point that is perhaps not
immediately obvious is to check that coker f1 and coker f2 are rank one projectivek ⊗Zp A-modules. To see this, note that by [Sta13, Tag 05BU], we can lift (L/π 0 L)η0
to a rank one summand U of the projective OK 0 ⊗Zp A-module L. Let Uη be thedirect summand of Lη obtained by projection of U to the η-eigenspace Lη ; thenthe projective OK ⊗Zp A-module Uη has rank one, as can be checked modulo π.
Similarly we may lift (L/π 0 L)η0 to a rank one summand V of L, and we let Vη0 be
the projection to the η 0 -part.
The natural map(3.7.6)

OK 0 ⊗OK (Uη ⊕ Vη0 ) → L

is an isomorphism, since both sides are projective OK 0 ⊗Zp A-modules of rank two,and the given map is an isomorphism modulo π 0 . It follows immediately from (3.7.6)
that Lη = Uη ⊕ π2 Vη0 and Lη0 = π1 Uη ⊕ Vη0 , so that coker f1 and coker f2 areprojective of rank one, as claimed.
Proposition 3.7.7. The morphism Mτloc → Mloc,Iw induces an isomorphism
0
Mτ,BT
→ MBT
loc,Iw . (We remind the reader that K /K is assumed totally ramified,
loc
and that τ is assumed to be a principal series inertial type.)

36

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTProof. We begin by constructing a morphism Mloc,Iw → Mτloc , inspired by thearguments of [RZ96, App. A]. We define an OK ⊗Zp A-module L by


(i)
(j)
σ −1
σ −1
L = ⊕σ ⊕ai=0
eσ L1 ⊕bj=0
eσ L2 ,(i)(j)

where the L1 ’s and L2 ’s are copies of L1 , L2 respectively.
We can upgrade the OK ⊗Zp A-module structure on L to that of an OK 0 ⊗Zp A(i)(i+1)
module by specifying how π 0 acts. If i < aσ −1, then we let π 0 : eσ L1 → eσ L1
be(j)0
the map induced by the identity on L1 , and if j < bσ − 1, then we let π : eσ L2 →(j+1)
(a −1)(0)eσ L 2be the map induced by the identity on L2 . We let π 0 : eσ L1 σ
→ eσ L2(bσ −1)(0)
0
be the map induced by f1 : L1 → L2 , and we let π : eσ L2→ eσ L1 be
the map induced by f2 : L2 → L1 . That this indeed gives L the structure of an
OK 0 ⊗Zp A-module follows from our assumption that f1 ◦ f2 = f2 ◦ f1 = π. We
give L a semilinear action of Gal(K 0 /K) = I(K 0 /K) by letting it act via (σ ◦ h)i · η(i)(j)
on each eσ L1 and via (σ ◦ h)j · η 0 on each eσ L2 .
We claim that L is a rank 2 projective OK 0 ⊗Zp A-module. Since coker f2 isprojective by assumption, we can choose a section to the k ⊗Zp A-linear morphism
L1 /π → coker f2 , with image U η , say. Similarly we choose a section to L2 /π →coker f1 with image V η0 . We choose lifts Uη , Vη0 of U η , V η0 to direct summands of
the OK ⊗Zp A-modules L1 , L2 respectively. There is a map of OK 0 ⊗Zp A-modules(3.7.8)

λ : OK 0 ⊗OK (Uη ⊕ Vη0 ) → L
(0)

(0)

induced by the map identifying Uη , Vη0 with their copies in L1 and L2 respectively.
The map λ is surjective modulo π 0 by construction, hence surjective by Nakayama’s
lemma. Regarding λ as a map of projective OK ⊗Zp A-modules of equal rank2e(K 0 /K), we deduce that λ is an isomorphism. Since the source of λ is a projective
OK 0 ⊗Zp A-module, the claim follows.
We now set


(i)
(j)
aσ −1
σ −1
L+ = ⊕σ ⊕i=0eσ (L1 )+ ⊕bj=0
eσ (L2 )+ ⊂ L
It is immediate from the construction that L+ is preserved by Gal(K 0 /K). The++
hypothesis that f1 , f2 and preserve L+
is an OK 0 ⊗Zp A1 , L2 implies that L+
submodule of L, while the hypothesis that each Li /Li is a projective A-module
implies the same for L/L+ . This completes the construction of our morphism
Mloc,Iw → Mτloc .
Just as in the proof of Proposition 3.7.2, the morphism Mloc,Iw → Mτloc followedby our morphism Mτloc → Mloc,Iw is the identity, while the composition in the otherorder is not, in general, naturally equivalent to the identity morphism. However,it follows immediately from Lemma 3.5.15 and the construction of L+ that ourmorphisms Mloc,Iw → Mτloc and Mτloc → Mloc,Iw respect the strong determinantτ,BTcondition, and so induce maps MBT
and Mτ,BT→ MBT
loc,Iw → Mloc
loc,Iw . To
loc
τ,BT
τ,BT
see that the composite Mloc → MBT→
M
isnaturallyequivalent
toloc,Iwloc+τ
the identity, suppose that (L, L ) ∈ Mloc (A) and observe that there is a natural
Gal(K 0 /K)-equivariant isomorphism of OK ⊗A Zp -modules


∼
(j)
bσ −1
σ −1
(3.7.9)⊕σ ⊕ai=0eσ L(i)→Lη ⊕j=0 eσ Lη 0

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS
0 i

37

0 j

(i) (π )(j) (π )induced by the maps Lη −→ (π 0 )i Lη and Lη0 −→ (π 0 )j Lη0 . The commutativityof the diagram
0 aσ −1(a −1) (π )eσ Lη σ

π0

π1


(0)
eσ Lη0/ eσ (π 0 )aσ −1 Lηid/ eσ Lη0

implies that the map in (3.7.9) is in fact an OK 0 ⊗Zp A-module isomorphism. The
map (3.7.9) induces an inclusion


(j) +
bσ −1
+
σ −1
⊕σ ⊕ai=0eσ (L(i)→ L+ .
η ) ⊕j=0 eσ (Lη 0 )If furthermore (L, L+ ) ∈ Mτ,BTthen this is an isomorphism because L+ satisfies
loc
condition (2) of Lemma 3.5.15.
Finally, we turn to the case of a cuspidal type. Let L as usual be a quadratic
2f
unramified extension of K, and set K 0 = L(π 1/(p −1) ). The field N continues todenote the maximal unramified extension of K in K 0 , so that N = L. Let τ be afcuspidal type, so that τ = η ⊕ η 0 , where η 6= η 0 but η 0 = η p .
Proposition 3.7.10. There is a morphism Mτ,BT→ MBT
loc,Iw which is representable
loc
by algebraic spaces and smooth. (We remind the reader that τ is now assumed to bea cuspidal inertial type.)
Proof. Let τ 0 be the type τ , considered as a (principal series) type for the totally ramified extension K 0 /N . Let c ∈ Gal(K 0 /K) be the unique element which
2f
fixes π 1/(p −1) but acts nontrivially on N . For any map α : X → Y of ON -modules
we write αc for the twist 1 ⊗ α : ON ⊗ON ,c X → ON ⊗ON ,c Y .
0
We may think of an object (L, L+ ) of Mτ,BT
as an object (L0 , (L0 )+ ) of Mτloc,BT
loc
equipped with the additional data of an isomorphism of OK 0 ⊗Zp A-modules θ :OK 0 ⊗OK 0 ,c L0 → L0 which is compatible with (L0 )+ , which satisfies θ ◦ θc = id, andwhich is compatible with the action of Gal(K 0 /N ) = I(K 0 /N ) in the sense thatfθ ◦ (1 ⊗ g) = g p ◦ θ.
Employing the isomorphism of Proposition 3.7.7, we think of (L0 , (L0 )+ ) as atuple (L01 , (L01 )+ , L02 , (L02 )+ , f1 , f2 ), where the L0i , (L0i )+ are ON ⊗Zp A-modules; by∼construction, the map θ induces isomorphisms θ1 : ON ⊗ON ,c L01 −→ L02 , θ2 :∼ON ⊗ON ,c L02 −→ L01 , which are compatible with (L01 )+ , (L02 )+ and f1 , f2 , andsatisfy θ1 ◦ θ2c = id.
(1)0Choose for
Peach embedding σ : k ,→ F an extension to an embedding σ : k ,→F, set e1 = σ eσ(1) , and write e2 = 1 − e1 . Then the map θ1 induces isomorphisms∼∼
θ11 : e1 L01 −→ e2 L02 and θ12 : e2 L01 −→ e1 L02 , while θ2 induces isomorphisms∼∼
θ21 : e1 L02 −→ e2 L01 and θ22 : e2 L02 −→ e1 L01 . The condition that θ1 ◦ θ2c = id−1
−1translates to θ22 = θ11
and θ21 = θ12, and compatibility with (L01 )+ , (L02 )+ implies∼∼
that θ11 , θ21 induce isomorphisms e1 (L01 )+ −→ e2 (L02 )+ and e1 (L02 )+ −→ e2 (L01 )+respectively.
Furthermore f1 , f2 induce maps e1 f1 : e1 L01 → e1 L02 , e1 g : e1 L02 → e1 L01 .
It follows that there is a map Mτ,BT→ Mloc,Iw , sending (L, L+ ) to the tuloc
0
0
0 +
0 +
ple (e1 L1 , e1 (L1 ) , e1 L2 , e1 (L2 ) , e1 f1 , e1 f2 ). To see that it respects the strong

38

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT+
determinant condition, one has to check that the conditions on L+
1 , L2 imply
0 +0 +those for e1 (L1 ) , e2 (L2 ) coincide, and this follows from the definitions (viaRemark 3.5.4). We therefore obtain a map Mτ,BT→ MBT
loc,Iw ,loc
Since this morphism is given by forgetting the data of e2 L01 , e2 L02 and the pairof isomorphisms θ11 , θ21 , it is evidently formally smooth. It is also a morphism
between $-adic formal algebraic stacks that are locally of finite presentation, andso is representable by algebraic spaces (by [Eme, Lem. 7.10]) and locally of finite
presentation (by [EG19b, Cor. 2.1.8] and [Sta13, Tag 06CX]). Thus this morphism
is in fact smooth.


3.8. Local models: local geometry. We now deduce our main results on thelocal structure of our moduli stacks from results in the literature on local models
for Shimura varieties.
Proposition 3.8.1. We can identify MBT
loc,Iw with the quotient of (the p-adic formal
completion of ) the naive local model for ResK/Qp GL2 (as defined in [PRS13, §2.4])
by a smooth group scheme over O.
fBT be the p-adic formal completion of the naive local model for
Proof. Let M
loc,Iw
ResK/Qp GL2 corresponding to a standard lattice chain L, as defined in [PRS13,§2.4]. By [RZ96, Prop. A.4], the automorphisms of the standard lattice chain L
are represented by a smooth group scheme PL over O. (This is in fact a parahoric
subgroup scheme of ResOK /Zp GL2 , and in particular it is affine.) Also by loc. cit.,
every lattice chain of type (L) is Zariski locally isomorphic to L. By comparingfBT is a PL -torsor over MBT for thethe two moduli problems, we see that M
loc,Iw
loc,Iw
Zariski topology and the proposition follows.

The following theorem describes the various regularity properties of local models.
Since we are working in the context of formal algebraic stacks, we use the terminology
developed in [Eme, §8] (see in particular [Eme, Rem. 8.21] and [Eme, Def. 8.35]).
Theorem 3.8.2. Suppose that d = 2 and that τ is a tame inertial type. Then
(1) Mτ,BTis residually Jacobson and analytically normal, and Cohen–Macaulay.
loc(2) The special fibre Mτ,BT,1
is reduced.
loc(3) Mτ,BT
is
flat
over
O.
loc
Proof. For scalar types, this follows from [Kis09, Prop. 2.2.2] by Lemma 3.7.2, andso we turn to studying the case of a non-scalar type. The properties in question
can be checked smooth locally (see [Eme, §8] for (1), and [Sta13, Tag 04YH] for (2);for (3), note that morphisms that are representable by algebraic spaces and smooth
are also flat, and take into account the final statement of [Eme, Lem. 8.34]), and soby Propositions 3.7.7 and 3.7.10 we reduce to checking the assertions of the theorem
for MBT
loc,Iw . Proposition 3.8.1 then reduces us to checking these assertions for the$-adic completion of the naive local model at Iwahori level for ResK/Qp GL2 .
Since this naive local model is a scheme of finite presentation over O, its special
fibre (i.e. its base-change to F) is Jacobson, and it is excellent; thus its $-adic
completion satisfies the properties of (1) if and only if the naive local model itself isnormal and Cohen–Macaulay. The special fibre of its $-adic completion is of course
just equal to its own special fibre, and so verifying (2) for the first of these special
fibres is equivalent to verifying it for the second. Finally, the $-adic completion

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

39

of an O-flat Noetherian ring is again O-flat, and so the $-adic completion of thenaive local model will be O-flat if the naive local model itself is.
All the properties of the naive local model that are described in the preceding
paragraph, other than the Cohen–Macaulay property, are contained in [PRS13, Thm.
2.17], if we can identify the naive local models at Iwahori level with the vertical
local models at Iwahori level, in the sense of [PR03, §8] (see also the discussion
above loc. cit. in [PRS13]).
The vertical local models are obtained by intersecting the preimages at Iwahori
level of the flat local models at hyperspecial level. Since the naive local models for the
Weil restriction of GL2 at hyperspecial level are already flat by a special case of [PR03,
Cor. 4.3], the naive local models at Iwahori level are identified with the verticalones and [PRS13, Thm. 2.17] applies to them directly. (To be precise, the resultsof [PR03] apply to restrictions of scalars ResF/F0 GL2 with F/F0 totally ramified.
However, thanks to the decomposition OK ⊗Zp A ∼= ⊕σ:W (k),→O OK ⊗W (k),σ A the
local model for ResK/Qp GL2 decomposes as a product of local models for totally
ramified extensions.)
Finally, Cohen–Macaulayness can be proved as in [Gör01, Prop. 4.24] and thediscussion immediately following. We thank U. Görtz for explaining this argument
to us. As in the previous paragraph we reduce to the case of local models at Iwahori
level for ResF/F0 GL2 with F/F0 totally ramified of degree e. In this setting theadmissible set M for the coweight µ = (e, 0) has precisely one element of length 0
and two elements of each length between 1 and e. Moreover, for elements x, y ∈ M
we have x < y in the Bruhat order if and only if `(x) < `(y). One checks easily
that M is e-Cohen–Macaulay in the sense of [Gör01, Def. 4.23], and we conclude
by [Gör01, Prop. 4.24]. Alternatively, this also follows from the much more generalrecent results of Haines–Richarz [HR19].

Corollary 3.8.3. Suppose that d = 2 and that τ is a tame inertial type. Then
(1) C τ,BT is analytically normal, and Cohen–Macaulay.
(2) The special fibre C τ,BT,1 is reduced.
(3) C τ,BT is flat over O.
Proof. This follows from Theorems 3.4.5 and 3.8.2, since all of these properties canbe verified smooth locally (as was already noted in the proof of the second of these
theorems).
Remark 3.8.4. There is another structural result about vertical local models that isproved in [PR03, §8] but which we haven’t incorporated into our preceding results,
namely, that the irreducible components of the special fibre are normal. Since
the notion of irreducible component is not étale local (and so in particular notsmooth local), this statement does not imply the corresponding statement for the
special fibres of Mτ,BTor C τ,BT . Rather, it implies the weaker, and somewhat more
loc
technical, statement that each of the analytic branches passing through each closed
point of the special fibre of these $-adic formal algebraic stacks is normal. We
won’t discuss this further here, since we don’t need this result.
3.9. Scheme-theoretic images. We continue to fix d = 2, h = 1, and we set
2f −1
K 0 = L(π 1/p), where L/K is the unramified quadratic extension, and π is a
uniformiser of K. (This is the choice of K 0 that we made before in the cuspidal
case, and contains the choice of K 0 that we made in the principal series case; since40

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

the category of Breuil–Kisin modules with descent data for the smaller extension isby Proposition 3.6.1 naturally a full subcategory of the category of Breuil–Kisin
modules with descent data for the larger extension, we can consider both principal
series and cuspidal types in this setting.)
Definition 3.9.1. For each a ≥ 1 we write Z dd,a and Z τ,a for the scheme-theoretic
images (in the sense of [EG19b, Defn. 3.2.8]) of the morphisms C dd,BT,a → Rdd,aτ
and C τ,BT,a → Rdd,a respectively. We write Z, Z for Z 1 , Z τ,1 respectively.
The following theorem records some basic properties of these scheme-theoretic
images. We refer to Appendix B for the notion of representations admitting a
potentially Barsotti–Tate lift of a given type, and for the definition of très ramifiéerepresentations.
Theorem 3.9.2.
(1) For each a ≥ 1, Z dd,a is an algebraic stack of finite
presentation over O/$a , and is a closed substack of Rdd,a . In turn, eachZ τ,a is a closed substack of Z dd,a , and thus in particular is an algebraic
stack of finite presentation over O/$a ; and Z dd,a is the union of the Z τ,a .
(2) The morphism C dd,BT,a → Rdd,a factors through a morphism C dd,BT,a →Z dd,a which is representable by algebraic spaces, scheme-theoretically dominant, and proper. Similarly, the morphism C τ,BT,a → Rdd,a factors througha morphism C τ,BT,a → Z τ,a which is representable by algebraic spaces,
scheme-theoretically dominant, and proper.
(3) The Fp -points of Z are naturally in bijection with the continuous representations r : GK → GL2 (Fp ) which are not a twist of a très ramifiée extensionof the trivial character by the mod p cyclotomic character. Similarly, the Fp τpoints of Z are naturally in bijection with the continuous representationsr : GK → GL2 (Fp ) which have a potentially Barsotti–Tate lift of type τ .
Proof. Part (1) follows easily from Theorem 3.1.12. Indeed, by [EG19b, Prop. 3.2.31]
we may think of Z dd,a as the scheme-theoretic image of the proper morphism ofalgebraic stacks C dd,BT,a → Rdd,aand similarly for each Z τ,a . The existence of the1
factorisations in (2) is then formal.
By [EG19b, Lem. 3.2.14], for each finite extension F0 /F, the F0 -points of Zτ
(respectively Z ) correspond to the étale ϕ-modules with descent data of theform M[1/u], where M is a Breuil–Kisin module of rank 2 with descent data andF-coefficients which satisfies the strong determinant condition (respectively, which
satisfies the strong determinant condition and is of type τ ). By Lemma 3.5.16 andCorollary 3.8.3, these precisely correspond to the Galois representations r : GK →GL2 (F) which admit potentially Barsotti–Tate lifts of some tame type (respectively,of type τ ). The result follows from Lemma B.5.

The thickenings C dd,BT,a ,→ C dd,BT,a+1 and Rdd,a ,→ Rdd,a+1 induce closed
immersions Z dd,a ,→ Z dd,a+1 . Similarly, the thickenings C τ,BT,a ,→ C τ,BT,a+1 giverise to closed immersions Z τ,a ,→ Z τ,a+1 .
Lemma 3.9.3. Fix a ≥ 1. Then the morphism Z dd,a ,→ Z dd,a+1 is a thickening,
and for each tame type τ, the morphism Z τ,a ,→ Z τ,a+1 is a thickening.

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

41

Proof. In each case, the claim of the lemma follows from the following more generalstatement: if
/ X0
X

Y


/ Y0

is a diagram of morphisms of algebraic stacks in which the upper horizontal arrow is athickening, the lower horizontal arrow is a closed immersion, and each of the verticalarrows is representable by algebraic spaces, quasi-compact, and scheme-theoreticallydominant, then the lower horizontal arrow is also a thickening.
Since the property of being a thickening may be checked smooth locally, andsince scheme-theoretic dominance of quasi-compact morphisms is preserved by flatbase-change, we may show this after pulling the entire diagram back over a smoothsurjective morphism V 0 → Y whose source is a scheme, and thus reduce to thecase in which the lower arrow is a morphism of schemes, and the upper arrowis a morphism of algebraic spaces. A surjecive étale morphism is also schemetheoretically dominant, and so pulling back the top arrow over a surjective étalemorphism U 0 → V 0 ×Y 0 X 0 whose source is a scheme, we finally reduce to consideringa diagram of morphisms of schemes
U

/ U0


V


/ V0

in which the top arrow is a thickening, the vertical arrows are quasi-compact andscheme-theoretically dominant, and the bottom arrow is a closed immersion.
Pulling back over an affine open subscheme of V 0 , and then pulling back the top
arrow over the disjoint union of the members of a finite affine open cover of thepreimage of this affine open in U 0 (note that this preimage is quasi-compact), we
further reduce to the case when all the schemes involved are affine. That is, we havea diagram of ring morphisms
/A
A0

B0


/B

in which the vertical arrows are injective, the horizontal arrows are surjective, andthe bottom arrow has nilpotent kernel. One immediately verifies that the top arrow
has nilpotent kernel as well.
We write C dd,BT := lima C dd,BT,a and Z dd := lima Z dd,a ; we then have evident−→−→
morphisms of Ind-algebraic stacks
C dd,BT → Z dd → Rddlying over Spf O, both representable by algebraic spaces, with the first being
furthermore proper and scheme-theoretically dominant in the sense of [Eme, Def.
6.13], and the second being a closed immersion.

42

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

Similarly, for each choice of tame type τ , we set C τ,BT = lima C τ,a , and Z τ :=−→lima Z τ,a . We again have morphisms−→
C τ,BT → Z τ → Rddof Ind-algebraic stacks over Spf O, both being representable by algebraic spaces, thefirst being proper and scheme-theoretically dominant, and the second being a closed
immersion. Note that by Corollary 3.5.13, C dd,BT is the disjoint union of the C τ,BT ,so it follows that Z dd,BT is the union (but not the disjoint union) of the Z τ .
Proposition 3.5.7 shows that C τ,BT is a $-adic formal algebraic stack of finite
presentation over Spf O. Each Z τ,a is an algebraic stack of finite presentation
over Spec O/$a by Theorem 3.9.2. Analogous remarks apply in the case of C dd,BTand Z dd .
Proposition 3.9.4. Z dd , and each Z τ , are $-adic formal algebraic stacks, of finite
presentation over Spf O.
Proof. We give the argument for Z dd , the argument for Z τ being identical. (Alternatively, this latter case follows from the former and the fact that the canonicalmorphism Z τ ,→ Z dd is a closed immersion.) That Z dd is a $-adic formal algebraic stack will follow from Proposition A.6 once we show that the morphism
C dd,BT → Z dd fits into the framework of Section A.5. It follows from Lemma A.3
that Z dd is a formal algebraic stack, and by construction it is locally Ind-finite typeover Spec O. Furthermore, since each Z dd,a is quasi-compact and quasi-separated(being of finite presentation over O/$a ), we see that Z dd is quasi-compact andquasi-separated. Thus Proposition A.6 indeed applies.
∼
The isomorphism Z dd −→ lima Z dd,a induces an isomorphism−→∼
Z dd ×O O/$b −→ lim Z dd,a ×O O/$b ,
−→
a

dd

for any fixed b ≥ 1. Since Z is quasi-compact and quasi-separated, so is Z dd ×O
O/$b , and thus this isomorphism factors through Z dd,a ×O O/$b for some a. Thus
the directed system Z dd,a ×O O/$b in a eventually stabilises, and so we see that∼

Z dd ×O O/$b −→ Z dd,a ×O O/$b
for sufficiently large values of a. Since Z dd,a is of finite presentation over O/$a ,we find that Z dd ×O O/$b is of finite presentation over O/$b . Consequently, we
conclude that Z dd is of finite presentation over Spf O, as claimed.
Remark 3.9.5. As observed in the general context of Subsection A.5, the thickening
Z dd,a ,→ Z dd ×O O/$a need not be an isomorphism a priori, and we have no reason
to expect that it is. Nevertheless, in Subsection 5.1 we will prove that this thickening
is generically an isomorphism for every value of a ≥ 1, and we will furthermore showthat each (Z dd,a )/F is generically reduced; see Proposition 5.1.2 and Remark 5.1.4
below. The proof of this result involves an application of Proposition A.11, anddepends on the detailed analysis of the irreducible components of the algebraic
stacks C τ,a and Z τ,a that we will make in Section 4.
We conclude this subsection by establishing some basic lemmas about the reduced
substacks underlying each of C τ,BT and Z τ .
Lemma 3.9.6. Let X be an algebraic stack over O/$a , and let Xred be the underlying reduced substack of X . Then Xred is a closed substack of X/F := X ×O/$a F.

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

43

Proof. The structural morphism X → Spec O/$a induces a natural morphism
Xred → (Spec O/$a )red = Spec F, so the natural morphism Xred → X factors
through X/F . Since the morphisms Xred → X and X/F → X are both closed
immersions, so is the morphism Xred → X/F .

Lemma 3.9.7. If f : X → Y is a quasi-compact morphism of algebraic stacks,and W is the scheme-theoretic image of f , then the scheme-theoretic image ofthe induced morphism of underlying reduced substacks fred : Xred → Yred is theunderlying reduced substack Wred .
Proof. Since the definitions of the scheme-theoretic image and of the underlying
reduced substack are both smooth local (in the former case see [EG19b, Rem.
3.1.5(3)], and in the latter case it follows immediately from the construction in [Sta13,
Tag 0509]), we immediately reduce to the case of schemes, which follows from [Sta13,
Tag 056B].

Lemma 3.9.8. For each a ≥ 1, C dd,BT,1 is the underlying reduced substackof C dd,BT,a , and Z dd,1 is the underlying reduced substack of Z dd,a ; consequently,
C dd,BT,1 is the underlying reduced substack of C dd,BT , and Z dd,1 is the underlying
reduced substack of Z dd . Similarly, for each tame type τ, C τ,BT,1 is the underlying
reduced substack of each C τ,BT,a , and of C τ,BT , while Z τ,1 is the underlying reduced
substack of each Z τ,a , and of Z τ .
Proof. The statements for the $-adic formal algebraic stacks follow directly fromthe corresponding statements for the various algebraic stacks modulo $a , and so
we focus on proving these latter statements, beginning with the case of C τ,BT,a .
Note that C τ,BT,1 = C τ,BT,a ×O/$a F is reduced by Corollary 3.8.3, so C dd,BT,1 =C dd,BT,a ×O/$a F is also reduced by Corollary 3.5.13. The claim follows for C dd,BT,aand C τ,BT,a from Lemma 3.9.6.
The claims for Z τ,a and Z dd,a are then immediate from Lemma 3.9.7, applied tothe morphisms C τ,BT,a → Z τ,a and C dd,BT,a → Z dd,a .

3.10. Versal rings and equidimensionality. We now show that C dd,BT and
Z dd,BT (and their substacks C τ,BT , Z τ ) are equidimensional, and compute theirdimensions, by making use of their versal rings. In [EG19b, §5] these versal rings
were constructed in a more general setting in terms of liftings of étale ϕ-modules;
in our particular setting, we will find it convenient to interpret them as Galois
deformation rings.
Fix a finite type point x : Spec F0 → Z τ,a , where F0 /F is a finite extension; wealso denote the induced finite type point of Rdd,a by x. Let r : GK → GL2 (F0 ) be
the Galois representation corresponding to x by Theorem 3.9.2 (3). Let E 0 be thecompositum of E and W (F0 )[1/p], with ring of integers OE 0 and residue field F0 .
As in Appendix C, we have the universal framed deformation OE 0 -algebra Rr ,
and we let Rr,0,τ
be the reduced and p-torsion free quotient of Rr whose Qp -points
correspond to the potentially Barsotti–Tate lifts of r of type τ . In this section wewill denote Rr,0,τ
by the more suggestive name Rrτ,BT . We recall, for instance from
[BG19, Thm. 3.3.8], that the ring Rrτ,BT [1/p] is regular.
As in Section 2.3, we write Rr|GK for the universal framed deformation OE 0 ∞algebra for r|GK∞ . By Lemma 2.3.3, we have a natural morphism(3.10.1)

Spf Rr|GK∞→ Rdd .

44

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

Lemma 3.10.2. The morphism (3.10.1) is versal (at x).
Proof. By definition, it suffices to show that if ρ : GK∞ → GLd (A) is a representation
with A a finite Artinian OE 0 -algebra, and if ρB : GK∞ → GLd (B) is a second
representation, with B a finite Artinian OE 0 -algebra admitting a surjection onto A,
such that the base change ρA of ρB to A is isomorphic to ρ (more concretely, sothat there exists M ∈ GLd (A) with ρ = M ρA M −1 ), then we may find ρ0 : GK∞ →GLd (B) which lifts ρ, and is isomorphic to ρB . This is straightforward: the naturalmorphism GLd (B) → GLd (A) is surjective, and so if M 0 is any lift of M to anelement of GLd (B), then we may set ρ0 = M 0 ρB (M 0 )−1 .
Definition 3.10.3. For any pro-Artinian OE 0 -algebra R with residue field F0 wed2/R denote the completion of (GL2 )/R along the closed subgroup of its special
let GLfibre given by the centraliser of r|GK∞ .
d2/R = Spf R ×O 0 GL
d2/O 0 . InRemark 3.10.4. For R as above we have GL
E
E
∼
d
d
deed, if R ∼
A
,
then
Spf
R
×
= lim
i
O
0 GL2/OE 0 = lim Spec Ai ×OE 0 GL2/OE 0 , and
E
←−i
−→i
d2/O 0 agrees with the completion of (GL2 )/A because Ai is a finite
Spec Ai ×OE0 GLiE
OE 0 -module.
d2/R has nice base-change properties more generally: if
It follows from this that GL
R → S is a morphism of pro-Artinian OE 0 -algebras each with residue field F0 , thend2/S ∼
d2/R . We apply this fact without
there is an isomorphism GL= Spf S ×Spf R GL
further comment in various arguments below.
d2/R
There is a pair of morphisms GL
r|

GK
∞

⇒ Spf Rr|GK , the first being simply∞

the projection to Spf Rr|GK , and the second being given by “change of framing”.
∞
d2/R
Composing such changes of framing endows GLwith the structure of ar|GK∞groupoid over Spf Rr|GK . Note that the two morphisms
∞

d2/R
GL
r|

GK
∞

⇒ Spf Rr|GK∞(3.10.1)−→ Rddcoincide, since changing the framing does not change the isomorphism class (asa Galois representation) of a deformation of r|GK∞ . Thus there is an induced
morphism of groupoids over Spf Rr|GK∞(3.10.5)

d2/R
GL
r|

GK
∞

→ Spf Rr|GK∞

×Rdd Spf Rr|GK .
∞Lemma 3.10.6. The morphism (3.10.5) is an isomorphism.
Proof. If A is an Artinian OE 0 -algebra, with residue field F0 , then a pair of A-valued
points of Spf Rr|GK map to the same point of Rdd if and only if they give rise to∞isomorphic deformations of ρ, once we forget the framings. But this precisely meansthat the second of them is obtained from the first by changing the framing via and2 .
A-valued point of GLIt follows from Lemma 3.10.2 that, for each a ≥ 1, the quotient Rr|GK /$a is a∞(non-Noetherian) versal ring for Rdd,a at x. By [EG19b, Lem. 3.2.16], for each a ≥ 1
a versal ring for Z τ,a at x is given by the scheme-theoretic image of the morphism(3.10.7)

C τ,BT,a ×Rdd,a Spf Rr|GK /$a → Spf Rr|GK /$a ,∞∞MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

45

in the sense that we now explain.
In general, the notion of scheme-theoretic image for morphisms of formal algebraic
stacks can be problematic; at the very least it should be handled with care. But
in this particular context, a definition is given in [EG19b, Def. 3.2.15]: we write
Rr|GK /$a as an inverse limit of Artinian local rings A, form the corresponding∞
scheme-theoretic images of the induced morphisms C τ,BT,a ×Rdd,a Spec A → Spec A,
and then take the inductive limit of these scheme-theoretic images; this is a formal
scheme, which is in fact of the form Spf Rτ,a for some quotient Rτ,a of Rr|GK /$a∞(where quotient should be understood in the sense of topological rings), and is bydefinition the scheme-theoretic image in question.
The closed immersions C τ,BT,a ,→ C τ,BT,a+1 induce corresponding closed immersions
C τ,BT,a ×Rdd,a Spf Rr|GK /$a → C τ,BT,a+1 ×Rdd,a+1 Spf Rr|GK /$a+1 ,∞∞and hence closed immersions of scheme-theoretic images Spf Rτ,a → Spf Rτ,a+1 ,corresponding to surjections Rτ,a+1 → Rτ,a . (Here we are using the fact that an
projective limit of surjections of finite Artin rings is surjective.) Thus we may form
the pro-Artinian ring lima Rτ,a . This projective limit is a quotient (again in the sense←−of topological rings) of Rr|GK , and the closed formal subscheme Spf(lim Rτ,a ) of←−∞
Spf Rr|GK is the scheme-theoretic image (computed in the sense described above)∞of the projection(3.10.8)

C τ,BT ×Rdd Spf Rr|GK∞→ Spf Rr|GK∞(This is a formal consequence of the construction of the Spf Rτ,a as scheme-theoretic
images, since any discrete Artinian quotient of Rr|GK is a discrete Artinian quotient∞of Rr|GK /$a , for some a ≥ 1.) It also follows formally (for example, by the same∞
argument as in the proof of [EG19b, Lem. 4.2.14]) that lim Rτ,a is a versal ring to←−Z τ at x. Our next aim is to identify this projective limit with Rrτ,BT .
Before we do this, we have to establish some preliminary facts related to thevarious objects and morphisms we have just introduced.
Lemma 3.10.9.
(1) Each of the rings Rτ,a is a complete local Noetherian ring, endowed with its
m-adic topology, and the same is true of the inverse limit lima Rτ,a .
←−(2) For each a ≥ 1, the morphism Spf Rτ,a → Spf Rr|GK induces an isomor∞phism∼

C τ,BT,a ×Rdd,a Spf Rτ,a −→ C τ,BT,a ×Rdd,a Spf Rr|GK /$a .
∞(3) For each a ≥ 1, the morphism Spf Rτ,a → Rdd,a is effective, i.e. may bepromoted (in a unique manner) to a morphism Spec Rτ,a → Rdd,a , and theinduced morphismC τ,a ×Rdd,a Spec Rτ,a → Spec Rτ,ais proper and scheme-theoretically dominant.
(4) Each transition morphism Spec Rτ,a ,→ Spec Rτ,a+1 is a thickening.
≤1
Proof. Recall that in Section 2.3.4 we defined a Noetherian quotient Rr|
G

of
K∞

[0,1]
Rr|GK , which is naturally identified with the framed deformation ring Rr
by
∞

46

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

Proposition 2.3.6. It follows from [EG19b, Lem. 5.4.15] (via an argument almost
identical to the one in the proof of [EG19b, Prop. 5.4.17]) that the morphism[0,1]≤1
Spf lim Rτ,a ,→ Spf Rr|GK factors through Spf Rr|= Spf Rr , and indeed
GK
←−
∞
∞
[0,1]
that lim Rτ,a is a quotient of Spf Rr ; this proves (1).
←−
It follows by the very construction of the Rτ,a that the morphism (3.10.7) factors
through the closed subscheme Spf Rτ,a of Spf Rr|GK /$a . The claim of (2) is a∞formal consequence of this.
We have already observed that the morphism Spf Rτ,a → Rdd,a factors throughτ,aZ . This latter stack is algebraic, and of finite type over O/$a . It follows
from [Sta13, Tag 07X8] that the morphism Spf Rτ,a → Z τ,a is effective. Taking
into account part (1) of the present lemma, we deduce from the theorem on formal
functions that the formal completion of the scheme-theoretic image of the projection
C τ,a ×Rdd,a Spec Rτ,a → Spec Rτ,aat the closed point of Spec Rτ,a coincides with the scheme-theoretic image of themorphism
C τ,a ×Rdd,a Spf Rτ,a → Spf Rτ,a .
Taking into account (2), we see that this latter scheme-theoretic image coincides
with Spf Rτ,a itself. This completes the proof of (3).
The claim of (4) follows from a consideration of the diagram
C τ,a ×Rdd,a Spec Rτ,a

/ C τ,a+1 ×Rdd,a+1 Spec Rτ,a+1


Spec Rτ,a/ Spec Rτ,a+1

just as in the proof of Lemma 3.9.3.

