MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY
MATTHEW EMERTON AND TOBY GEEContents
Lecture 1. Introduction
Lecture 2. Stacks, etc.
Lecture 3. Definitions related to étale ϕ-modules
Lecture 4. Moduli stacks of ϕ-modules
Lecture 5. Stacks of (ϕ, Γ)-modules
Lecture 6. Crystalline and semistable moduli stacks
Lecture 7. The Herr Complex, and families of extensions
Lecture 8. Crystalline Lifts
Lecture 9. Geometric Breuil–Mézard
Lecture 10. Bernstein Centers, Moduli Spaces, and the Categorical p-adic
Langlands program
References

2
8
11
17
21
26
33
40
45
49
56

This survey article is based on 10 lectures that we gave at the September 2019
Hausdorff School in Bonn on “the Emerton–Gee stack and related topics”, andthe sections correspond to the original lectures. We have retained the somewhat
informal style of some of these lectures, but have filled them out with further details.
We hope that the first 9 lectures can serve as an extended introduction to [EG22];
the reader might first wish to read the actual introduction to [EG22], which inparticular motivates the results described here. We also refer to [EG22] for any
definitions not given in these lectures. The 10th lecture is a speculative look ahead
to connections between our stacks and the p-adic local Langlands correspondence,and in particular explains some of our work in progress with Andrea Dotto [DEG].
We would like to thank the organisers (Johannes Anschütz, Arthur-César Le
Bras, and Andreas Mihatsch) and participants of the Hausdorff school for the many
helpful questions and corrections resulting from our lectures. We would especiallylike to thank Ashwin Iyengar for transcribing M.E.’s lectures at the Hausdorff
school, and for allowing us to use his LaTeX files as a basis for some sections of thisarticle. Some material in Lecture 8 is based on the notes for a series of lectures that
TG gave at the ENS Lyon in June 2018, and we are grateful to the organisers (Laurent Berger, Gabriel Dospinescu, Philippe Gille, Wieslawa Niziol, Vincent Pilloni,
Sandra Rozensztajn, Amaury Thuillier) of the thematic trimester on “Algebraic
Groups and Geometrization of the Langlands Program” for the invitation to speak
The first author was supported in part by the NSF grants DMS-1601871, DMS-1902307, andDMS-2201242. The second author was supported in part by a Leverhulme Prize, EPSRC grant
EP/L025485/1, Marie Curie Career Integration Grant 303605, and by ERC Starting Grant 306326.
1

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M. EMERTON AND T. GEE

there, and the encouragement to write up notes for the lectures. We would also
like to thank Matteo Tamiozzo for some comments on this material.
TG would like to thank the organisers (Najmuddin Fakhruddin, Eknath Ghate,
Arvind Nair, C. S. Rajan, and Sandeep Varma) of the International Colloquium on
Arithmetic Geometry at TIFR in January 2020, for the invitation to speak at theconference, for their hospitality, and for the invitation to submit this article to theproceedings, without which it is unlikely that we would have completed it. Bothauthors thank the anonymous referee for their helpful comments and corrections.
Lecture 1. Introduction
As the title suggests, the paper [EG22] is about moduli stacks of (ϕ, Γ)-modules.
In fact, we’re really interested in the moduli of (local) (p-adic and mod p) Galois
representations, so we’ll first give some background on this problem, and the complications that arise. The theory of (ϕ, Γ)-modules and the construction of ourmoduli stacks will be the topic of subsequent lectures.
1.1. Galois representations. Let p be prime, and let K/Qp be a finite extension.
We will be interested in p-adic and mod p representations of the absolute Galois
group GK := Gal(K/K). In the mod p case, this means that we will fix some d ≥ 1
and study continuous homomorphisms
GK := Gal(K/K) → GLd (Fp ).
Note these factor through GLd (Fq ) for some q. Similarly we could look at continuous homomorphisms
GK → GLd (Qp );these will factor through GLd (E) for some finite E/Qp , and using compactness,they can be conjugated so as to factor through GLd (OE ).
We want to arrange these representations into algebraic families (this is moreor-less what it means to construct a moduli space of d-dimensional representationsof GK ). This was done by Carl Wang-Erickson in [WE18], but the topologicalnature of GK means that these families are “smaller” than one might hope. (Wediscuss the relationship between our stacks and Wang-Erickson’s in Lecture 10.2.)
We will elaborate on this, before turning to our “larger” families (which are definedby passing from GK -representations to (ϕ, Γ)-modules).
1.1.1. Example. The first example of an “algebraic family” we might consider is afamily of unramified characters×burx : GK → GK /IK ∼= FrobZ → Fp
×which sends Frobenius to x ∈ Fp (note this is well-defined because any ele×

ment of Fp lives in F×
q for some q, so we define the unramified character by×b
Z  Z/(q − 1)Z → Fq ). But this doesn’t quite come together as an algebraic
family: if we wanted to make this algebraic, we would let Fp [x, x−1 ]× be our ring×of coefficients and take specializations to Fp . We can define a homomorphism
FrobZ → Fp [x, x−1 ]× taking Frob 7→ x, but this doesn’t extend continuously to
FrobZ . So this gives the first basic obstruction to constructing the right moduli
space.
b

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

3

As we will explain in the rest of this section, to get around the problem in thebexample, you could consider the Weil group, which is a “decompletion” of FrobZ ;but ultimately, this only works well when either d = 1 or we work with coefficientrings in which p is invertible, so when we work with p-adic coefficients in the cased > 1 we have to use integral p-adic Hodge theory (and specifically the theory of(ϕ, Γ)-modules), as in [EG22].
1.2. Weil–Deligne Case. We begin by recalling the “Weil–Deligne formalism”,
which is a standard and effective approach to dealing with topological issues in thetheory of `-adic Galois representations of GK , when ` 6= p. As we will see, it is of
more dubious merit in the p-adic context. The ideas we explain are dealt with inmuch more detail (and for representations into more general groups than GLd ) in
[DHKM20] and [Zhu20].
Note the local Galois group GK has a tame quotient and pro-p wild inertia
subgroup:
PK ,→ GK  Gtame= (Z n Z[1/q])∧K
where 1 ∈ Z acts by multiplication by q on Z[1/q], and we take the profinite
completion. We can then define the “Weil–Deligne group” WDK as usual, whichfits into the following diagram:
0

PK

WDK

Z n Z[1/q]

0

0

PK

GK

(Z n Z[1/q])∧0

Then for any open and finite index subgroup Q ≤ PK , WDK /Q is a finitely presented discrete group, and
WDK = lim WDK /Q.
←−
Q

So we have essentially “decompleted and discretized” the tame part, but we areremembering the topology on the wild part of the Galois group. Furthermore,
GK /Q = (WDK /Q)∧ , and since representations of WDK /Q with values in finite
rings do extend over the profinite completion, this seems like a good first approachto defining a moduli space.1
1.2.1. Definition. Let VQ → Spec Z denote the scheme parameterizing representations ρ : WDK /Q → GLd .
Note that WDK /Q is a finitely presented group, so it’s easy to find a finite
presentation for VQ as an affine scheme. If Q0 ⊆ Q, then there is a natural closed
immersion VQ ,→ VQ0 (this is easy to check using the moduli description), so we
can study the Ind-scheme
V := lim VQ .
−→Q

By construction, any continuous ρ : GK → GLd (F) (for F any finite field) factors
through a finite quotient of GK , and thus a finite quotient of WDK , and thus
1One technical point is that this definition of WD , and thus the consequent definition of aK
moduli space, depend on choices: namely, a choice of a lift of Frobenius and also a choice of agenerator of tame inertia. We don’t dwell on this issue here, but note that Scholze and Zhu have
explained how to make the same construction more canonically (involving no auxiliary choices) inthe case of `-adic representations when ` 6= p [Zhu20, Props. 3.1.10, 3.1.11].

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M. EMERTON AND T. GEE

factors through one of the WDK /Q, so gives an F-valued point of V . Conversely
a representation WDK /Q → GLd (F) extends continuously to GK /Q and thus to
GK . Therefore
V (F) = {continuous ρ : GK → GLd (F)} .
If Rρ denotes the universal lifting ring of ρ one can check that there is a naturalmap
Spf Rρ → V.
In fact, this map is versal to V at ρ, and so V is some sort of geometric object
that joins together all the formal deformation theory of the various ρ. The nextquestion, then, is exactly what sort of geometric object is V ? While the VQ are
finitely presented affine schemes, their colimit V is more infinitary, and we need to
study what sort of geometry we can get when we take this sort of colimit.
So what does V look like? It could involve taking a countable disjoint union ofvarieties: this is still geometric, and it’s locally of finite presentation with countably
many components (recall each of the VQ are finitely presented). A slightly more
complicated possibility is taking a not-necessarily disjoint union of varieties, eachof which is an irreducible (though not necessarily connected) component of theirunion. This isn’t of finite presentation, but it is still a scheme, and it’s still quite
geometric.

Figure 1. An infinite union of connected components

Figure 2. An infinite union of irreducible components
But it could fail to be a scheme! For instance, take A1F and glue a copy of A1Fpp
to each closed point.

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

5

Figure 3. An Ind-scheme which is not a scheme
We refer the reader to [EG21, §4.3] for many more examples of Ind-schemes thataren’t schemes.
So if V is of one of the first two types, and is still a scheme, then we’d be ingood shape because we can do geometry, but otherwise we haven’t done much tosimplify our original problem. In fact, it really depends on the characteristic of ourcoefficient field: in characteristic p we will see that things behave badly, while incharacteristic 0, we will see that we get a well-behaved formal scheme.
1.3. Analysis of V when d = 1 and K = Qp . For simplicity, assume p > 2. In
this case, we use local class field theory to replace GQp with
b
d
×
×
Z
Gab
Qp = Qp = Zp × p .

Taking the Weil–Deligne subgroup gives
×
×
Z
WDab
Qp = Qp = Zp × p .

In this case, the pro-p wild ramification part is a copy of Zp , embedded in WDabQp
via 1 + pZp ,→ Z×
p . So we choose a sequence Q = Qn so thatn+1 ×
) × pZ ∼WDab= Z/pn Z × Z/(p − 1)Z × pZ .
Qp /Qn = (Z/pThusn

VQn = Spec Z[x]/(xp − 1) × Spec Z[x]/(xp−1 − 1) × Gm,Z .
(1) To see what happens away from p, we can invert p on this scheme and see
what we get:VZ[1/p] = lim VQn ,Z[1/p]−→nn= (lim Spec Z[1/p][x]/(xp − 1)) × Spec Z[1/p][x]/(xp−1 − 1) × Gm,Z[1/p] .
−→nn

If we now base change to k = Q or k = Fl for ` 6= p, then (xp − 1) is
separable, so we are just adding more and more closed points as we go
further along the directed system, and so we basically get infinitely many
copies of Spec Z[x]/(xp−1 − 1) × Gm , indexed by pth roots of unity in k.
(2) But instead if we base change to Zp , then the closed immersions VQn ,Zp ,→VQn+1 ,Zp are actually nilpotent thickenings, and in fact we end up with
VZp = Spf Zp [[T ]] × Spec Z[x]/(xp−1 − 1) × Gm .

6

M. EMERTON AND T. GEENow Spf Zp [[T ]] is a nice Noetherian formal scheme; be warned that thisis specific to dimension 1. Indeed already in dimension 2 the situation
becomes much more complicated, as we will see shortly.
Before turning to the case of dimensions 2 and higher, we give some more examples of Ind-schemes. First, here’s a formal scheme that isn’t Noetherian: Take A1Fp
and add a extra formal direction at each closed point on the line (one can do thisfinitely many times and then take a filtered colimit over the finite steps). This isan affine formal scheme Spf A for A some topological ring, but A is not Noetherian.
More precisely, we let Fp = {a0 , a1 , . . . , } and set A := limj Aj where←−
Aj = Fp [x] ×Fp Fp [[x − a0 ]] ×Fp · · · ×Fp Fp [[x − aj ]].
2
Now here’s something even worse. Consider the subsheaf of AF
given by takingp1
a “horizontal” AF
and adding a “vertical” A1F through each closed point, as inpp

Figure 3, and then formally thickening each of the vertical lines (in A2F ) in thephorizontal direction, as in Figure 4.

Figure 4. An Ind-scheme with thickened vertical lines
Then this is not even a formal scheme, and it’s actually hard to tell this apartfrom A2 , because it has the same closed points and versal rings.
We could even take the above construction and then delete the point at theorigin. Call this V? ; we will return to it shortly.
1.4. Analysis of V in general. For general d and general K, VQ /Z[1/p] is reduced, local complete intersection (so Cohen–Macaulay), and flat/Z[1/p] of relative
dimension d2 [DHKM20, Zhu20], and
VQ /Z[1/p] ,→ VQ0 /Z[1/p]is just adding connected components. Note in particular that the local deformation
rings of mod ` residual representations are just computed using this variety bylooking at the complete local rings at the stalks of the corresponding points.
Now let’s switch back to the characteristic p setting. Let K = Qp , d = 2, andassume that p > 2. Let’s just study the behaviour of V in characteristic p; let’s also
fix the determinant of our two-dimensional Galois representations to be ω i , whereω denotes the mod p cyclotomic character, and we choose 1 ≤ i ≤ p − 2. We write
det=ω iV/F
for the resulting Ind-scheme.
p
i

det=ω
Let’s begin by considering the Fp -valued points of V/F; these correspond topρ : WDQp → GL2 (Fp ) of determinant ω i . There is one irreducible such ρ up toQisomorphism, namely IndQp2 ω2i . Because of the choice of framing, this will notpMODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