Lemma 3.10.10.
(1) The projection C τ,BT ×Rdd Spf Rr|GK → Spf Rr|GK factors through a mor∞∞phism C τ,BT ×Rdd Spf Rr|GK → Spf(lim Rτ,a ), which is scheme-theoretically←−∞dominant in the sense that its scheme-theoretic image (computed in the manner described above) is equal to its target.
(2) There is a projective morphism of schemes Xr → Spec(lim Rτ,a ), which is←−
uniquely determined, up to unique isomorphism, by the requirement that its
m-adic completion (where m denotes the maximal ideal of lim Rτ,a ) may be←−
identified with the morphism C τ,BT ×Rdd Spf Rr|GK → Spf(lim Rτ,a ) of (1).
←−∞Proof. Part (1) follows formally from the various constructions and definitions of
the objects involved (just like part (2) of Lemma 3.10.9).
We now consider the morphism
C τ,BT ×Rdd Spf Rr|GK∞→ Spf(lim Rτ,a ).
←−Once we recall that lim Rτ,a is Noetherian, by Lemma 3.10.9 (1), it follows exactly as←−in the proof of [Kis09, Prop. 2.1.10] (which treats the case that τ is the trivial type),
via an application of formal GAGA [Gro61, Thm. 5.4.5], that this morphism arises as
the formal completion along the maximal ideal of lim Rτ,a of a projective morphism←−

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

47

Xr → Spec(lim Rτ,a ) (and Xr is unique up to unique isomorphism, by [Gro61, Thm.
←−
5.4.1]).
We next establish various properties of the scheme Xr constructed in the previous
br to denote the fibre productlemma. To ease notation going forward, we write X
τ,BTC
×Rdd Spf Rr|GK (which is reasonable, since this fibre product is isomorphic∞
to the formal completion of Xr ).
Lemma 3.10.11. The scheme Xr is Noetherian, normal, and flat over OE 0 .
Proof. Since Xr is projective over the Noetherian ring lim Rτ,a , it is Noetherian. The←−
other claimed properties of Xr will be deduced from the corresponding properties
of C τ,BT that are proved in Corollary 3.8.3.
To this end, we first note that, since the morphism C τ,BT → Rdd factors
through Z τ , it follows (for example as in the proof of [EG19b, Lem. 3.2.16]) that
we have isomorphisms∼

C τ,BT ×Z τ Spf(lim Rτ,a ) −→ C τ,BT ×Z τ Z τ ×Rdd Spf Rr|GK
←−
∞
∼
τ,BT
−→ C
×Rdd Spf Rr|GK

∞

br .
=: X

br with the fibre product C τ,BT ×Z τ Spf(lim Rτ,a ).
In summary, we may identify X←−
br is analytically normal. To see this, let Spf B → X
br be
We now show that X
a morphism whose source is a Noetherian affine formal algebraic space, which isrepresentable by algebraic spaces and smooth. We must show that the completionbn is normal, for each maximal ideal n of B. In fact, it suffices to verify this for someB
br , and so without loss of generality we may
collection of such Spf B which cover X
choose our B as follows: first, choose a collection of morphisms Spf A → C τ,BT whose
sources are Noetherian affine formal algebraic spaces, and which are representableby algebraic spaces and smooth, which, taken together, cover C τ,BT . Next, for eachsuch A, choose a collection of morphismsbrSpf B → Spf A ×C τ,BT X
whose sources are Noetherian affine formal algebraic spaces, and which are representable by algebraic spaces and smooth, which, taken together, cover the fibre
product. Altogether (considering all such B associated to all such A), the composite
morphisms
br → Xbr
Spf B → Spf A ×C τ,BT X
br .
are representable by algebraic spaces and smooth, and cover X
Now, let n be a maximal ideal in one of these rings B, lying over a maximal
ideal m in the corresponding ring A. The extension of residue fields A/m → B/n isfinite, and each of these fields is finite over F0 . Enlarging F0 sufficiently, we may
assume that in fact each of these residue fields coincides with F0 . (On the level ofrings, this amounts to forming various tensor products of the form – ⊗W (F0 ) W (F00 ),which doesn’t affect the question of normality.) The morphism Spf Bn → Spf An
is then seen to be smooth in the sense of [Sta13, Tag 06HG], i.e., it satisfies theinfinitesimal lifting property for finite Artinian O0 -algebras with residue field F0 :br above as a fibre product, and the fact that
this follows from the identification of Xτ,aτ
Spf(lim R ) → Z is versal at the closed point x. Thus Spf Bn is a formal power←−
series ring over Spf Am , by [Sta13, Tag 06HL], and hence Spf Bn is indeed normal,

48

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

since Spf Am is so, by Corollary 3.8.3. By Lemma 3.10.14 below, this implies that
br is normal.
the algebraization Xr of X
We next claim that the morphism
Spf(lim Rτ,a ) → Z τ←−
is a flat morphism of formal algebraic stacks, in the sense of [Eme, Def. 8.35].
br → C τ,BT is also flat. Since
Given this, we find that the base-changed morphism X
τ,BTCorollary 3.8.3 shows that C
is flat over OE 0 , we conclude that the same is true
br . Again, by Lemma 3.10.14, this implies that the algebraization Xr is also flatof X
over OE 0 .
It remains to show the claimed flatness. To this end, we note first that for eacha ≥ 1, the morphism(3.10.12)(3.10.13)

Spf Rτ,a → Z τ,a

is a versal morphism from a complete Noetherian local ring to an algebraic stack
which is locally of finite type over O/$a . We already observed in the proof of
Lemma 3.10.9 (3) that (3.10.13) is effective, i.e. can be promoted to a morphism
Spec Rτ,a → Z τ,a . It then follows from [Sta13, Tag 0DR2] that this latter morphism
is flat, and thus that (3.10.13) is flat in the sense of [Eme, Def. 8.35]. It follows
easily that the morphism (3.10.12) is also flat: use the fact that a morphism of$-adically complete local Noetherian O-algebras which becomes flat upon reduction
modulo $a , for each a ≥ 1, is itself flat, which follows from (for example) [Sta13,
Tag 0523].
The following lemma is standard, and is presumably well-known. We sketch theproof, since we don’t know a reference.
Lemma 3.10.14. If S is a complete Noetherian local O-algebra and Y → Spec S is
a proper morphism of schemes, then Y is flat over Spec O (resp. normal) if and only
Yb (the mS -adic completion of Y ) is flat over Spf O (resp. is analytically normal).
Proof. The properties of Y that are in question can be tested by considering thevarious local rings OY,y , as y runs over the points of Y ; namely, we have to consider
whether or not these rings are flat over O, or normal. Since any point y specializes
to a closed point y0 of Y , so that OY,y is a localization of OY,y0 , and thus O-flat(resp. normal) if OY,y0 is, it suffices to consider the rings OY,y0 for closed points y0
of Y . Note also that since Y is proper over Spec S, any closed point of Y lies overthe closed point of Spec S.
Now let Spec A be an affine neighbourhood of a closed point y0 of Y ; let m be thecorresponding maximal ideal of A. As we noted, m lies over mS , and so gives riseb the mS -adic completion of A; and any maximalb of A,b := mA
to a maximal ideal mbb
ideal of A contains mS A, and so arises from a maximal ideal of A in this manner∼ bb(since A/mS −→ A/m
S ). Write Am to denote the m-adic completion of A (which
b Then A
b is faithfully flat overbmaps isomorphically to the m-adic completion of A).
bm is.
the localization Am = OY,y0 , and hence Am is flat over O if and only if A
Consequently we see that Y is flat over O if and only if, for each affine open subset
b becomes flat over O afterSpec A of Y , the corresponding mS -adic completion A
completing at each of its maximal ideals. Another application of faithful flatness ofcompletions of Noetherian local rings shows that this holds if and only if each such

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

49

b is flat over O after localizing at each of its maximal ideals, which holds if and onlyA
b is flat over O. This is precisely what it means for Yb to be flat over O.
each such A
The proof that analytic normality of Yb implies that Y is normal is similar.
b at each ofIndeed, analytic normality by definition means that the completion of A
its maximal ideals is normal. This completion is faithfully flat over the localizationof Spec A at its corresponding maximal ideal, and so [Sta13, Tag 033G] implies that
this localization is also normal. The discussion of the first paragraph then implies
that Y is normal. For the converse direction, we have to deduce normality of thebm from the normality of the corresponding localizations Am . This
completions A
follows from that fact that Y is an excellent scheme (being of finite type over thecomplete local ring S), so that each A is an excellent ring [Sta13, Tag 0C23].
[0,1]

Proposition 3.10.15. The projective morphism Xr → Spec Rr
a projective and scheme-theoretically dominant morphism

factors throughXr → Spec Rrτ,BT(3.10.16)

which becomes an isomorphism after inverting $.
Proof. We begin by showing the existence of (3.10.16), and that it induces a bijection
on closed points after inverting $. Since Xr is O-flat, by Lemma 3.10.11, it suffices
to show that the induced morphism[0,1]Spec E ×O Xr → Spec Rr[1/$]

factors through a morphism(3.10.17)

Spec E ×O Xr → Spec Rrτ,BT [1/$],

which induces a bijection on closed points.
This can be proved in exactly the same way as [Kis09, Prop. 2.4.8], which
treats the case that τ is trivial. Indeed, the computation of the Dcris of a Galois
representation in the proof of [Kis09, Prop. 2.4.8] goes over essentially unchanged
to the case of a Galois representation coming from C τ,BT , and finite type points ofSpec Rrτ,BT [1/$] yield p-divisible groups and thus Breuil–Kisin modules exactly asin the proof of [Kis09, Prop. 2.4.8] (bearing in mind Lemma 3.5.16 above). The
tame descent data comes along for the ride.
The morphism (3.10.17) is a projective morphism whose target is Jacobson, andwhich induces a bijection on closed points. It is thus proper and quasi-finite, andhence finite. Its source is reduced (being even normal, by Lemma 3.10.11), and its
target is normal (as it is even regular, as we noted above). A finite morphism whose
source is reduced, whose target is normal and Noetherian, and which induces abijection on finite type points, is indeed an isomorphism. (The connected components
of a normal scheme are integral, and so base-changing over the connected componentsof the target, we may assume that the target is integral. The source is a union offinitely many irreducible components, each of which has closed image in the target.
Since the morphism is surjective on finite type points, it is surjective, and thus
one of these closed images coincides with the target. The injectivity on finite typepoints then shows that the source is also irreducible, and thus integral, as it isreduced. It follows from [Sta13, Tag 0AB1] that the morphism is an isomorphism.)Thus (3.10.17) is an isomorphism. Finally, since Rrτ,BT is also flat over O (by its
definition), this implies that (3.10.16) is scheme-theoretically dominant.
50

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTCorollary 3.10.18. lim Rτ,a = Rrτ,BT ; thus Rrτ,BT is a versal ring to Z τ at x.
←−
Proof. The theorem on formal functions shows that if we write the scheme-theoretic
image of (3.10.16) in the form Spec B, for some quotient B of Rr|GK , then the∞scheme-theoretic image of the morphism (3.10.8) coincides with Spf B. The corollary then follows from Proposition 3.10.15, which shows that (3.10.16) is schemetheoretically dominant.
Proposition 3.10.19. The algebraic stacks Z dd,a and Z τ,a are equidimensional ofdimension [K : Qp ].
Proof. Let x be a finite type point of Z τ,a , defined over some finite extension F0of F, and corresponding to a Galois representation r with coefficients in F0 . By
Corollary 3.10.18 the ring Rrτ,BT coincides with the versal ring lima Rτ,a at x of the←−∼$-adic formal algebraic stack Z τ , and so Spf Rτ,a −→ Spf Rrτ,BT ×Z τ Z τ,a . Since
Z τ is a $-adic formal algebraic stack, the natural morphism Z τ,1 → Z τ ×Spf O F isa thickening, and thus the same is true of the morphism Spf Rτ,1 → Spf Rrτ,BT /$obtained by pulling the former morphism back over Spf Rrτ,BT /$.
Since Rrτ,BT is flat over OE 0 and equidimensional of dimension 5 + [K : Qp ], it
follows that Rτ,1 is equidimensional of dimension 4 + [K : Qp ]. The same is thentrue of each Rτ,a , since these are thickenings of Rτ,1 , by Lemma 3.10.9 (4).
We have a versal morphism Spf Rτ,a → Z τ,a at the finite type point x of Z τ,a .
It follows from Lemma 3.10.6 that∼d2/ Spf Rτ,a −→GL
Spf Rτ,a ×Z τ,a Spf Rτ,a .
To find the dimension of Z τ,a it suffices to compute its dimension at finite
type points (cf. [Sta13, Tag 0DRX], recalling the definition of the dimension of analgebraic stack, [Sta13, Tag 0AFP]). It follows from [EG17, Lem. 2.40] applied tod2/ Spf Rτ,a ] of Zbxτ,a , together with Remark 3.10.4, thatthe presentation [Spf Rτ,a /GLZ τ,a is equidimensional of dimension [K : Qp ]. Since Z dd,a is the union of the Z τ,aby Theorem 3.9.2, Z dd,a is also equidimensional of dimension [K : Qp ] by [Sta13,
Tag 0DRZ].

Proposition 3.10.20. The algebraic stacks C τ,BT,a are equidimensional of dimension [K : Qp ].
Proof. Let x0 be a finite type point of C τ,BT,a , defined over some finite extension F0of F, lying over the finite type point x of Z τ,a . Let r be the Galois representation
with coefficients in F0 corresponding to x, and recall that Xr denotes a projective
br over Spf Rr
Spec Rrτ,BT -scheme whose pull-back X
is isomorphic to C τ,BT ×Rdd
|GK∞
Spf Rr|GK . The point x0 gives rise to a closed point x̃ of Xr (of which x0 is the∞bX ,x̃ denote the complete local ring
image under the morphism Xr → C τ,BT ). Let OrbX ,x̃ → C τ,BT is versal at x̃,to Xr at the point x̃; then the natural morphism Spf Ora0
bX ,x̃ /$ is a versal ring for the point x of C τ,BT,a .
so that Or
The isomorphism (3.10.5) induces (after pulling back over C τ,BT ) an isomorphism∼ b
d2 b −→br ,GL
Xr ×C τ,BT X/Xr

and thence an isomorphism
d2 b
GL

∼ b
b
/OXr ,x̃ −→ OXr ,x̃ ×C τ,BT OXr ,x̃ .

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

51

Since Rτ,BT is equidimensional of dimension 5 + [K : Qp ], it follows from Proposition 3.10.15 that Xr is equidimensional of dimension 5 + [K : Qp ], and thusbX ,x̃ /$a is(taking into account the flatness statement of Lemma 3.10.11) that Orequidimensional of dimension 4 + [K : Qp ]. As in the proof of Proposition 3.10.19,
an application of [EG17, Lem. 2.40] shows that dimx0 C τ,BT,a is equal to [K : Qp ].
Since x0 was an arbitrary finite type point, the result follows.

3.11. The Dieudonné stack. We now specialise the choice of K 0 in the following
way. Choose a tame inertial type τ = η ⊕ η 0 . Fix a uniformiser π of K. If τ is a tamefprincipal series type, we take K 0 = K(π 1/(p −1) ), while if τ is a tame cuspidal type,2f
we let L be an unramified quadratic extension of K, and set K 0 = L(π 1/(p −1) ).
Let N be the maximal unramified extension of K in K 0 . In either case K 0 /K is a
Galois extension; in the principal series case, we have e0 = (pf − 1)e, f 0 = f , andin the cuspidal case we have e0 = (p2f − 1)e, f 0 = 2f . We refer to this choice ofextension as the standard choice (for the fixed type τ and uniformiser π).
For the rest of this section we assume that η 6= η 0 (we will not need to consider
Dieudonné modules for scalar types).
Let M be a Breuil–Kisin module with A-coefficients and descent data of type τand height at most 1, and let D := M/uM be its corresponding Dieudonné module
as in Definition 2.2.1. If we write Di := ei D, then this Dieudonné module is given
by rank two projective modules Dj over A (j = 0, . . . , f 0 − 1) with linear maps
F : Dj → Dj+1 and V : Dj → Dj−1 (subscripts understood modulo f 0 ) such thatF V = V F = p.
Now, I(K 0 /K) is abelian of order prime to p, so we can write D = Dη ⊕ Dη0 ,where Dη is the submodule on which I(K 0 /K) acts via η. Since Mη is obtained fromthe projective SA -module M by applying a projector, each Dη,j is an invertible
A-module, and F, V induce linear maps F : Dη,j → Dη,j+1 and V : Dη,j+1 → Dη,jsuch that F V = V F = p.
We can of course apply the same construction with η 0 in the place of η, obtaining a
Dieudonné module Dη0 . We now prove some lemmas relating these various Dieudonnémodules. We will need to make use of a variant of the strong determinant condition,so we begin by discussing this and its relationship to the strong determinant conditionof Subsection 3.5.
Definition 3.11.1. Let (L, L+ ) be a pair consisting of a rank two projective
OK 0 ⊗Zp A-module L, and an OK 0 ⊗Zp A-submodule L+ ⊂ L, such that Zariski
locally on Spec A, L+ is a direct summand of L as an A-module.
Then we say that the pair (L, L+ ) satisfies the Kottwitz determinant condition
over K 0 if for all a ∈ OK 0 , we haveY
detA (a|L+ ) =ψ(a)ψ:K 0 ,→E

as polynomial functions on OK 0 in the sense of [Kot92, §5].
There is a finite type stack MK 0 ,det over Spec O, with MK 0 ,det (Spec A) being thegroupoid of pairs (L, L+ ) as above which satisfy the Kottwitz determinant condition
over K 0 . As we have seen above, by a result of Pappas–Rapoport, this stack is flatover Spec O (see [Kis09, Prop. 2.2.2]).

52

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

Lemma 3.11.2. If A is an E-algebra, then a pair (L, L+ ) as in Definition 3.11.1
satisfies the Kottwitz determinant condition over K 0 if and only if L+ is a rank one
projective OK 0 ⊗Zp A-module.
Q
Proof. We may write OK 0 ⊗Zp A = K 0 ⊗Qp A ∼= ψ:K 0 ,→E A, where the embeddingψ : K 0 ,→ E corresponds to an idempotent eψ ∈ K 0 ⊗Qp A. Decomposing L+
as ⊕ψ eψ L+ , the left-hand side of the Kottwitz determinant condition becomesQ
Q+
rkA eψ L+. It follows that the Kottwitz determinantψ detA (a|eψ L ) =ψ ψ(a)
condition is satisfied if and only if the projective A-module eψ L+ has rank one forall ψ, which is equivalent to L+ being a rank one projective K 0 ⊗Qp A-module, as
required.
Proposition 3.11.3. If M is an object of C τ,BT (A), then the pair
(M/E(u)M, im ΦM /E(u)M)
satisfies the Kottwitz determinant condition for K 0 .
0fixme DS: I think thisargument doesn’t makesense as written because
C τ,BT has noE-algebra-points. TG:
probably we want tohave A a finite (?) flat
O-algebra instead?

Proof. Let C τ,BT be the closed substack of C τ consisting of those M for which
the pair (M/E(u)M, im ΦM /E(u)M) satisfies the Kottwitz determinant condition
0
for K 0 . We need to show that C τ,BT is a closed substack of C τ,BT . Since C τ,BT isflat over Spf O by Corollary 3.8.3, it is enough to show that if A is an E-algebra,
0then C τ,BT (A) = C τ,BT (A).
To see this, let M be an object of C τ (A). By Lemma 3.11.2, M is an object
0of C τ,BT (A) if and only if im ΦM /E(u)M is a rank one projective K 0 ⊗Qp A-module.
Similarly, M is an object of C τ,BT (A) if and only if for each ξ, (im ΦM )ξ /E(u)Mξ
is a rank one projective N ⊗Qp A-module. Since
im ΦM /E(u)M = ⊕ξ (im ΦM )ξ /E(u)Mξ ,the equivalence of these two conditions is clear.

Lemma 3.11.4. If (L, L+ ) is an object of MK 0 ,det (A) (i.e. satisfies the Kottwitz
V2
V2
determinant condition over K 0 ), then the morphism OK 0 ⊗Z A L+ → OK 0 ⊗Z A Lpp
induced by the inclusion L+ ⊂ L is identically zero.
Remark 3.11.5. Note that, although L+ need not be locally free over OK 0 ⊗Zp A, its
exterior square is nevertheless defined, so that the statement of the lemma makes
sense.
Proof of Lemma 3.11.4. Since MK 0 ,det is O-flat, it is enough to treat the case thatV2
A is O-flat. In this case L, and thus also
L, are O-flat. Given this additional
assumption, it suffices to prove that the morphism of the lemma becomes zeroafter tensoring with Qp over Zp . This morphism may naturally be identified withV2
V2
the morphism K 0 ⊗Z A L+ → K 0 ⊗Z A L induced by the injection Qp ⊗Zp L+ ,→pp
Qp ⊗Zp L. Locally on Spec A, this is the embedding of a free K 0 ⊗Zp A-module ofV2
rank one as a direct summand of a free K 0 ⊗Zp A-module of rank two. ThusofV2
the source in fact vanishes, and hence so doesof the embedding.

V
V
V
2
2 ∗
2
Lemma 3.11.6. If M is an object of C τ,BT (A), thenΦM :ϕ M ,→
M
is exactly divisible by E(u), i.e. can be written as E(u) times an isomorphism of
SA -modules.

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

53

Proof.
It follows from Proposition 3.11.3 and Lemmaof
V2
V2 3.11.4 that the reduction
V2 ∗
ϕ M→
ΦM modulo E(u) vanishes, so we can think ofΦM as a morphism
V2
E(u)M. We need to show that the cokernel X of this morphism vanishes.
Since im ΦM ⊇ E(u)M, X is a finitely generated A-module, so that in order toprove that it vanishes, it is enough to prove that X/pX = 0.
Since the formation of cokernels is compatible with base change, this means that
we can (and do) assume that A is an F-algebra. Since the special fibre C τ,BT isof finite type over F, we can and do assume that A is furthermore of finite typeover F. The special fibre of C τ,BT is reduced by Corollary 3.8.3, so we may assume
that A is reduced, and it is therefore enough to prove that X vanishes modulo eachmaximal ideal of A. Since the residue fields at such maximal ideals are finite, weare reduced to the case that A is a finite field, when the result follows from [Kis09,
Lem. 2.5.1].
Lemma 3.11.7. There is a canonical isomorphism∼“(F ⊗ F )/p” : Dη,j ⊗A Dη0 ,j −→ Dη,j+1 ⊗A Dη0 ,j+1 ,characterised by the fact that it is compatible with change of scalars, and thatp · “(F ⊗ F )/p” = F ⊗ F.
Proof. Since C BT is flat over O, we see that in the universal case, the formulap · “(F ⊗ F )/p” = F ⊗ F
uniquely determines the isomorphism “(F ⊗ F )/p” (if it exists). Since any Breuil–Kisin module with descent data is obtained from the universal case by change of
scalars, we see that the isomorphism “(F ⊗ F )/p” is indeed characterised by theproperties stated in the lemma, provided that it exists.
To check that the isomorphism exists, we can again consider the universal case,and hence assume that A is a flat O-algebra. In this case, it suffices to check thatthe morphism F ⊗ F : Dη,j ⊗A Dη0 ,j → Dη,j+1 ⊗A Dη,j+1 is divisible by p, and thatthe formula (F ⊗ F )/p is indeed an isomorphism. Noting that the direct sum overj = 0, . . . , f 0 − 1 ofmay
V2these morphisms
V2 ∗V2 be identified with the reduction modulo uof the morphism
ΦM :
ϕ M→
M, this follows from Lemma 3.11.6.
The isomorphism “(F ⊗ F )/p” of the preceding lemma may be rewritten as an
isomorphism of invertible A-modules(3.11.8)∼

HomA (Dη,j , Dη,j+1 ) −→ HomA (Dη0 ,j+1 , Dη0 ,j ).

Lemma 3.11.9. The isomorphism (3.11.8) takes F to V .
Proof. The claim of the lemma is equivalent to showing that the composite
id ⊗VDη,j ⊗A Dη0 ,j+1 −→ Dη,j ⊗A Dη0 ,j“(F ⊗F )/p”−→

Dη,j+1 ⊗A Dη0 ,j+1

coincides with the morphism F ⊗ id . It suffices to check this in the universal case,and thus we may assume that p is a non-zero divisor in A, and hence verify therequired identity of morphisms after multiplying each of them by p. The identity to
be verified then becomes?

(F ⊗ F ) ◦ (id ⊗V ) = p(F ⊗ id),which follows immediately from the formula F V = p.

54

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

We now consider the moduli stacks classifying the Dieudonné modules withthe properties we have just established, and the maps from the moduli stacks of
Breuil–Kisin modules to these stacks.
Suppose first that we are in the principal series case. Then there is a moduli
stack classifying the data of the Dη,j together with the F and V , namely the stackhDη := Spec W (k)[X0 , Y0 , . . . , Xf −1 , Yf −1 ]/(Xj Yj − p)j=0,...,f −1 ) /Gfm ,where the f copies of Gm act as follows:−1(u0 , . . . , uf −1 ) · (Xj , Yj ) 7→ (uj u−1
j+1 Xj , uj+1 uj Yj ).

To see this, recall that the stack[point/Gm ]classifies line bundles, so the f copies of Gm in Dη correspond to f line bundles,
which are the line bundles Dη,j (j = 0, . . . , f − 1). If we locally trivialise these line
bundles, then the maps F : Dη,j → Dη,j+1 and V : Dη,j+1 → Dη,j act by scalars,
which we denote by Xj and Yj respectively. The f copies of Gm are then encoding
possible changes of trivialisation, by units uj , which induce the indicated changes
on the Xj ’s and Yj ’s.
There is then a natural map
C τ → Dη ,classifying the Dieudonné modules underlying the Breuil–Kisin modules with descent
data.
There is a more geometric way to think about what Dη classifies. To begin
with, we just rephrase what we’ve already indicated: it represents the functor
which associates to a W (k)-scheme the groupoid whose objects are f -tuples ofline bundles (Dη,j )j=0,...,f −1 equipped with morphisms Xj : Dη,j → Dη,j+1 andYj : Dη,j+1 → Dη,j such that Yj Xj = p. (Morphisms in the groupoid are just
isomorphisms between collections of such data.) Equivalently, we can think of this−1
as giving the line bundle Dη,0 , and then the f line bundles Dj := Dη,j+1 ⊗ Dη,j,−1−1equipped with sections Xj ∈ Dj and Yj ∈ Dj whose product in Dj ⊗ Dj = O(the trivial line bundle) is equal to the element p. Note that it superficially looks
like we are remembering f + 1 line bundles, rather than f , but this is illusory, since∼in fact D0 ⊗ · · · ⊗ Df −1 is trivial; indeed, the isomorphism D0 ⊗ · · · ⊗ Df −1 −→ O
is part of the data we should remember.
It will be helpful to introduce another stack, the stack Gη of η-gauges. This
classifies f -tuples of line bundles Dj (j = 0, . . . , f − 1) equipped with sections
Xj ∈ Dj and Yj ∈ Dj−1 . Explicitly, it can be written as the quotient stackhGη := Spec W (k)[X0 , Y0 , . . . , Xf −1 , Yf −1 ]/(Xj Yj − p)j=0,...,f −1 ) /Gfm ,where the f copies of Gm act as follows:(v0 , . . . , vf −1 ) · (Xj , Yj ) 7→ (vj Xj , vj−1 Yj ).
There is a natural morphism of stacks Dη → Gη given by forgetting forgetting D0 and∼
the isomorphism D0 ⊗ D1 ⊗ · · · ⊗ Df −1 −→ O. In terms of the explicit descriptions
via quotient stacks, we have a morphism Gfm → Gfm given by (uj )j=0,...,f −1 7→(uj u−1j+1 )j=0,...,f −1 , which is compatible with the actions of these two groups on

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

55

Spec W (k)[(Xj , Yj )j=0,...,f −1 ]/(Xj Yj − p)j=0,...,f −1 , and we are just considering themap from the quotient by the first Gfm to the quotient by the second Gfm .
Composing our morphism C τ → Dη with the forgetful morphism Dη → Gη , weobtain a morphism C τ → Gη .
We now turn to the case that τ is a cuspidal type. In this case our Dieudonnémodules have unramified as well as inertial descent data; accordingly, we let ϕf2f
denote the element of Gal(K 0 /K) which acts trivially on π 1/(p −1) and non-trivially∼on L. Then the descent data of ϕf induces isomorphisms Dj −→ Dj+f , which arecompatible with the F, V , and which identify Dη,j with Dη0 ,f +j .
If we choose local trivialisations of the line bundles Dη,0 , . . . , Dη,f , then the maps
F : Dη,j → Dη,j+1 and V : Dη,j+1 → Dη,j for 0 ≤ j ≤ f − 1 are given by scalars Xj
and Yj respectively. The identification of Dη,j and Dη0 ,f +j given by ϕf identifies−1Dη,j ⊗ Dη,j+1
with Dη0 ,f +j ⊗ Dη−1
0 ,f +j+1 , which via the isomorphsim (3.11.8) is−1
identified with Dη,f +j+1 ⊗ Dη,f+j . It follows that for 0 ≤ j ≤ f − 2 the data ofDη,j , Dη,j+1 and Dη,f +j recursively determines Dη,f +j+1 . From Lemma 3.11.9
we see, again recursively for 0 ≤ j ≤ f − 2, that there are unique trivialisationsof Dη,f +1 , . . . , Dη,2f −1 such that F : Dη,f +j → Dη,f +j+1 is given by Yj , andV : Dη,f +j+1 → Dη,f +j is given by Xj . Furthermore, there is some unit α suchthat F : Dη,2f −1 → Dη,0 is given by αYf −1 , and V : Dη,0 → Dη,2f −1 is givenby α−1 Xf −1 . Note that the map F 2f : Dη,0 → Dη,0 is precisely pf α.
Consequently, we see that the data of the Dη,j (together with the F, V ) is classified
by the stackhDη := Spec W (k)[X0 , Y0 , . . . , Xf −1 , Yf −1 ]/(Xj Yj − p)j=0,...,f −1 ) × Gm /Gfm+1 ,where the f + 1 copies of Gm act as follows:−1(u0 , . . . , uf −1 , uf ) · ((Xj , Yj ), α) 7→ ((uj u−1
j+1 Xj , uj+1 uj Yj ), α).

We again definehGη := Spec W (k)[X0 , Y0 , . . . , Xf −1 , Yf −1 ]/(Xj Yj − p)j=0,...,f −1 ) /Gfm ,where the f copies of Gm act as(v0 , . . . , vf −1 ) · (Xj , Yj ) 7→ (vj Xj , vj−1 Yj ).
There are again natural morphisms of stacks C τ → Dη → Gη , where the second
morphism is given in terms of the explicit descriptions via quotient stacks as follows:
we have a morphism Gfm+1 → Gfm given by (uj )j=0,...,f 7→ (uj u−1
j+1 )j=0,...,f −1 , andthe morphism Dη → Gη is the obvious one which forgets the factor of Gm comingfrom α.
For our analysis of the irreducible components of the stacks C τ,BT,1 at the endof Section 4, it will be useful to have a more directly geometric interpretationof a morphism S → Gη , in the case that the source is a flat W (k)-scheme, or,more generally, a flat p-adic formal algebraic stack over Spf W (k). In order todo this we will need some basic material on effective Cartier divisors for (formal)
algebraic stacks; while it is presumably possible to develop this theory in considerable
generality, we only need a very special case, and we limit ourselves to this setting.
The property of a closed subscheme being an effective Cartier divisor is notpreserved under arbitrary pull-back, but it is preserved under flat pull-back. Moreprecisely, we have the following result.

56

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

Lemma 3.11.10. If X is a scheme, and Z is a closed subscheme of X, then thefollowing are equivalent:(1) Z is an effective Cartier divisor on X.
(2) For any flat morphism of schemes U → X, the pull-back Z ×X U is aneffective Cartier divisor on U .
(3) For some fpqc covering {Xi → X} of X, each of the pull-backs Z ×X Xi isan effective Cartier divisor on Xi .
Proof. Since Z is an effective Cartier divisor if and only if its ideal sheaf IZ is
an invertible sheaf on X, this follows from the fact that the invertibility of aquasi-coherent sheaf is a local property in the fpqc topology.
Lemma 3.11.11. If A is a Noetherian adic topological ring, then pull-back under thenatural morphism Spf A → Spec A induces a bijection between the closed subschemesof Spec A and the closed subspaces of Spf A.
Proof. It follows from [Sta13, Tag 0ANQ] that closed immersions Z → Spf A arenecessarily of the form Spf B → Spf A, and correspond to continuous morphisms
A → B, for some complete linearly topologized ring B, which are taut (in thesense of [Sta13, Tag 0AMX]), have closed kernel, and dense image. Since A is
adic, it admits a countable basis of neighbourhoods of the origin, and so it follows
from [Sta13, Tag 0APT] (recalling also [Sta13, Tag 0AMV]) that A → B is surjective.
Because any ideal of definition I of A is finitely generated, it follows from [Sta13, Tag
0APU] that B is endowed with the I-adic topology. Finally, since A is Noetherian,any ideal in A is I-adically closed. Thus closed immersions Spf B → Spf A are
determined by giving the kernel of the corresponding morphism A → B, which canbe arbitrary. The same is true of closed immersions Spec B → Spec A, and so thelemma follows.

Definition 3.11.12. If A is a Noetherian adic topological ring, then we say that aclosed subspace of Spf A is an effective Cartier divisor on Spf A if the corresponding
closed subscheme of Spec A is an effective Cartier divisor on Spec A.
Lemma 3.11.13. Let Spf B → Spf A be a flat adic morphism of Noetherian affine
formal algebraic spaces. If Z ,→ Spf A is a Cartier divisor, then Z ×Spf A Spf B ,→Spf B is a Cartier divisor. Conversely, if Spf B → Spf A is furthermore surjective,
and if Z ,→ Spf A is a closed subspace for which the base-change Z ×Spf A Spf B ,→Spf B is a Cartier divisor, then Z is a Cartier divisor on Spf A.
Proof. The morphism Spf B → Spf A corresponds to an adic flat morphism A → B([Sta13, Tag 0AN0] and [Eme, Lem. 8.18]) and hence is induced by a flat morphism
Spec B → Spec A, which is furthermore faithfully flat if and only if Spf B → Spf A
is surjective (again by [Eme, Lem. 8.18]). The present lemma thus follows from
Lemma 3.11.10.
The preceding lemma justifies the following definition.
Definition 3.11.14. We say that a closed substack Z of a locally Noetherian
formal algebraic stack X is an effective Cartier divisor on X if for any morphism
U → X whose source is a Noetherian affine formal algebraic space, and which isrepresentable by algebraic spaces and flat, the pull-back Z ×X U is an effective
Cartier divisor on U .