7

i

det=ω
correspond to a literal single closed point of V/F, but rather to an SL2 orbit.2pThis orbit will be (the underlying reduced subscheme of) a connected component
det=ω iof V/F.
p
Any reducible ρ will be an extension of mod p characters of WDQp , and any suchcharacter is an unramified twist of a power of ω. Any family of such extensions will
induce a corresponding family of semi-simplifications (more precisely, of associatedpseudo-characters), and the powers of the cyclotomic character that appear will be
locally constant, and thus constant on a given connected component of the family.
×
We will restrict attention to those ρ for which ρss = urλ−1 ω i ⊕urλ , for some λ ∈ Fp ,and in fact to those ρ which are an extension of urλ by urλ−1 ω i (rather than anextension in the opposite order). Any such non-split extension is unique up toisomorphism (equivalently, the corresponding Ext1 is one-dimensional) unless i = 1
and λ = ±1; in this latter case there is a one-dimensional space of peu ramifiéeextensions, and it is the ρ classified by this space that we will consider.
There is actually a p-adic Hodge-theoretic framework that describes exactly
these ρ that we are singling out (the irreducible ρ, and the particular reducible ρ that
we have described); namely, they are Fontaine–Laffaille with Hodge–Tate weights{0, i} (and with determinant ω i ). And what we will now describe is the Ind-scheme
FL,det=ω i
V/Fwhich classifies families of WDQp -representations over finite type Fp p
algebras A that are two-dimensional, have determinant fixed to be ω i , and which are
Fontaine–Laffaille with Hodge weights {0, i} when pushed forward to any Artinian
quotient of A.
Actually, to avoid grappling with the details of framings, we will describe theFL,det=ω i
quotient stack3 [V/F
/SL2 ]. To state the answer, recall the subsheaf V? ofp
2
A/Fp described above. There is an action of Gm on this substack obtained by
restricting the following action of Gm on A2 :a · (x, y) = (x, a2 y).
1.4.1. Proposition. There is an isomorphism of stacks
i∼FL,det=ω[V/F/SL2 ] −→ [V? /Gm ].
p

Before sketching the proof, we note that V? is rather nasty. It is Zariski dense in
A2 \ {0} — indeed, it has the same set of Fp -points, and the same versal rings atthese points, as A2 \ {0} — but it is disconnected, and it looks more like a formal
scheme than an honest scheme like A2 \ {0} . Given this, it makes sense to ask: can
one study some related but different moduli problem so as to actually get A2 \ {0}(or, rather, the stack [(A2 \ {0})/Gm ]) as the answer? Well, you can! If you study
families of Fontaine–Laffaille modules (the “linear algebra perspective” in p-adic
Hodge theory) instead of representations then this is exactly what you get. The
inclusioni∼FL,det=ω[V/F/SL2 ] −→ [V? /Gm ] ,→ [(A2 \ {0})/Gm ]p∼2
Qp -representations ∧ ρ −→2“Fixing the determinant” means choosing an isomorphism of WD

ω i , and so the “change of frame” group is reduced from GL2 to SL2 .
3See the next lecture for more about stacks!

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M. EMERTON AND T. GEE

identifies the source with the locus in the Fontaine–Laffaille moduli stack over which
the Fontaine–Laffaille functor from Fontaine–Laffaille modules to Galois representations (or, more generally, to WDQp -representations) can be applied so as to obtaingenuine WDQp -representations.
Sketch of proof of Prop. 1.4.1. As already noted, both the source and the targetof the claimed isomorphism are disconnected: the source is the disjoint union of
the reducible
 and irreducible loci, while the target is the disjoint union of [ V? ∩(Gm × A1 ) /Gm ] and its complement (which equals [(Spf Fp [[x]] × Gm )/Gm ] =[Spf Fp [[x]]/µ2 ]). We check the isomorphism on each connected component separately. On the irreducible locus, it amounts to the fact that the mod p Fontaine–Q
Laffaille deformation ring of IndQp2 ω2i is isomorphic to Fp [[x]]. On the reducibleplocus, there is more to check, but the key point is that if ρ is a non-split extension of×
urλ by urλ−1 ω i , for some λ ∈ Fp , which is trivial on some given open subgroup Q
of PQp , then the degree of λ over Fp is bounded in terms of the index [PQp : Q]. iThis example of V FL,det=ω is illustrative of the overall behaviour of V : wheni
working over Fp or Zp , it is nasty as soon as d > 1. In the special case of V FL,det=ω ,the moduli space of Fontaine–Laffaille modules served as a much more pleasant
improvement. Fontaine–Laffaille theory doesn’t apply to general mod p or p-adic
Galois representations (even for Qp , but even more so if K/Qp is ramified). But
Fontaine’s theory of (ϕ, Γ)-modules does! And in fact the moduli space V does sit
inside a moduli space of (ϕ, Γ)-modules, and as we will see, this latter space is a
lot nicer. Thus we are motivated to define moduli stacks of (ϕ, Γ)-modules and doa thorough study of their geometry.
Lecture 2. Stacks, etc.
In this lecture we give a very brief introduction to the idea of a formal algebraic
stack; details can be found in [Eme].
2.1. Functors of points. There’s a tension between rings and topological spaces
in algebraic geometry, going back to Diophantus and Descartes. Grothendieck’s
formulation encompasses both, by giving you a topological space, but also giving
you a sheaf of rings. You then solve equations by studying morphisms of schemes.
To study algebraic spaces and stacks, it is helpful to take the functorial pointof view, which encompasses the theory of schemes, but gives you a framework toextend it. This perspective starts with the following observation.
2.1.1. Lemma (Yoneda, Grothendieck). A scheme X can be thought of as a (pre)sheafX
Sch −→ Set by taking Y 7→ Hom(Y, X). In fact it’s a Zariski/étale/fppf sheaf.
But we can consider more general sheaves. For instance consider a system X1 ,→X2 ,→ . . . with closed immersions as transition maps, and define a sheaf
X := lim Xi−→
iThis is an Ind-scheme. Often, this will not actually be a scheme: e.g. embed apoint in a line in a plane, in 3-space, etc. This gives you some “infinite dimensional
affine space”, which is not a scheme: note the identity map from this space to itself

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

9

doesn’t factor through one of the finite steps, whereas a map from a quasi-compact
scheme does always factor in this way.
This last remark illustrates one way that finiteness/quasi-compactness assumptions are used in our theory; they help us to detect finite-dimensional parts ofpossibly infinite-dimensional objects.
2.2. Algebraic Spaces. Another example of a more general sheaf is an algebraic
space. If X is an fppf sheaf, and U is a scheme and U → X is a morphism, thenwe say that the morphism U → X is “representable by schemes” if in any fibre
product diagram
T ×X UU

T
X
for which T is a scheme, then T ×X U is also a scheme. It usually suffices to checkthis for a certain subclass of T (e.g. affine schemes). You could also require thatthe morphism U → X is surjective or étale, which are both properties that one candefine via base change to schemes.
2.2.1. Definition. An algebraic space is an fppf sheaf X which admits a map U → X
whose source is a scheme and which is representable by schemes, surjective, andétale.
There is an obvious notion of an Ind-algebraic space.
2.3. Formal algebraic spaces. We want to be able to talk about formal algebraic
spaces, and formal algebraic stacks.
Suppose A is a complete topological ring with a countable basis of open neighborhoods of 0 consisting of ideals In . Assume that every A/In+1 → A/In is a
nilpotent thickening. Then we define
Spf A = lim Spec A/In .
−→For instance if A = Zp and In = pn , then we get Spf Zp . This is a proper subsheaf ofSpec Zp . Interestingly, properties like flatness, reducedness, Cohen–Macaulayness
need the ring A, and can’t be seen on the quotients A/In . For instance, Zp is
reduced, while each Z/pn Z for n > 1 is not.
2.3.1. Definition. A formal algebraic space is an fppf sheaf X which admits a map
F
i Ui → X for which each Ui is an affine formal scheme, and such that the map isrepresentable by algebraic spaces, étale and surjective.
Then if X = lim Xi , where the Xi are algebraic spaces and the maps are nilpotent−→
thickenings, one can show that X is a formal algebraic space. Vice versa, you canwrite a formal algebraic space using an Ind-construction.
2.4. Stacks. Stacks are “sheaves” of groupoids. Some (small) groupoids are contractible, and are equivalent to their underlying sets, but some are not. For example, consider the category with one object x and two morphisms {1x , σ} whereσ 2 = 1x : this is not contractible. The moral is that in sets, equality is a property,
but in higher category theory and the theory of stacks, equality is exhibited by anisomorphism which is some extra data.

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M. EMERTON AND T. GEE

Really we want to say that a stack is a “2-sheaf”, which means a 2-functor fromthe category of schemes to the 2-category of groupoids, satisfying a 2-categoricalanalogue of the sheaf condition. This is technically possible to do, but is a bitcomplicated once you start trying to work out all the coherence conditions youneed, and usually involves making some kind of choices of pullbacks.
On the other hand, one way to avoid this is to use the formalism of categories
fibered in groupoids, which [Sta] does, and which we will do. For example, a morphism of stacks is a fully faithful embedding of categories fibered in groupoids. An
occasionally annoying terminological issue is that “isomorphism of stacks” means
an equivalence of categories fibered in groupoids. From now on we will typicallydenote stacks by calligraphic letters X , and schemes (or more generally sheaves) byroman letters X.
There are some interesting things you get from the “extra data” involved inhaving to choose an isomorphism. For example, the diagonal morphism ∆ : X →X × X is a monomorphism if and only if X is equivalent to something that lands insets (in other words, if and only if X can be regarded as being a usual sheaf of sets).
In general, the diagonal really tells you something about isomorphisms between
objects in the groupoids. For instance, if T is a scheme and f : T → X × X is amorphism, induced by the pair of morphisms f1 , f2 : T ⇒ X , then the pull-back of∆ along f will be the Isom sheaf classifying isomorphisms between the elements f1
and f2 of the groupoid X (T ).
2.5. Algebraic and Formal Algebraic Stacks. We make the following definitions.
2.5.1. Definition. We say that an fppf stack X is an algebraic stack if there exists amorphism U → X whose source is a scheme, and which is representable by algebraic
spaces, surjective, and smooth. Note that in this case asking for the morphism tobe étale is strictly stronger than asking for smooth, and defines the notion of a
Deligne–Mumford stack, which we will not make use of.
2.5.2. Example. Recall the schemes VQ from the previous lecture, whose A-valued
points correspond to representations WDK /Q → GLd (A). Usually in Galois representation theory, we want to study representations up to isomorphism, which meansthat we should quotient out by the conjugation action of GLd . This motivates theconsideration of the quotient stack [VQ / GLd ], which is an algebraic stack, and is
the moduli stack of d-dimensional WDK /Q-representations.
We can also form [V / GLd ] = limQ [VQ / GLd ], which is an Ind-algebraic stack. If−→
we work over Z[1/p], then we saw in the last lecture that V becomes a usual scheme(though not of finite type), and so then [V / GLd ] is again an algebraic stack.
If we work over Zp , then we’ve seen that V, and so also [V / GLd ], is not avery good object. As we’ve already said, we will replace this moduli stack of
WDK -representations by a moduli stack of étale (ϕ, Γ)-modules. This won’t be an
algebraic stack, but it will be a formal algebraic stack, in the following sense.
2.5.3. Definition. FWe say that an fppf stack X is a formal algebraic stack if there
exists a morphism i Ui → X which is representable by algebraic spaces, surjective,
and smooth, with the Ui being affine formal schemes.

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

11

If X is qcqs, then we may write X = lim Xi , where Xi are algebraic stacks and−→the transition maps are thickenings. The converse is true if lim is countably indexed−→
and the transition maps are thickenings.
If f : X → Y is a morphism of stacks that is representable by algebraic stacks(in the obvious sense), you can ascribe geometric properties to f . In particular, if∆ : Y → Y × Y is representable by algebraic spaces, and X is an algebraic stack,then any morphism X → Y is representable by algebraic stacks.
Now imagine X is an algebraic stack, locally of finite type over an excellent
locally Noetherian scheme S, and consider a map X → F, where F is an fppf
stack for which ∆F is representable by algebraic spaces. Then as remarked in theprevious paragraph, the morphism X → F is representable by algebraic stacks. In
particular, it makes sense to ask that X → F be proper. In practice, X will bea stack of Breuil–Kisin modules of bounded height, and F will be a stack of étale
ϕ-modules.
We would like to form the “scheme-theoretic image” of F in X . Morally, thisamounts to contracting the proper equivalence relation F ×X F on F. In general
you wouldn’t expect contracting a proper equivalence relation to give you somethingalgebraic. However, we have the following theorem, the main result of [EG21].
2.5.4. Theorem (Scheme Theoretic Images). Assume given an fppf sheaf F which
is limit preserving and whose diagonal is representable by algebraic spaces. Supposethat X is an algebraic stack, and that X → F is a proper morphism. Assume further
that F admits versal rings at all finite type points, and that these rings satisfy thefollowing effectivity property with respect to the image of X : if Spf R → F isa morphism realizing the pro-Artinian local ring R as a versal ring to F, if wepull X back over this morphism to obtain a proper map XR → Spf R, and if welet Spf S ,→ Spf R be the scheme-theoretic image of this map, then the composite
morphism Spf S → Spf R → F factors through Spec S.
Then there exists an algebraic closed substack Z ,→ F such that X → F factors
through a morphism X → Z which is proper and scheme theoretically dominant.
Furthermore, the closed substack Z is uniquely determined by these properties, andwe refer to it as the scheme-theoretic image of the morphism X → F.
We will use this theorem repeatedly in our context to construct the Ind-algebraic
stacks that we want.
Lecture 3. Definitions related to étale ϕ-modules
In this lecture we give the definitions of the various kinds of ϕ-modules withcoefficients that we make use of; in the following lecture, we will begin to use thesedefinitions to define our moduli stacks, and to prove their basic properties.
3.1. Rings. We begin with some material from [EG22, §2.1]. We fix a finite extension K of Qp with ring of integers OK and residue field k, and regard K as a
subfield of some fixed algebraic closure Qp of Qp . We let C denote the completionof Qp . It is a perfectoid field, whose tilt C[ is a complete non-archimedean valued
perfect field of characteristic p. If F is a perfectoid closed subfield of C, then its
tilt F [ is a closed, and perfect, subfield of C[ . We will only need to consider thefollowing two examples of F (other than C itself), which arise from the theories of(ϕ, Γ)-modules and Breuil–Kisin modules.