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We consider the W (k)-scheme Spec W (k)[X, Y ]/(XY − p), which we endow witha Gm -action via u · (X, Y ) := (uX, u−1 Y ). There is an obvious morphism
Spec W (k)[X, Y ]/(XY − p) → Spec W (k)[X] = A1
given by (X, Y ) → X, which is Gm -equivariant (for the action of Gm on A1 given
by u · X := uX), and so induces a morphism
(3.11.15)[ Spec W (k)[X, Y ]/(XY − p) /Gm ] → [A1 /Gm ].
Lemma 3.11.16. If X is a locally Noetherian p-adic formal algebraic stack which
is furthermore flat over Spf W (k), then the groupoid of morphisms
X → [Spec W (k)[X, Y ]/(XY − p)/Gm ]is in fact a setoid, and is equivalent to the set of effective Cartier divisors on X thatare contained in the effective Cartier divisor (Spec k) ×Spf W (k) X on X .
Proof. Essentially by definition (and taking into account [Eme, Lem. 8.18]), it
suffices to prove this in the case when X = Spf B, where B is a flat Noetherian adic
W (k)-algebra admitting (p) as an ideal of definition. In this case, the restriction
map[Spec W (k)[X, Y ]/(XY −p)/Gm ](Spec B) → [Spec W (k)[X, Y ]/(XY −p)/Gm ](Spf B)
is an equivalence of groupoids. Indeed, the essential surjectivity follows from the(standard and easily verified) fact that if {Mi } is a compatible family of locally free
B/pi B-modules of rank one, then M := lim Mi is a locally free B-module of rank←−
one, for which each of the natural morphisms M/pi M → Mi is an isomorphism.
The full faithfulness follows from the fact that a locally free B-module of rank one isp-adically complete, and so is recovered as the inverse limit of its compatible familyof quotients {M/pi M }.
We are therefore reduced to the same statement with X = Spec B. The composite morphism Spec B → [A1 /Gm ] induced by (3.11.15) corresponds to giving apair (D, X) where D is a line bundle on Spec B, and X is a global section of D−1 .
Indeed, giving a morphism Spec B → [A1 /Gm ] is equivalent to giving a Gm -torsor
P → Spec B, together with a Gm -equivariant morphism P → A1 . Giving a Gm torsor P over Spec B is equivalent to giving an invertible sheaf D on Spec B (theassociated Gm -torsor is then obtained by deleting the zero section from the line bundle D → X corresponding to D), and giving a Gm -equivariant morphism P → A1
is equivalent to giving a global section of D−1 .
It follows that giving a morphism Spec B → [Spec W (k)[X, Y ]/(XY − p)/Gm ]corresponds to giving a line bundle D and sections X ∈ D−1 , Y ∈ D satisfying
XY = p. To say that B is flat over W (k) is just to say that p is a regular element on B,
and so we see that X (resp. Y ) is a regular section of D−1 (resp. D). Again, since p isa regular element on B, we see that Y is uniquely determined by X and the equation
XY = p, and so giving a morphism Spec B → [Spec W (k)[X, Y ]/(XY − p)/Gm ]is equivalent to giving a line bundle D and a regular section X of D−1 , such that∼pB ⊂ X ⊗B D ⊂ D−1 ⊗B D −→ B; this last condition guarantees the existence ofthe (then uniquely determined) Y .
Now giving a line bundle D on Spec B and a regular section X ∈ D−1 is the sameas giving the zero locus D of X, which is a Cartier divisor on Spec B. (There is a
canonical isomorphism (D, X) ∼= ID , 1 , where ID denotes the ideal sheaf of D.)

58

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTThe condition that pB ⊂ X ⊗B D is equivalent to the condition that p ∈ ID , i.e.
that D be contained in Spec B/pB, and we are done.
Lemma 3.11.17. If S is a locally Noetherian p-adic formal algebraic stack whichis flat over W (k), then giving a morphism S → Gη over W (k) is equivalent to giving
a collection of effective Cartier divisors Dj on S (j = 0, . . . , f − 1), with each Dj
contained in the Cartier divisor S cut out by the equation p = 0 on S (i.e. the special
fibre of S).
Proof. This follows immediately from Lemma 3.11.16, by the definition of Gη .



4. Extensions of rank one Breuil–Kisin modules with descent data
The goal of this section is to construct certain universal families of extensionsof rank one Breuil–Kisin modules over F with descent data, and to use these todescribe the generic behaviour of the various irreducible components of the special
fibres of C τ,BT and Z τ .
In Subsection 4.1 we present some generalities on extensions of Breuil–Kisin
modules. In Subsection 4.3 we explain how to construct our desired families ofextensions. In Subsection 4.4 we recall the fundamental computations related toextensions of rank one Breuil–Kisin modules from [DS15], to which the results ofSubsection 4.3 will be applied.
We assume throughout this section that [K 0 : K] is not divisible by p; since weare assuming throughout the paper that K 0 /K is tamely ramified, this is equivalent
to assuming that K 0 does not contain an unramified extension of K of degree p. In
our final applications K 0 /K will contain unramified extensions of degree at most 2,
and p will be odd, so this assumption will be satisfied. (In fact, we specialize tosuch a context begining in Subsection 4.7.)
4.1. Extensions of Breuil–Kisin modules with descent data. When discussing the general theory of extensions of Breuil–Kisin modules, it is convenient toembed the category of Breuil–Kisin modules in a larger category which is abelian,
contains enough injectives and projectives, and is closed under passing to arbitrary
limits and colimits. The simplest way to obtain such a category is as the categoryof modules over some ring, and so we briefly recall how a Breuil–Kisin modulewith A-coefficients and descent data can be interpreted as a module over a certain
A-algebra.
Let SA [F ] denote the twisted polynomial ring over SA , in which the variable F
obeys the following commutation relation with respect to elements s ∈ SA :F · s = ϕ(s) · F.
Let SA [F, Gal(K 0 /K)] denote the twisted group ring over SA [F ], in which theelements g ∈ Gal(K 0 /K) commute with F , and obey the following commutation
relations with elements s ∈ SA :g · s = g(s) · g.
One immediately confirms that giving a left SA [F, Gal(K 0 /K)]-module M is equivalent to equipping the underlying SA -module M with a ϕ-linear morphism ϕ : M →M and a semi-linear action of Gal(K 0 /K) which commutes with ϕ.
In particular, if we let K(A) denote the category of left SA [F, Gal(K 0 /K)]modules, then a Breuil–Kisin module with descent data from K 0 to K may naturally

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59

be regarded as an object of K(A). In the following lemma, we record the factthat extensions of Breuil–Kisin modules with descent data may be computed as
extensions in the category K(A).
Lemma 4.1.1. If 0 → M0 → M → M00 → 0 is a short exact sequence in K(A),such that M0 (resp. M00 ) is a Breuil–Kisin module with descent data of rank d0 andheight at most h0 (resp. of rank d00 and height at most h00 ), then M is a Breuil–Kisin
module with descent data of rank d0 + d00 and height at most h0 + h00 .
More generally, if E(u)h ∈ AnnSA (coker ΦM0 ) AnnSA (coker ΦM00 ), then M is a
Breuil–Kisin module with descent data of height at most h.
Proof. Note that since ΦM0 [1/E(u)] and ΦM00 [1/E(u)] are both isomorphisms byassumption, it follows from the snake lemma that ΦM [1/E(u)] is isomorphism.
Similarly we have a short exact sequence of SA -modules
0 → coker ΦM0 → coker ΦM → coker ΦM00 → 0.
The claims about the height and rank of M follow immediately.



We now turn to giving an explicit description of the functors Exti (M, – ) for a
Breuil–Kisin module with descent data M.
Definition 4.1.2. Let M be a Breuil–Kisin module with A-coefficients and descent•data (of some height). If N is any object of K(A), then we let CM(N) denote thecomplex
HomSA [Gal(K 0 /K)] (M, N) → HomSA [Gal(K 0 /K)] (ϕ∗ M, N),with differential being given by
α 7→ ΦN ◦ ϕ∗ α − α ◦ ΦM .
0
1
Also let Φ∗M denote the map CM(N) → CM(N) given by α 7→ α ◦ ΦM . When M
is clear from the context we will usually suppress it from the notation and write
simply C • (N).

Each C i (N) is naturally an S0A -module. The formation of C • (N) is evidently
functorial in N, and is also exact in N, since M, and hence also ϕ∗ M, is projective0
over SA , and since1Gal(K/K) has prime-to-p order. Thus the cohomology functors0••
H C (–) and H C (–) form a δ-functor on K(A).
Lemma 4.1.3. There is a natural isomorphism
HomK(A) (M, – ) ∼= H 0 C • (– ) .
Proof. This is immediate.

It follows from this lemma and a standard dimension shifting argument (or,equivalently, the theory of δ-functors) that there is an embedding of functors
(4.1.4)
Ext1K(A) (M, – ) ,→ H 1 C • (–) .
Lemma 4.1.5. The embedding of functors (4.1.4) is an isomorphism.
Proof. We first describe the embedding (4.1.4) explicitly. Suppose that
0→N→E→M→0
is an extension in K(A). Since M is projective over SA , and since Gal(K 0 /K) isof prime-to-p order, we split this short exact sequence over the twisted group ring60

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTSA [Gal(K 0 /K)], say via some element σ ∈ HomSA [Gal(K 0 /K)] (M, E). This splitting
is well-defined up to the addition of an element α ∈ HomSA [Gal(K 0 /K)] (M, N).
This splitting is a homomorphism in K(A) if and only if the elementΦE ◦ ϕ∗ σ − σ ◦ ΦM ∈ HomSA [Gal(K 0 /K)] (ϕ∗ M, N)
vanishes. If we replace σ by σ + α, then this element is replaced by(ΦE ◦ ϕ∗ σ − σ ◦ ΦM ) + (ΦN ◦ ϕ∗ α − α ◦ ΦM ).

Thus the coset of ΦE ◦ ϕ∗ σ − σ ◦ ΦM in H 1 C • (N) is well-defined, independent of
the choice of σ, and this coset is the image of the class of the extension E under theembedding(4.1.6)
Ext1K(A) (M, N) ,→ H 1 C • (N)(up to a possible overall sign, which we ignore, since it doesn’t affect the claim ofthe lemma).
Now, given any element ν ∈ HomSA [Gal(K 0 /K)] (ϕ∗ M, N), we may give theSA [Gal(K 0 /K)]-module E := N ⊕ M the structure of a SA [F, Gal(K 0 /K)]-module
as follows: we need to define a ϕ-linear morphism E → E, or equivalently a linear
morphism ΦE : ϕ∗ E → E. We do this by setting


ΦN
ν
ΦE :=
.
0 ΦM
Then E is an extension of M by N, and if we let σ denote the obvious embedding of
M into E, then one computes thatν = ΦE ◦ ϕ∗ σ − σ ◦ ΦM .
This shows that (4.1.6) is an isomorphism, as claimed.



Another dimension shifting argument, taking into account the preceding lemma,shows that Ext2K(A) (M, – ) embeds into H 2 C • (–) . Since the target of this embedding vanishes, we find that the same is true of the source. This yields the following
corollary.
Corollary 4.1.7. If M is a Breuil–Kisin module with A-coefficients and descentdata, then Ext2K(A) (M, – ) = 0.
We summarise the above discussion in the following corollary.
Corollary 4.1.8. If M is a Breuil–Kisin module with A-coefficients and descentdata, and N is an object of K(A), then we have a natural short exact sequence
0 → HomK(A) (M, N) → C 0 (N) → C 1 (N) → Ext1K(A) (M, N) → 0.
The following lemma records the behaviour of these complexes with respect tobase change.
Lemma 4.1.9. Suppose that M, N are Breuil–Kisin modules with descent data
and A-coefficients, that B is an A-algebra, and that Q is a B-module. Then the•b A Q) and C • b (N ⊗b A Q) coincide, the former complex formed
complexes CM(N ⊗M ⊗A B
with respect to K(A) and the latter with respect to K(B).
Proof. Indeed, there is a natural isomorphismb A Q) ∼b A B, N ⊗b A Q),HomSA [Gal(K 0 /K)] (M, N ⊗= HomSB [Gal(K 0 /K)] (M ⊗and similarly with ϕ∗ M in place of M.

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The following slightly technical lemma is crucial for establishing finiteness properties, and also base-change properties, of Exts of Breuil–Kisin modules.
Lemma 4.1.10. Let A be a O/$a -algebra for some a ≥ 1, suppose that M is a
Breuil–Kisin module with descent data and A-coefficients, of height at most h, andsuppose that N is a u-adically complete, u-torsion free object of K(A).
Let C • be the complex defined in Definition 4.1.2, and write δ for its differential.
Suppose that Q is an A-module with the property that C i ⊗A Q is v-torsion free for
i = 0, 1 and v-adically separated for i = 0.
Then:(1) For any integer M ≥ (eah + 1)/(p − 1), ker(δ ⊗ idQ ) ∩ v M C 0 ⊗A Q = 0.
(2) For any integer N ≥ (peah + 1)/(p − 1), δ ⊗ idQ induces an isomorphism∼(Φ∗M )−1 (v N C 1 ⊗A Q) −→ v N (C 1 ⊗A Q).
Consequently, for N as in (2) the natural morphism of complexes of A-modules δ⊗idQδ⊗idQ[C 0 ⊗A Q −→ C 1 ⊗A Q] → [C 0 ⊗A Q/ (Φ∗M )−1 (v N C 1 ⊗A Q) −→ C 1 ⊗A Q/v N C 1 ⊗A Q]
is a quasi-isomorphism.
Since we are assuming that the C i ⊗A Q are v-torsion free, the expressionv r C i (N)⊗A Q may be interpreted as denoting either v r C i (N)⊗A Q or v r C i (N) ⊗A
Q, the two being naturally isomorphic.
Remark 4.1.11. Before giving the proof of Lemma 4.1.10, we observe that the
hypotheses on the C i ⊗A Q are satisfied if either Q = A, or else N is a projective
SA -module and Q is a finitely generated B-module for some finitely generated
A-algebra B. (Indeed C 1 ⊗A Q is v-adically separated as well in these cases.)(1) Since M is projective of finite rank over A[[u]], and since N is u-adicallycomplete and u-torsion free, each C i is v-adically separated and v-torsion free. In
particular the hypothesis on Q is always satisfied by Q = A. (In fact since N isu-adically complete it also follows that the C i are v-adically complete. Here weuse that Gal(K 0 /K) has order prime to p to see that C 0 is an S0A -module direct
summand of HomSA (M, N), and similarly for C 1 .)(2) Suppose N is a projective SA -module. Then the C i are projective S0A -modules,
again using that Gal(K 0 /K) has order prime to p. Since each C i (N)/vC i (N) is
A-flat, it follows that C i (N) ⊗A Q is v-torsion free. If furthermore B is a finitely
generated A-algebra, and Q is a finitely generated B-module, then the C i (N) ⊗A Q
are v-adically separated (being finitely generated modules over the ring A[[v]] ⊗A B,
which is a finitely generated algebra over the Noetherian ring A[[v]], and hence isitself Noetherian).
0

Proof of Lemma 4.1.10. Since pa = 0 in A, there exists H(u) ∈ SA with ue ah =0
E(u)h H(u) in SA . Thus the image of ΦM contains ue ah M = v eah M, and there
exists a map Υ : M → ϕ∗ M such that ΦM ◦ Υ is multiplication by v eah .
We begin with (1). Suppose that f ∈ ker(δ ⊗ idQ ) ∩ v M C 0 ⊗A Q. Since C 0 ⊗A Q
is v-adically separated, it is enough, applying induction on M , to show that f ∈v M +1 C 0 ⊗A Q. Since f ∈ ker(δ ⊗ idQ ), we have f ◦ ΦM = ΦN ◦ ϕ∗ f . Sincef ∈ v M C 0 ⊗A Q, we have f ◦ ΦM = ΦN ◦ ϕ∗ f ∈ v pM C 1 ⊗A Q. Precomposingwith Υ gives v eah f ∈ v pM C 0 ⊗A Q. Since C 0 ⊗A Q is v-torsion free, it follows thatf ∈ v pM −eah C 0 ⊗A Q ⊆ uM +1 C 0 ⊗A Q, as required.

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

We now move on to (2). Set M = N − eah. By precomposing with Υ we seethat α ◦ ΦM ∈ v N C 1 ⊗A Q implies α ∈ v M C 0 ⊗A Q; from this, together with theinequality pM ≥ N , it is straightforward to check that(Φ∗M )−1 (v N C 1 ⊗A Q) = (δ ⊗ idQ )−1 (v N C 1 ⊗A Q) ∩ v M C 0 ⊗A Q.
Note that M satisfies the condition in (1). To complete the proof we will show thatfor any M as in (1) and any N ≥ M + eah the map δ induces an isomorphism∼(δ ⊗ idQ )−1 (v N C 1 ⊗A Q) ∩ v M C 0 ⊗A Q −→ v N C 1 ⊗A Q.
By (1), δ ⊗ idQ induces an injection (δ ⊗ idQ )−1 (v N C 1 ⊗A Q) ∩ v M C 0 ⊗A Q ,→v N C 1 ⊗A Q, so it is enough to show that (δ ⊗ idQ )(v M C 0 ⊗A Q) ⊇ v N C 1 ⊗A Q.
Equivalently, we need to show thatv N C 1 ⊗A Q → (C 1 ⊗A Q)/(δ ⊗ idQ ) v M C 0 ⊗A Q)
is identically zero. Since the formation of cokernels is compatible with tensor
products, we see that this morphism is obtained by tensoring the corresponding
morphismv N C 1 → C 1 /δ v M C 0with Q over A, so we are reduced to the case Q = A. (Recall from Remark 4.1.11(1)that the hypotheses of the Lemma are satisfied in this case, and that C 1 is v-adicallyseparated.)
We claim that for any g ∈ v N C 1 , we can find an f ∈ v N −eah C 0 such that δ(f ) −g ∈ v p(N −eah) C 1 . Admitting the claim, given any g ∈ v N C 1 , we may find h ∈ v M C 0with δ(h) = g by successive approximation in the following way: Set h0 = f for f as
in the claim; then h0 ∈ v N −eah C 0 ⊆ v M C 0 , and δ(h0 )−g ∈ v p(N −eah) C 1 ⊆ v N +1 C 1 .
Applying the claim again with N replaced by N + 1, and g replaced by g − δ(h0 ), we
find f ∈ v N +1−eah C 0 ⊆ v M +1 C 0 with δ(f ) − g + δ(h0 ) ∈ v p(N +1−eah) C 1 ⊆ v N +1 C 1 .
Setting h1 = h0 + f , and proceeding inductively, we obtain a Cauchy sequence
converging (in the v-adically complete A[[v]]-module C 0 ) to the required element h.
It remains to prove the claim. Since δ(f ) = ΦN ◦ ϕ∗ f − f ◦ ΦM , and since if
f ∈ v N −eah C 0 then ΦN ◦ ϕ∗ f ∈ v p(N −eah) C 1 , it is enough to show that we can findan f ∈ v N −eah C 0 with f ◦ ΦM = −g. Since ΦM is injective, the map Υ ◦ ΦM is also
multiplication by v eah , and so it suffices to take f with v eah f = −g ◦ Υ ∈ v N C 0 . 
Corollary 4.1.12. Let A be a Noetherian O/$a -algebra, and let M, N be Breuil–Kisin modules with descent data and A-coefficients. If B is a finitely generated
A-algebra, and Q is a finitely generated B-module, then the natural morphism ofcomplexes of B-modules
δ⊗idQ

δ

b A Q) −→ C 1 (N ⊗b A Q)][C 0 (N) ⊗A Q −→ C 1 (N) ⊗A Q] → [C 0 (N ⊗is a quasi-isomorphism.
b A Q)
Proof. By Remarks 4.1.11 and 2.1.8(2) we can apply Lemma 4.1.10 to both C i (N ⊗iand C (N) ⊗A Q, and we see that it is enough to show that the natural morphismof complexes

 δ


[ C 0 (N) ⊗A Q /(Φ∗M ⊗ idQ )−1 v N C 1 (N) ⊗A Q −→ C 1 (N) ⊗A Q / v N C 1 (N) ⊗A Q ] δ 1
b A Q)/ Φ∗M )−1 (v N C 1 (N ⊗b A Q) →b A Q)/v N C 1 (N ⊗b A Q)][C 0 (N ⊗C (N ⊗

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

63

is a quasi-isomorphism. In fact, it is even an isomorphism.

Proposition 4.1.13. Let A be a O/$a -algebra for some a ≥ 1, and let M, N be
Breuil–Kisin modules with descent data and A-coefficients. Then Ext1K(A) (M, N)
and Ext1K(A) (M, N/ui N) for i ≥ 1 are finitely presented A-modules.
If furthermore A is Noetherian, then HomK(A) (M, N) and HomK(A) (M, N/ui N)
for i ≥ 1 are also finitely presented (equivalently, finitely generated) A-modules.
Proof. The statements for N/ui N follow easily from those for N, by considering
the short exact sequence 0 → ui N → N → N/ui N → 0 in K(A) and applying
Corollary 4.1.7. By Corollary 4.1.8, it is enough to consider the cohomology of thecomplex C • . By Lemma 4.1.10 with Q = A, the cohomology of C • agrees with thecohomology of the induced complex
C 0 / (Φ∗M )−1 (v N C 1 )) → C 1 /v N C 1 ,for an appropriately chosen value of N . It follows that for an appropriately chosen
value of N , Ext1K(A) (M, N) can be computed as the cokernel of the induced morphism
C 0 /v N C 0 → C 1 /v N C 1 .
Under our hypothesis on N, C 0 /v N C 0 and C 1 /v N C 1 are finitely generated
projective A-modules, and thus finitely presented. It follows that Ext1K(A) (M, N) is
finitely presented.
In the case that A is furthermore assumed to be Noetherian, it is enoughto notethat since v N C 0 ⊆ (Φ∗M )−1 (v N C 1 ), the quotient C 0 / (Φ∗M )−1 (v N C 1 ) is a finitely
generated A-module.
Proposition 4.1.14. Let A be a O/$a -algebra for some a ≥ 1, and let M and N
be Breuil–Kisin modules with descent data and A-coefficients. Let B be an Ab AB → N ⊗b A B be a morphism of Breuil–Kisin modules
algebra, and let fB : M ⊗with B-coefficients.
Then there is a finite type A-subalgebra B 0 of B and a morphism of Breuil–Kisin
b AB0 → N ⊗b A B 0 such that fB is the base change of fB 0 .
modules fB 0 : M ⊗Proof. By Lemmas 4.1.3 and 4.1.9 (the latter applied with Q = B) we can and dob A B) → C 1 (N ⊗b A B), thethink of fB as being an element of the kernel of δ : C 0 (N ⊗••complex C here and throughout this proof denoting CM as usual.
Fix N as in Lemma 4.1.10, and write f B for the corresponding element of0b A B)/v N = (C 0 (N)/v N )⊗A B (this equality following easily from the assumpC (N ⊗tion that M and N are projective SA -modules of finite rank). Since C 0 (N)/v N is a
projective A-module of finite rank, it follows that for some finite type A-subalgebrab A B 0 )/v N such thatB 0 of B, there is an element f B 0 ∈ (C 0 (N)/v N ) ⊗A B 0 = C 0 (N ⊗0
f B 0 ⊗B B = f B . Denote also by f B 0 the induced element ofb A B 0 )/ Φ∗M )−1 (v N C 1 (N ⊗b A B 0 )).
C 0 (N ⊗By Lemma 4.1.10 (and Lemma 4.1.3) we have a commutative diagram with exactrows
 δ/ H 0 (C • (N ⊗/ C 0 (N ⊗/ C 1 (N ⊗b A B 0 ))b A B 0 )/ (Φ∗M )−1 (v N C 1 (N ⊗b A B 0 ))b A B 0 )/v N00
/ H 0 (C • (N ⊗b A B))


/ C 0 (N ⊗b A B)/ (Φ∗ )−1 (v N C 1 (N ⊗b A B))Mδ
/ C 1 (N ⊗b A B)/v N

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

b B 0 B. By a diagram chase we only needin which the vertical arrows are induced by ⊗to show that δ(f B 0 ) = 0. Since δ(fB ) = 0, it is enough to show that the right handvertical arrow is an injection. This arrow can be rewritten as the tensor productof the injection of A-algebras B 0 ,→ B with the flat (even projective of finite rank)
A-module C 1 (N)/v N , so the result follows.

We have the following key base-change result for Ext1 ’s of Breuil–Kisin modules
with descent data.
Proposition 4.1.15. Suppose that M and N are Breuil–Kisin modules with descent data and coefficients in a O/$a -algebra A. Then for any A-algebra B, and∼for any B-module Q, there are natural isomorphisms Ext1K(A) (M, N) ⊗A Q −→∼b A B, N ⊗b A B) ⊗B Q −→ Ext1K(B) (M ⊗b A B, N ⊗b A Q).
Ext1K(B) (M ⊗Proof. We first prove the lemma in the case of an A-module Q. It follows from
Lemmas 4.1.5 and 4.1.10 that we may compute Ext1K(A) (M, N) as the cokernel ofthe morphismδC 0 (N)/v N C 0 (N) −→ C 1 (N)/v N C 1 (N),for some sufficiently large value of N (not depending on N), and hence that we may
compute Ext1K(A) (M, N) ⊗A Q as the cokernel of the morphismδ
C 0 (N)/v N C 0 (N) ⊗A Q −→ C 1 (N)/v N C 1 (N) ⊗A Q.
b A Q) as the cokernel of the morphism
We may similarly compute Ext1K(A) (M, N ⊗δb A Q)/v N C 0 (N ⊗b A Q) −→ C 1 (N ⊗b A Q)/v N C 1 (N ⊗b A Q).
C 0 (N ⊗b A Q satisfies the necessary hypotheses for Lemma 4.1.10(Remark 2.1.8 (2) shows that N ⊗to apply.) Once we note that the natural morphismb A Q)/v N C i (N ⊗b A Q)C i (N)/v N C i (N) ⊗A Q → C i (N ⊗is an isomorphism for i = 0 and 1 (because M is a finitely generated projective
SA -module), we obtain the desired isomorphism∼b A Q).
Ext1K(A) (M, N) ⊗A Q −→ Ext1K(A) (M, N ⊗If B is an A-algebra, and Q is a B-module, then by Lemma 4.1.9 there is anatural isomorphism∼b A Q) −→ Ext1K(B) (M ⊗b A B, N ⊗b A Q);
Ext1K(A) (M, N ⊗combined with the preceding base-change result, this yields one of our claimed
isomorphisms, namely∼

b A B, N ⊗b A Q).
Ext1K(A) (M, N) ⊗A Q −→ Ext1K(B) (M ⊗Taking Q to be B itself, we then obtain an isomorphism∼b A B, N ⊗b A B).
Ext1K(A) (M, N) ⊗A B −→ Ext1K(B) (M ⊗This allows us to identify Ext1K(A) (M, N) ⊗A Q, which is naturally isomorphic tob A B, N ⊗b A B) ⊗B Q, yielding theExt1K(A) (M, N) ⊗A B ⊗B Q, with Ext1K(B) (M ⊗second claimed isomorphism.
MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

65

In contrast to the situation for extensions (cf. Proposition 4.1.15), the formationof homomorphisms between Breuil–Kisin modules is in general not compatible with
arbitrary base-change, as the following example shows.
Example 4.1.16. Take A = (Z/pZ)[x±1 , y ±1 ], and let Mx be the free Breuil–Kisin module of rank one and A-coefficients with ϕ(e) = xe for some generator e of Mx . Similarly define My with ϕ(e0 ) = ye0 for some generator e0 of
My . Then HomK(A) (Mx , My ) = 0. On the other hand, if B = A/(x − y)∼b A B and My ⊗b A B are isomorphic, so that HomK(B) (Mx ⊗B,
b My ⊗B)bthen Mx ⊗6=
HomK(A) (Mx , My ) ⊗A B.
However, it is possible to establish such a compatibility in some settings. Corolb A B, N ⊗b A B)
lary 4.1.18, which gives a criterion for the vanishing of HomK(B) (M ⊗for any A-algebra B, is a first example of a result in this direction. Lemma 4.1.20
deals with flat base change, and Lemma 4.1.21, which will be important in Section 4.3, proves that formation of homomorphisms is compatible with base-changeover a dense open subscheme of Spec A.
Proposition 4.1.17. Suppose that A is a Noetherian O/$a -algebra, and that M
and N are objects of K(A) that are finitely generated over SA (or, equivalently, overA[[u]]). Consider the following conditions:b A B, N ⊗b A B) = 0 for any finite type A-algebra B.
(1) HomK(B) (M ⊗(2) HomK(κ(m)) M ⊗A κ(m), N ⊗A κ(m) = 0 for each maximal ideal m of A.
(3) HomK(A) (M, N ⊗A Q) = 0 for any finitely generated A-module Q.
Then we have (1) =⇒ (2) ⇐⇒ (3). If A is furthermore Jacobson, then all three
conditions are equivalent.
Proof. If m is a maximal ideal of A, then κ(m) is certainly a finite type A-algebra,
and so evidently (1) implies (2). It is even a finitely generated A-module, and so
also (2) follows from (3).
We next prove that (2) implies (3). To this end, recall that if A is any ring,and M is any A-module, then M injects into the product of its localizations at all
maximal ideals. If A is Noetherian, and M is finitely generated, then, by combining
this fact with the Artin–Rees Lemma, we see that M embeds into the product ofits completions at all maximal ideals. Another way to express this is that, if I runs
over all cofinite length ideals in A (i.e. all ideals for which A/I is finite length), thenM embeds into the projective limit of the quotients M/IM (the point being that
this projective limit is the same as the product over all m-adic completions). We
are going to apply this observation with A replaced by SA , and with M taken to
be N ⊗A Q for some finitely generated A-module Q.
In A[[u]], one sees that u lies in the Jacobson radical (because 1 + f u is invertible
in A[[u]] for every f ∈ A[[u]]), and thus in every maximal ideal, and so the maximal
ideals of A[[u]] are of the form (m, u), where m runs over the maximal ideals of A.
Thus the ideals of the form (I, un ), where I is a cofinite length ideal in A, are cofinal
in all cofinite length ideals in A[[u]]. Since SA is finite over A[[u]], we see that theideals (I, un ) in SA are also cofinal in all cofinite length ideals in A[[u]]. Since A[[u]],and hence SA , is furthermore Noetherian when A is, we see that if Q is a finitely
generated A-module, and N is a finitely generated SA -module, then N ⊗A (Q/IQ)
is u-adically complete, for any cofinite length ideal I in A, and hence equal to thelimit over n of N ⊗A Q/(I, un ). Putting this together with the observation of the

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

preceding paragraph, we see that the natural morphism
N ⊗A Q → lim N ⊗A (Q/IQ)←−
I(where I runs over all cofinite length ideals of A) is an embedding. The induced
morphism
HomK(A) (M, N ⊗A Q) → lim HomK(A) (M, N ⊗A (Q/IQ))←−
I

is then evidently also an embedding.
Thus, to conclude that HomK(A) (M, N ⊗A Q) vanishes, it suffices to show that
HomK(A) (M, N ⊗A (Q/IQ)) vanishes for each cofinite length ideal I in A. An easy
induction on the length of A/I reduces this to showing that
 HomK(A) M, N⊗A κ(m) ,or, equivalently, HomK(κ(m)) M ⊗A κ(m), N ⊗A κ(m) , vanishes for each maximal
ideal m. Since this is the hypothesis of (2), we see that indeed (2) implies (3).
It remains to show that (3) implies (1) when A is Jacobson. Applying the result“(2) implies (3)” (with A replaced by B, and taking Q in (3) to be B itself as ab A B and N ⊗b A B, we see that it suffices to prove the vanishing of
B-module) to M ⊗b A B) ⊗B κ(n), (N ⊗b A B) ⊗B κ(n) = HomK(A) M, N ⊗b A κ(n)HomK(B) (M ⊗for each maximal ideal n of B. Since A is Jacobson, the field κ(n) is in fact a finitelybgenerated A-module, hence N ⊗κ(n)= N ⊗A κ(n), and so the desired vanishing is a
special case of (3).
Corollary 4.1.18. If A is a Noetherian and Jacobson O/$a -algebra, and if M and
N are Breuil–Kisin modules with descent data and A-coefficients, then the following
three conditions are equivalent:b A B, N ⊗b A B) = 0 for any A-algebra B.
(1) HomK(B) (M ⊗(2) HomK(κ(m)) M ⊗A κ(m), N ⊗A κ(m) = 0 for each maximal ideal m of A.
(3) HomK(A) (M, N ⊗A Q) = 0 for any finitely generated A-module Q.
b A B, N ⊗b A B)
Proof. By Proposition 4.1.17, we need only prove that if HomK(B) (M ⊗vanishes for all finitely generated A-algebras B, then it vanishes for all A-algebras B.
This is immediate from Proposition 4.1.14.
Corollary 4.1.19. Suppose that M and N are Breuil–Kisin modules with descent data and coefficients in a Noetherian
O/$a -algebra A, and that furthermore

HomK(A) M ⊗A κ(m), N ⊗A κ(m) vanishes for each maximal ideal m of A. Thenthe A-module Ext1K(A) (M, N) is projective of finite rank.
Proof. By Proposition 4.1.13, in order to prove that Ext1K(A) (M, N) is projectiveof finite rank over A, it suffices to prove that it is flat over A. For this, it suffices
to show that Q 7→ Ext1K(A) (M, N) ⊗A Q is exact when applied to finitely generated
A-modules Q. Proposition 4.1.15 (together with Remark 2.1.8 (1)) allows us to
identify this functor with the functor Q 7→ Ext1K(A) (M, N ⊗A Q). Note that thefunctor Q 7→ N ⊗A Q is an exact functor of Q, since SA is a flat A-module (as A is
Noetherian; see Remark 2.1.4(3)). Thus, taking into account Corollary 4.1.7, we seethat it suffices to show that HomK(A) (M, N ⊗A Q) = 0 for each finitely generated Amodule Q, under the hypothesis that HomK(A) M⊗A κ(m), N⊗A κ(m) = 0 for eachmaximal ideal m of A. This is the implication (2) =⇒ (3) of Proposition 4.1.17. MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

67

Lemma 4.1.20. Suppose that M is a Breuil–Kisin modules with descent data andcoefficients in a Noetherian O/$a -algebra A. Suppose that N is either a Breuil–Kisin
module with A-coefficients, or that N = N0 /uN N0 , where N0 a Breuil–Kisin modulewith A-coefficients and N ≥ 1. Then, if B is a finitely generated flat A-algebra, wehave a natural isomorphism∼b A B, N ⊗b A B) −→HomK(B) (M ⊗HomK(A) (M, N) ⊗A B.
Proof. By Corollary 4.1.8 and the flatness of B, we have a left exact sequence
0 → HomK(A) (M, N) ⊗A B → C 0 (N) ⊗A B → C 1 (N) ⊗A B
and therefore (applying Corollary 4.1.12 to treat the case that N is projective) aleft exact sequenceb A B) → C 1 (N ⊗b A B).
0 → HomK(A) (M, N) ⊗A B → C 0 (N ⊗The result follows from Corollary 4.1.8 and Lemma 4.1.9.