12

M. EMERTON AND T. GEE

3.1.1. Example (The cyclotomic case). We write K(ζp∞ ) to denote the extensionof K obtained by adjoining all p-power roots of unity. It is an infinite degree Galois
extension of K, whose Galois group is naturally identified with an open subgroupof Z×
p . We let Kcyc denote the unique subextension of K(ζp∞ ) whose Galois groupover K is isomorphic to Zp (so Kcyc is the “cyclotomic Zp -extension” of K). If web cyc denote the closure of Kcyc in C, then Kb cyc is a perfectoid subfield of C.
let K
3.1.2. Example (The Kummer case). If we choose a uniformizer π of K, as well as ancompatible system of p-power roots π 1/p of π (here, “compatible” has the obvious
n+1n∞meaning, namely that (π 1/p )p = π 1/p ), then we define K∞ = K(π 1/p ) :=Sn1/pb ∞ denote the closure of K∞ in C, then Kb ∞ is again a). If we let Kn K(πperfectoid subfield of C.
b ⊗L Qp is an algebraic closure of L,b so
3.1.3. Remark. Let L = Kcyc or K∞ . Then Lbthat the absolute Galois groups of L and L are canonically identified. The actionof GL on Qp extends to an action on C, and by a theorem of Ax–Tate–Sen [Ax70],b
we have CGL = L.
If OF denotes the ring of integers in F (with F = C or one of the two possibilities
just discussed), then OF[ is the ring of integers in F [ . Since F [ and OF[ are perfect,
we may form their rings of (truncated) Witt vectors W (OF[ ), W (F [ ), Wa (F [ ),[).
Wa (OF[ ) (where a ≥ 1 is an integer). We write Ainf := W (OCWe always consider these rings of Witt vectors as topological rings with the socalled weak topology, which admits the following description: if x is any elementof OF[ of positive valuation, and if [x] denotes the Teichmüller lift of x, then we
endow Wa (OF[ ) with the [x]-adic topology, so that W (OF[ ) is then endowed withthe (p, [x])-adic topology. The topology on Wa (F [ ) is then characterized by thefact that Wa (OF[ ) is an open subring (and the topology on W (F [ ) is the inverse
limit topology).
While we could formulate all of our results in terms of ϕ-modules over theserings of Witt vectors, for the purposes of proving our structural results about thestacks (and for connecting to the usual theories of (ϕ, Γ)-modules and Breuil–Kisin
modules), we need to consider various smaller (in particular Noetherian) subrings.
These come from the Fontaine–Wintenberger theory of the field of norms, but wewill skip over this and go straight to the definitions.
Firstly, in the Kummer case, we set S = W (k)[[u]], with an endomorphism ϕdetermined by the properties that it is semilinear for the Frobenius on W (k), andsatisfies ϕ(u) = up , and let OE be the p-adic completion of S[1/u]. The choice of∞[compatible system of p-power roots of π gives an element π 1/p ∈ OK
, and thered∞is a continuous ϕ-equivariant embedding[
S ,→ W (OKd)∞

sending u 7→ [πbedding

1/p∞]. This embedding extends to a continuous ϕ-equivariant em[dOE ,→ W ((K∞ ) ).

e K := Gal(K(ζp∞ )/K), andNow consider the cyclotomic case, where we write Γe K ,→ Z×
the cyclotomic character induces an embedding χ : Γ
p . Consequently,

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

13

eK ∼
there is an isomorphism Γ= ΓK × ∆, where ΓK ∼= Zp and ∆ is finite. We have∆Kcyc = (K(ζp∞ )) . We will usually write Γ for ΓK from now on.
We now make a simplifying assumption for the purposes of exposition:
assume that K/Qp is unramified. We will keep this assumption in place for the
rest of this lecture and the following two lectures. (All of our results hold for general
K/Qp , but there is a subtlety: for general K/Qp , there is no ϕ-stable analogue ofthe ring A+
K defined in the next paragraph. This means that in many arguments
in [EG22] we reduce to the unramified case by somewhat technical although essentially straightforward arguments. We want to avoid these complications in theselectures.)If we choose a compatible system of pn th roots of 1, then these give rise in the[\usual way to an element ε ∈ (K(ζp∞ )) . There is then a continuous embedding
W (k)[[T ]] ,→ W (O[ \
)
∞
K(ζp

)

(the source being endowed with its (p, T )-adic topology, and the target with its
weak topology), defined via T 7→ [ε] − 1. We denote the image of this embeddingby (A0K )+ . This embedding extends to an embedding[\ )) ,→ W (K(ζ\W (k)((Tp∞ ))(here the source is the p-adic completion of the Laurent series ring W (k)((T ))),
0whose image we denote by A0K . Write TK∈ A0K for the image of T . The actions
00
e K on T ∈ A are given by the explicit formulaeof ϕ and γ ∈ ΓKK(3.1.4)00 p
ϕ(TK) = (1 + TK
) − 1,(3.1.5)00 χ(γ)γ(1 + TK
) = (1 + TK),

0 ∆e K → Z×
where χ : Γ
p again denotes the cyclotomic character. We set AK := (AK ) ,+0
TK = tr A0K /AK (TK ), and AK = W (k)[[TK ]]; then we have AK = W (k)((TK ))∧ ,++and A+
K is (ϕ, ΓK )-stable. We have ϕ(TK ) ∈ TK AK , and g(TK ) ∈ TK AK for
all g ∈ ΓK . From now on we will often write T for TK .