Lemma 4.1.21. Suppose that M is a Breuil–Kisin module with descent data andcoefficients in a Noetherian O/$a -algebra A which is furthermore a domain. Suppose
also that N is either a Breuil–Kisin module with A-coefficients, or that N = N0 /uN N0 ,where N0 is a Breuil–Kisin module with A-coefficients and N ≥ 1. Then there isb A Af andsome nonzero f ∈ A with the following property: writing MAf = M ⊗b A Af , then for any finitely generated Af -algebra B, and any finitely
NAf = N ⊗generated B-module Q, there are natural isomorphisms∼

b Af B, NAf ⊗b Af B) ⊗B Q
HomK(Af ) (MAf , NAf ) ⊗Af Q −→ HomK(B) (MAf ⊗∼b Af B, NAf ⊗b Af Q).
−→ HomK(B) (MAf ⊗Proof of Lemma 4.1.21. Note that since A is Noetherian, by Remark 2.1.4(3) we
see that N is A-flat. By Corollary 4.1.8 we have an exact sequence
0 → HomK(A) (M, N) → C 0 (N) → C 1 (N) → Ext1K(A) (M, N) → 0.
Since by assumption M is a projective SA -module, and N is a flat A-module, theC i (N) are also flat A-modules.
By Proposition 4.1.13, Ext1K(A) (M, N) is a finitely generated A-module, so by
the generic freeness theorem [Sta13, Tag 051R] there is some nonzero f ∈ A suchthat Ext1K(A) (M, N)f is free over Af .
Since localisation is exact, we obtain an exact sequence
0 → HomK(Af ) (M, N)f → C 0 (N)f → C 1 (N)f → Ext1K(A) (M, N)f → 0
and therefore (applying Corollary 4.1.12 to treat the case that N is a Breuil–Kisin
module) an exact sequence
0 → HomK(Af ) (MAf , NAf ) → C 0 (NAf ) → C 1 (NAf ) → Ext1K(A) (M, N)f → 0.
Since the last three terms are flat over Af , this sequence remains exact upon
tensoring over Af with Q. Applying Corollary 4.1.12 again to treat the case that N
is a Breuil–Kisin module, we see that in particular we have a left exact sequenceb Af Q) → C 1 (NAf ⊗b Af Q),
0 → HomK(Af ) (MAf , NAf ) ⊗Af Q → C 0 (NAf ⊗and Corollary 4.1.8 together with Lemma 4.1.9 yield one of the desired isomorphisms,
namely∼b Af B, NAf ⊗b Af Q).
HomK(Af ) (MAf , NAf ) ⊗Af Q −→ HomK(B) (MAf ⊗68

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

If we consider the case when Q = B, we obtain an isomorphism∼b Af B, NAf ⊗b Af B).
HomK(Af ) (MAf , NAf ) ⊗Af B −→ HomK(B) (MAf ⊗Rewriting the tensor product – ⊗Af Q as – ⊗Af B ⊗B Q, we then find that∼b Af B, NAf ⊗b Af B) ⊗B Q −→ HomK(B) (MAf ⊗b Af B, NAf ⊗b Af Q),
HomK(B) (MAf ⊗which gives the second desired isomorphism.

Variants on the preceding result may be proved using other versions of the genericfreeness theorem.
Example 4.1.22. Returning to the setting of Example 4.1.16, one can check using
Corollary 4.1.18 that the conclusion of Lemma 4.1.21 (for M = Mx and N = My )holds with f = x − y. In this case all of the resulting Hom groups vanish (cf. also theproof of Lemma 4.3.7). It then follows from Corollary 4.1.19 that Ext1K(A) (M, N)f
is projective over Af , so that the proof of Lemma 4.1.21 even goes through withthis choice of f .
As well as considering homomorphisms and extensions of Breuil–Kisin modules,
we need to consider the homomorphisms and extensions of their associated étale ϕmodules; recall that the passage to associated étale ϕ-modules amounts to invertingu, and so we briefly discuss this process in the general context of the category K(A).
We let K(A)[1/u] denote the full subcategory of K(A) consisting of objects on
which multiplication by u is invertible. We may equally well regard it as thecategory of left SA [1/u][F, Gal(K 0 /K)]-modules (this notation being interpreted in
the evident manner). There are natural isomorphisms (of bi-modules)(4.1.23)∼

SA [1/u] ⊗SA SA [F, Gal(K 0 /K)] −→ SA [1/u][F, Gal(K 0 /K)]

and(4.1.24)∼

SA [F, Gal(K 0 /K)] ⊗SA SA [1/u] −→ SA [1/u][F, Gal(K 0 /K)].

Thus (since SA → SA [1/u] is a flat morphism of commutative rings) the morphismof rings SA [F, Gal(K 0 /K)] → SA [1/u][F, Gal(K 0 /K)] is both left and right flat.
If M is an object of K(A), then we see from (4.1.23) that M[1/u] := SA [1/u] ⊗SA∼
M −→ SA [1/u][F, Gal(K 0 /K)]⊗SA [F,Gal(K 0 /K)] M is naturally an object of K(A)[1/u].
Our preceding remarks about flatness show that M 7→ M[1/u] is an exact functor
K(A) → K(A)[1/u].
Lemma 4.1.25.
(1) If M and N are objects of K(A)[1/u], then there is anatural isomorphism∼

ExtiK(A)[1/u] (M, N ) −→ ExtiK(A) (M, N ).
(2) If M is an object of K(A) and N is an object of K(A)[1/u], then there is anatural isomorphism∼

ExtiK(A) (M, N ) −→ ExtiK(A) (M[1/u], N ),for all i ≥ 0.
Proof. The morphism of (1) can be understood in various ways; for example, by
thinking in terms of Yoneda Exts, and recalling that K(A)[1/u] is a full subcategory of K(A). If instead we think in terms of projective resolutions, we can begin
with a projective resolution P• → M in K(A), and then consider the induced

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69∼

projective resolution P• [1/u] of M [1/u]. Noting that M [1/u] −→ M for any object∼
M of K(A)[1/u], we then find (via tensor adjunction) that HomK(A) (P• , N ) −→•HomK(A)[1/u] (P [1/u], N ), which induces the desired isomorphism of Ext’s by passing to cohomology.
Taking into account the isomorphism of (1), the claim of (2) is a general fact
about tensoring over a flat ring map (as can again be seen by considering projective
resolutions).

Remark 4.1.26. The preceding lemma is fact an automatic consequence of theabstract categorical properties of our situation: the functor M 7→ M[1/u] is left
adjoint to the inclusion K(A)[1/u] ⊂ K(A), and restricts to (a functor naturally
equivalent to) the identity functor on K(A)[1/u].
The following lemma expresses the Hom between étale ϕ-modules arising from
Breuil–Kisin modules in terms of a certain direct limit.
Lemma 4.1.27. Suppose that M is a Breuil–Kisin module with descent data ina Noetherian O/$a -algebra A, and that N is an object of K(A) which is finitely
generated and u-torsion free as an SA -module. Then there is a natural isomorphism∼lim HomK(A) (ui M, N) −→ HomK(A)[1/u] (M[1/u], N[1/u]),−→iwhere the transition maps are induced by the inclusions ui+1 M ⊂ ui M.
Remark 4.1.28. Note that since N is u-torsion free, the transition maps in thecolimit are injections, so the colimit is just an increasing union.
Proof. There are compatible injections HomK(A) (ui M, N) → HomK(A)[1/u] (M[1/u], N[1/u]),taking f 0 ∈ HomK(A) (ui M, N) to f ∈ HomK(A) (M, N[1/u]) where f (m) = u−i f 0 (ui m).
Conversely, given f ∈ HomK(A) (M, N[1/u]), there is some i such that f (M) ⊂ u−i N,
as required.

We have the following analogue of Proposition 4.1.17.
Corollary 4.1.29. Suppose that M and N are Breuil–Kisin modules with descentdata in a Noetherian O/$a -algebra A. Consider the following conditions:b A B)[1/u], (N ⊗b A B)[1/u] = 0 for any finite type A(1) HomK(B)[1/u] (M ⊗algebra B.
(2) HomK(κ(m))[1/u] (M⊗A κ(m))[1/u], (N⊗A κ(m))[1/u] = 0 for each maximal
ideal m of A.
(3) HomK(A)[1/u] M[1/u], (N ⊗A Q)[1/u] = 0 for any finitely generated Amodule Q.
Then we have (1) =⇒ (2) ⇐⇒ (3). If A is furthermore Jacobson, then all three
conditions are equivalent.
Proof. By Lemma 4.1.27, the three conditions are respectively equivalent to thefollowing conditions.
b A B), N ⊗b A B = 0 for any finite type A-algebra B and all(10 ) HomK(B) ui (M ⊗i ≥ 0.
(20 ) HomK(κ(m)) ui (M ⊗A κ(m)), N ⊗A κ(m) = 0 for each maximal ideal m of
A and all i ≥ 0.
(30 ) HomK(A) ui M, N ⊗A Q = 0 for any finitely generated A-module Q and alli ≥ 0.

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Since M is projective, the first two conditions are in turn equivalent tob A B, N ⊗b A B = 0 for any finite type A-algebra B and all(100 ) HomK(B) (ui M) ⊗i ≥ 0.
(200 ) HomK(κ(m)) (ui M) ⊗A κ(m), N ⊗A κ(m) = 0 for each maximal ideal m of
A and all i ≥ 0.
The result then follows from Proposition 4.1.17.

Definition 4.1.30. If M and N are objects of K(A), then we defineker-Ext1K(A) (M, N) := ker Ext1K(A) (M, N) → Ext1K(A) (M[1/u], N[1/u]) .
The point of this definition is to capture, in the setting of Lemma 2.3.3, thenon-split extensions of Breuil–Kisin modules whose underlying extension of Galois
representations is split.
Suppose now that M is a Breuil–Kisin module. The exact sequence in K(A)
0 → N → N[1/u] → N[1/u]/N → 0
gives an exact sequence of complexes0/ C 0 (N)

/ C 0 (N[1/u])/ C 0 (N[1/u]/N)

/0

0


/ C 1 (N)
/ C 1 (N[1/u])/ C 1 (N[1/u]/N)/ 0.

It follows from Corollary 4.1.8, Lemma 4.1.25(2), and the snake lemma that we havean exact sequence(4.1.31)0 → HomK(A) (M, N) → HomK(A) (M, N[1/u])→ HomK(A) (M, N[1/u]/N) → ker-Ext1K(A) (M, N) → 0.

Lemma 4.1.32. If M, N are Breuil–Kisin modules with descent data and coefficients in a Noetherian O/$a -algebra A, and N has height at most h, then f (M) is
killed by ui for any f ∈ HomK(A) (M, N[1/u]/N) and any i ≥ be0 ah/(p − 1)c.
Proof. Suppose that f is an element of HomK(A) (M, N[1/u]/N). Then f (M) is a
finitely generated submodule of N[1/u]/N, and it therefore killed by ui for somei ≥ 0. Choosing i to be the exponent of f (M) (that is, choosing i to be minimal), it
follows that (ϕ∗ f )(ϕ∗ M) has exponent precisely ip. (From the choice of i, we seethat ui−1 f (M) is nonzero but killed by u, i.e., it is just a W (k 0 ) ⊗ A-module, andso its pullback by ϕ : SA → SA has exponent precisely p. Then by the flatnessof ϕ : SA → SA we have uip−1 (ϕ∗ f )(ϕ∗ M) = up−1 ϕ∗ (ui−1 f (M)) 6= 0.)0
We claim that ui+e ah (ϕ∗ f )(ϕ∗ M) = 0; admitting this, we deduce that i + e0 ah ≥ip, as required. To see the claim, take x ∈ ϕ∗ M, so that ΦN ((ui ϕ∗ f )(x)) =ui f (ΦM (x)) = 0. It is therefore enough to show that the kernel ofΦN : ϕ∗ N[1/u]/ϕ∗ N → N[1/u]/N

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

71

0

is killed by ue ah ; but this follows immediately from an application of the snake
lemma to the commutative diagram0/ ϕ∗ NΦN

0
/N/ ϕ∗ N[1/u]ΦN/ ϕ∗ N[1/u]/ϕ∗ N

/0

ΦN


/ N[1/u]/ N[1/u]/N/0

together with the assumption that N has height at most h and an argument as in
the first line of the proof of Lemma 4.1.10.
Lemma 4.1.33. If M, N are Breuil–Kisin modules with descent data and coefficients in a Noetherian O/$a -algebra A, and N has height at most h, then for any
i ≥ be0 ah/(p − 1)c we have an exact sequence
0 → HomK(A) (ui M, ui N) → HomK(A) (ui M, N)→ HomK(A) (ui M, N/ui N) → ker-Ext1K(A) (M, N) → 0.
Proof. Comparing Lemma 4.1.32 with the proof of Lemma 4.1.27, we see thatthe direct limit in that proof has stabilised at i, and we obtain an isomorphism∼
HomK(A) (M, N[1/u]) → HomK(A) (ui M, N) sending a map f to f 0 : ui m 7→ ui f (m).
The same formula evidently identifies HomK(A) (M, N) with HomK(A) (ui M, ui N)
and HomK(A) (M, N[1/u]/N) with HomK(A) (ui M, N[1/u]/ui N). But any map in
the latter group has image contained in N/ui N (by Lemma 4.1.32 applied to
HomK(A) (M, N[1/u]/N), together with the identification in the previous sentence),
so that HomK(A) (ui M, N[1/u]/ui N) = HomK(A) (ui M, N/ui N).

Proposition 4.1.34. Let M and N be Breuil–Kisin modules with descent data andcoefficients in a Noetherian O/$a -domain A. Then there is some nonzero f ∈ A
b A Af and NAf = N ⊗b A Af , thenwith the following property: if we write MAf = M ⊗if B is any finitely generated Af -algebra, and if Q is any finitely generated B-module,
we have natural isomorphisms∼

b Af B, N ⊗b Af B) ⊗B Q
ker-Ext1K(Af ) (M, N) ⊗Af Q −→ ker-Ext1K(Af ) (MAf ⊗∼b Af B, N ⊗b Af Q).
−→ ker-Ext1K(Af ) (MAf ⊗Proof. In view of Lemma 4.1.33, this follows from Lemma 4.1.21, with M there
being our ui M, and N being each of N, N/ui N in turn.
The following result will be crucial in our investigation of the decomposition of
C dd,1 and Rdd,1 into irreducible components.
Proposition 4.1.35. Suppose that M and N are Breuil–Kisin modules with descent data and coefficients in a Noetherian O/$a -algebra A which is furthermore a domain, and suppose that HomK(A) M ⊗A κ(m), N ⊗A κ(m) vanishes
for each maximal ideal m of A. Then there is some nonzero f ∈ A with theb A Af and NAf = N ⊗b A Af , then forfollowing property: if we write MAf = M ⊗1bb Af B),any finitely generated Af -algebra B, each of ker-ExtK(B) (MAf ⊗Af B, NAf ⊗1b Af B, NAf ⊗b Af B), and
ExtK(B) (MAf ⊗b Af B, NAf ⊗b Af B)/ ker-Ext1K(A ) (MAf ⊗b Af B, NAf ⊗b Af B)
Ext1K(B) (MAf ⊗f
is a finitely generated projective B-module.

72

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTProof. Choose f as in Proposition 4.1.34, let B be a finitely generated Af -algebra,
and let Q be a finitely generated B-module. By Propositions 4.1.15 and 4.1.34, themorphismb Af B, NAf ⊗b Af B)⊗B Q → Ext1K(B) (MAf ⊗b Af B, NAf ⊗b Af B)⊗B Q
ker-Ext1K(B) (MAf ⊗is naturally identified with the morphismb Af B, NAf ⊗b Af Q) → Ext1K(B) (MAf ⊗b Af B, NAf ⊗b Af Q);
ker-Ext1K(B) (MAf ⊗in particular, it is injective. By Proposition 4.1.15 and Corollary 4.1.19 we see thatb Af B, NAf ⊗b Af B) is a finitely generated projective B-module; hence
Ext1K(B) (MAf ⊗it is also flat. Combining this with the injectivity just proved, we find thatb Af B, NAf ⊗b Af B)/ ker-Ext1K(B) (MAf ⊗b Af B, NAf ⊗b Af B) = 0
Tor1B Q, Ext1K(B) (M ⊗for every finitely generated B-module Q, and thus thatb Af B, NAf ⊗b Af B)/ ker-Ext1K(B) (MAf ⊗b Af B, NAf ⊗b Af B)
Ext1K(B) (MAf ⊗is a finitely generated flat, and therefore finitely generated projective, B-module.
b Af B, NAf ⊗b Af B) is a direct summand of the finitely genThus ker-Ext1K(B) (MAf ⊗1
b Af B, NAf ⊗b Af B), and so is itself aerated projective B-module ExtK(B) (MAf ⊗finitely generated projective B-module.

4.2. Families of extensions. Let M and N be Breuil–Kisin modules with descentdata and A-coefficients, so that Ext1K(A) (M, N) is an A-module. Suppose that
ψ : V → Ext1K(A) (M, N) is a homomorphism of A-modules whose source is a
projective A-module of finite rank. Then we may regard ψ as an element of
Ext1K(A) (M, N) ⊗A V ∨ = Ext1K(A) (M, N ⊗A V ∨ ),and in this way ψ corresponds to an extension(4.2.1)

0 → N ⊗A V ∨ → E → M → 0,

which we refer to as the family of extensions of M by N parametrised by V (orby ψ, if we want to emphasise our choice of homomorphism). We let Ev denotethe pushforward of E under the morphism N ⊗A V ∨ → N given by evaluation onv ∈ V . In the special case that Ext1K(A) (M, N) itself is a projective A-module offinite rank, we can let V be Ext1K(A) (M, N) and take ψ be the identity map; in thiscase we refer to (4.2.1) as the universal extension of M by N. The reason for thisterminology is as follows: if v ∈ Ext1K(A) (M, N), then Ev is the extension of M by
N corresponding to the element v.
Let B := A[V ∨ ] denote the symmetric algebra over A generated by V ∨ . The
short exact sequence (4.2.1) is a short exact sequence of Breuil–Kisin modules withdescent data, and so forming its u-adically completed tensor product with B overA, we obtain a short exact sequenceb AB → E ⊗b AB → M ⊗b AB → 0
0 → N ⊗A V ∨ ⊗of Breuil–Kisin modules with descent data over B (see Lemma 2.1.6). Pushing thisshort exact sequence forward under the natural map
b A B = V ∨ ⊗A B → B
V∨⊗MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

73

induced by the inclusion of V ∨ in B and the multiplication map B ⊗A B → B, we
obtain a short exact sequencee → M⊗b AB → E
b AB → 0(4.2.2)0 → N⊗of Breuil–Kisin modules with descent data over B, which we call the family ofextensions of M by N parametrised by Spec B (which we note is (the total space of)the vector bundle over Spec A corresponding to the projective A-module V ).
If αv : B → A is the morphism induced by the evaluation map V ∨ → A given bysome element v ∈ V , then base-changing (4.2.2) by αv , we recover the short exact
sequence
0 → N → Ev → M → 0.
More generally, suppose that A is a O/$a -algebra for some a ≥ 1, and let C be any
A-algebra. Suppose that αṽ : B → C is the morphism induced by the evaluation map
V ∨ → C corresponding to some element ṽ ∈ C ⊗A V . Then base-changing (4.2.2)
by αṽ yields a short exact sequencee⊗b AC → E
b BC → M ⊗b A C → 0,
0 → N⊗
whose associated extension class corresponds to the image of ṽ under the naturalb A C, N ⊗A C), themorphism C ⊗A V → C ⊗A Ext1K(A) (M, N) ∼= Ext1K(C) (M ⊗first arrow being induced by ψ and the second arrow being the isomorphism of
Proposition 4.1.15.
4.2.3. The functor represented by a universal family. We now suppose that thering A and the Breuil–Kisin modules M and N have the following properties:
Assumption 4.2.4. Let A be a Noetherian and Jacobson O/$a -algebra for somea ≥ 1, and assume that for each maximal ideal m of A, we have that
HomK(κ(m)) M ⊗A κ(m), N ⊗A κ(m) = HomK(κ(m)) N ⊗A κ(m), M ⊗A κ(m) = 0.
By Corollary 4.1.19, this assumption implies in particular that V := Ext1K(A) (M, N)
is projective of finite rank, and so we may form Spec B := Spec A[V ∨ ], whichparametrised the universal family of extensions. We are then able to give thefollowing precise description of the functor represented by Spec B.
Proposition 4.2.5. The scheme Spec B represents the functor which, to any O/$a algebra C, associates the set of isomorphism classes of tuples (α, E, ι, π), where α isa morphism α : Spec C → Spec A, E is a Breuil–Kisin module with descent data andcoefficients in C, and ι and π are morphisms α∗ N → E and E → α∗ M respectively,ιπwith the property that 0 → α∗ N → E → α∗ M → 0 is short exact.
Proof. We have already seen that giving a morphism Spec C → Spec B is equivalent
to giving the composite morphism α : Spec C → Spec B → Spec A, together withan extension class [E] ∈ Ext1K(C) (α∗ M, α∗ N). Thus to prove the proposition, we
just have to show that any automorphism of E which restricts to the identity onα∗ N and induces the identity on α∗ M is itself the identity on E. This follows from
Corollary 4.1.18, together with Assumption 4.2.4.

Fix an integer h ≥ 0 so that E(u)h ∈ AnnSA (coker ΦM ) AnnSA (coker ΦN ), sothat by Lemma 4.1.1, every Breuil–Kisin module parametrised by Spec B has heightat most h. There is a natural action of Gm ×O Gm on Spec B, given by rescaling eachof ι and π. There is also an evident forgetful morphism Spec B → Spec A ×O C dd,a ,74

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

given by forgetting ι and π, which is evidently invariant under the Gm ×O Gm -action.
(Here and below, C dd,a denotes the moduli stack defined in Section 3.9 for our fixed
choice of h and for d equal to the sum of the ranks of M and N.) We thus obtain amorphism(4.2.6)

Spec B ×O Gm ×O Gm → Spec B ×Spec A×O C dd,a Spec B.

Corollary 4.2.7. Suppose that AutK(C) (α∗ M) = AutK(C) (α∗ N) = C × for any
morphism α : Spec C → Spec A. Then the morphism (4.2.6) is an isomorphism, andconsequently the induced morphism[Spec B/Gm ×O Gm ] → Spec A ×O C dd,a
is a finite type monomorphism.
Proof. By Proposition 4.2.5, a morphism
Spec C → Spec B ×Spec A×O C dd,a Spec B
corresponds to an isomorphism class of tuples (α, β : E → E0 , ι, ι0 , π, π 0 ), where• α is a morphism α : Spec C → Spec A,• β : E → E0 is an isomorphism of Breuil–Kisin modules with descent dataand coefficients in C,• ι : α∗ N → E and π : E → α∗ M are morphisms with the property thatιπ

0 → α∗ N → E → α∗ M → 0
is short exact,• ι0 : α∗ N → E0 and π 0 : E0 → α∗ M are morphisms with the property thatπ0

ι00 → α∗ N → E0 → α∗ M → 0
is short exact.
Assumption 4.2.4 and Corollary 4.1.18 together show that HomK(C) (α∗ N, α∗ M) = 0.
ιβIt follows that the composite α∗ N → E → E0 factors through ι0 , and the induced
endomorphism of α∗ N is injective. Reversing the roles of E and E0 , we see that it isin fact an automorphism of α∗ N, and it follows easily that β also induces an automorphism of α∗ M. Again, Assumption 4.2.4 and Proposition 4.1.18 together showthat HomK(C) (α∗ M, α∗ N) = 0, from which it follows easily that β is determined by
the automorphisms of α∗ M and α∗ N that it induces.
Since AutK(C) (α∗ M) = AutK(C) (α∗ N) = C × by assumption, we see that β ◦ ι, ι0and π, π 0 ◦ β differ only by the action of Gm ×O Gm , so the first claim of thecorollary follows. The claim regarding the monomorphism is immediate from
Lemma 4.2.8 below. Finally, note that [Spec B/Gm ×O Gm ] is of finite typeover Spec A, while C dd,a has finite type diagonal. It follows that the morphism[Spec B/Gm ×O Gm ] → Spec A ×O C dd,a is of finite type, as required.

Lemma 4.2.8. Let X be a scheme over a base scheme S, let G be a smooth affinegroup scheme over S, and let ρ : X ×S G → X be a (right) action of G on X. Let
X → Y be a G-equivariant morphism, whose target is an algebraic stack over S on
which G acts trivially. Then the induced morphism[X/G] → Y
is a monomorphism if and only if the natural morphism
X ×S G → X ×Y X

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

75(induced by the morphisms pr1 , ρ : X ×S G → X) is an isomorphism.
Proof. We have a Cartesian diagram as follows.
/ X ×Y X
X ×S G[X/G]/ [X/G] ×Y [X/G]

The morphism [X/G] → Y is a monomorphism if and only if the bottom horizontal
morphism of this square is an isomorphism; since the right hand vertical arrow is asmooth surjection, this is the case if and only if the top horizontal morphism is an
isomorphism, as required.

4.3. Families of extensions of rank one Breuil–Kisin modules. In this section we construct universal families of extensions of rank one Breuil–Kisin modules.
We will use these rank two families to study our moduli spaces of Breuil–Kisin modules, and the corresponding spaces of étale ϕ-modules. We show how to compute thedimensions of these universal families; in the subsequent sections, we will combine
these results with explicit calculations to determine the irreducible components ofour moduli spaces. In particular, we will show that each irreducible component hasa dense open substack given by a family of extensions.
4.3.1. Universal unramified twists. Fix a free Breuil–Kisin module with descent data
M over F, and write Φi for ΦM,i : ϕ∗ (Mi−1 ) → Mi . (Here we are using the notationof Section 2.1, so that Mi = ei M is cut out by the idempotent ei of Section 1.7.)
We will construct the “universal unramified twist” of M.
Definition 4.3.2. If Λ is an F-algebra, and if λ ∈ Λ× , then we define MΛ,λto be the free Breuil–Kisin module with descent data and Λ-coefficients whoseb F Λ (so the usual base change
underlying SΛ [Gal(K 0 /K)]-module is equal to M ⊗∗of M to Λ), and for which ΦMΛ,λ : ϕ MΛ,λ → MΛ,λ is defined via the f 0 -tuple(λΦ0 , Φ1 , . . . , Φf 0 −1 ). We refer to MΛ,λ as the unramified twist of M by λ over Λ.
If M is a free étale ϕ-module with descent data, then we define MΛ,λ in theanalogous fashion. If we write X = Spec Λ, then we will sometimes write MX,λ(resp. MX,λ ) for MΛ,λ (resp. MΛ,λ ).
As usual, we write Gm := Spec F[x, x−1 ]. We may then form the rank one
Breuil–Kisin module with descent data MGm ,x , which is the universal instance of anunramified twist: given λ ∈ Λ× , there is a corresponding morphism Spec Λ → Gm×determined by the requirement that x ∈ Γ(Gm , OG
) pulls-back to λ, and MX,λ ismobtained by pulling back MGm ,x under this morphism (that is, by base changing
under the corresponding ring homomorphism F[x, x−1 ] → Λ).
Lemma 4.3.3. If MΛ is a Breuil–Kisin module of rank one with Λ-coefficients,
then EndK(Λ) (M) = Λ. Similarly, if MΛ is a étale ϕ-module of rank one withΛ-coefficients, then EndK(Λ) (MΛ ) = Λ.
Proof. We give the proof for MΛ , the argument for MΛ being essentially identical.
One reduces easily to the case where MΛ is free. Since an endomorphism ψ of MΛ isin particular an endomorphism of the underlying SΛ [1/u]-module, we see that there
is some λ ∈ SΛ [1/u] such that ψ is given by multiplication by λ. The commutation
relation with ΦMΛ means that we must have ϕ(λ) = λ, so that certainly (considering

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

the powers of u in λ of lowest negative and positive degrees) λ ∈ W (k 0 ) ⊗Zp Λ,and in fact λ ∈ Λ. Conversely, multiplication by any element of Λ is evidently an
endomorphism of MΛ , as required.

Lemma 4.3.4. Let κ be a field of characteristic p, and let Mκ , Nκ be étale ϕmodules of rank one with κ-coefficients and descent data. Then any nonzero elementof HomK(κ) (Mκ , Nκ ) is an isomorphism.
Proof. Since κ((u)) is a field, it is enough to show that if one of the induced maps
Mκ,i → Nκ,i is nonzero, then they all are; but this follows from the commutation
relation with ϕ.
Lemma 4.3.5. If λ, λ0 ∈ Λ× and MΛ,λ ∼= MΛ,λ0 (as Breuil–Kisin modules withdescent data over Λ), then λ = λ0 . Similarly, if MΛ,λ ∼= MΛ,λ0 , then λ = λ0 .
Proof. Again, we give the proof for M , the argument for M being essentially identical.
Write Mi = F((u))mi , and write Φi (1 ⊗ mi−1 ) = θi mi , where θi =6 0. There areµi ∈ Λ[[u]][1/u] such that the given isomorphism MΛ,λ ∼= MΛ,λ0 takes mi to µi mi .
The commutation relation between the given isomorphism and ΦM imposes thecondition
λi µi θi mi = λ0i ϕ(µi−1 )θi mi
where λi (resp. λ0i ) equals 1 unless i = 0, when it equals λ (resp. λ0 ).
0Thus we have µi = (λ0i /λi )ϕ(µi−1 ), so that in particular µ0 = (λ0 /λ)ϕf (µ0 ).
Considering the powers of u in µ0 of lowest negative and positive degrees we conclude
0that µ0 ∈ W (k 0 ) ⊗ Λ; but then µ0 = ϕf (µ0 ), so that λ0 = λ, as required.

Remark 4.3.6. If M has height at most h, and we let C (temporarily) denote themoduli stack of rank one Breuil–Kisin modules of height at most h with F-coefficientsand descent data then Lemma 4.3.5 can be interpreted as saying that the morphism
Gm → C that classifies MGm ,x is a monomorphism, i.e. the diagonal morphism
Gm → Gm ×C Gm is an isomorphism. Similarly, the morphism Gm → R (where
we temporarily let R denote the moduli stack of rank one étale ϕ-modules with
F-coefficients and descent data) that classifies MGm ,x is a monomorphism.
Now choose another rank one Breuil–Kisin module with descent data N over F.
Let (x, y) denote the standard coordinates on Gm ×F Gm , and consider the rank
one Breuil–Kisin modules with descent data MGm ×F Gm ,x and NGm ×F Gm ,y over
Gm ×F Gm .
Lemma 4.3.7. There is a non-empty irreducible affine open subset Spec Adist of
Gm ×F Gm whose finite type points are exactly the maximal ideals m of Gm ×F Gmsuch that
HomK(κ(m)) Mκ(m),x̄ [1/u], Nκ(m),ȳ [1/u] = 0(where we have written x̄ and ȳ to denote the images of x and y in κ(m)× ).
Furthermore, if R is any finite-type Adist -algebra, and if m is any maximal idealof R, then

HomK(κ(m)) Mκ(m),x̄ , Nκ(m),ȳ = HomK(κ(m)) Mκ(m),x̄ [1/u], Nκ(m),ȳ [1/u] = 0,
and also
HomK(κ(m)) Nκ(m),ȳ , Mκ(m),x̄ = HomK(κ(m)) Nκ(m),ȳ [1/u], Mκ(m),x̄ [1/u] = 0.
In particular, Assumption 4.2.4 is satisfied by MAdist ,x and NAdist ,y .

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77

Proof. If Hom Mκ(m),x̄ [1/u], Nκ(m),ȳ [1/u]) = 0 for all maximal ideals m of F[x, y, x−1 , y −1 ],then we are done: Spec Adist = Gm × Gm . Otherwise, we see that for some finite
extension F0 /F and some a, a0 ∈ F0 , we have a non-zero morphism MF0 ,a [1/u] →NF0 ,a0 [1/u]. By Lemma 4.3.4, this morphism must in fact be an isomorphism. Since
M and N are both defined over F, we furthermore see that the ratio a0 /a lies in F.
We then let Spec Adist be the affine open subset of Gm ×F Gm where a0 x =6 ay; theclaimed property of Spec Adist then follows easily from Lemma 4.3.5.
For the remaining statements of the lemma, note that if m is a maximal ideal ina finite type Adist -algebra, then its pull-back to Adist is again a maximal ideal m0 ofAdist (since Adist is Jacobson), and the vanishing of
HomK(κ(m)) Mκ(m),x̄ [1/u], Nκ(m),ȳ [1/u]follows from the corresponding statement for κ(m0 ), together with Lemma 4.1.20.
Inverting u induces an embedding
HomK(κ(m)) Mκ(m),x̄ , Nκ(m),ȳ ,→ HomK(κ(m)) Mκ(m),x̄ [1/u], Nκ(m),ȳ [1/u] ,and so certainly the vanishing of the target implies the vanishing of the source.
The statements in which the roles of M and N are reversed follow from Lemma 4.3.4.
Define T := Ext1K(Gm ×F Gm ) MGm ×F Gm ,x , MGm ×F Gm ,y ); it follows from Proposition 4.1.13 that T is finitely generated over F[x, x−1 , y, y −1 ], while Proposition 4.1.15 shows that TAdist := T ⊗F[x±1 ,y±1 ] Adist is naturally isomorphic to
Ext1K(Adist ) MAdist ,x , NAdist ,y . (Here and elsewhere we abuse notation by writing x,
y for x|Adist , y|Adist .) Corollary 4.1.19 and Lemma 4.3.7 show that TAdist is in facta finitely generated projective Adist -module. If, for any Adist -algebra B, we write∼
TB := TAdist ⊗Adist B −→ T ⊗F[x±1 ,y±1 ] B, then Proposition 4.1.15 again shows that∼
TB −→ Ext1K(B) MB,x , NB,y .
By Propositions 4.1.34 and 4.1.35, together with Lemma 4.3.7, there is a nonempty(so dense) affine open subset Spec Ak-free of Spec Adist with the properties that
UAk-free := ker-Ext1K(Ak-free ) (MAk-free ,x , NAk-free ,y )andTAk-free /UAk-free∼−→ Ext1K(Ak-free ) (MAk-free ,x , NAk-free ,y )/ ker-Ext1K(Ak-free ) (MAk-free ,x , NAk-free ,y )are finitely generated and projective over Ak-free , and furthermore so that for all
finitely generated Ak-free -algebras B, the formation of ker-Ext1K(B) (MB,x , NB,y ) and
Ext1K(B) (MB,x , NB,y )/ ker-Ext1K(B) (MB,x , NB,y ) is compatible with base change
from UAk-free and TAk-free /UAk-free respectively.
We choose a finite rank projective module V over F[x, x−1 , y, y −1 ] admitting asurjection V → T . Thus, if we write VAdist := V ⊗F[x±1 ,y±1 ] Adist , then the induced
morphism VAdist → TAdist is a (split) surjection of Adist -modules.
Following the prescription of Subsection 4.2, we form the symmetric algee overbra B twist := F[x±1 , y ±1 ][V ∨ ], and construct the family of extensions E
Spec B twist . We may similarly form the symmetric algebras B dist := Adist [TA∨dist ] ande dist and Ee k-freeB k-free := Ak-free [TA∨k-free ], and construct the families of extensions E
dist
k-freeover Spec B
and Spec B
respectively. Since TAk-free /UAk-free is projective, the78

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

natural morphism TA∨k-free → UA∨k-free is surjective, and hence C k-free := A[UA∨k-free ] isa quotient of B k-free ; geometrically, Spec C k-free is a subbundle of the vector bundle
Spec B k-free over Spec A.
We write X := Spec B k-free \ Spec C k-free ; it is an open subscheme of the vector
e 0 to X is the universal family of extensions
bundle Spec B k-free . The restriction of E
over A which do not split after inverting u.
Remark 4.3.8. Since Spec Adist and Spec Ak-free are irreducible, each of the vector
bundles Spec B dist and Spec B k-free is also irreducible. In particular, Spec B k-free is
Zariski dense in Spec B dist , and if X is non-empty, then it is Zariski dense in eachof Spec B k-free and Spec B dist . Similarly, Spec B twist ×Gm ×F Gm Spec Adist is Zariski
dense in Spec B twist .
The surjection VAdist → TAdist induces a surjection of vector bundles π : Spec B twist ×Gm ×F Gm
Spec Adist → Spec B dist over Spec Adist , and there is a natural isomorphism∼ ee dist −→b F[x±1 ,y±1 ] Adist .
(4.3.9)π∗ E
E⊗
e is classified by a morphism
The rank two Breuil–Kisin module with descent data E
twist
dd,1
ξ : Spec B→C; similarly, the rank two Breuil–Kisin module with descente dist is classified by a morphism ξ dist : Spec B dist → C dd,1 . If we write ξAdist fordata Ethe restriction of ξ to the open subset Spec B twist ×Gm ×F Gm Spec Adist of Spec B twist ,then the isomorphism (4.3.9) shows that ξ dist ◦ π = ξAdist . We also write ξ k-free forthe restriction of ξ dist to Spec B k-free , and ξX for the restriction of ξ k-free to X.
Lemma 4.3.10. The scheme-theoretic images (in the sense of [EG19b, Def. 3.1.4])of ξ : Spec B twist → C dd,1 , ξ dist : Spec B dist → C dd,1 , and ξ k-free : Spec B k-free →C dd,1 all coincide; in particular, the scheme-theoretic image of ξ is independentof the choice of surjection V → T , and the scheme-theoretic image of ξ k-free isindependent of the choice of Ak-free . If X is non-empty, then the scheme-theoretic
image of ξX : X → C dd,1 also coincides with these other scheme-theoretic images,
and is independent of the choice of Ak-free .
Proof. This follows from the various observations about Zariski density made in
Remark 4.3.8.
Definition 4.3.11. We let C(M, N) denote the scheme-theoretic image of ξ dist :Spec B dist → C dd,1 , and we let Z(M, N) denote the scheme-theoretic image of thecomposite ξ dist : Spec B dist → C dd,1 → Z dd,1 . Equivalently, Z(M, N) is the schemetheoretic image of the composite Spec B dist → C dd,1 → Rdd,1 (cf. [EG19b, Prop.
3.2.31]), and the scheme-theoretic image of C(M, N) under the morphism C dd,1 →Z dd,1 . (Note that Lemma 4.3.10 provides various other alternative descriptions ofC(M, N) (and therefore also Z(M, N)) as a scheme-theoretic image.)
Remark 4.3.12. Note that C(M, N) and Z(M, N) are both reduced (because they
are each defined as a scheme-theoretic image of Spec B dist , which is reduced bydefinition).
As well as scheme-theoretic images, as in the preceding Lemma and Definition,
we will need to consider images of underlying topological spaces.
Lemma 4.3.13. The image of the morphism on underlying topological spaces| Spec B twist | → |C dd,1 | induced by ξ is a constructible subset of |C dd,1 |, and isindependent of the choice of V .

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79

Proof. The fact that the image of | Spec B twist | is a constructible subset of |C dd,1 |follows from the fact that ξ is a morphism of finite presentation between Noetherian
stacks; see [Ryd11, App. D]. Suppose now that V 0 is another choice of finite rank
projective F[x, x−1 , y, y −1 ]-module surjecting onto T . Choose a finite rank projective
module surjecting onto each of V and V 0 , compatible with the given surjections ofeach these sheaves onto T . (E.g. one could take W = V ⊕ V 0 .) Thus it suffices toprove the independence claim of the lemma in the case when V 0 admits a surjection
onto V (compatible with the maps of each of V and V 0 onto T ). If we write
B 0 := F[x±1 , y ±1 ][(V 0 )∨ ], then the natural morphism Spec B 0 → Spec B twist is asurjection, and the morphism ξ 0 : Spec B 0 → C dd,1 is the composite of this surjection
with the morphism ξ. Thus indeed the images of | Spec B 0 | and of | Spec B twist |coincide as subsets of |C dd,1 |.
Definition 4.3.14. We write |C(M, N)| to denote the constructible subset of |C dd,1 |described in Lemma 4.3.13.
Remark 4.3.15. We caution the reader that we don’t define a substack C(M, N)
of C dd,1 . Rather, we have defined a closed substack C(M, N) of C dd,1 , and aconstructible subset |C(M, N)| of |C dd,1 |. It follows from the constructions that
|C(M, N)| is the closure in |C dd,1 | of |C(M, N)|.
As in Subsection 4.2, there is a natural action of Gm ×F Gm on T , and hence oneach of Spec B dist , Spec B k-free and X, given by the action of Gm as automorphisms
on each of MGm ×F Gm ,x and NGm ×F Gm ,y (which induces a corresponding action on
T , hence on TAdist and TAk-free , and hence on Spec B dist and Spec B k-free ). Thus wemay form the corresponding quotient stacks [Spec B dist /Gm ×F Gm ] and [X/Gm ×F
Gm ], each of which admits a natural morphism to C dd,1 .
Remark 4.3.16. Note that we are making use of two independent copies of Gm ×F Gm ;one parameterises the different unramified twists of M and N, and the other the
automorphisms of (the pullbacks of) M and N.
Definition 4.3.17. We say that the pair (M, N) is strict if Spec Adist = Gm ×F Gm .
Before stating and proving the main result of this subsection, we prove somelemmas (the first two of which amount to recollections of standard — and simple —facts).
Lemma 4.3.18. If X → Y is a morphism of stacks over S, with X algebraic and offinite type over S, and Y having diagonal which is representable by algebraic spaces
and of finite type, then X ×Y X is an algebraic stack of finite type over S.
Proof. The fact that X ×Y X is an algebraic stack follows from [Sta13, Tag 04TF].
Since composites of morphisms of finite type are of finite type, in order to showthat X ×Y X is of finite type over S, it suffices to show that the natural morphism
X ×Y X → X ×S X is of finite type. Since this morphism is the base-change of the
diagonal morphism Y → Y ×S Y, this follows by assumption.

Lemma 4.3.19. The following conditions
are equivalent:(1) ker-Ext1K(κ(m)) Mκ(m),x̄ , Nκ(m),ȳ = 0 for all maximal ideals m of Ak-free .
(2) UAk-free = 0.
(3) Spec C k-free is the trivial vector bundle over Spec Ak-free .

80

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTProof. Conditions (2) and (3) are equivalent by definition. Since the formationof ker-Ext1K(Ak-free ) (MAk-free ,x , NAk-free ,y ) is compatible with base change, and since
Ak-free is Jacobson, (1) is equivalent to the assumption that
ker-Ext1K(Ak-free ) (MAk-free ,x , NAk-free ,y ) = 0,i.e. that UAk-free = 0, as required.



Lemma 4.3.20. If the equivalent conditions of Lemma 4.3.19 hold, then the naturalmorphism
Spec B k-free ×Spec Ak-free ×F C dd,1 Spec B k-free→ Spec B k-free ×Spec Ak-free ×F Rdd,1 Spec B k-free
is an isomorphism.
Proof. Since C dd,1 → Rdd,1 is separated (being proper) and representable, thediagonal morphism C dd,1 → C dd,1 ×Rdd,1 C dd,1 is a closed immersion, and hence themorphism in the statement of the lemma is a closed immersion. Thus, in order toshow that it is an isomorphism, it suffices to show that it induces a surjection on
R-valued points, for any F-algebra R. Since the source and target are of finite typeover F, by Lemma 4.3.18, we may in fact restrict attention to finite type R-algebras.
A morphism Spec R → Spec B k-free ×Spec Ak-free ×F C dd,1 Spec B k-free corresponds
to an isomorphism class of tuples (α, β : E → E0 , ι, ι0 , π, π 0 ), where• α is a morphism α : Spec R → Spec Ak-free ,• β : E → E0 is an isomorphism of Breuil–Kisin modules with descent dataand coefficients in R,• ι : α∗ N → E, ι0 : α∗ N → E0 , π : E → α∗ M and π 0 : E0 → α∗ M areιπmorphisms with the properties that 0 → α∗ N → E → α∗ M → 0 and
ι0

π0

ι0

π0

0 → α∗ N → E0 → α∗ M → 0 are both short exact.
Similarly, a morphism Spec R → Spec B k-free ×Spec Ak-free ×F Rdd,1 Spec B k-free corresponds to an isomorphism class of tuples (α, E, E0 , β, ι, ι0 , π, π 0 ), where• α is a morphism α : Spec R → Spec Ak-free ,• E and E0 are Breuil–Kisin modules with descent data and coefficients in
R, and β is an isomorphism β : E[1/u] → E0 [1/u] of etale ϕ-modules withdescent data and coefficients in R,• ι : α∗ N → E, ι0 : α∗ N → E0 , π : E → α∗ M and π 0 : E0 → α∗ M areιπmorphisms with the properties that 0 → α∗ N → E → α∗ M → 0 and
0 → α∗ N → E0 → α∗ M → 0 are both short exact.
Thus to prove the claimed surjectivity, we have to show that, given a tuple(α, E, E0 , β, ι, ι0 , π, π 0 ) associated to a morphism Spec R → Spec B k-free ×Spec Ak-free ×F Rdd,1
Spec B k-free , the isomorphism β restricts to an isomorphism E → E0 .
By Lemma 4.3.19, the natural map Ext1 (α∗ M, α∗ N) → Ext1K(R) (α∗ M[1/u], α∗ N[1/u])
is injective; so the Breuil–Kisin modules E and E0 are isomorphic. Arguing as in
the proof of Corollary 4.2.7, we see that β is equivalent to the data of an R-pointof Gm ×O Gm , corresponding to the automorphisms of α∗ M[1/u] and α∗ N[1/u]that it induces. These restrict to automorphisms of α∗ M and α∗ N, so that (again
by the proof of Corollary 4.2.7) β indeed restricts to an isomorphism E → E0 , asrequired.
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We now present the main result of this subsection.
Proposition 4.3.21. (1) The morphism ξ dist induces a morphism(4.3.22)[Spec B dist /Gm ×F Gm ] → C dd,1 ,which is representable by algebraic spaces, of finite type, and unramified, whose fibres
over finite type points are of degree ≤ 2. In the strict case, this induced morphism
is in fact a monomorphism, while in general, the restriction ξX of ξ dist induces afinite type monomorphism[X/Gm ×F Gm ] ,→ C dd,1 .
(2) If ker-Ext1K(κ(m)) Mκ(m),x̄ , Nκ(m),ȳ = 0 for all maximal ideals m of Ak-free ,then the composite morphism(4.3.23)(4.3.24)[Spec B k-free /Gm ×F Gm ] → C dd,1 → Rdd,1

is a representable by algebraic spaces, of finite type, and unramified, with fibres ofdegree ≤ 2. In the strict case, this induced morphism is in fact a monomorphism,
while in general, the composite morphism(4.3.25)[X/Gm ×F Gm ] ,→ C dd,1 → Rdd,1

is a finite type monomorphism.
Remark 4.3.26. The failure of (4.3.22) to be a monomorphism in general is due,effectively, to the possibility that an extension E of some MR,x by NR,y and an
extension E0 of some MR,x0 by NR,y0 might be isomorphic as Breuil–Kisin modules
while nevertheless (x, y) 6= (x0 , y 0 ). As we will see in the proof, whenever thishappens the map NΛ,y → E → E0 → MΛ,x0 is nonzero, and then E0 ⊗R κ(m)[1/u] issplit for some maximal ideal m of R. This explains why, to obtain a monomorphism,
we can restrict either to the strict case or to the substack of extensions that are
non-split after inverting u.
Remark 4.3.27. We have stated this proposition in the strongest form that weare able to prove, but in fact its full strength is not required in the subsequent
applications. In particular, we don’t need the precise bounds on the degrees of thefibres.
Proof of Proposition 4.3.21. By Corollary 4.2.7 (which we can apply because Assumption 4.2.4 is satisfied, by Lemma 4.3.7) the natural morphism [Spec B dist /Gm ×F
Gm ] → Spec Adist ×F C dd,1 is a finite type monomorphism, and hence so is its restriction to the open substack [X/Gm ×F Gm ] of its source.
Let us momentarily write X to denote either [Spec B dist /Gm ×F Gm ] or [X/Gm ×F
Gm ]. To show that the finite type morphism X → C dd,1 is representable byalgebraic spaces, resp. unramified, resp. a monomorphism, it suffices to show thatthe corresponding diagonal morphism X → X ×C dd,1 X is a monomorphism, resp.
étale, resp. an isomorphism.
Now since X → Spec Adist ×F C dd,1 is a monomorphism, the diagonal morphism
X → X ×Spec Adist ×F C dd,1 X is an isomorphism, and so it is equivalent to show thatthe morphism of products
X ×Spec Adist ×F C dd,1 X → X ×C dd,1 X

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

is a monomorphism, resp. étale, resp. an isomorphism. This is in turn equivalent toshowing the corresponding properties for the morphisms(4.3.28)

Spec B dist ×Spec Adist ×C dd,1 Spec B dist → Spec B dist ×C dd,1 Spec B distor
X ×Spec Adist ×C dd,1 X → X ×C dd,1 X.

(4.3.29)

Now each of these morphisms is a base-change of the diagonal Spec Adist →Spec Adist ×F Spec Adist , which is a closed immersion (affine schemes being separated), and so is itself a closed immersion. In particular, it is a monomorphism,and so we have proved the representability by algebraic spaces of each of (4.3.22)
and (4.3.23). Since the source and target of each of these monomorphisms is of finite
type over F, by Lemma 4.3.18, in order to show that either of these monomorphisms
is an isomorphism, it suffices to show that it induces a surjection on R-valued
points, for arbitrary finite type F-algebras R. Similarly, to check that the closed
immersion (4.3.28) is étale, it suffices to verify that it is formally smooth, and forthis it suffices to verify that it satisfies the infinitesimal lifting property with respect
to square zero thickenings of finite type F-algebras.
A morphism Spec R → Spec B dist ×C dd,1 Spec B dist corresponds to an isomorphismclass of tuples (α, α0 , β : E → E0 , ι, ι0 , π, π 0 ), where• α, α0 are morphisms α, α0 : Spec R → Spec Adist ,• β : E → E0 is an isomorphism of Breuil–Kisin modules with descent dataand coefficients in R,• ι : α∗ N → E, ι0 : (α0 )∗ N → E0 , π : E → α∗ M and π 0 : E0 → (α0 )∗ Mιπ
are morphisms with the properties that 0 → α∗ N → E → α∗ M → 0 andι0π0

0 → (α0 )∗ N → E0 → (α0 )∗ M → 0 are both short exact.
We begin by proving that (4.3.28) satisfies the infinitesimal lifting criterion (when
R is a finite type F-algebra). Thus we assume given a square-zero ideal I ⊂ R, suchthat the induced morphism
Spec R/I → Spec B dist ×C dd,1 Spec B dist
factors through Spec B dist ×Spec Adist ×F C dd,1 Spec B dist . In terms of the data (α, α0 , β :E → E0 , ι, ι0 , π, π 0 ), we are assuming that α and α0 coincide when restricted
to Spec R/I, and we must show that α and α0 themselves coincide.
To this end, we consider the composite(4.3.30)ιβπ0α∗ N → E → E0 → (α0 )∗ M.

If we can show the vanishing of this morphism, then by reversing the roles of E and
E0 , we will similarly deduce the vanishing of π ◦ β −1 ◦ ι0 , from which we can conclude
that β induces an isomorphism between α∗ N and (α0 )∗ N. Consequently, it also
induces an isomorphism between α∗ M and (α0 )∗ M, so it follows from Lemma 4.3.5
that α = α0 , as required.
We show the vanishing of (4.3.30). Suppose to the contrary that it doesn’t
vanish, so that we have a non-zero morphism α∗ N → (α0 )∗ M. It follows from
Proposition 4.1.17 that, for some maximal ideal m of R, there exists a non-zeromorphismα∗ (N) ⊗R κ(m)→(α0 )∗ (M) ⊗R κ(m).

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By assumption α and α0 coincide modulo I. Since I 2 = 0, there is an inclusionI ⊂ m, and so in particular we find that∼(α0 )∗ (M) ⊗R κ(m) −→ α∗ (M) ⊗R κ(m).
Thus there exists a non-zero morphismα∗ (N) ⊗R κ(m)→α∗ (M) ⊗R κ(m).
Then, by Lemma 4.3.4, after inverting u we obtain an isomorphism∼α∗ (N) ⊗R κ(m)[1/u]−→α∗ (M) ⊗R κ(m)[1/u],
contradicting the assumption that α maps Spec R into Spec Adist . This completes
the proof that (4.3.28) is formally smooth, and hence that (4.3.22) is unramified.
We next show that, in the strict case, the closed immersion (4.3.28) is an
isomorphism, and thus that (4.3.22) is actually a monomorphism. As noted above, itsuffices to show that (4.3.28) induces a surjection on R-valued points for finite typeF-algebras R, which in terms of the data (α, α0 , β : E → E0 , ι, ι0 , π, π 0 ), amounts toshowing that necessarily α = α0 . Arguing just as we did above, it suffices show thevanishing of (4.3.30).
Again, we suppose for the sake of contradiction that (4.3.30) does not vanish.
It then follows from Proposition 4.1.17 that for some maximal ideal m of R there
exists a non-zero morphismα∗ (N) ⊗R κ(m)→(α0 )∗ (M) ⊗R κ(m).
Then, by Lemma 4.3.4, after inverting u we obtain an isomorphism(4.3.31)∼α∗ (N) ⊗R κ(m)[1/u] −→ (α0 )∗ (M) ⊗R κ(m)[1/u].

In the strict case, such an isomorphism cannot exist by assumption, and thus (4.3.30)
must vanish.
We now turn to proving that (4.3.29) is an isomorphism. Just as in the precedingarguments, it suffices to show that (4.3.30) vanishes, and if not then we obtainan isomorphism (4.3.31). Since we are considering points of X × X, we are giventhat the induced extension E0 ⊗R κ(m)[1/u] is non-split, so that the base changeof the morphism (4.3.30) from R[[u]] to κ(m)((u)) must vanish. Consequently thecomposite β ◦ ι induces a non-zero morphism α∗ (N) ⊗R κ(m)[1/u] → (α0 )∗ (N) ⊗R
κ(m)[1/u], which, by Lemma 4.3.4, must in fact be an isomorphism. Comparing thisisomorphism with the isomorphism (4.3.31), we find that (α0 )∗ (N) ⊗R κ(m)[1/u]and (α0 )∗ (M) ⊗R κ(m)[1/u] are isomorphic, contradicting the fact that α0 maps
Spec R to Spec Adist . Thus in fact the composite (4.3.30) must vanish, and we havecompleted the proof that (4.3.23) is a monomorphism.
To complete the proof of part (1) of the proposition, we have to show thatthe fibres of (4.3.22) are of degree at most 2. We have already observed that[Spec B dist /Gm ×F Gm ] → Spec Adist ×F C dd,1 is a monomorphism, so it is enoughto check that given a finite extension F0 /F and an isomorphism class of tuples(α, α0 , β : E → E0 , ι, ι0 , π, π 0 ), where• α, α0 are distinct morphisms α, α0 : Spec F0 → Spec Adist ,• β : E → E0 is an isomorphism of Breuil–Kisin modules with descent dataand coefficients in F0 ,

84

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT• ι : α∗ N → E, ι0 : (α0 )∗ N → E0 , π : E → α∗ M and π 0 : E0 → (α0 )∗ Mιπ
are morphisms with the properties that 0 → α∗ N → E → α∗ M → 0 andι0π0

0 → (α0 )∗ N → E0 → (α0 )∗ M → 0 are both short exact.
then α0 is determined by the data of α and E. To see this, note that since we areassuming that α0 6= α, the arguments above show that (4.3.30) does not vanish, so∼that (since F0 is a field), we have an isomorphism α∗ N[1/u] −→ (α0 )∗ M[1/u]. Since
we are over Adist , it follows that E[1/u] ∼= E0 [1/u] is split, and that we also have∼∗0 ∗an isomorphism α M[1/u] −→ (α ) N[1/u]. Thus if α00 is another possible choice∼∼for α0 , we have (α00 )∗ M[1/u] −→ (α0 )∗ M[1/u] and (α00 )∗ N[1/u] −→ (α0 )∗ N[1/u],
00
0whence α = α by Lemma 4.3.5, as required.
We turn to proving (2), and thus assume that
ker-Ext1K(κ(m)) Mκ(m),x̄ , Nκ(m),ȳ = 0
for all maximal ideals m of Ak-free .
Lemma 4.3.20 shows thatSpec B k-free ×Spec Ak-free ×F C dd,1 Spec B k-free → Spec B k-free ×Spec Ak-free ×F Rdd,1 Spec B k-free
is an isomorphism, from which we deduce that[Spec B k-free /Gm ×F Gm ] → Spec Ak-free ×F Rdd,1
is a monomorphism. Using this as input, the claims of (2) may be proved in an
essentially identical fashion to those of (1).
Corollary 4.3.32. The dimension of C(M, N) is equal to the rank of TAdist as aprojective Adist -module. If
ker-Ext1K(κ(m)) Mκ(m),x̄ , Nκ(m),ȳ = 0
for all maximal ideals m of Ak-free , then the dimension of Z(M, N) is also equal tothis rank, while if
ker-Ext1K(κ(m)) Mκ(m),x̄ , Nκ(m),ȳ 6= 0
for all maximal ideals m of Ak-free , then the dimension of Z(M, N) is strictly lessthan this rank.
Proof. The dimension of [Spec B dist /Gm ×F Gm ] is equal to the rank of TAdist (it is
the quotient by a two-dimensional group of a vector bundle over a two-dimensionalbase of rank equal to the rank of TAdist ). By Lemma 4.3.10, C(M, N) is the schemetheoretic image of the morphism [Spec B dist /Gm ×F Gm ] → C dd,1 provided by
Proposition 4.3.21(1), which (by that proposition) is representable by algebraic
spaces and unramified. Since such a morphism is locally quasi-finite (in fact, in
this particular case, we have shown that the fibres of this morphism have degree at
most 2), [Sta13, Tag 0DS6] ensures that
 C(M, N) has the claimed dimension.
If ker-Ext1K(κ(m)) Mκ(m),x̄ , Nκ(m),ȳ = 0 for all maximal ideals m of Ak-free , thenan identical argument using Proposition 4.3.21(2) implies the claim regarding thedimension of Z(M, N).
Finally, suppose that
ker-Ext1K(κ(m)) Mκ(m),x̄ , Nκ(m),ȳ 6= 0
for all maximal ideals m of Ak-free . Then the composite [Spec B k-free /Gm ×F Gm ] →C dd,1 → Rdd,1 has the property that for every point t in the source, the fibre over

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85

the image of t has a positive dimensional fibre. [Sta13, Tag 0DS6] then implies theremaining claim of the present lemma.

4.4. Rank one modules over finite fields, and their extensions. We now
wish to apply the results of the previous subsections to study the geometry of ourvarious moduli stacks. In order to do this, it will be convenient for us to have an
explicit description of the rank one Breuil–Kisin modules of height at most one withdescent data over a finite field of characteristic p, and of their possible extensions.
Many of the results in this section are proved (for p > 2) in [DS15, §1] in the context
of Breuil modules, and in those cases it is possible simply to translate the relevant
statements to the Breuil–Kisin module context.
Assume from now on that e(K 0 /K) is divisible by pf − 1, so that we are in
the setting of [DS15, Remark 1.7]. (Note that the parallel in [DS15] of our field
extension K 0 /K, with ramification and inertial indices e0 , f 0 and e, f respectively, isthe extension K/L with indices e, f and e0 , f 0 respectively.)
Let F be a finite subfield of Fp containing the image of some (so all) embedding(s)k 0 ,→ Fp . Recall that for each g ∈ Gal(K 0 /K) we write g(π 0 )/π 0 = h(g) withh(g) ∈ µe(K 0 /K) (K 0 ) ⊂ W (k 0 ). We abuse notation and denote the image of h(g) in
k 0 again by h(g), so that we obtain a map h : Gal(K 0 /K) → (k 0 )× . Note that h
restricts to a character on the inertia subgroup I(K 0 /K), and is itself a character
when e(K 0 /K) = pf − 1.
Lemma 4.4.1. Every rank one Breuil–Kisin module of height at most one withdescent data and F-coefficients is isomorphic to one of the modules M(r, a, c) definedby:• M(r, a, c)i = F[[u]] · mi ,• ΦM(r,a,c),i(1 P⊗ mi−1 ) = ai uri mi ,P
• ĝ( i mi ) = i h(g)ci mi for all g ∈ Gal(K 0 /K),
where ai ∈ F× , ri ∈ {0, . . . , e0 } and ci ∈ Z/e(K 0 /K) are sequences satisfying
pci−1 ≡ ci + ri (mod e(K 0 /K)), the sums in the third bullet point run from 0 to
f 0 − 1, and the ri , ci , ai are periodic with period dividing f .
Furthermore, two such modules M(r, a, c) and M(s, b, d) are isomorphic if and
Qf −1
Qf −1
only if ri = si and ci = di for all i, and i=0 ai = i=0 bi .
Proof. The proof is elementary; see e.g. [Sav08, Thm. 2.1, Thm. 3.5] for proofs ofanalogous results.
P
We will sometimes refer to the element m = i mi ∈ M(r, a, c) as the standard
generator of M(r, a, c).
Remark 4.4.2. When p > 2 many of the results in this section (such as the above)
can be obtained by translating [DS15, Lem. 1.3, Cor. 1.8] from the Breuil module
context to the Breuil–Kisin module context. We briefly recall the dictionary between
these two categories (cf. [Kis09, §1.1.10]). If A is a finite local Zp -algebra, write
SA = S ⊗Zp A, where S is Breuil’s ring. We regard SA as a SA -algebra via u 7→ u,
and we let ϕ : SA → SA be the composite of this map with ϕ on SA . Then
given a Breuil–Kisin module of height at most 1 with descent data M, we set
M := SA ⊗ϕ,SA M. We have a map 1 ⊗ ϕM : SA ⊗ϕ,SA M → SA ⊗SA M, and weset
Fil1 M := {x ∈ M : (1 ⊗ ϕM )(x) ∈ Fil1 SA ⊗SA M ⊂ SA ⊗SA M}

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

and define ϕ1 : Fil1 M → M as the composite
1⊗ϕM

ϕ1 ⊗1

Fil1 M −→ Fil1 SA ⊗SA M −→ SA ⊗ϕ,SA M = M.
Finally, we define ĝ on M via ĝ(s ⊗ m) = g(s) ⊗ ĝ(m). One checks without difficultythat this makes M a strongly divisible module with descent data (cf. the proofsof [Kis09, Proposition 1.1.11, Lemma 1.2.4]).
In the correspondence described above, the Breuil–Kisin module M((ri ), (ai ), (ci ))corresponds to the Breuil module M((e0 − ri ), (ai ), (pci−1 )) of [DS15, Lem. 1.3].
Definition 4.4.3. If M = M(r, a, c) is a rank one Breuil–Kisin module as described00
in the preceding lemma, we set αi (M) := (pf −1 ri−f 0 +1 + · · · + ri )/(pf − 1) (equivf −1falently, (pri−f +1 + · · · + ri )/(p − 1)). We may abbreviate αi (M) simply as αiwhen M is clear from the context.
It follows easily from the congruence ri ≡ pci−1 − ci (mod e(K 0 /K)) together
with the hypothesis that pf − 1 | e(K 0 /K) that αi ∈ Z for all i. Note that the αi ’s
are the unique solution to the system of equations pαi−1 − αi = ri for all i. Notealso that (pf − 1)(ci − αi ) ≡ 0 (mod e(K 0 /K)), so that hci −αi is a character withimage in k × .
Lemma 4.4.4. We have T (M(r, a, c)) = σi ◦ hci −αi · urQf −1 ai |GK∞ , where urλi=0is the unramified character of GK sending geometric Frobenius to λ.
Proof. Set N = M(0, (ai ), 0), so that N is effectively a Breuil–Kisin module without
descent data. Then for N this result follows from the second paragraph of theproof [GLS14, Lem. 6.3]. (Note that the functor TS of loc. cit. is dual to ourfunctor T ; cf. [Fon90, A 1.2.7]. Note also that the fact that
P the base field isunramified in loc. cit. does not change the calculation.) If n =ni is the standard
0
generator of N as in Lemma 4.4.1, let γ ∈ Zun
⊗
(k
⊗
F)
be
an
element so that
Zp
Fp
p
ϕ=1
γm ∈ (OEd.
nr ⊗S[1/u] N[1/u])
Now for M as in the statement of the lemma it is straightforward to verify that
γ

0
fX
−1

ϕ=1
,
[π 0 ]−αi ⊗ mi ∈ (OEd
nr ⊗S[1/u] M[1/u])

i=0

and the result follows.



One immediately deduces the following.
Corollary 4.4.5. Let M = M(r, a, c) and N = M(s, b, d) be rank one Breuil–Kisin modules with descent data as above. We have T (M) = T (N) if and only if
Qf −1
ci − αi (M) ≡ di − αi (N) (mod e(K 0 /K)) for some i (hence for all i) and i=0 ai =Qf −1
i=0 bi .
Lemma 4.4.6. In the notation of the previous Corollary, there is a nonzero map
M → N (equivalently, dimF HomK(F) (M, N) = 1) if and only if T (M) = T (N) andαi (M) ≥ αi (N) for each i.
Proof. The proof is essentially the same as that of [DS15, Lem. 1.6]. (Indeed, whenp > 2 this lemma can once again be proved by translating directly from [DS15] tothe Breuil–Kisin module context.)
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Using the material of Section 4.1, one can compute Ext1 (M, N) for any pair ofrank one Breuil–Kisin modules M, N of height at most one. We begin with thefollowing explicit description of the complex C • (N) of Section 4.1.
Definition 4.4.7. We write C0u = C0u (M, N) ⊂ F((u))Z/f Z for the space of f tuples (µi ) such that each nonzero term of µi has degree congruent to ci − di(mod e(K 0 /K)), and set C0 = C0u ∩ F[[u]]Z/f Z .
We further define C1u = C1u (M, N) ⊂ F((u))Z/f Z to be the space of f -tuples (hi )such that each nonzero term of hi has degree congruent to ri +ci −di (mod e(K 0 /K)),and set C1 = C1u ∩ F[[u]]Z/f Z . There is a map ∂ : C0u → C1u defined by∂(µi ) = (−ai uri µi + bi ϕ(µi−1 )usi )Evidently this restricts to a map ∂ : C0 → C1 .
Lemma 4.4.8. There is an isomorphism of complexes
∂

∼

[C0 −
→ C1 ] → C • (N)in which (µi ) ∈ C0 is sent to the map mi 7→ µi ni in C 0 (N), and (hi ) ∈ C1 is sent to
the map (1 ⊗ mi−1 ) 7→ hi ni in C 1 (N).
Proof. Each element of HomSF (M, N) has the form mi 7→ µi ni for some f 0 -tuple(µi )i∈Z/f 0 Z of elements of F[[u]]. The condition that this map is Gal(K 0 /K)equivariant is easily seen to be equivalent to the conditions that (µi ) is periodic withperiod dividing f , and that each nonzero term of µi has degree congruent to ci − di(mod e(K 0 /K)). (For the former consider the action of a lift to g ∈ Gal(K 0 /K)
satisfying h(g) = 1 of a generator of Gal(k 0 /k), and for the latter consider the actionof I(K 0 /K); cf. the proof of [DS15, Lem. 1.5].) It follows that the map C0 → C 0 (N)in the statement of the Lemma is an isomorphism. An essentially identical argumentshows that the given map C1 → C 1 (N) is an isomorphism.
To conclude, it suffices to observe that if α ∈ C 0 (N) is given by mi 7→ µi ni with(µi )i ∈ C0 then δ(α) ∈ C 1 (N) is the map given by(1 ⊗ mi−1 ) 7→ (−ai uri µi + bi ϕ(µi−1 )usi )ni ,which follows by a direct calculation.

It follows from Corollary 4.1.8 that Ext1K(F) (M, N) ∼= coker ∂. If h ∈ C1 , we write
P(h) for the element of Ext1K(F) (M, N) represented by h under this isomorphism.
Remark 4.4.9. Let M = M(r, a, c) and N = M(s, b, d) be rank one Breuil–Kisin modules with descent data as in Lemma 4.4.1. It follows from the proof of Lemma 4.1.5,
and in particular the description of the map (4.1.6) found there, that the extension
P(h) is given by the formulas• Pi = F[[u]] · mi + F[[u]] · ni ,si
• ΦP,iai uri mi + hi ni .
P(1 ⊗ ni−1
P) = bi uci ni , ΦP,iP(1 ⊗ mi−1 ) d=i P
0• ĝ( i mi ) = i h(g) mi , ĝ( i ni ) = h(g)i ni for all g ∈ Gal(K /K).
From this description it is easy to see that the extension P(h) has height at most 1
0
if and only if each hi is divisible by uri +si −e .
Theorem 4.4.10. The dimension of Ext1K(F) (M, N) is given by the formula

f
−1 
X
0
∆+
# j ∈ [0, ri ) : j ≡ ri + ci − di (mod e(K /K))
i=0

88

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

where ∆ = dimF HomK(F) (M, N) is 1 if there is a nonzero map M → N and
0 otherwise, while the subspace consisting of extensions of height at most 1 hasdimension

f
−1 
X
0
0
∆+
# j ∈ [max(0, ri + si − e ), ri ) : j ≡ ri + ci − di (mod e(K /K)) .
i=0

Proof. When p > 2, this result (for extensions of height at most 1) can be obtainedby translating [DS15, Thm. 1.11] from Breuil modules to Breuil–Kisin modules. We
argue in the same spirit as [DS15] using the generalities of Section 4.1.
Choose N as in Lemma 4.1.10(2). For brevity we write C • in lieu of C • (N).
We now use the description of C • provided by Lemma 4.4.8. As we have noted,
C 0 consists of the maps mi 7→ µi ni with (µi ) ∈ C0 . Since (ϕ∗M )−1 (v N C 1 ) contains
precisely the maps mi 7→ µi ni in C 0 such that v N | uri µi , we compute thatdimF C 0 / (ϕ∗M )−1 (v N C 1 ) is the quantity

f
−1 
X
Nf −
# j ∈ [e(K 0 /K)N − ri , e(K 0 /K)N ) : j ≡ ci − di (mod e(K 0 /K)) .
i=0

We have dimF C 1 /v N C 1 = N f , so our formula for the dimension of Ext1K(F) (M, N)
now follows from Lemma 4.1.10.