3.1.6. Remark. We could equally well work with the rings A0K and the theoryeof (ϕ, Γ)-modules,
but it is often convenient to work with the procyclic group Γand not have to carry around the finite group ∆.
3.1.7. Remark. Note that in both the Kummer and cyclotomic settings, we areconsidering rings which are abstractly a power series ring over the Witt vectors,
equipped with a lift of Frobenius. This will mean that various foundational partsof the theory can be developed in parallel for the two cases.
3.2. Coefficients. We now consider coefficients. Recall that if A is a p-adicallycomplete Zp -algebra, then A is said to be topologically of finite type over Zp if
it can be written as a quotient of a restricted formal power series ring in finitely
many variables Zp hhX1 , . . . , Xn ii; equivalently, if and only if A/p is a finite typeFp -algebra ([FK18, §0, Prop. 8.4.2]). In particular, if A is a Z/pa -algebra forsome a ≥ 1, then A is topologically of finite type over Zp if and only if it is of finite
type over Z/pa . Our coefficient rings will always be assumed to be topologicallyof finite type over Zp , and they will usually also be (finite type) Z/pa -algebras forsome a ≥ 1.

14

M. EMERTON AND T. GEE

3.2.1. Remark. Our stacks are all limit preserving, so that values on A-valued points(for any A) are determined by their values on A which are (topologically) of finite
type. It therefore does not cause us any problems to restrict to coefficients A of thiskind. In addition many of our arguments with ϕ-modules require this assumption
on A, and we do not know if the results hold without it.
Our various coefficient rings are all defined by taking completed tensor products.
[
More precisely, if a ≥ 1, let v denote an element of the maximal ideal of Wa (OC)[
whose image in OC is non-zero. We then set

[
[ b
[
Wa (OC)A = Wa (OC) ⊗Zp A := lim Wa (OC) ⊗Zp A /v i←−i(so that the indicated completion is the v-adic completion). Note that any two[[ bchoices of v induce the same topology on Wa (OC) ⊗Zp A, so that Wa (OC) ⊗Zp Ais well-defined independent of the choice of v. We then define[b Zp A := Wa (OCWa (C[ )A = Wa (C[ ) ⊗)A [1/v];

this ring is again well-defined independently of the choice of v. We set
[
[ b
[
W (OC
)A = W (OC) ⊗Zp A := lim Wa (OC
)A ,
←−
a

and similarlyb Zp A := lim Wa (C[ )A .
W (C[ )A = W (C[ ) ⊗←−aIn keeping with our notation above, we will usually write[
Ainf,A := W (OC)A .

We define the imperfect coefficient rings in the Kummer and cyclotomic casesin the same way, but we can be more explicit: we have SA = (W (k) ⊗Zp A)[[u]],and OE,A is the p-adic completion of SA [1/u], and similarly for A+
K,A and AK,A .

3.2.2. A digression on flatness and completion. Since Ainf is not Noetherian, we
have to be a bit careful when taking completions, and for example it does notseem to be obvious that the natural map SA → Ainf,A is injective. The following
is [EG22, Prop. 2.2.12], and is proved using [FGK11, §5] and [BMS18, Rem. 4.31].
3.2.3. Proposition. Suppose that A → B is a flat homomorphism of finite type
Z/pa Z-algebras for some a ≥ 1. Then all the maps in the following diagram are
flat. Furthermore the vertical arrows are all injections, while the horizontal arrows
are all faithfully flat (and so in particular also injections). If A → B is furthermore
faithfully flat, then the same is true of the diagonal arrows.

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY</ Ainf,B o
9

A+
K,B

/ Ainf,A o

A+
K,A


AK,A

< SB

SA

/ W (C[ )B o
9


AK,B
;

15


/ W (C[ )A o


OE,B
<

OE,A

It is straightforward to show that ϕ extends to a continuous endomorphism[)A , and so on; and the natural action of GK on these rings isof Ainf,A , W (OCcontinuous, as is the action of Γ on A+
K,A and AK,A .
3.3. ϕ-modules, Breuil–Kisin(–Fargues) modules, (ϕ, Γ)-modules, and Galois representations. Let R be a Zp -algebra, equipped with a ring endomorphism ϕ, which is congruent to the (p-power) Frobenius modulo p. If M is an
R-module, we writeϕ∗ M := R ⊗R,ϕ M.
3.3.1. Definition. An étale ϕ-module over R is a finite R-module M , equippedwith a ϕ-semilinear endomorphism ϕM : M → M , which has the property that theinduced R-linear morphism
1⊗ϕMΦM : ϕ∗ M −→ M
is an isomorphism. A morphism of étale ϕ-modules is a morphism of the underlying
R-modules which commutes with the morphisms ΦM . We say that M is projective(resp. free) if it is projective of constant rank (resp. free of constant rank) as an
R-module.
We apply this in particular with R = AK,A or OE,A . In the former case, anétale (ϕ, Γ)-module is an étale ϕ-module over AK,A , equipped with a commuting
continuous semilinear action of Γ. In both cases there is a relationship with Galois
representations as follows; the case without coefficients is due to Fontaine (and was
the motivation for introducing (ϕ, Γ)-modules in the first place), and the version
with coefficients is due to Dee [Dee01].
b ur denote the p-adic completion of the ring of integers of the maximal
Let A
Kunramified extension of AK [1/p] in W (C[ )[1/p]; this is preserved by the naturalb ur as usual. Then if A is anactions of ϕ and GK on W (C[ )[1/p]. Define A
K,AArtinian Zp -algebra, we have an equivalence of categories between the category offinite projective étale (ϕ, Γ)-modules M with A-coefficients, and the category offinite free A-modules V with a continuous action of GK , given by the functors
b ur ⊗A V )GKcyc ,V 7→ (AK,Ab ur ⊗A
M 7→ (AM )ϕ=1 .
K,A
K,A