Remark 4.4.11. One can show exactly as in [DS15] that each element of Ext1K(F) (M, N)
can be written uniquely in the form P(h) for h ∈ C1 with deg(hi ) < ri , except thatwhen there exists a nonzero morphism M → N, the polynomials hi for f | i may
also have a term of degree α0 (M) − α0 (N) + r0 in common. Since we will not needthis fact we omit the proof.
4.5. Extensions of shape J. We now begin the work of showing, for each nonscalar tame type τ , that C τ,BT,1 has 2f irreducible components, indexed by thesubsets J of {0, 1, . . . , f −1}. We will also describe the irreducible components of Z τ,1 .
The proof of this hinges on examining the extensions considered in Theorem 4.4.10,
and then applying the results of Subsection 4.3. We will show that most of thesefamilies of extensions have positive codimension in C τ,BT,1 , and are thus negligible
from the point of view of determining irreducible components. By a base change
argument, we will also be able to show that we can neglect the irreducible Breuil–Kisin modules. The rest of Section 4 is devoted to establishing the necessary bounds
on the dimension of the various families of extensions, and to studying the map
from C τ,BT,1 to Rdd,1 .
We now introduce notation that we will use for the remainder of the paper. We
fix a tame inertial type τ = η ⊕ η 0 with coefficients in Qp . We allow the case of
scalar types (that is, the case η = η 0 ). Assume that the subfield F of Fp is large
enough so that the reductions modulo mZp of η and η 0 (which by abuse of notation
we continue to denote η, η 0 ) have image in F. We also fix a uniformiser π of K.
Remark 4.5.1. We stress that when we write τ = η ⊕ η 0 , we are implicitly orderingη, η 0 . Much of the notation in this section depends on distinguishing η, η 0 , as dosome of the constructions later in paper (in particular, the map to the Dieudonnéstack of Section 4.9).
As in Subsection 3.11, we make the following “standard choice” for the extenf
sion K 0 /K: if τ is a tame principal series type, we take K 0 = K(π 1/(p −1) ), while

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

89

if τ is a tame cuspidal type, we let L be an unramified quadratic extension of K,
2f
and set K 0 = L(π 1/(p −1) ). In either case K 0 /K is a Galois extension and η, η 0both factor through I(K 0 /K). In the principal series case, we have e0 = (pf − 1)e,f 0 = f , and in the cuspidal case we have e0 = (p2f − 1)e, f 0 = 2f . Either way, we
0have e(K 0 /K) = pf − 1.
In either case, it follows from Lemma 4.4.1 that a Breuil–Kisin module of rank
one with descent data from K 0 to K is described by the data of the quantities
ri , ai , ci for 0 ≤ i ≤ f − 1, and similarly from Lemma 4.4.8 that extensions between
two such Breuil–Kisin modules are described by the hi for 0 ≤ i ≤ f − 1. This
common description will enable us to treat the principal series and cuspidal cases
largely in parallel.
×ab
The character h|IK of Section 4.4 is identified via the Artin map OL→ ILab = IK
×with the reduction map OL → (k 0 )× . Thus for each σ ∈ Hom(k 0 , Fp ) the map σ◦h|IL0
is the fundamental character ωσ defined in Section 1.7. Define ki , ki0 ∈ Z/(pf − 1)Z0
for all i by the formulas η = σi ◦ hki |I(K 0 /K) and η 0 = σi ◦ hki |I(K 0 /K) . In particular
we have ki = pi k0 , ki0 = pi k00 for all i.
Definition 4.5.2. Let M = M(r, a, c) and N = M(s, b, d) be Breuil–Kisin modulesof rank one with descent data. We say that the pair (M, N) has type τ provided
that for all i:• the multisets {ci , di } and {ki , ki0 } are equal, and• ri + si = e0 .
Lemma 4.5.3. The following are equivalent.
(1) The pair (M, N) has type τ .
(2) Some element of Ext1K(F) (M, N) of height at most one satisfies the strong
determinant condition and is of type τ .
(3) Every element of Ext1K(F) (M, N) has height at most one, satisfies the strong
determinant condition, and is of type τ .
(Accordingly, we will sometimes refer to the condition that ri + si = e0 for all i as
the determinant condition.)
Proof. Suppose first that the pair (M, N) has type τ . The last sentence of Remark 4.4.9 shows that every element of Ext1K(F) (M, N) has height at most one. Let
P be such an element. The condition on the multisets {ci , di } guarantees that P
has unmixed type τ . By Proposition 3.5.12 we see that dimF (imP,i /E(u)Pi )η̃ isindependent of η̃. From the condition that ri + si = e0 we know that the sum overall η̃ of these dimensions is equal to e0 ; since they are all equal, each is equal to e,
and Lemma 3.5.11 tells us that P satisfies the strong determinant condition. This
proves that (1) implies (3).
Certainly (3) implies (2), so it remains to check that (2) implies (1). Supposethat P ∈ Ext1K(F) (M, N) has height at most one, satisfies the strong determinantcondition, and has type τ . The condition that {ci , di } = {ki , ki0 } follows from P
having type τ , and the condition that ri + si = e0 follows from the last part ofLemma 3.5.11.

Definition 4.5.4. If (M, N) is a pair of type τ (resp. P is an extension of type τ ),we define the shape of (M, N) (resp. of P) to be the subset J := {i | ci = ki } ⊆ Z/f 0 Z,
unless τ is scalar, in which case we define the shape to be the subset ∅. (Equivalently,
J is in all cases the complement in Z/f 0 Z of the set {i | ci = ki0 }.)90

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

Observe that in the cuspidal case the equality ci = ci+f means that i ∈ J if andonly if i + f 6∈ J, so that the set J is determined by its intersection with any fconsecutive integers modulo f 0 = 2f .
In the cuspidal case we will say that a subset J ⊆ Z/f 0 Z is a shape if it satisfies
i ∈ J if and only if i + f 6∈ J; in the principal series case, we may refer to any subset
J ⊆ Z/f 0 Z as a shape.
We define the refined shape of the pair (M, N) (resp. of P) to consist of its shape−1
J, together with the f -tuple of invariants r := (ri )fi=0. If (J, r) is a refined shapethat arises from some pair (or extension) of type τ , then we refer to (J, r) as arefined shape for τ .
We say the pair (i − 1, i) is a transition for J if i − 1 ∈ J, i 6∈ J or vice-versa.
(In the first case we sometimes say that the pair (i − 1, i) is a transition out of J,
and in the latter case a transition into J.) Implicit in many of our arguments below
is the observation that in the cuspidal case (i − 1, i) is a transition if and only if(i + f − 1, i + f ) is a transition.
4.5.5. An explicit description of refined shapes. The refined shapes for τ admit
an explicit description. If P is of shape J, for some fixed J ⊆ Z/f 0 Z then, sinceci , di are fixed, we see that the ri and si appearing in P are determined modulo00e(K 0 /K) = pf − 1. Furthermore, we see that ri + si ≡ 0 (mod pf − 1), so thatthese values are consistent with the determinant condition; conversely, if we make0
any choice of the ri in the given residue class modulo (pf − 1), then the si aredetermined by the determinant condition, and the imposed values are consistent
with the descent data. There are of course only finitely many choices for the ri , andso there are only finitely many possible refined shapes for τ .
To make this precise, recall that we have the congruenceri ≡ pci−1 − ci0(mod pf − 1).

We will write [n] for the least non-negative residue class of n modulo e(K 0 /K) =0
pf − 1.
If both i − 1 and i lie in J, then we have ci−1 = ki−1 and ci = ki . The first0of these implies that pci−1 = ki , and therefore ri ≡ 0 (mod pf − 1). The sameconclusion holds if neither i − 1 and i lie in J. Therefore if (i − 1, i) is not a transition
we may write0

ri = (pf − 1)yiand0

si = (pf − 1)(e − yi ).

with 0 ≤ yi ≤ e.
Now suppose instead that (i − 1, i) is a transition. (In particular the type τ
is not scalar.) This time pci−1 = di (instead of pci−1 = ci ), so that ri ≡ di − ci0(mod pf − 1). In this case we write0

ri = (pf − 1)yi − [ci − di ]and0

si = (pf − 1)(e + 1 − yi ) − [di − ci ]with 1 ≤ yi ≤ e.
Conversely, for fixed shape J one checks that each choice of integers yi in theranges described above gives rise to a refined shape for τ .
If (i − 1, i) is not a transition and (hi ) ∈ C1u (M, N) then non-zero terms of hi
0have degree congruent to ri + ci − di ≡ ci − di (mod pf − 1). If instead (i − 1, i) isa transition and (hi ) ∈ C1u (M, N) then non-zero terms of hi have degree congruent

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

91

0

to ri + ci − di ≡ 0 (mod pf − 1). In either case, comparing with the preceding
paragraphs we see that #{j ∈ [0, ri ) : j ≡ ri + ci − di (mod e(K 0 /K))} is exactly yi .
By Theorem 4.4.10, we conclude that for a fixed choice of the ri the dimension
Pf −1of the corresponding Ext1 is ∆ + i=0 yi (with ∆ as in the statement of loc. cit.).
−1 
We say that the refined shape J, (ri )fi=0
is maximal if the ri are chosen to be
maximal subject to the above conditions, or equivalently if the yi are all chosen tobe e; for each shape J, there is a unique maximal refined shape (J, r).
4.5.6. The sets Pτ . To each tame type τ we now associate a set Pτ , which will be asubset of the set of shapes in Z/f 0 Z. (In Appendix B we will recall, following [Dia07],that the set Pτ parameterises the Jordan–Hölder factors of the reduction mod p ofσ(τ ).)
Qf 0 −1
We write η(η 0 )−1 = j=0 (σj ◦ h)γj for uniquely defined integers 0 ≤ γj ≤ p − 1
not all equal to p − 1, so that
(4.5.6)[ki − ki0 ] =

0
fX
−1

pj γi−jj=00with subscripts taken modulo f .
If τ is scalar then we set Pτ = {∅}. Otherwise we let Pτ be the collection ofshapes J ⊆ Z/f 0 Z satisfying the conditions:• if i − 1 ∈ J and i ∈/ J then γi 6= p − 1, and• if i − 1 ∈/ J and i ∈ J then γi 6= 0.
When τ is a cuspidal type, so that η 0 = η q , the integers γj satisfy γi+f = p − 1 − γifor all i; thus the condition that if (i − 1, i) is a transition out of J then γi 6= p − 1translates exactly into the condition that if (i + f − 1, i + f ) is a transition into J
then γi+f 6= 0.
4.5.7. Moduli stacks of extensions. We now apply the constructions of stacks andtopological spaces of Definitions 4.3.11 and 4.3.14 to the families of extensions
considered in Section 4.5.
Definition 4.5.8. If (J, r) is a refined shape for τ , then we let M(J, r) := M(r, 1, c)and let N(J, r) := M(s, 1, d), where c, d, and s are determined from J, r, and τ
according to the discussion of (4.5.5); for instance we take ci = ki when i ∈ J andci = ki0 when i 6∈ J. For the unique maximal shape (J, r) refining J, we write simply
M(J) and N(J).
Definition 4.5.9. If (J, r) is a refined shape for τ , then following Definition
4.3.11,
we may construct the reduced closed substack C M(J, r), N(J, r) of C τ,BT,1 , aswell as the reduced closed substack Z M(J, r), N(J, r) of Z τ,1 . We introduce thenotation C(J, r) and Z(J, r) for these two stacks, and note that (by definition)
Z(J, r) is the scheme-theoretic image of C(J, r) under the morphism C τ,BT,1 → Z τ,1 .
Theorem 4.5.10. If (J, r) is any refined shape for τ , then dim C(J, r) ≤ [K : Qp ],with equality if and only if (J, r) is maximal.
Proof. This follows from Corollary 4.3.32, Theorem 4.4.10, and Proposition 4.1.15.
(See also the discussion following Definition 4.5.4, and note that over Spec Adist , wehave ∆ = 0 by definition.)

92

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTDefinition 4.5.11. If J ⊆ Z/f 0 Z is a shape, and if r is chosen so that (J, r) is
a maximal refined shape for τ , then we write C(J) to denote the closed substackC(J, r) of C τ,BT,1 , and Z(J) to denote the closed substack Z(J, r) of Z τ,1 . Again,
we note that by definition Z(J) is the scheme-theoretic image of C(J) in Z τ,1 .
We will see later that the C(J) are precisely the irreducible components of C τ,BT,1 ;in particular, their finite type points can correspond to irreducible Galois representations. While we do not need it in the sequel, we note the following definition
and result, describing the underlying topological spaces of the loci of reducible
Breuil–Kisin modules of fixed refined shape.
Definition 4.5.12. For each refined type (J, r), we write |C(J, r)τ | for the constructible subset |C(M(J, r), N(J, r))| of |C τ,BT,1 | of Definition 4.3.14 (where M(J, r),N(J, r) are the Breuil–Kisin modules of Definition 4.5.8). We write |Z(J, r)τ | forthe image of |C(J, r)τ | in |Z τ,1 | (which is again a constructible subset).
Lemma 4.5.13. The Fp -points of |C(J, r)τ | are precisely the reducible Breuil–Kisin
modules with Fp -coefficients of type τ and refined shape (J, r).
Proof. This is immediate from the definition.



4.6. ker-Ext1 and vertical components. In this section we will establish somebasic facts about ker-Ext1K(F) (M, N), and use these results to study the imagesof our irreducible components in Z τ,1 . Let M = M(r, a, c) and N = M(s, b, c) be
Breuil–Kisin modules as in Section 4.4.
Recall from (4.1.31) that the dimension of ker-ExtK(F) (M, N) is bounded aboveby the dimension of HomK(F) (M, N[1/u]/N); more precisely, by Lemma 2.3.3 wefind in this setting that
(4.6.1)

dimF ker-Ext1K(F) (M, N) = dimF HomK(A) (M, N[1/u]/N)−(dimF HomF[GK ] (T (M), T (N)) − dimF HomK(A) (M, N)).

A map f : M → N[1/u]/N has the form f (mi ) = µi ni for some f 0 -tuple ofelements µi ∈ F((u))/F[[u]]. By the same argument as in the first paragraph of theproof of Lemma 4.4.8, such a map belongs to C 0 (N[1/u]/N) (i.e., it is Gal(K 0 /K)equivariant) if and only if the µi are periodic with period dividing f , and eachnonzero term of µi has degree congruent to ci − di (mod e(K 0 /K)). One computesthat δ(f )(1 ⊗ mi−1 ) = (usi ϕ(µi−1 ) − uri µi )ni and so f ∈ C 0 (N[1/u]/N) lies in
HomK(F) (M, N[1/u]/N) precisely when
(4.6.2)ai uri µi = bi ϕ(µi−1 )usi

for all i.
Remark 4.6.3. Let f ∈ HomK(F) (M, N[1/u]/N) be given as above. Choose any
lifting µ̃i of µi to F((u)). Then (with notation as in Definition 4.4.7) the tuple (µ̃i )is an element of C0u , and we define hi = ∂(µ̃i ). Then hi lies in F[[u]] for all i, so that(hi ) ∈ C1 , and a comparison with Lemma 4.4.8 shows that f maps to the extension
class in ker-Ext1K(F) (M, N) represented by P(h).
Recall that Lemma 4.1.32 implies that nonzero terms appearing in µi have degree
at least −be0 /(p − 1)c. From this we obtain the following trivial bound on ker-Ext.
Lemma 4.6.4. We have dimF ker-Ext1K(F) (M, N) ≤ de/(p − 1)ef .

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

93

Proof. The degrees of nonzero terms of µi all lie in a single congruence class moduloe(K 0 /K), and are bounded below by −e0 /(p − 1). Therefore there are at mostde/(p − 1)e nonzero terms, and since the µi are periodic with period dividing f thelemma follows.

Remark 4.6.5. It follows directly from Corollary 4.6.4 that if p > 3 and e 6= 1
then we have dimF ker-Ext1K(F) (M, N) ≤ [K : Qp ]/2, for then de/(p − 1)e ≤ e/2.
Moreover these inequalities are strict if e > 2.
We will require a more precise computation of ker-Ext1K(F) (M, N) in the settingof Section 4.5 where the pair (M, N) has maximal refined shape (J, r). We now
return to that setting and its notation.
Let τ be a tame type. We will find the following notation to be helpful. We let
γi∗ = γi if i − 1 6∈ J, and γi∗ = p − 1 − γi if i − 1 ∈ J. (Here the integers γi are as in
Section 4.5.6. In the case of scalar types this means that we have γi∗ = 0 for all i.)00
Since p[ki−1 − ki−1
] − [ki − ki0 ] = (pf − 1)γi , an elementary but useful calculationshows that
(4.6.6)0p[di−1 − ci−1 ] − [ci − di ] = γi∗ (pf − 1),

when (i−1, i) is a transition, and that in this case γi∗ = 0 if and only if [di−1 −ci−1 ] <0
pf −1 . Similarly, if τ is not a scalar type and (i − 1, i) is not a transition then
(4.6.7)0p[di−1 − ci−1 ] + [ci − di ] = (γi∗ + 1)(pf − 1).

The main computational result of this section is the following.
Proposition 4.6.8. Let (J, r) be any maximal refined shape for τ , and suppose thatthe pair (M, N) has refined shape (J, r). Then dimF ker-Ext1K(F) (M, N) is equal to#{0 ≤ i < f : the pair (i − 1, i) is a transition and γi∗ = 0},Q
Q
except that when e = 1, i ai = i bi , and the quantity displayed above is f , thenthe dimension of ker-Ext1K(F) (M, N) is equal to f − 1.
Proof. The argument has two parts. First we show that dimF HomK(F) (M, N[1/u]/N)is precisely the displayed quantity in the statement of the Proposition; then wecheck that dimF HomF[GK ] (T (M), T (N)) − dimF HomK(F) (M, N) is equal to 1 inthe exceptional case of the statement, and 0 otherwise. The result then follows
from (4.6.1).
0
Let f : mi 7→ µi ni be an element of C 0 (N[1/u]/N). Since ue kills µi , and all0
nonzero terms of µi have degree congruent to ci − di (mod pf − 1), certainly allnonzero terms of µi have degree at least −e0 + [ci − di ]. On the other hand since
the shape (J, r) is maximal we have ri = e0 − [ci − di ] when (i − 1, i) is a transition
and ri = e0 otherwise. In either case uri kills µi , so that (4.6.2) becomes simply thecondition that usi kills ϕ(µi−1 ).
If (i−1, i) is not a transition then si = 0, and we conclude that µi−1 = 0. Suppose
instead that (i − 1, i) is a transition, so that si = [ci − di ]. Then all nonzero terms
0of µi−1 have degree at least −si /p > −pf −1 > −e(K 0 /K). Since those terms must0
have degree congruent to ci−1 − di−1 (mod pf − 1), it follows that µi−1 has at most0
one nonzero term (of degree −[di−1 − ci−1 ]), and this only if [di−1 − ci−1 ] < pf −1 ,or equivalently γi∗ = 0 (as noted above). Conversely if γi∗ = 0 then∗usi ϕ(u−[di−1 −ci−1 ] ) = u[ci −di ]−p[di−1 −ci−1 ] = u−γi (p

f0

−1)

94

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

vanishes in F((u))/F[[u]]. We conclude that µi−1 may have a single nonzero term if
and only if (i − 1, i) is a transition and γi∗ = 0, and this completes the first part of
the argument.
Turn now to the second part. Looking at Corollary 4.4.5 and Lemma 4.4.6,
to compare HomF[GK ] (T (M), T (N)) and HomK(F) (M, N) we need to compute thequantities αi (M) − αi (N). By definition this quantity is equal to
0(4.6.9)fX01
pf −j (ri+j − si+j ) .
0f
p − 1 j=1

Suppose first that τ is non-scalar. When (i + j − 1, i + j) is a transition, we have0
ri+j − si+j = (e − 1)(pf − 1) + [di+j − ci+j ] − [ci+j − di+j ], and otherwise we00have ri+j − si+j = e(pf − 1) = (e − 1)(pf − 1) + [di+j − ci+j ] + [ci+j − di+j ].
0
Substituting these expressions into (4.6.9), adding and subtracting pf 01−1 pf [di − ci ],and regrouping gives
0

0
f
X
0
pf − 11−[di − ci ] + (e − 1) ·+ f0
pf −j (p[di+j−1 − ci+j−1 ] ∓ [ci+j − di+j ]) ,p−1
p − 1 j=1

where the sign is − if (i + j − 1, i + j) is a transition and + if not. Applying theformulas (4.6.6) and (4.6.7) we conclude that
0

0
f
X 0
pf − 1 X f 0 −j ∗(4.6.10) αi (M) − αi (N) = −[di − ci ] + (e − 1) ·+p
γi+j +
pf −j
p−1
j=1

j∈Si

where the set Si consists of 1 ≤ j ≤ f such that (i + j − 1, i + j) is not a transition.
Finally, a moment’s inspection shows that the same formula still holds if τ is scalar(recalling that J = ∅ in that case).
Suppose that we are in the exceptional case of the proposition, so that e = 1,γi∗ = 0 for all i, and every pair (i − 1, i) is a transition.
The
Q
Q formula (4.6.10)
gives αi (M) − αi (N) = −[di − ci ]. Since also i ai =i bi the conditionsof Corollary 4.4.5 are satisfied, so that T (M) = T (N); but on the other hand
αi (M) < αi (N), so that by Lemma 4.4.6 there are no nonzero maps M → N, anddimF HomF[GK ] (T (M), T (N)) − dimF HomK(F) (M, N) = 1.
Q If instead
Q we are not in the exceptional case of the proposition, then either
a
6
=
i i
i bi , or else (4.6.10) gives αi (M) − αi (N) > −[di − ci ] for all i. Supposethat T (M) ∼= T (N). It follows from Corollary 4.4.5 that αi (M) − αi (N) ≡ −[di − ci ](mod e(K 0 /K)). Combined with the previous inequality we deduce that αi (M) −αi (N) > 0, and Lemma 4.4.6 guarantees the existence of a nonzero map M → N. We
deduce that in any event dimF HomF[GK ] (T (M), T (N)) = dimF HomK(F) (M, N),
completing the proof.
Corollary 4.6.11. Let (J, r) be any maximal refined shape for τ , and suppose thatthe pair (M, N) has refined shape (J, r). Then dimF ker-Ext1K(F) (M, N) = 0 if andonly if J ∈ Pτ .
Proof. This is immediate from Proposition 4.6.8, comparing the definition of γi∗with the defining condition on elements of Pτ , and noting that the exceptional casein Proposition 4.6.8 can occur only if f is even (so in particular f − 1 6= 0 in these
exceptional cases).
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95

Recall that Z(J) is by definition the scheme-theoretic image of C(J) in Z τ,1 . In
the remainder of this section, we show that the Z(J) with J ∈ Pτ are pairwise
distinct irreducible components of Z τ,1 . In Section 4.8 below we will show that theyin fact exhaust the irreducible components of Z τ,1 .
Theorem 4.6.12. Z(J) has dimension at most [K : Qp ], with equality occurring if
and only if J ∈ Pτ . Consequently, the Z(J) with J ∈ Pτ are irreducible componentsof Z τ,1 .
Proof. The first part is immediate from Corollary 4.3.32, Proposition 4.1.15, Corollary 4.6.11 and Theorem 4.5.10. Since Z τ,1 is equidimensional of dimension [K : Qp ]by Proposition 3.10.19, and the Z(J) are closed and irreducible by construction,
the second part follows from the first together with [Sta13, Tag 0DS2].

We also note the following result.
Proposition 4.6.13. If J ∈ Pτ , then there is a dense open substack U of C(J) suchthat the canonical morphism C(J) → Z(J) restricts to an open immersion on U.
Proof. This follows from Proposition 4.3.21 and Corollary 4.6.11.

For later use, we note the following computation. Recall that we write N(J) =N(J, r) for the maximal shape (J, r) refining J, and that τ = η ⊕ η 0 .
Proposition 4.6.14. For each shape J we have
 0
−1
fY
−1
T (N(J)) ∼(σi ◦ h)ti  |GK∞
=η·
i=0

where

(
γi + δJ c (i)ti =0

if i − 1 ∈ J
if i − 1 6∈ J.

Here δJ c is the characteristic function of the complement of J in Z/f 0 Z, and we are
abusing notation by writing η for the function σi ◦ hki , which agrees with η on IK .
In particular the map J 7→ T (N(J)) is injective on Pτ .
Remark 4.6.15. In the cuspidal case it is not a priori clear that the formula in
Proposition 4.6.14 gives a character of GK∞ (rather than a character only whenrestricted to GL∞ ), but this is an elementary (if somewhat painful) calculation using
the definition of the γi ’s and the relation γi + γi+f = p − 1.
Proof. We begin by explaining how the final statement follows from the rest of the
Proposition. First observe that if J ∈ Pτ then 0 ≤ ti ≤ p − 1 for all i. Indeed theonly possibility for a contradiction would be if γi = p − 1 and i 6∈ J, but then thedefinition of Pτ requires that we cannot have i − 1 ∈ J. Next, note that we never
have ti = p − 1 for all i. Indeed, this would require J = Z/f 0 Z and γi = p − 1 for
all i, but by definition the γi are not all equal to p − 1.
The observations in the previous paragraph imply that (for J ∈ Pτ ) the character
T (N(J)) uniquely determines the integers ti , and so it remains to show that the
integers ti determine the set J. If ti = 0 for all i, then either J = ∅ or J = Z/f 0 Z(for otherwise there is a transition out of J, and δJ c (i) 6= 0 for some i − 1 ∈ J).
But if J = Z/f 0 Z then γi = 0 for all i and τ is scalar; but for scalar types we haveZ/f 0 Z 6∈ Pτ , a contradiction. Thus ti = 0 for all i implies J = ∅.

96

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTFor the rest of this part of the argument, we may therefore suppose ti 6= 0 forsome i, which forces i − 1 ∈ J. The entire set J will then be determined by recursion
if we can show that knowledge of ti along with whether or not i ∈ J, determines
whether or not i − 1 ∈ J. Given the defining formula for ti , the only possible
ambiguity is if ti = 0 and γi + δJ c (i) = 0, so that γi = 0 and i ∈ J. But thedefinition of Pτ requires i − 1 ∈ J in this case. This completes the proof.
We now turn to proving the formula for T (N(J)). We will use Lemma 4.4.4 applied
at i = 0, for which we have to compute α0 − d0 writing α0 = α0 (N). Recall that
we have already computed α0 (M(J)) − α0 (N(J)) in the proof of Proposition 4.6.8.
0Since α0 (M(J)) + α0 (N(J)) = e(pf − 1)/(p − 1), taking the difference between
these formulas gives0

2α0 = [d0 − c0 ] −fX0pf −j γj∗ +

X

0

pf −j

j∈S0c

j=1

where S0c consists of those 1 ≤ j ≤ f such that (j − 1, j) is a transition. Subtract
Pf 0
0
2[d0 ] from both sides, and add the expression −[k0 − k00 ] + j=1 pf −j γj (which
vanishes by definition) to the right-hand side. Note that [d0 − c0 ] − [k0 − k00 ] − 2[d0 ]is equal to −2[k0 ] if 0 6∈ J, and to e(K 0 /K) − 2[k0 − k00 ] − 2[k00 ] if 0 ∈ J. Since
γj − γj∗ = 2γj − (p − 1) if j − 1 ∈ J and is 0 otherwise, the preceding expression
rearranges to give (after dividing by 2)0α0 − [d0 ] = −κ0 +

X
j−1∈J

p

f 0 −jγj +X
j−1∈J,j6∈J

f 0 −jp= −κ0 +fX0

pf −j tj

j=1

where κ0 = [k0 ] if 0 6∈ J and κ0 = [k0 − k00 ] + [k00 ] if 0 ∈ J. Since in either case
κ0 ≡ k0 (mod e(K 0 /K)) the result now follows from Lemma 4.4.4.
Definition 4.6.16. Let r : GK → GL2 (F0 ) be representation. Then we say that a
Breuil–Kisin module M with F0 -coefficients is a Breuil–Kisin model of r of type τ
if M is an F0 -point of C τ,BT,1 , and TF0 (M) ∼= r|GK∞ .
Theorem 4.6.17. The Z(J), with J ∈ Pτ , are pairwise distinct closed substacksof Z τ,1 . For each J ∈ Pτ , there is a dense set of finite type points of Z(J) with theproperty that the corresponding Galois representations have σ J as a Serre weight,
and which furthermore admit a unique Breuil–Kisin model of type τ .
Proof. Recall from Definition 4.3.11 that Z(J) is defined to be the scheme-theoretic
image of a morphism Spec B dist → Z dd,1 . As in the proof of Lemma 4.3.13, since
the source and target of this morphism are of finite presentation over F, its image isa dense constructible subset of its scheme-theoretic image, and so contains a dense
open subset, which we may interpret as a dense open substack U of Z(J). From
the definition of B dist , the finite type points of U correspond to reducible Galois
representations admitting a model of type τ and refined shape (J, r), for which (J, r)
is maximal.
That the Z(J) are pairwise distinct is immediate from the above and Proposition 4.6.14. Combining this observation with Theorem 4.5.10, we see that by deleting
the intersections of Z(J) with the Z(J 0 , r0 ) for all refined shapes (J 0 , r0 ) 6= (J, r),we obtain a dense open substack U 0 whose finite type points have the property

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

97

that every Breuil–Kisin model of type τ of the corresponding Galois representation has shape (J, r). The unicity of such a Breuil–Kisin model then follows from
Corollary 4.6.11.
It remains to show that every such Galois representation r has σ J as a Serreweight. Suppose first that τ is a principal series type. We claim that (writing
σ J = σ~t,~s ⊗ (η 0 ◦ det) as in Appendix B) we have0

T (N(J))|IK = η |IK

fY−1

ωσsii+ti .

i=0To see this, note that by Proposition 4.6.14 it is enough to show that η|IK =Qf −1
η 0 |IK i=0 ωσsii+2ti , which follows by comparing the central characters of σ J andσ(τ ) (or from a direct computation with the quantities si , ti ).
Since det r|IK = ηη 0 ε−1 , we have!
Qf −1 si +ti
ωσi
∗
0
i=0
∼
Qf −1
.
r|IK = η |IK ⊗ε−1 i=0 ωσtii
0
The result then follows from Lemma B.6, using Lemma B.5(2) and the fact that thefibre of the morphism C τ,BT,1 → Rdd,1 above r is nonempty to see that r is not trèsramifiée.
The argument in the cuspidal case proceeds analogously, noting that if the characeI =ter θ (as in Appendix B) corresponds to θe under local class field theory then θ|
K
0
Q
Q
f
−1
f
−1
t
si
e I )2
η 0 i=0 ωσi0 , and that from central characters we have ηη 0 = (θ|K
i=0 ωσi . 
iRemark 4.6.18. With more work, we could use the results of [GLS15] and our results
on dimensions of families of extensions to strengthen Theorem 4.6.17, showingthat there is a dense set of finite type points of Z(J) with the property that thecorresponding Galois representations have σ J as their unique Serre weight. In fact,
we will prove this as part of our work on the geometric Breuil–Mézard conjecture, and
it is an immediate consequence of Theorem 5.2.2 below (which uses Theorem 4.6.17
as an input).
4.7. Irreducible Galois representations. We now show that the points of C τ,BT,1
which are irreducible (that is, cannot be written as an extension of rank one Breuil–Kisin modules) lie in a closed substack of positive codimension. We begin with thefollowing useful observation.
Lemma 4.7.1. The rank two Breuil–Kisin modules with descent data and Fp coefficients which are irreducible (that is, which cannot be written as an extensionof rank 1 Breuil–Kisin modules with descent data) are precisely those whose corresponding étale ϕ-modules are irreducible, or equivalently whose corresponding
GK -representations are irreducible.
Proof. Let M be a Breuil–Kisin module with descent data corresponding to a
τ,BT,1
finite type point of Cdd, let M = M[1/u], and let ρ be the GK -representation
corresponding to M . As noted in the proof of Lemma 2.3.5, ρ is reducible if and only
if ρ|GK∞ is reducible, and by Lemma 2.3.3, this is equivalent to M being reducible.
That this is in turn equivalent to M being reducible may be proved in the same way
as [GLS14, Lem. 5.5].


98

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

Recall that L/K denotes the unramified quadratic extension; then the irreducible
representations ρ : GK → GL2 (Fp ) are all induced from characters of GL . Bearing
in mind Lemma 4.7.1, this means that we can study irreducible Breuil–Kisin modules
via a consideration of base-change of Breuil–Kisin modules from K to L, and ourprevious study of reducible Breuil–Kisin modules. Since this will require us toconsider Breuil–Kisin modules (and moduli stacks thereof) over both K and L, we
will have to introduce additional notation in order to indicate over which of the two
fields we might be working. We do this simply by adding a subscript ‘K’ or ‘L’ to
our current notation. We will also omit other decorations which are being held fixedτ
throughout the present discussion. Thus we write CK
to denote the moduli stack
τ|Lτ,BT,1that was previously denoted C, and CL to denote the corresponding moduli
stack for Breuil–Kisin modules over L, with the type taken to be the restriction
τ|L of τ from K to L. (Note that whether τ is principal series or cuspidal, therestriction τ|L is principal series.)
As usual we fix a uniformiser π of K, which we also take to be our fixed uniformiser
2fof L. Also, throughout this section we take K 0 = L(π 1/(p −1) ), so that K 0 /L is thestandard choice of extension for τ and π regarded as a type and uniformiser for L.
If P is a Breuil–Kisin module with descent data from K 0 to L, then we let P(f ) be
the Breuil–Kisin module W (k 0 ) ⊗Gal(k0 /k),W (k0 ) P, where the pullback is given by the
non-trivial automorphism of k 0 /k. In particular, we have M(r, a, c)(f ) = M(r0 , a0 , c0 )where ri0 = ri+f , a0i = ai+f , and c0i = ci+f .
We let σ denote the non-trivial automorphism of L over K, and write G :=Gal(L/K) = hσi, a cyclic group of order two. There is an action α of G on CL
defined via ασ : P 7→ P(f ) . More precisely, this induces an action of G := hσi on
τ
CL|L in the strict4 sense thatασ ◦ ασ = idC τ|L .
L

We now define the fixed point stack for this action.
τDefinition 4.7.2. We let the fixed point stack (CL|L )G denote the stack whose Aτ
valued points consist of an A-valued point M of CL|L , together with an isomorphism∼ı : M −→ M(f ) which satisfies the cocycle condition that the composite
ı

ı(f )

M −→ M(f ) −→ (M(f ) )(f ) = M
is equal to the identity morphism idM .
τWe now give another description of (CL|L )G , in terms of various fibre products,
which is technically useful. This alternate description involves two steps. In the firststep, we define fixed points of the automorphism ασ , without imposing the additional
condition that the fixed point data be compatible with the relation σ 2 = 1 in G.
Namely, we defineτττ
(CL|L )ασ := CL|L τ × τ CL|L
|L

CL

|L

×CL

4From a 2-categorical perspective, it is natural to relax the notion of group action on a stack so
as to allow natural transformations, rather than literal equalities, when relating multiplicationin the group to the compositions of the corresponding equivalences of categories arising in thedefinition of an action. An action in which actual equalities hold is then called strict. Since ouraction is strict, we are spared from having to consider the various 2-categorical aspects of thesituation that would otherwise arise.

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS
τ

τ

99

τ

where the first morphism CL|L → CL|L × CL|L is the diagonal, and the second isτ
id ×ασ . Working through the definitions, one finds that an A-valued point of (CL|L )αστ|Lconsists of a pair (M, M0 ) of objects of CL over A, equipped with isomorphisms∼∼
α : M −→ M0 and β : M −→ (M0 )(f ) . The morphism(M, M0 , α, β) 7→ (M, ı),τwhere ı := (α−1 )(f ) ◦ β : M → M(f ) , induces an isomorphism between (CL|L )ασ andτthe stack classifying points M of CL|L equipped with an isomorphism ı : M → M(f ) .
(However, no cocycle condition has been imposed on ı.)τLet IC τ|L denote the inertia stack of CL|L . We define a morphism
Lτ(CL|L )ασ → IC τ|LL

via(M, ı) 7→ (M, ı(f ) ◦ ı),
where, as in Definition 4.7.2, we regard the composite ı(f ) ◦ ı as an automorphismof M via the identity (M(f ) )(f ) = M. Of course, we also have the identity section
τe : CL|L → IC τ|L . We now define
L

τ

τ

τ

(CL|L )G := (CL|L )ασ × CL|L .
I τ|L
C

L

τ|

If we use the description of (CL L )ασ as classifying pairs (M, ı), then (just unwinding
definitions) this fibre product classifies tuples (M, ı, M0 , α), where α is an isomor∼
phism M −→ M0 which furthermore identifies ı(f ) ◦ ı with idM0 . Forgetting M0τ|and α then induces an isomorphism between (CL L )G , as defined via the above fibre
product, and the stack defined in Definition 4.7.2.
τ
To compare this fixed point stack to CK, we make the following observations.
Given a Breuil–Kisin module with descent data from K 0 to K, we obtain a Breuil–Kisin module with descent data from K 0 to L via the obvious forgetful map.
Conversely, given a Breuil–Kisin module P with descent data from K 0 to L, theadditional data required to enrich this to a Breuil–Kisin module with descent datafrom K 0 to K can be described as follows as follows: let θ ∈ Gal(K 0 /K) denote the2funique element which fixes π 1/(p −1) and acts nontrivially on L. Then to enrich
the descent data on P to descent data from K 0 to K, it is necessary and sufficient
to give an additive map θ̂ : P → P satisfying θ̂(sm) = θ(s)θ̂(m) for all s ∈ SF andfm ∈ P, and such that θ̂ĝ θ̂ = ĝ p for all g ∈ Gal(K 0 /L).
In turn, the data of the additive map θ̂ : P → P is equivalent to giving thedata of the map θ(θ̂) : P → P(f ) obtained by composing θ̂ with the Frobenius
on L/K. The defining properties of θ̂ are equivalent to asking that this map is
an isomorphism of Breuil–Kisin modules with descent data satisfying the cocycleττ
condition of Definition 4.7.2; accordingly, we have a natural morphism CK→ (CL|L )G ,and a restriction morphism(4.7.3)

ττ
CK→ CL|L .

The following simple lemma summarises the basic facts about base-change in thesituation we are considering.
∼ττLemma 4.7.4. There is an isomorphism CK−→ (CL|L )G .

100

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTProof. This follows immediately from the preceding discussion.



Remark 4.7.5. In the proof of Theorem 4.7.9 we will make use of the following
analogue of Lemma 4.7.4 for étale ϕ-modules. Write RK , RL for the moduli
stacks of Definition 3.1.4, i.e. for the moduli stacks of rank 2 étale ϕ-modules withdescent data respectively to K or to L. Then we have an action of G on RL
defined via M 7→ M (f ) := W (k 0 ) ⊗Gal(k0 /k),W (k0 ) M , and we define the fixed point
stack (RL )G exactly as in Definition 4.7.2: namely an A-valued point of (RL )G∼
consists of an A-valued point M of RL , together with an isomorphism ι : M −→ M (f )satisfying the cocycle condition. The preceding discussion goes through in this∼setting, and shows that there is an isomorphism RK −→ (RL )G .
τ|LττWe also note that the morphisms CK → CL and CK → RK induce a monomorphism(4.7.6)ττ
CK,→ CL|L ×RL RK

One way to see this is to rewrite this morphism (using the previous discussion) as amorphismττ(CL|L )G → CL|L ×RL (RL )G ,and note that the descent data via G on an object classified by the source of thismorphism is determined by the induced descent data on its image in (RL )G .
τ
We now use the Lemma 4.7.4 to study the locus of finite type points of CK
which correspond to irreducible Breuil–Kisin modules. Any irreducible Breuil–Kisin
module over K becomes reducible when restricted to L, and so may be described as
an extension
0 → N → P → M → 0,
where M and N are Breuil–Kisin modules of rank one with descent data from K 0
to L, and P is additionally equipped with an isomorphism P ∼= P(f ) , satisfying thecocycle condition of Definition 4.7.2.
Note that the characters T (M), T (N) of GL∞ are distinct and cannot be extended
to characters of GK . Indeed, this condition is plainly necessary for an extension P
to arise as the base change of an irreducible Breuil–Kisin module (see the proof
of Lemma 2.3.5). Conversely, if T (M), T (N) of GL∞ are distinct and cannot be
extended to characters of GK , then for any P ∈ Ext1K(F) (M, N) whose descent data
can be enriched to give descent data from K 0 to K, this enrichment is necessarily
irreducible. In particular, the existence of such a P implies that the descent dataon M and N cannot be enriched to give descent data from K 0 to K.
We additionally have the following observation.

Lemma 4.7.7. If M, N are such that there is an extension
0→N→P→M→0
whose descent data can be enriched to give an irreducible Breuil–Kisin module
over K, then there exists a nonzero map N → M(f ) .
θ̂Proof. The composition N → P −→ P → M, in which first and last arrows are
the natural inclusions and projections, must be nonzero (or else θ̂ would givedescent data on N from K 0 to K). It is not itself a map of Breuil–Kisin modules,
because θ̂ is semilinear, but is a map of Breuil–Kisin modules when viewed as amap N → M(f ) .
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101

We now consider (for our fixed M, N, and working over L rather than over K)
the scheme Spec B dist as in Subsection 4.3. Following Lemma 4.7.7, we assume that
there exists a nonzero map N → M(f ) . The observations made above show that weare in the strict case, and thus that Spec Adist = Gm × Gm and that furthermore
we may (and do) set V = T . We consider the fibre product with the restriction
morphism (4.7.3)
τ
Y (M, N) := Spec B dist ×C τ|L CK.
LLet Gm ,→ Gm × Gm be the diagonal closed immersion, and let (Spec B dist )|Gm
denote the pull-back of Spec B dist along this closed immersion. By Lemma 4.7.7, theprojection Y (M, N) → Spec B dist factors through (Spec B dist )|Gm , and combining
this with Lemma 4.7.4 we see that Y (M, N) may also be described as the fibre
productτ(Spec B dist )|Gm ×C τ|L (CL|L )G .
L

Recalling the warning of Remark 4.3.16, Proposition 4.3.21 now shows that there
is a monomorphismτ[(Spec B dist )|Gm /Gm × Gm ] ,→ CL|L ,and thus, by Lemma 4.2.8, that there is an isomorphism∼(Spec B dist )|Gm ×C τ|L (Spec B dist )|Gm −→ (Spec B dist )|Gm × Gm × Gm .
L

(An inspection of the proof of Proposition 4.3.21 shows that in fact this result is moreor-less proved directly, as the key step in proving the proposition.) An elementary
manipulation with fibre products then shows that there is an isomorphism∼

Y (M, N) ×(C τ|L )G Y (M, N) −→ Y (M, N) × Gm × Gm ,Land thus, by another application of Lemma 4.2.8, we find that there is a monomorphism(4.7.8)τ[Y (M, N)/Gm × Gm ] ,→ (CL|L )G .

We define Cirred to be the union over all such pairs (M, N) of the scheme-theoreticτimages of the various projections Y (M, N) → (CL|L )G . Note that this image dependson (M, N) up to simultaneous unramified twists of M and N, and there are only
finitely many such pairs (M, N) up to such unramified twist. By definition, Cirred isττa closed substack of CK
which contains every finite type point of CK
corresponding
to an irreducible Breuil–Kisin module.
The following is the main result of this section.
τ
Theorem 4.7.9. The closed substack Cirred of CK= C τ,BT,1 , which contains everyτfinite type point of CK corresponding to an irreducible Breuil–Kisin module, hasdimension strictly less than [K : Qp ].

Proof. As noted above, there are only finitely many pairs (M, N) up to unramified
twist, so it is enough to show that for each of them, the scheme-theoretic image ofthe monomorphism (4.7.8) has dimension less than [K : Qp ].
By [Sta13, Tag 0DS6], to prove the present theorem, it then suffices to show that
dim Y (M, N) ≤ [K : Qp ] + 1 (since dim Gm × Gm = 2). To establish this, it suffices
to show, for each point x ∈ Gm (F0 ), where F0 is a finite extension of F, that thedimension of the fibre Y (M, N)x is bounded by [K : Qp ]. After relabelling, as we

102

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

may, the field F0 as F and the Breuil–Kisin modules Mx and Nx as M and N, we
may suppose that in fact F0 = F and x = 1.
Manipulating the fibre product appearing in the definition of Y (M, N), we findthat
(4.7.10)τ
Y (M, N)1 = Ext1K(F) (M, N) ×C τ|N CK,
L

where the fibre product is taken with respect to the morphism Ext1K(F) (M, N) → CLτ
that associates the corresponding rank two extension to an extension of rank one
Breuil–Kisin modules, and the restriction morphism (4.7.3).
In order to bound the dimension of Y (M, N)1 , it will be easier to first embed it into another, larger, fibre product, which we now introduce. Namely, themonomorphism (4.7.6) induces a monomorphism
Y (M, N)1 ,→ Y 0 (M, N)1 := Ext1K(F) (M, N) ×RL RK .
Any finite type point of this fibre product lies over a fixed isomorphism class of finite
type points of RK (corresponding to some fixed irreducible Galois representation); we
let P be a choice of such a point. The restriction of P then lies in a fixed isomorphismclass of finite type points of RL (namely, the isomorphism class of the direct sum
M[1/u] ⊕ N[1/u] ∼= M[1/u] ⊕ M(f ) [1/u]). Thus the projection Y 0 (M, N)1 → RKfactors through the residual gerbe of P , while the morphism Y 0 (M, N)1 → RL
factors through the residual gerbe of M[1/u] ⊕ N[1/u] ∼= M[1/u] ⊕ M(f ) [1/u]. Since
P corresponds via Lemma 2.3.3 to an irreducible Galois representation, we find that
Aut(P ) = Gm . Since M[1/u] ⊕ N[1/u] corresponds via Lemma 2.3.3 to the direct
sum of two non-isomorphic Galois characters, we find that Aut(M[1/u] ⊕ N[1/u]) =Gm × Gm .
Thus we obtain monomorphisms(4.7.11)

Y (M, N)1 ,→ Y 0 (M, N)1,→ Ext1K(F) (M, N) ×[Spec F 0 //Gm ×Gm ] [Spec F 0 //Gm ] ∼= Ext1K(F) (M, N) × Gm .
In Proposition 4.7.12 we obtain a description of the image of Y (M, N)1 under thismonomorphism which allows us to bound its dimension by [K : Qp ], as required. 
We now prove the bound on the dimension of Y (M, N)1 that we used in the proofof Theorem 4.7.9. Before establishing this bound, we make some further remarks.
To begin with, we remind the reader that we are working with Breuil–Kisin modules,étale ϕ-modules, etc., over L rather than K, so that e.g. the structure parametersof M, N are periodic modulo f 0 = 2f (not modulo f ), and the pair (M, N) has
type τ |L . We will readily apply various pieces of notation that were introduced
above in the context of the field K, adapted in the obvious manner to the context
of the field L. (This applies in particular to the notation C1u , C0u , etc. introduced in
Definition 4.4.7.)
We write m, n for the standard generators of M and N. The existence of thenonzeromap N → M(f ) implies that αi (N) ≥ αi+f (M) for all i, and also that
Q
Q
a
=
i i
i bi . Thanks to the latter we will lose no generality by assuming thatai = bi = 1 for all i. Let m̃ be the standard generator for M(f ) . The map
N → M(f ) will (up to a scalar) have the form ni 7→ uxi m̃i for integers xi satisfying
pxi−1 − xi = si − ri+f for all i; thus xi = αi (N) − αi+f (M) for all i. Since the0characters T (M) and T (N) are conjugate we must have xi ≡ di − ci+f (mod pf − 1)

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

103

for all i (cf. Lemma 4.4.4). Moreover, the strong determinant condition si + ri = e0
for all i implies that xi = xi+f .
We stress that we make no claims about the optimality of the following result;
we merely prove “just what we need” for our applications. Indeed the estimates of[Hel09, Car17] suggest that improvement should be possible.
Proposition 4.7.12. We have dim Y (M, N)1 ≤ [K : Qp ].
Remark 4.7.13. Since the image of Y (M, N)1 in Ext1K(F) (M, N) lies in ker-Ext1K(F) (M, N)
with fibres that can be seen to have dimension at most one, many cases of Proposition 4.7.12 will already follow from Remark 4.6.5 (applied with L in place of K).
Proof of Proposition 4.7.12. Let P = P(h) be an element of Ext1K(F) (M, N) whose
e bedescent data can be enriched to give descent data from K 0 to K, and let P
such an enrichment. By Lemma 4.7.7 (and the discussion preceding that lemma)
the étale ϕ-module P[ u1 ] is isomorphic to M[ u1 ] ⊕ M(f ) [ u1 ]. All extensions of the
GL∞ -representation T (M[ u1 ] ⊕ M(f ) [ u1 ]) to a representation of GK∞ are isomorphic(and given by the induction of T (M[ u1 ]) to GK∞ ), so the same is true of theétale ϕ-modules with descent data from K 0 to K that enrich the descent data on
M[ u1 ] ⊕ M(f ) [ u1 ]. One such enrichment, which we denote P , has θ̂ that interchanges
e 1 ] is isomorphic to P .
m and m̃. Thus P[u
As in the proof of Lemma 4.7.7, the hypothesis that T (M) ∼6 T (N) implies that=any non-zero map (equivalently, isomorphism) of étale ϕ-modules with descent datae 1 ] → P takes the submodule N[ 1 ] to M(f ) [ 1 ]. We may scale the map λ so
λ : P[uuuthat it restricts to the map ni → uxi m̃i on N. Then there is an element ξ ∈ F× sothat λ induces multiplication by ξ on the common quotients M[ u1 ]. That is, themap λ may be assumed to have the form
 
 x
 
ni
u i 0
m̃i
(4.7.14)
7→
mi
νi ξ
mi
0

for some (νi ) ∈ F((u))f . The condition that the map λ commutes with the descentdata from K 0 to L is seen to be equivalent to the condition that nonzero terms in0
νi have degree congruent to ci − di + xi (mod pf − 1); or equivalently, if we defineµi := νi u−xi for all i, that the tuple µ = (µi ) is an element of the set C0u = C0u (M, N)
of Definition 4.4.7.
The condition that λ commutes with ϕ can be checked to give

 
 
ni−1
usi
0
ni
ϕ
=
.
mi−1
ϕ(νi−1 )uri+f −xi − νi uri −xi urimi
The extension P is of the form P(h), for some h ∈ C1 as in Definition 4.4.7. The
lower-left entry of the first matrix on the right-hand side of the above equation mustthen be hi . Since ri+f − xi = si − pxi−1 , the resulting condition can be rewritten as
hi = ϕ(µi−1 )usi − µi uri ,or equivalently that h = ∂(µ). Comparing with Remark 4.6.3, we recover the factthat the extension class of P is an element of ker-Ext1K(F) (M, N), and the tuple µdetermines an element of the space H defined as follows.
Definition 4.7.15. The map ∂ : C0u → C1u induces a map C0u /C0 → C1u /∂(C0 ), which
we also denote ∂. We let H ⊂ C0u /C0 denote the subspace consisting of elements µsuch that ∂(µ) ∈ C1 /∂(C0 ).

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By the discussion following Lemma 4.4.8, an element µ ∈ H determines anextension P(∂(µ)). Indeed, Remark 4.6.3 and the proof of (4.1.31) taken together
show that there is a natural isomorphism, in the style of Lemma 4.4.8, between themorphism ∂ : H → C1 /∂(C0 ) and the connection map HomK(F) (M, N[1/u]/N) →Ext1K(F) (M, N), with im ∂ corresponding to ker-Ext1K(F) (M, N).
Conversely, let h be an element of ∂(C0u ) ∩ C1 , and set νi = uxi µi . The condition
e with descent data from K 0 to K and ξ ∈ F×that there is a Breuil–Kisin module P
1e ] → P defined as above is an isomorphism is precisely the condition
such that λ : P[uthat the map θ̂ on P pulls back via λ to a map that preserves P. One computesthat this pullback is
 



ni
−νi+f
uxi
ni+f
−1
θ̂
=ξ
mi
(ξ 2 − νi νi+f )u−xi νi
mi+f
recalling that xi = xi+f .
We deduce that θ̂ preserves P precisely when the νi are integral and νi νi+f ≡ ξ 2(mod uxi ) for all i. For i with xi = 0 the latter condition is automatic given theformer, which is equivalent to the condition that µi and µi+f are both integral. If
instead xi > 0, then we have the nontrivial condition νi+f ≡ ξ 2 νi−1 (mod uxi ); inother words that µi , µi+f have u-adic valuation exactly −xi , and their principal
parts determine one another via the equation µi+f ≡ ξ 2 (u2xi µi )−1 (mod 1).
Let Gm,ξ be the multiplicative group with parameter ξ. We now (using thenotation of Definition 4.7.15) define H0 ⊂ C0u /C0 × Gm,ξ to be the subvariety
consisting of the pairs (µ, ξ) with exactly the preceding properties; that is, we
regard C0u /C0 as an Ind-affine space in the obvious way, and define H0 to be thepairs (µ, ξ) satisfying• if xi = 0 then vali µ = vali+f µ = ∞, and• if xi > 0 then vali µ = vali+f µ = −xi and µi+f ≡ ξ 2 (u2xi µi )−1 (mod u0 )where we write vali µ for the u-adic valuation of µi , putting vali µ = ∞ when µi is
integral.
Putting all this together with (4.7.10), we find that the map
H0 ∩ (H × Gm,ξ ) → Y (M, N)1
e is a well-defined surjection, where P = P(∂(µ)), Pesending (µ, ξ) to the pair (P, P)
is the enrichment of P to a Breuil–Kisin module with descent data from K 0 to K in
which θ̂ is pulled back to P from P via the map λ as in (4.7.14). (Note that Y (M, N)1
is reduced and of finite type, for example by (4.7.11), so the surjectivity can bechecked on Fp -points.) In particular dim Y (M, N)1 ≤ dim H0 .
Note that H0 will be empty if for some i we have xi > 0 but xi + ci − di 6≡ 0
0(mod pf − 1) (so that νi cannot be a u-adic unit). Otherwise, the dimension
Pf −10
of H0 is easily computed to be D = 1 + i=0 dxi /(pf − 1)e (indeed if d is theD−dnumber of nonzero xi ’s, then H0 ∼), and since xi ≤ e0 /(p − 1) we= Gd+1m × Ga0find that H has dimension at most 1 + de/(p − 1)ef . This establishes the bounddim Y (M, N)1 ≤ 1 + de/(p − 1)ef .
Since p > 2 this bound already establishes the theorem when e > 1. If insteade = 1 the above bound gives dim Y (M, N) ≤ [K : Qp ] + 1. Suppose for thesake of contradiction that equality holds. This is only possible if H0 ∼= Gfm+1 ,0(i)0H ⊂ H × Gm,ξ , and xi = [di − ci ] > 0 for all i. Define µ ∈ Cu to be the element

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

105

such that µi = u−[di −ci ] , and µj = 0 for j 6= i. Let F00 /F be any finite extension suchP
that #F00 > 3. For each nonzero z ∈ F00 define µz = j6=i,i+f µ(i) +zµ(i) +z −1 µ(i+f ) ,so that (µz , 1) is an element of H0 (F00 ). Since H0 ⊂ H × Gm,ξ and H is linear,
the differences between the µz for varying z lie in H(F00 ), and (e.g. by consideringµ1 − µ−1 and µ1 − µz for any z ∈ F00 with z 6= z −1 ) we deduce that each µ(i) lies in
H. In particular each ∂(µ(i) ) lies in C1 .
If (i − 1, i) were not a transition then (since e = 1) we would have either ri = 0 or
si = 0. The former would contradict ∂(µ(i) ) ∈ C1 (since the ith component of ∂(µ(i) )would be u−[di −ci ] , of negative degree), and similarly the latter would contradict∂(µ(i−1) ) ∈ C1 . Thus (i − 1, i) is a transition for all i. In fact the same observations
show more precisely that ri ≥ xi = [di − ci ] and si ≥ pxi−1 = p[di−1 − ci−1 ].
Summing these inequalities and subtracting e0 we obtain 0 ≥ p[di−1 − ci−1 ] − [ci − di ],and comparing with (4.6.6) shows that we must also have γi∗ = 0 for all i. Since
e = 1 and (i − 1, i) is a transition for all i the refined shape of the pair (M, N) is
automatically maximal; but then we are in the exceptional case of Proposition 4.6.8,
which (recalling the proof of that Proposition) implies that T (M) ∼= T (N). This is
the desired contradiction.

4.8. Irreducible components. We can now use our results on families of extensions of characters to classify the irreducible components of the stacks C τ,BT,1and Z τ,1 . In Section 5 we will combine these results with results coming from
Taylor–Wiles patching (in particular the results of [GK14, EG14], which we combine
in Appendix C) to describe the closed points of each irreducible component of Z τ,1in terms of the weight part of Serre’s conjecture.
Corollary 4.8.1. Each irreducible component of C τ,BT,1 is of the form C(J) forsome J; conversely, each C(J) is an irreducible component of C τ,BT,1 .
Remark 4.8.2. Note that at this point we have not established that different sets Jgive distinct irreducible components C(J); we will prove this in Section 4.9 below
by a consideration of Dieudonné modules.
Proof of Corollary 4.8.1. By Proposition 3.10.20, C τ,BT,1 is equidimensional of dimension [K : Qp ]. By construction, the C(J) are irreducible substacks of C τ,BT,1 ,and by Theorem 4.5.10 they also have dimension [K : Qp ], so they are in fact
irreducible components by [Sta13, Tag 0DS2].
By Theorem 4.7.9 and Theorem 4.5.10, we see that there is a closed substack
Csmall of C τ,BT,1 of dimension strictly less than [K : Qp ], with the property that every
finite type point of C τ,BT,1 is a point of at least one of the C(J) or of Csmall (or both).
(Indeed, we can take Csmall to be the union of the stack Cirred of Theorem 4.7.9 andthe stacks C(J, r) for non-maximal shapes (J, r).) Since C τ,BT,1 is equidimensionalof dimension [K : Qp ], it follows that the C(J) exhaust the irreducible components
of C τ,BT,1 , as required.

We now deduce a classification of the irreducible components of Z τ,1 ; Theorem 5.2.2 below is a considerable refinement of this, giving a precise description of
the finite type points of the irreducible components in terms of the weight part of
Serre’s conjecture.
Corollary 4.8.3. The irreducible components of Z τ,1 are precisely the Z(J) for J ∈Pτ , and if J 6= J 0 then Z(J) 6= Z(J 0 ).

106

A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITTProof. By Theorem 4.6.12, if J ∈ Pτ then Z(J) is an irreducible component of Z τ,1 .
Furthermore, these Z(J) are pairwise distinct by Theorem 4.6.17.
Since the morphism C τ,BT,1 → Z τ,1 is scheme-theoretically dominant, it follows
from Corollary 4.8.1 that each irreducible component of Z τ,1 is dominated bysome C(J). Applying Theorem 4.6.12 again, we see that if J ∈/ Pτ then C(J) does
not dominate an irreducible component, as required.

4.9. Dieudonné modules and the morphism to the gauge stack. We now
study the images of the irreducible components C(J) in the gauge stack Gη ; thisamounts to computing the Dieudonné modules and Galois representations associatedto the extensions of Breuil–Kisin modules that we considered in Section 4. Suppose
throughout this subsection that τ is a non-scalar type, and that (J, r) is a maximal
refined shape. Recall that in the cuspidal case this entails that i ∈ J if and only if
i + f 6∈ J.
Lemma 4.9.1. Let P ∈ Ext1K(F) (M, N) be an extension of type τ and refined shape(J, r). Then for i ∈ Z/f 0 Z we have F = 0 on D(P)η,i−1 if i ∈ J, while V = 0 on
D(P)η,i if i ∈/ J.
Proof. Recall that D(P) = P/uP. Let wi be the image of mi in D(P) if i ∈ J, andlet wi be the image of ni in D(P) if i 6∈ J. It follows easily from the definitions that
D(P)η,i is generated over F by wi .
Recall that the actions of F, V on D(P) are as specified in Definition 2.2.1. In
particular F is induced by ϕ, while V is c−1 V mod u where V is the unique map on
P satisfying V ◦ ϕ = E(u), and c = E(0). For the Breuil–Kisin module P, we haveϕ(ni−1 ) = bi usi ni ,ϕ(mi−1 ) = ai uri mi + hi ni ,0and so one checks (using that E(u) = ue in F) that
−1 −1siV(mi ) = a−1
i u mi−1 − ai bi hi ni−1 ,ri
V(ni ) = b−1
i u ni−1 .

From Definition 4.5.4 and the discussion immediately following it, we recall that
if (i − 1, i) is not a transition then ri = e0 , si = 0, and hi is divisible by u (the latter0
because nonzero terms of hi have degrees congruent to ri + ci − di (mod pf − 1),and ci 6≡ di since τ is non-scalar). On the other hand if (i − 1, i) is a transition,
0
then ri , si > 0, and nonzero terms of hi have degrees divisible by pf − 1; in thatcase we write h0i for the constant coefficient of hi , and we remark that h0i does notvanish identically on Ext1K(F) (M, N).
Suppose, for instance, that i − 1 ∈ J and i ∈ J. Then wi−1 and wi are the images0in D(P) of mi−1 and mi . From the above formulas we see that uri = ue and hi areboth divisible by u, while on the other hand usi = 1. We deduce that F (wi−1 ) = 0
and V (wi ) = c−1 a−1
i wi−1 . Computing along similar lines, it is easy to check thefollowing four cases.
(1) i − 1 ∈ J, i ∈ J. Then F (wi−1 ) = 0 and V (wi ) = c−1 a−1
i wi−1 .
(2) i − 1 ∈/ J, i ∈/ J. Then F (wi−1 ) = bi wi , V (wi ) = 0.
(3) i − 1 ∈ J, i ∈/ J. Then F (wi−1 ) = h0i wi , V (wi ) = 0.
−1 0(4) i − 1 ∈/ J, i ∈ J. Then F (wi−1 ) = 0, V (wi ) = −c−1 a−1
i bi hi wi−1 .
In particular, if i ∈ J then F (wi ) = 0, while if i ∈/ J then V (wi+1 ) = 0.

Since C τ,BT is flat over O by Corollary 3.8.3, it follows from Lemma 3.11.17 thatthe natural morphism C τ,BT → Gη is determined by an f -tuple of effective Cartier

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107

divisors {Dj }0≤j<f lying in the special fibre C τ,BT,1 . Concretely, Dj is the zero locusof Xj , which is the zero locus of F : Dη,j → Dη,j+1 . The zero locus of Yj (which is
the zero locus of V : Dη,j+1 → Dη,j ) is another Cartier divisor Dj0 . Since C τ,BT,1
is reduced, we conclude that each of Dj and Dj0 is simply a union of irreducible
components of C τ,BT,1 , each component appearing precisely once in precisely one ofeither Dj or Dj0 .
Proposition 4.9.2. Dj is equal to the union of the irreducible components C(J) of
C τ,BT,1 for those J that contain j + 1.
Proof. Lemma 4.9.1 shows that if j + 1 ∈ J, then Xj = 0, while if j + 1 ∈/ J, thenYj = 0. In the latter case, by an inspection of case (3) of the proof of Lemma 4.9.1,
we have Xj = 0 if and only if j ∈ J and h0j+1 = 0. Since h0j+1 does not vanish
identically on an irreducible component, we see that the irreducible components on
which Xj vanishes identically are precisely those for which j + 1 ∈ J, as claimed. 
Theorem 4.9.3. The algebraic stack C τ,BT,1 has precisely 2f irreducible components,
namely the irreducible substacks C(J).
Proof. By Corollary 4.8.1, we need only show that if J 6= J 0 then C(J) 6= C(J 0 ); but
this is immediate from Proposition 4.9.2.

5. Moduli stacks of Galois representations and the geometric
Breuil–Mézard conjecture
We now make a more detailed study of the stacks Z dd and Z τ . In particular, we
prove a “geometric Breuil–Mézard” result, showing in particular that the finite typepoints of each irreducible components are precisely described by the weight partof Serre’s conjecture. We also prove a new result on the structure of potentially
Barsotti–Tate deformation rings, Proposition 5.1.1, showing that their special fibres
are generically reduced.
5.1. Generic reducedness of Spec Rrτ,BT /$. We return to the setting of Subsection 3.10: that is, we fix a finite type point Spec F0 → Z τ,a , where F0 /F is a
finite extension, and let r : GK → GL2 (F0 ) be the corresponding Galois representation. It follows from Corollary 3.10.18 that Spec Rτ,a is a closed subscheme ofSpec Rrτ,BT /$a , but we have no reason to believe that equality holds. It follows from
Lemma 3.9.8, together with Lemma 5.1.6 below, that Spec Rτ,1 is the underlying
reduced subscheme of Spec Rrτ,BT /$, so that equality holds in the case a = 1 if
and only if Spec Rrτ,BT /$ is reduced. Again, we have no reason to believe that
this holds in general, but the main result of this section is Proposition 5.1.1 below,
showing that Spec Rrτ,BT /$ is generically reduced. We will use this in the proofof our geometric Breuil–Mézard result below. (Recall that a scheme is genericallyreduced if it contains an open reduced subscheme whose underlying topologicalspace is dense. In the case of a Noetherian affine scheme Spec A, this is equivalent
to requiring that the localisation of A at each of its minimal primes is reduced.)
Proposition 5.1.1. For any tame type τ , the scheme Spec Rrτ,BT /$ is genericallyreduced, with underlying reduced subscheme Spec Rτ,1 .
We will deduce Proposition 5.1.1 from the following global statement.

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Proposition 5.1.2. Let τ be a tame type. There is a dense open substack U of Z τsuch that U/F is reduced.
Proof. The proposition will follow from an application of Proposition A.11, and the
key to this application will be to find a candidate open substack U 1 of Z τ,1 , which
we will do using our study of the irreducible components of C τ,BT,1 and Z τ,1 .
Recall that, for each J ∈ Pτ , we let Z(J) denote the scheme-theoretic image ofC(J) under the proper morphism C τ,BT,1 → Z τ,1 . Each Z(J) is a closed substack ofZ τ,1 , and so, if we let V(J) be the complement in Z τ,1 of the union of the Z(J 0 ) for all
J 0 6= J, J 0 ∈ Pτ , then V(J) is a dense open open substack of Z τ,1 , by Corollary 4.8.3.
The preimage W(J) of V(J) in C τ,BT,1 is therefore a dense open substack of C(J).
Possibly shrinking W(J) further, we may suppose by Proposition 4.6.13 that themorphism W(J) → Z τ,1 is a monomorphism.
The complement |C(J)| \ |W(J)| is a closed subset of |C(J)|, and thus of |C τ,BT,1 |,and its image under the proper morphism C τ,BT,1 → Z τ,1 is a closed subset of|Z τ,BT,1 |, which is (e.g. for dimension reasons) a proper closed subset of |Z(J)|; so if
we let U(J) be the complement in V(J) of this image, then U(J) is open and densein Z(J), and the morphism C τ,BT,1 ×Z τ,1 U(J) → U(J) is a monomorphism. Set U 1 =∪J U(J). Since the U(J) are pairwise disjoint by construction, C τ,BT,1 ×Z τ,1 U 1 → U 1
is again a monomorphism. By construction (taking into account Corollary 4.8.3),
U 1 is dense in Z τ,1 .
Now let U denote the open substack of Z τ corresponding to U 1 . Since |Z τ | =τ,1|Z |, we see that U is dense in Z τ . We have seen in the previous paragraphthat the statement of Proposition A.11 (5) holds (taking a = 1, X = C τ,BT ,and Y = Z); so Proposition A.11 implies that, for each a ≥ 1, the closed immersion
U ×Z τ Z τ,a ,→ U ×O O/$a is an isomorphism.
In particular, since the closed immersion U 1 = U ×Z τ Z τ,1 → U/F is an isomorphism, we may regard U/F as an open substack of Z τ,1 . Since Z τ,1 is reduced,
by Lemma 3.9.8, so is its open substack U/F . This completes the proof of the
proposition.

Corollary 5.1.3. Let τ be a tame type. There is a dense open substack U of Z τ∼such that we have an isomorphism C τ,BT ×Z τ U −→ U, as well as isomorphisms∼∼

U ×Z τ C τ,BT,a −→ U ×Z τ Z τ,a −→ U ×O O/$a ,for each a ≥ 1.
Proof. This follows from Proposition 5.1.2 and Proposition A.11.



Remark 5.1.4. More colloquially, Corollary 5.1.3 shows that for each tame type τ ,there is an open dense substack U of Z τ consisting of Galois representations whichhave a unique Breuil–Kisin model of type τ .
Lemma 5.1.5. If U is an open substack of Z τ satisfying the conditions of Proposiτtion 5.1.2, and if T → Z/F
is a smooth morphism whose source is a scheme, thenτ
T ×Z/FU/F is reduced, and is a dense open subscheme of T .
τ,1Proof. Since Z/F
is a Noetherian algebraic stack (being of finite presentation overSpec F), the open immersion
τU/F → Z/F

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109

τis quasi-compact ([Sta13, Tag 0CPM]). Since T → Z/F
is flat (being smooth, byτ U
assumption), the pullback T ×Z/F
→
T
is
an
open
immersion
with dense image;/F
here we use the fact that for a quasi-compact morphism, the property of being
scheme-theoretically dominant is preserved by flat base-change, together with the factthat an open immersion with dense image induces a scheme-theoretically dominantmorphism after passing to underlying reduced substacks. Since the source of thismorphism is smooth over the reduced algebraic stack U/F , it is itself reduced. 

The following result is standard, but we recall the proof for the sake of completeness.
Lemma 5.1.6. Let T be a Noetherian scheme, all of whose local rings at finite typepoints are G-rings. If T is reduced (resp. generically reduced), then so are all of its
complete local rings at finite type points.
Proof. Let t be a finite type point of T , and write A := OT,t . Then A is a (generically)b is also (generically)
reduced local G-ring, and we need to show that its completion Abb
reduced. Let b
p be a (minimal) prime of A; since A → A is (faithfully) flat, bp liesover a (minimal) prime p of A by the going-down theorem.
bbp is reduced.
Then Ap is reduced by assumption, and we need to show that A
bbp is regular.
By [Sta13, Tag 07QK], it is enough to show that the morphism A → A
b are G-rings (the latter by [Sta13, Tag 07PS]), so the composite
Both A and A
b → (Abbp )b
A→A
is a composite of regular morphisms, and is thus a regular morphism by [Sta13, Tag
07QI].
bbp )b, so thisThis composite factors through the natural morphism Ap → (Amorphism is also regular. Factoring it as the composite
bbp → (Abbp )b,Ap → Abbp is regular, as required.
it follows from [Sta13, Tag 07NT] that Ap → AProof of Proposition 5.1.1. By Corollary 3.10.18, we have a versal morphismτSpf Rrτ,BT /$ → Z/F.
τ
Since Z/F
is an algebraic stack of finite presentation over F (as Z τ is a $-adic
formal algebraic stack of finite presentation over Spf O), we may apply [Sta13, Tag
τ0DR0] to this morphism so as to find a smooth morphism V → Z/Fwith source a0
finite type O/$-scheme, and a point v ∈ V with residue field F , such that there
τ,BTτ
bV,v ∼
is an isomorphism O.
= Rr /$, compatible with the given morphism to Z/F
Proposition 5.1.2 and Lemma 5.1.5 taken together show that V is generically reduced,
and so the result follows from Lemma 5.1.6.