16

M. EMERTON AND T. GEE

b ur with respect to the mA -adic topology), we canTaking limits (and forming A
K,Aextend this to the case that A is complete local Noetherian, or A = Fp ; this isimportant later on for identifying the versal rings of our moduli stacks with Galois
deformation rings. Note though that for more general A (e.g. A = Fp [X]), it
is not the case that there is an equivalence between (ϕ, Γ)-modules and Galois
representations (cf. the discussion in Lecture 1.4).
There is also a version of this in the Kummer setting: if OEd
ur is the p-adic
completion of the ring of integers in the maximal unramified extension of Frac(OE )in W (C[ )[1/p], then for A as above, we have an equivalence of categories betweenétale ϕ-modules over OE,A and GK∞ -representations, given bybd
V 7→ (O⊗ V )GK∞ ,E ur ,A A
bd⊗OE,A M )ϕ=1 .
M 7→ (OE ur ,A
In the next few lectures we will define moduli stacks of étale ϕ-modules and étale(ϕ, Γ)-modules. One of the key tools in proving the basic properties of these stacks
is to study the corresponding stacks of finite height ϕ-modules over A+
K,A or SA .
We’ll do this more generally in the next lecture, but for now we’ll just have thefollowing definition, which is important in the crystalline and semistable theory.
(Actually, because K/Qp is assumed unramified, the finite height modules overA+
K,A have an intrinsic utility for describing crystalline representations in terms ofWach modules. However, this doesn’t extend to general K/Qp , or to the semistablecase, so we will not make any use of this property of Wach modules in these lectures.)3.3.2. Definition. Let E(u) be the minimal polynomial over W (k) of our fixed
uniformiser π of K. We define a (projective) Breuil–Kisin module (resp. a Breuil–Kisin–Fargues module) of height at most h with A-coefficients to be a finitely
generated projective SA -module (resp. Ainf,A -module) M, equipped with a ϕsemilinear morphism ϕ : M → M, with the property that the corresponding
morphism ΦM : ϕ∗ M → M is injective, with cokernel killed by E(u)h . If M is
a Breuil–Kisin module then Ainf,A ⊗SA M is a Breuil–Kisin–Fargues module.
Note that our Breuil–Kisin(–Fargues) modules are “effective”, i.e. ϕ-stable. Thismeans that we will end up only considering non-negative Hodge–Tate weights; thegeneral case follows by twisting by a sufficiently large power of E(u), so this isharmless.
The connection between Breuil–Kisin(–Fargues) modules and crystalline/semistable
Galois representations will be explained in Lecture 6.
3.4. Almost Galois descent. Let F be a closed perfectoid subfield of C. The
following is [EG22, Thm. 2.4.1], which is proved by a somewhat elaborate argument
making use of almost Galois descent.
3.4.1. Theorem. Let A be a finite type Z/pa -algebra, for some a ≥ 1. The inclusion
W (F [ )A → W (C[ )A is a faithfully flat morphism of Noetherian rings, and thefunctor M 7→ W (C[ )A ⊗W (F [ )A M induces an equivalence between the category offinitely generated projective W (F [ )A -modules and the category of finitely generated
projective W (C[ )A -modules endowed with a continuous semilinear GF -action. A
quasi-inverse functor is given by N 7→ N GF .

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

17

We will find this result very useful later on, because it lets us combine theKummer and cyclotomic settings. The key extra ingredient (which also takes somework) is that when we have the additional structure of a ϕ-module, we can descendb cyc or F = K
b ∞ ) to thefrom the perfect coefficient ring W (F [ )A (with F = K
imperfect coefficient rings AK,A and OE,A respectively, in the following way.
3.4.2. Definition. Let A be a p-adically complete Zp -algebra. An étale (ϕ, GK )module with A-coefficients (resp. an étale (ϕ, GKcyc )-module with A-coefficients,
resp. an étale (ϕ, GK∞ )-module with A-coefficients) is by definition a finitely generated W (C[ )A -module M equipped with an isomorphism of W (C[ )A -modules∼

ΦM : ϕ∗ M −→ M,
and a W (C[ )A -semilinear action of GK (resp. GKcyc , resp. GK∞ ), which is continuous and commutes with ΦM . We say that M is projective if it is projective ofconstant rank as a W (C[ )A -module.
The following is [EG22, Prop. 2.7.8].
3.4.3. Proposition. Let A be a finite type Z/pa -algebra for some a ≥ 1.
(1) The functor M 7→ W (C[ )A ⊗AK,A M is an equivalence between the categoryof finite projective étale ϕ-modules over AK,A and the category of finite
projective étale (ϕ, GKcyc )-modules with A-coefficients.
It induces an equivalence of categories between the category of finite projective étale (ϕ, ΓK )-modules with A-coefficients and the category of finite
projective étale (ϕ, GK )-modules with A-coefficients.
(2) The functor M 7→ W (C[ )A ⊗OE,A M is an equivalence of categories between
the category of finite projective étale ϕ-modules over OE,A and the categoryof finite projective étale (ϕ, GK∞ )-modules with A-coefficients.
Lecture 4. Moduli stacks of ϕ-modules
We now define our first moduli stacks, using the objects introduced in the previous lecture. We maintain our assumption (made purely for the purposes of exposition) that K/Qp is unramified.
4.1. Results of Pappas–Rapoport. We begin our discussion of stacks with ϕmodules; the more complicated case of (ϕ, Γ)-modules will be built on this. These
stacks were first studied by Pappas–Rapoport [PR09] in the context of Breuil–Kisin modules; they showed (among other things) that there are algebraic stacks of
Breuil–Kisin modules, and that the morphisms to the stacks of étale ϕ-modules are
well-behaved. These results were built upon in [EG21, §5], which showed that thestack of étale ϕ-modules is itself well-behaved, and in particular is Ind-algebraic.
We begin by recalling the results of Pappas–Rapoport (in a slightly more generalcontext, where ϕ is not necessarily given by ϕ(u) = up ; this makes some arguments
messier but doesn’t change any of the key points).
4.1.1. Situation. Fix a finite extension k/Fp and write A+ := W (k)[[T ]]. Write A
for the p-adic completion of A+ [1/T ].
If A is a p-adically complete Zp -algebra, we write A+
A := (W (k) ⊗Zp A)[[T ]]; we+equip AA with its (p, T )-adic topology, so that it is a topological A-algebra (where A
has the p-adic topology). Let AA be the p-adic completion of A+
A [1/T ], which weregard as a topological A-algebra by declaring A+
to
be
an
open
subalgebra.
A