5.2. The geometric Breuil–Mézard conjecture. We now study the irreducible
components of Z dd,1 . We do this by a slightly indirect method, defining certain formal sums of these irreducible components which we then compute via the geometric
Breuil–Mézard conjecture, and in particular the results of Appendix C.
By Lemma 3.9.8 and Proposition 3.10.19, Z dd,1 is reduced and equidimensional,
and each Z τ,1 is a union of some of its irreducible components. Let K(Z dd,1 ) be thefree abelian group generated by the irreducible components of Z dd,1 . We say that

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

an element of K(Z dd,1 ) is effective if the multiplicity of each irreducible component
is nonnegative. We say that an element of K(Z dd,1 ) is reduced and effective if themultiplicity of each irreducible component is 0 or 1.
Let x be a finite type point of Z dd,1 , corresponding to a representation r : GK →GL2 (F0 ). As in Section 3.10, there is a quotient Rx of the framed deformation ring
[0,1]
Rr /$ which is a versal ring to Z dd,1 at x; each Rτ,1 is a quotient of Rx . Indeed,since Z τ,1 is a union of irreducible components of Z dd,1 , Spec Rτ,1 is a union of
irreducible components of Spec Rx .
Let K(Rx ) be the free abelian group generated by the irreducible components ofSpec Rx . By [Sta13, Tag 0DRB,Tag 0DRD], there is a natural multiplicity-preserving
surjection from the set of irreducible components of Spec Rx to the set of irreducible
components of Z dd,1 which contain x. Using this surjection, we can define a grouphomomorphism
K(Z dd,1 ) → K(Rx )in the following way: we send any irreducible component Z of Z dd,1 which contains x
to the formal sum of the irreducible components of Spec Rx in the preimage of Z
under this surjection, and we send every other irreducible component to 0.
Lemma 5.2.1. An element T of K(Z dd,1 ) is effective if and only if for every finite
type point x of Z dd,1 , the image of T in K(Rx ) is effective. We have T = 0 if andonly if its image is 0 in every K(Rx ).
Proof. The “only
P if” direction is trivial, so we need only consider the “if” implication.
Write T = Z aZ Z, where the sum runs over the irreducible components Z of
Z dd,1 , and the aZ are integers.
Suppose first that the image of T in K(Rx ) is effective; we then have to show thateach aZ is nonnegative. To see this, fix an irreducible component Z, and choose x
to be a finite type point of Z dd,1 which is contained in Z and in no other irreducible
component of Z dd,1 . Then the image of T in K(Rx ) is equal to aZ times the sumof the irreducible components of Spec Rx . By hypothesis, this must be effective,
which implies that aZ is nonnegative, as required.
Finally, if the image of T in K(Rx ) is 0, then aZ = 0; so if this holds for all x,then T = 0.
For each tame type τ , we let Z(τ ) denote the formal sum of the irreducible
components of Z τ,1 , considered as an element of K(Z dd,1 ). By Lemma C.1, for
eachP
non-Steinberg Serre weight σ of GL2 (k), there are integers nτ (σ) such thatσ = τ nτ (σ)σ(τ ) in the Grothendieck group of mod p representations of GL2 (k),where the τ run over the tame types. We set
X
Z(σ) :=nτ (σ)Z(τ ) ∈ K(Z dd,1 ).
τ

The integers nτ (σ) are not necessarily unique, but it follows from the following
result that Z(σ) is independent of the choice of nτ (σ), and is reduced and effective.
Theorem 5.2.2.
(1) Each Z(σ) is an irreducible component of Z dd,1 .
(2) The finite type points of Z(σ) are precisely the representations r : GK →GL2 (F0 ) having σ as a Serre weight. P(3) For each tame type τ , we have Z(τ ) = σ∈JH(σ(τ )) Z(σ).

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111(4) Every irreducible component of Z dd,1 is of the form Z(σ) for some unique
Serre weight σ.
(5) For each tame type τ , and each J ∈ Pτ , we have Z(σ J ) = Z(J).
Proof. Let x be a finite type point of Z dd,1 corresponding to r : GK → GL2 (F0 ),and write Z(σ)x , Z(τ )x for the images in K(Rx ) of Z(σ) and Z(τ ) respectively.
Each Spec Rτ,1 is a closed subscheme of Spec R , the universal framed deformation
OE 0 -algebra for r, so we may regard the Z(τ )x as formal sums (with multiplicities)of irreducible subschemes of Spec R /π.
By definition, Z(τ )x is just the underlying cycle of Spec Rτ,1 . By Proposition 5.1.1,
this is equal to the underlying cycle of Spec Rrτ,BT /$. Consequently, Z(σ)x is thecycle denoted by Cσ in Appendix C. It follows from Theorem C.4 that:• Z(σ)x is effective, and is nonzero precisely
P when σ is a Serre weight for r.
• For each tame type τ , we have Z(τ )x = σ∈JH(σ(τ )) Z(σ)x .
Applying Lemma 5.2.1, we see that each Z(σ) is effective, and that (3) holds. Since
Z τ,1 is reduced, Z(τ ) is reduced and effective, so it follows from (3) that each Z(σ) is
reduced and effective. Since x is a finite type point of Z(σ) if and only if Z(σ)x 6= 0,
we have also proved (2).
Since every irreducible component of Z dd,1 is an irreducible component ofsome Z τ,1 , in order to prove (1) and (4) it suffices to show that for each τ , every irreducible component of Z τ,1 is of the form Z(σ J ) for some J, and that each Z(σ J ) is
irreducible. Now, by Corollary 4.8.3, we know that Z τ,1 has exactly #Pτ irreducible
components, namely the Z(J 0 ) for J 0 ∈ Pτ . On the other hand, the Z(σ J ) are
reduced and effective, and since there certainly exist representations admitting σ J
as their unique Serre weight, it follows from (2) that for each J, there must be
a J 0 ∈ Pτ such that Z(J 0 ) contributes to Z(σ J ), but not to any Z(σ J 00 ) for J 00 6= J.
Since Z(τ ) is reduced and effective, and the sum in (3) is over #Pτ weights σ,
it follows that we in fact have Z(σ J ) = Z(J 0 ). This proves (1) and (4), and toprove (5), it only remains to show that J 0 = J. To see this, note that by (2),Z(σ J ) = Z(J 0 ) has a dense open substack whose finite type points have σ J as
their unique non-Steinberg Serre weight (namely the complement of the union of
the Z(σ 0 ) for all σ 0 =6 σ J ). By Theorem 4.6.17, it also has a dense open substack
whose finite type points have σ J 0 as a Serre weight. Considering any finite typepoint in the intersection of these dense open substacks, we see that σ J = σ J 0 , sothat J = J 0 , as required.

Appendix A. Formal algebraic stacks
In this appendix we briefly recall some basic definitions and facts concerning
formal algebraic stacks; our primary reference is [Eme]. We then develop some
simple geometric lemmas which will be applied in the main body of the paper.
We first recall the definition of a formal algebraic stack [Eme, Def. 5.3].
Definition A.1. An fppf stack in groupoids X over a scheme S is called a formal
algebraic stack if there is a morphism U → X , whose domain U is a formal algebraic
space over S (in the sense of [Sta13, Tag 0AIL]), and which is representable byalgebraic spaces, smooth, and surjective.
We will be primarily interested in the case when S = Spec O, where, as in themain body of the paper, O is the ring of integers in a finite extension E of Qp . We

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let $ denote a uniformiser of O, and let Spf O denote the affine formal scheme (oraffine formal algebraic space, in the terminology of [Sta13]) obtained by $-adicallycompleting Spec O.
Among all the formal algebraic stacks over Spec O, we single out the $-adic
formal algebraic stacks as being of particular interest. The following definition is aparticular case of [Eme, Def. 7.6].
Definition A.2. A formal algebraic stack X over Spec O is called $-adic if thecanonical map X → Spec O factors through Spf O, and if the induced map X →Spf O is algebraic, i.e. representable by algebraic stacks (in the sense of [Sta13, Tag
06CF] and [Eme, Def. 3.1]).
We refer to [Eme] for the various other notions related to formal algebraic stacks
that we employ. We also recall the following key lemma, which allows us to recognisecertain Ind-algebraic stacks as being formal algebraic stacks [Eme, Lem. 6.3].
Lemma A.3. If X1 ,→ X2 ,→ · · · ,→ Xn ,→ · · · is a sequence of finite orderthickenings of algebraic stacks, then limn Xn is a formal algebraic stack.
−→A.4. Open and closed substacks. Suppose first that X is an algebraic stack oversome scheme S. An open substack of X is then, by definition ([Sta13, Tag 04YM]),
a strictly full substack X 0 such that the natural morphism X 0 → X is an open
immersion; that is, it is representable by algebraic spaces, and an open immersion
on all pullbacks to algebraic spaces. A closed substack is defined in the analogous
way. The stack X has an underlying topological space |X |, and the open subsets of|X | are in natural bijection with the open substacks of X by [Sta13, Tag 06FJ].
We now note that the preceding definitions of open and closed substacks in fact
apply perfectly well to a stack over S which is not assumed to be algebraic (seee.g. [Eme, Def. 3.26]). In particular, we can apply it in the case when X is a formal
algebraic stack.
We make an important observation: if X ,→ X 0 is a morphism of stacks over S
which is representable by algebraic spaces and is a thickening (in the sense of [Sta13,
Tag 0BPN]), then pull-back under this morphism induces a bijection between open
substacks of X 0 and open substacks of X . (If U → X is an open immersion, and T
is an S-scheme, we define
U 0 (T ) := {T → X 0 | the base-changed morphism X ×X 0 T → X factors through U}.
We leave it to the reader to check that U 0 is an open substack of X 0 , and that
U 7→ U 0 and U 0 7→ X ×X 0 U 0 are mutually inverse; see also [Eme, Lem. 3.41], where
this result is established in the context of a more general statement about the
topological invariance of the étale site.)
Finally, we say that an open substack U of a formal algebraic stack X is dense if
its underlying topological space |U| is dense in |X |. Note that this need not imply
that it is scheme-theoretically dense (as is already the case if X is a non-reduced
scheme).

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113

A.5. A geometric situation. We suppose given a commutative diagram of morphisms of formal algebraic stacks
X

/Y
" 
Spf O

We suppose that each of X and Y is quasi-compact and quasi-separated, and that thehorizontal arrow is scheme-theoretically dominant, in the sense of [Eme, Def. 6.13].
We furthermore suppose that the morphism X → Spf O realises X as a finite type$-adic formal algebraic stack.
Concretely, if we write X a := X ×O O/$a , then each X a is an algebraic stack,∼locally of finite type over Spec O/$a , and there is an isomorphism lima X a −→ X .
−→Furthermore, the assumption that the horizontal arrow is scheme-theoreticallydominant means that we may find an isomorphism Y ∼Y a , with each Y a
= lim
−→a
being a quasi-compact and quasi-separated algebraic stack, and with the transition
morphisms being thickenings, such that the morphism X → Y is induced by acompatible family of morphisms X a → Y a , each of which is scheme-theoreticallydominant. (The Y a are uniquely determined by the requirement that for all b ≥ alarge enough so that the morphism X a → Y factors through Y ⊗O O/$b , Y a is thescheme-theoretic image of the morphism X a → Y ⊗O O/$b . In particular, Y a is a
closed substack of Y ×O O/$a .)It is often the case, in the preceding situation, that Y is also a $-adic formal
algebraic stack. For example, we have the following result. (Note that the usual graphargument shows that the morphism X → Y is necessarily algebraic, i.e. representableby algebraic stacks, in the sense of [Sta13, Tag 06CF] and [Eme, Def. 3.1]. Thus itmakes sense to speak of it being proper, following [Eme, Def. 3.11].)
Proposition A.6. Suppose that the morphism X → Y is proper, and that Y is
locally Ind-finite type over Spec O (in the sense of [Eme, Rem. 8.30]). Then Y is a$-adic formal algebraic stack.
Proof. This is an application of [Eme, Prop. 10.5].



A key point is that, because the formation of scheme-theoretic images is not
generally compatible with non-flat base-change, the closed immersion(A.7)

Y a ,→ Y ×O O/$a

is typically not an isomorphism, even if Y is a $-adic formal algebraic stack. Our
goal in the remainder of this discussion is to give a criterion (involving the morphism
X → Y) on an open substack U ,→ Y which guarantees that the closed immersion
U ×Y Y a ,→ U ×O O/$a induced by (A.7) is an isomorphism.
We begin by establishing a simple lemma. For any a ≥ 1, we have the 2commutative diagram
(A.8)

Xa

/ Ya


X


/Y

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

Similarly, if b ≥ a ≥ 1, then we have the 2-commutative diagram
/ Ya
(A.9)
Xa

Xb


/ Yb

Lemma A.10. Each of the diagrams (A.8) and (A.9) is 2-Cartesian.
Proof. We may embed the diagram (A.8) in the larger 2-commutative diagramXa/ Y a (A.7) / Y ⊗O O/$a


X


/Y


Y

Since the outer rectangle is manifestly 2-Cartesian, and since (A.7) is a closed
immersion (and thus a monomorphism), we conclude that (A.8) is indeed 2-Cartesian.
A similar argument shows that (A.9) is 2-Cartesian.

We next note that, since each of the closed immersions Y a ,→ Y is a thickening,
giving an open substack U ,→ Y is equivalent to giving an open substack U a ,→ Y a
for some, or equivalently, every, choice of a ≥ 1; the two pieces of data are related∼by the formulas U a := U ×Y Y a and lima U a −→ U.
−→
Proposition A.11. Suppose that X → Y is proper. If U is an open substack of Y,
then the following conditions are equivalent:(1) The morphism X ×Y U → U is a monomorphism.
(2) The morphism X ×Y U → U is an isomorphism.
(3) For every a ≥ 1, the morphism X a ×Y a U a → U a is a monomorphism.
(4) For every a ≥ 1, the morphism X a ×Y a U a → U a is an isomorphism.
(5) For some a ≥ 1, the morphism X a ×Y a U a → U a is a monomorphism.
(6) For some a ≥ 1, the morphism X a ×Y a U a → U a is an isomorphism.
Furthermore, if these equivalent conditions hold, then the closed immersion U a ,→U ×O O/$a is an isomorphism, for each a ≥ 1.
Proof. The key point is that Lemma A.10 implies that the diagram/ Ua
X a ×Y a U a
X ×Y U/U

is 2-Cartesian, for any a ≥ 1, and similarly, that if b ≥ a ≥ 1, then the diagram/ Ua
X a ×Y a U a
X b ×Y b U b/ Ub

is 2-Cartesian. Since the vertical arrows of this latter diagram are finite orderthickenings, we find (by applying the analogue of [Sta13, Tag 09ZZ] for algebraic
stacks, whose straightforward deduction from that result we leave to the reader) that

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115

the top horizontal arrow is a monomorphism if and only if the bottom horizontal
arrow is. This shows the equivalence of (3) and (5). Since the morphism X ×Y U → U
is obtained as the inductive limit of the various morphisms X a ×Y a U a → U a , wefind that (3) implies (1) (by applying e.g. [Eme, Lem. 4.11 (1)], which shows that theinductive limit of monomorphisms is a monomorphism), and also that (4) implies (2)(the inductive limit of isomorphisms being again an isomorphism).
Conversely, if (1) holds, then the base-changed morphism
X ×Y (U ×O O/$a ) → U ×O O/$a
is a monomorphism. The source of this morphism admits an alternative description
as X a ×Y U, which the 2-Cartesian diagram at the beginning of the proof allows us
to identify with X a ×Y a U a . Thus we obtain a monomorphism
X a ×Y a U a ,→ U ×O O/ $a .
Since this monomorphism factors through the closed immersion U a ,→ U ×O
O/$a , we find that each of the morphisms of (3) is a monomorphism; thus (1)implies (3). Similarly, (2) implies (4), and also implies that the closed immersion
U a ,→ U ×O O/$a is an isomorphism, for each a ≥ 1.
Since clearly (4) implies (6), while (6) implies (5), to complete the proof of the
proposition, it suffices to show that (5) implies (6). Suppose then that X a ×Y a
U a → U a is a monomorphism. Since U a ,→ Y a is an open immersion, it is inparticular flat. Since X a → Y a is scheme-theoretically dominant and quasi-compact(being proper), any flat base-change of this morphism is again scheme-theoreticallydominant, as well as being proper. Thus we see that X a ×Y a U a → U a is a schemetheoretically dominant proper monomorphism, i.e. a scheme-theoretically dominantclosed immersion, i.e. an isomorphism, as required.

Appendix B. Serre weights and tame types
We begin by recalling some results from [Dia07] on the Jordan–Hölder factors of
the reductions modulo p of lattices in principal series and cuspidal representationsof GL2 (k), following [EGS15, §3] (but with slightly different normalisations thanthose of loc. cit.).
Let τ be a tame inertial type. Recall from Section 1.7 that we associate a
representation σ(τ ) of GL2 (OK ) to τ as follows: if τ ' η ⊕ η 0 is a tame principal
GL (O )
series type, then we set σ(τ ) := IndI 2 K η 0 ⊗ η, while if τ = η ⊕ η q is a tamecuspidal type, then σ(τ ) is the inflation to GL2 (OK ) of the cuspidal representationof GL2 (k) denoted by Θ(η) in [Dia07]. (Here we have identified η, η 0 with theircomposites with ArtK .)Write σ(τ ) for the semisimplification of the reduction modulo p of (a GL2 (OK )stable O-lattice in) σ(τ ). The action of GL2 (OK ) on σ(τ ) factors through GL2 (k),so the Jordan–Hölder factors JH(σ(τ )) of σ(τ ) are Serre weights. By the resultsof [Dia07], these Jordan–Hölder factors of σ(τ ) are pairwise non-isomorphic, andare parametrised by the set Pτ (defined in Section 4.5.6) in a fashion that we now
recall.
Suppose first that τ = η ⊕ η 0 is a tame principal series type. Set f 0 = f in thiscase. We define 0 ≤ γi ≤ p − 1 (for i ∈ Z/f Z) to be the unique integers not all equal
Qf −1
to p − 1 such that η(η 0 )−1 = i=0 ωσγii . If instead τ = η ⊕ η 0 is a cuspidal type,set f 0 = 2f . We define 0 ≤ γi ≤ p − 1 (for i ∈ Z/f 0 Z) to be the unique integers

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Qf 0 −1such that η(η 0 )−1 = i=0 ωσγi0 . Here the σi0 are the embeddings l → F, where l isithe quadratic extension of k, σ00 is a fixed choice of embedding extending σ0 , and0(σi+1)p = σi0 for all i.
If τ is scalar then we set Pτ = {∅}. Otherwise we have η =6 η 0 , and as in0
Section 4.5.6 we let Pτ be the collection of subsets J ⊂ Z/f Z satisfying theconditions:• if i − 1 ∈ J and i ∈/ J then γi 6= p − 1, and• if i − 1 ∈/ J and i ∈ J then γi 6= 0
and, in the cuspidal case, satisfying the further condition that i ∈ J if and only if
i + f 6∈ J.
The Jordan–Hölder factors of σ(τ ) are by definition Serre weights, and areparametrised by Pτ as follows (see [EGS15, §3.2, 3.3]). For any J ⊆ Z/f 0 Z, we letδJ denote the characteristic function of J, and if J ∈ Pτ we define sJ,i by(
p − 1 − γi − δJ c (i) if i − 1 ∈ J
sJ,i =
γi − δJ (i)if i − 1 ∈/ J,and we set tJ,i = γi + δJ c (i) if i − 1 ∈ J and 0 otherwise.
In the principal series case we let σ(τ )J := σ~t,~s ⊗ η 0 ◦ det; the σ(τ )J are precisely
the Jordan–Hölder factors of σ(τ ).
In the cuspidal case, one checks that sJ,i = sJ,i+f for all i, and also that the
Qf 0 −1character η 0 · i=0 (σi0 )tJ,i : l× → F× factors as θ ◦ Nl/k where Nl/k is the norm
map. We let σ(τ )J := σ 0,~s ⊗ θ ◦ det; the σ(τ )J are precisely the Jordan–Hölder
factors of σ(τ ).
Remark B.1. The parameterisations above are easily deduced from those given in
GL (O )
[EGS15, §3.2, 3.3] for the Jordan–Hölder factors of the representations IndI 2 K η 0 ⊗η and Θ(η). (Note that there is a minor mistake in [EGS15, §3.1]: since the conventions of [EGS15] regarding the inertial Langlands correspondence agree with thoseof [GK14], the explicit identification of σ(τ ) with a principal series or cuspidal typein [EGS15, §3.1] is missing a dual. The explicit parameterisation we are using here
is of course independent of this issue.
This mistake has the unfortunate effect that various explicit formulae in [EGS15,§7] need to be modified in a more or less obvious fashion; note that since σ(τ ) is self
dual up to twist, all formulae can be fixed by making twists and/or exchanging ηand η 0 . In particular, the definition of the strongly divisible module before [EGS15,
Rem. 7.3.2] is incorrect as written, and can be fixed by either reversing the rolesof η, η 0 or changing the definition of the quantity c(j) defined there.)
Remark B.2. In the cuspidal case, write η in the form (σ00 )(q+1)b+1+c where 0 ≤ b ≤q − 2, 0 ≤ c ≤ q − 1. Set t0J,i = tJ,i+f for integers 1 ≤ i ≤ f . Then one can check(q+1)b+δJ (0)that σ(τ )J = σ~t0 ,~s ⊗ (σ0◦ det).

We now recall some facts about the set of Serre weights W (r) associated to arepresentation r : GK → GL2 (Fp ).
Definition B.3. We say that a crystalline representation r : GK → GL2 (Qp )has type σ~t,~s provided that for each embedding σj : k ,→ F there is an embeddingσ
ej : K ,→ Qp lifting σj such that the σej -labeled Hodge–Tate weights of r are{−sj − tj , 1 − tj }, and the remaining (e − 1)f pairs of Hodge–Tate weights of r are

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117

all {0, 1}. (In particular the representations of type σ~0,~0 (the trivial weight) are thesame as those of Hodge type 0.)
Definition B.4. Given a representation r : GK → GL2 (Fp ) we define W (r) to be
the set of Serre weights σ such that r has a crystalline lift of type σ.
It follows easily from the formula σ~∨= σ −~s−~t,~s that σ ∈ W (r) if and only if σ ∨t,~sis in the set of Serre weights associated to r∨ in [GLS15, Defn. 4.1.3].
There are several definitions of the set W (r) in the literature, which by the
papers [BLGG13, GK14, GLS15] are known to be equivalent (up to normalisation).
While the preceding definition is perhaps the most compact, it is the description of
W (r) via the Breuil–Mézard conjecture that appears to be the most amenable togeneralisation; see Theorem C.4 for an instance of this description, and [GHS18] formuch more discussion.
Recall that r is très ramifiée if it is a twist of an extension of the trivial characterby the mod p cyclotomic character, and if furthermore the splitting field of its
1/p
1/p
×
projective image is not of the form K(α1 , . . . , αs ) for some α1 , . . . , αs ∈ OK.
Lemma B.5.
(1) If τ is a tame type, then r has a potentially Barsotti–Tatelift of type τ if and only if W (r) ∩ JH(σ(τ )) 6= 0.
(2) The following conditions are equivalent:(a) r admits a potentially Barsotti–Tate lift of some tame type.
(b) W (r) contains a non-Steinberg Serre weight.
(c) r is not très ramifiée.
Proof. (1) By the main result of [GLS15], and bearing in mind the differences between
our conventions and those of [GK14] as recalled in Section 1.7, we have σ ∈ W (r)if and only if σ ∨ ∈ W BT (r), where W BT (r) is the set of weights defined in [GK14,§3]. By [GK14, Cor. 3.5.6] (bearing in mind once again the differences between ourconventions and those of [GK14]), it follows that we have W (r) ∩ JH(σ(τ )) 6= 0
if and only if e(R,0,τ /π) 6= 0 in the notation of loc. cit., and by definition r has
a potentially Barsotti–Tate lift of type τ if and only if R,0,τ 6= 0. It follows
from [EG14, Prop. 4.1.2] that R,0,τ 6= 0 if and only if e(R,0,τ /π) 6= 0, as required.
(2) By part (1), condition (a) is equivalent to W (r) containing a Serre weightoccurring as a Jordan–Hölder factor of σ(τ ) for some tame type τ . It is easily seen(either by inspection, or by Lemma C.1 below) that the Serre weights occurring as
Jordan–Hölder factors of the σ(τ ) are precisely the non-Steinberg Serre weights,
so (a) and (b) are equivalent.
Suppose that (a) holds; then r becomes finite flat over a tame extension. However
the restriction to a tame extension of a très ramifiée representation is still trèsramifiée, and therefore not finite flat, so (c) also holds. Conversely, suppose forthe sake of contradiction that (c) holds, but that (b) does not hold, i.e. that W (r)consists of a single Steinberg weight.
Twisting, we can without loss of generality assume that W (r) = {σ~0,p−1~ }.
By [GK14, Cor. A.5] we can globalise r, and then the hypothesis that W (r) contains
σ~0,p−1~ implies that it has a semistable lift of Hodge type 0. If this lift were in fact
crystalline, then W (r) would also contain the weight σ~0,~0 by (1). So this lift is
not crystalline, and in particular the monodromy operator N on the corresponding
weakly admissible module is nonzero. But then ker(N ) is a free filtered submoduleof rank 1, and since the lift has Hodge type 0, ker(N ) is in fact a weakly admissible

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submodule. It follows that the lift is an unramified twist of an extension of ε−1
by the trivial character, so that r is an unramified twist of an extension of ε−1 by
the trivial character. But we are assuming that (c) holds, so r is finite flat, so thatby (1), W (r) contains the weight σ~0,~0 , a contradiction.

Lemma B.6. Suppose that σ~t,~s is a non-Steinberg Serre weight. Suppose that r :GK → GL2 (Fp ) is a reducible representation satisfying!
Qf −1 si +tiω∗
i=0 σiQf −1
r|IK ∼,=
0ε−1 i=0 ωσtii
and that r is not très ramifiée. Then σ~t,~s ∈ W (r).
Proof. Write r as an extension of characters χ by χ0 . It is straightforward fromthe classification of crystalline characters as in [GHS18, Lem. 5.1.6] that there exist
crystalline lifts χ, χ0 of χ, χ0 so that χ, χ0 have Hodge–Tate weights 1 − tj and−sj − tj respectively at one embedding lifting each σj and Hodge–Tate weights 1
and 0 respectively at the others. In the case that r is not the twist of an extensionof ε−1 by 1 the result follows because the corresponding Hf1 (GK , χ0 ⊗ χ−1 ) agrees
with the full H 1 (GK , χ0 ⊗ χ−1 ) (as a consequence of the usual dimension formulasfor Hf1 , [Nek93, Prop. 1.24]).
If r is twist of an extension of ε−1 by 1, the assumption that σ~t,~s is non-Steinberg
implies sj = 0 for all j. The hypothesis that r is not très ramifiée guarantees
Qf −1
that r ⊗ i=0 ωσ−ti i is finite flat, so has a Barsotti–Tate lift, and we deduce that
σ~t,~0 ∈ W (r).
Appendix C. The geometric Breuil–Mézard Conjecture forpotentially Barsotti–Tate representations
In this appendix, by combining the methods of [EG14] and [GK14] we prove a
special case of the geometric Breuil–Mézard conjecture [EG14, Conj. 4.2.1]. This
result is “globalised” in Section 5.
Let K/Qp be a finite extension, and let E/Qp be another finite extension, withring of integers O, uniformiser $, and residue field F. We assume that E issufficiently large, and in particular that E contains K. Let r : GK → GL2 (F)be a continuous representation, and let Rr be the universal framed deformation
O-algebra for r. For each tame type τ , let Rr,0,τ
be the reduced and p-torsion freequotient of Rr whose Qp -points correspond to the potentially Barsotti–Tate liftsof r of type τ . In Section 5 we denote this ring by Rrτ,BT , but we use the more
cumbersome notation Rr,0,τ
here to make it easier for the reader to refer to [EG14]and [GK14].

By [EG14, Prop. 4.1.2], Rr,0,τ
/$ is zero if r has no potentially Barsotti–Tatelifts of type τ , and otherwise it is equidimensional of dimension 4 + [K : Qp ]. EachSpec Rr,0,τ
/$ is a closed subscheme of Spec Rr /$, and we write Z(Rr,0,τ
/$) forthe corresponding cycle, as in [EG14, Defn. 2.2.5]. (This is a formal sum of theirreducible components of Spec Rr,0,τ
/$, weighted by the multiplicities with which
they occur.)MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

119

Lemma C.1. If σ is a non-Steinberg
Serre weight of GL2 (k), then there areP
integers nτ (σ) such that σ = τ nτ (σ)σ(τ ) in the Grothendieck group of mod p
representations of GL2 (k), where the τ run over the tame types.
Proof. This is an immediate consequence of the surjectivity of the natural map
from the Grothendieck group of Qp -representations of GL2 (k) to the Grothendieckgroup of Fp -representations of GL2 (k) [Ser77, §III, Thm. 33], together with theobservation that the reduction of the Steinberg representation of GL2 (k) is precisely
σ~0,p−1~ .
Let σ be a non-Steinberg Serre weight of GL2 (k), so that by Lemma C.1 we can
write
X
(C.2)
σ=
nτ (σ)σ(τ )τin the Grothendieck group of mod p representations of GL2 (k). Note that the
integers nτ (σ) are not uniquely determined; however, all our constructions elsewherein this paper will be (non-obviously!) independent of the choice of the nτ (σ). We
also writeXσ(τ ) =mσ (τ )σ;σ

since σ(τ ) is multiplicity-free, each mσ (τ ) is equal to 0 or 1. Then!
X X
σ=nτ (σ)mσ0 (τ ) σ 0 ,σ0τand therefore
(C.3)

X

nτ (σ)mσ0 (τ ) = δσ,σ0 .

τFor each non-Steinberg Serre weight σ, we set
X

Cσ :=
nτ (σ)Z(Rr,0,τ
/$),τ

where the sum ranges over the tame types τ , and the integers nτ (σ) are as in (C.2).
By definition this is a formal sum with (possibly negative) multiplicities of irreducible subschemes of Spec Rr /$; recall that we say that it is effective if all of themultiplicities are non-negative.
Theorem C.4. Let σ be a non-Steinberg Serre weight. Then the cycle Cσ is effective,
and is nonzero precisely when σ ∈ W (r). It is independent of the choice of integersnτ (σ) satisfying (C.2). For each tame type τ , we haveX

Z(Rr,0,τ/$) =Cσ .
σ∈JH(σ(τ ))Proof. We will argue exactly as in the proof of [EG14, Thm. 5.5.2] (taking n = 2),
and we freely use the notation and definitions of [EG14]. Since p > 2, we have p - n
and thus a suitable globalisation ρ exists provided that [EG14, Conj. A.3] holds
for r. Exactly as in the proof of [EG14, Thm. 5.5.4], this follows from the proofof Theorem A.1.2 of [GK14] (which shows that r has a potentially Barsotti–Tatelift) and Lemma 4.4.1 of op.cit. (which shows that any potentially Barsotti–Taterepresentation is potentially diagonalizable). These same results also show that the

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A. CARAIANI, M. EMERTON, T. GEE, AND D. SAVITT

equivalent conditions of [EG14, Lem. 5.5.1] hold in the case that λv = 0 for all v,
and in particular in the case that λv = 0 and τv is tame for all v, which is all that
we will require.
By [EG14, Lem. 5.5.1(5)], we see that for each choice of tame types τv , we haveX Y
0mσv (τv )Z⊗(C.5)
Z(R∞ /$) =(ρ).
v|p σ v
⊗v|p σ v v|pNow, by definition we have
(C.6)
Y

Z(R̄∞ /$) =Z(Rr,0,τ
/$) × Z(F[[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]]).
v
v|p

Fix a non-Steinberg
Serre weight σ = ⊗v σ v , and sum over all choices of types τv ,Q
weighted by v|p nτv (σ v ). We obtainY
XY/$) × Z(F[[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]])nτv (σ v )Z(Rr,0,τ
v
τ

v|p

v|p

=

XY
τ

nτv (σ v )X Y0
mσ0v (τv )Z⊗0 (ρ)
v|p σ v

⊗v|p σ 0v v|pv|pwhich by (C.3) simplifies to
Y
0
(ρ).
(C.7)
Cσv × Z(F[[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]]) = Z⊗v|p σ vv|p0
Since Z⊗(ρ) is effective by definition (as it is defined as a positive multiple ofv|p σ vQ
the support cycle of a patched module), this shows that every v|p Cσv is effective.
We conclude that either every Cσ is effective, or that every −Cσ is effective.
Substituting (C.7) and (C.6) into (C.5), we see that
Y

(C.8)
Z(Rr,0,τ
/$) × Z(F[[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]])
v
v|p




=

X

Y

v|p

Cσv  × Z(F[[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]],σ∈JH(σ(τ ))Pand we deduce that either Z(Rr,0,τ
/$) = σ mσ (τ )Cσ for all τ , or Z(Rr,0,τ
/$) =P− σ mσ (τ )Cσ for all τ .
/$) is effective, the second possibility holds if and only if
Since each Z(Rr,0,τ
every −Cσ is effective (since either all the −Cσ are effective, or all the Cσ areeffective). It remainsPto show that this possibility leads to a contradiction. Now,
if Z(Rr,0,τ
/$) = − σ mσ (τ )Cσ for all τ , then substituting into the definitionP/$), we obtain
Cσ = τ nτ (σ)Z(Rr,0,τ
!
X X
Cσ =
nτ (σ)mσ0 (τ ) (−Cσ0 ) ,σ0τand applying (C.3), we obtain Cσ = −Cσ , so that Cσ = 0 for all σ. Thus all the Cσ
are effective, as claimed.

MODULI STACKS OF TWO-DIMENSIONAL GALOIS REPRESENTATIONS

121

0Since Z⊗(ρ) by definition depends only on (the global choices in the Taylor–v|p σ vWiles method, and) ⊗v|p σ v , and not on the particular choice of the nτ (σ), it follows
from (C.7) that Cσ is also independent of this choice.
0
Finally, note that by definition Z⊗(ρ) is nonzero precisely when σ v is inv|p σ vBT
the set W (r) defined in [GK14, §3]; but by the main result of [GLS15], this isprecisely the set W (r).
Remark C.8.1. As we do not use wildly ramified types elsewhere in the paper, wehave restricted the statement of Theorem C.4 to the case of tame types; but thestatement admits a natural extension to the case of wildly ramified inertial types(with some components now occurring with multiplicity greater than one), and theproof goes through unchanged in this more general setting.
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Email address: a.caraiani@imperial.ac.uk
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Email address: emerton@math.uchicago.edu

Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago,
IL 60637, USA
Email address: toby.gee@imperial.ac.uk
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Email address: savitt@math.jhu.edu
Department of Mathematics, Johns Hopkins University