18

M. EMERTON AND T. GEE

Let ϕ be a ring endomorphism of A which is congruent to the (p-power) Frobenius endomorphism modulo p, and satisfies ϕ(A+ ) ⊆ A+ .
By [EG21, Lem. 5.2.2 and 5.2.5] and [EG22, Lem. 3.2.5, 3.2.6], ϕ is faithfully flat,and induces the usual Frobenius on W (k), and it extends uniquely to an A-linear
continuous endomorphism of A+
A and AA .
Note in particular that the formation of SA , OE,A , A+
K,A and AK,A above areparticular instances of this construction.
Fix a polynomial F ∈ W (k)[T ] which is congruent to a positive power of T
modulo p. (For example, if we are working in the Breuil–Kisin setting, we willtake T = u, ϕ(u) = up , and F = E(u).)
In order to know that the various categories in groupoids that we will define
are actually stacks, we use the following result of Drinfeld [Dri06, Thm. 3.11](see [EG21, Thm. 5.1.18] for the precise statements given here).
4.1.2. Theorem. The following notions are local for the fpqc topology on Spec A.
(1) A finitely generated projective AA -module.
(2) A projective AA -module of rank d.
(3) A finitely generated projective AA -module which is fpqc locally free of rank d.
(4) A finitely generated projective A+
A -module.
(5) A projective A+-module
of
rank
d.
A
(6) A finitely generated projective A+
A -module which is fpqc locally free of rank d.
4.1.3. Remark. More precisely, saying that the notion of a finitely generated projective AA -module is local for the fpqc topology on Spec A means the following (andthe meanings of the other statements in Theorem 4.1.2 are entirely analogous):
If A0 is any faithfully flat A-algebra, set A00 := A0 ⊗A A0 . Then the categoryof finitely generated projective AA -modules is canonically equivalent to the category of finitely generated projective AA0 -modules M 0 which are equipped with an
isomorphism∼
M 0 ⊗AA0 ,a7→1⊗a AA00 −→ M 0 ⊗AA0 ,a7→a⊗1 AA00
which satisfies the usual cocycle condition.
If we fix integers a, d ≥ 1, then by Theorem 4.1.2 we may define an fpqc stack ingroupoids Rad over Spec Z/pa as follows: For any Z/pa -algebra A, we define Rad (A)to be the groupoid of étale ϕ-modules M over AA which are projective of rank d.
There is a closely related version of this considered in [PR09], namely Rad,free ,where we demand that M is furthermore fpqc-locally free. (In fact, we don’t knowwhether or not this is a consequence of M being projective, so we don’t knowwhether Rad,free = Rad .) In either case, if A → B is a morphism of Zp -algebras, andM is an object of Rad (A), then the pull-back of M to Rad (B) is defined to be thetensor product AB ⊗AA M .
From now on we regard Rad as an fppf stack over Z/pa . By [Sta, Tag 04WV],
we may also regard the stack Rad as an fppf stack over Zp , and as a varies, we may
form the 2-colimit R := lima Rad , which is again an fppf stack over Zp .
−→The following definition generalises that of a Breuil–Kisin module.
4.1.4. Definition. Let h be a non-negative integer. A ϕ-module of F -height at
most h over A+
A is a pair (M, ϕM ) consisting of a finitely generated T -torsion freeA+-moduleM,
and a ϕ-semilinear map ϕM : M → M, with the further properties
AMODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

19

that if we writeΦM := 1 ⊗ ϕM : ϕ∗ M → M,then ΦM is injective, and the cokernel of ΦM is killed by F h .
A ϕ-module of finite F -height over A+
A is a ϕ-module of F -height at most hfor some h ≥ 0. A morphism of ϕ-modules is a morphism of the underlying A+
Amodules which commutes with the morphisms ΦM .
We say that a ϕ-module of finite F -height is projective of rank d if it is a finitely
generated projective A+
A -module of constant rank d.
If we fix integers a, d ≥ 1 and an integer h ≥ 0, then we may define an fpqc stackain groupoids Cd,hover Spec Z/pa as follows: For any Z/pa -algebra A, we defineaCd,h (A) to be the groupoid of ϕ-modules of F -height at most h over A+
A which areprojective of rank d.
aJust as for the stack Rad , we may and do also regard the stack Cd,has an fppf stackaover Zp , and we then, allowing a to vary, define Cd,h := lima Cd,h , obtaining an fppf−→astack over Spf Zp . There are canonical morphisms Cd,h→ Rad and Cd,h → Rd given+
by tensoring with AA over A+
A . One can show that any projective AA -module iseven Zariski locally free, so these morphisms factor through the Pappas–Rapoport
versions Rad,free (resp. Rd,free ). Reassuringly, we have the following lemma [EG22,
Lem. 3.1.3].
4.1.5. Lemma. If A is a p-adically complete Zp -algebra, then there is a canonicalequivalence between the groupoid of morphisms Spf A → Rd and the groupoid ofrank d étale ϕ-modules over AA ; and there is a canonical equivalence between thegroupoid of morphisms Spf A → Cd,h and the groupoid of ϕ-modules of rank d andF -height at most h over A+
A.
The following is essentially [PR09, Thm. 2.1 (a), Cor. 2.6].
4.1.6. Theorem.
a(1) The stack Cd,his an algebraic stack of finite presentation over Spec Z/pa ,with affine diagonal.
a(2) The morphism Cd,h→ Rad,free is representable by algebraic spaces, proper,
and of finite presentation.
(3) The diagonal morphism ∆ : Rad,free → Rad,free ×Z/pa Rad,free is representableby algebraic spaces, affine, and of finite presentation.
We very briefly indicate the main ideas of the proof. One of the key points isthe following [PR09, Prop. 2.2] (see [EG21, Lem. 5.2.9] for this version). Assume
that A is a Z/pa Z-algebra. For n ≥ 0, write
Un = 1 + T n Md (A+A ),Vn = {A ∈ GLd (AA ) | A, A−1 ∈ T −n Md (A+A )}.
4.1.7. Lemma. For each m ≥ 0 there is an n(m) ≥ 0 (implicitly depending alsoon a) such that if n ≥ n(m), then:(1) For each g ∈ Un , B ∈ Vm , there is a unique h ∈ Un such that g −1 Bϕ(g) =h−1 B.
(2) For each h ∈ Un , B ∈ Vm there is a unique g ∈ Un such that g −1 Bϕ(g) =h−1 B.

20

M. EMERTON AND T. GEE

In each case the uniqueness statement is easy, and the existence is proved bya (slightly tricky) successive approximation argument. Given the lemma, the firstpart of Theorem 4.1.6 follows from standard results about the affine Grassmannian:
the point is that working locally, the data of a finite height A+
A -module is the dataof the matrix in Md (A+)
ofthecorresponding
linearizedϕ,and the finite heightA
condition implies that this is in Vn for some n depending on a and h. Changesof basis correspond to ϕ-conjugacy, and Lemma 4.1.7 lets us replace ϕ-conjugacy
by left multiplication provided we work in a sufficiently deep congruence subgroup,and this is enough to get the result.
The rest of Theorem 4.1.6 is also proved by reducing to explicit statements about
matrices over AA . The key point in each case is to obtain a bound on the T -adic
poles in some matrix entries, and the bound on the height h gives a bound on how
different the denominators of g and ϕ(g) can be (for some matrix g), which can beplayed off against the fact that the poles of ϕ(g) have approximately p times theorder of the poles of g.
We then deduce that Theorem 4.1.6 holds with Rad,free replaced by Rad ; roughly
speaking the idea is to regard a projective AA -module as a pair consisting of anfpqc-locally free projective AA -module of higher rank, together with an idempotent.
We also make use of the following technical statement [EG21, Lem. 5.1.23]: if M isa finitely generated projective A((u))-module, then there exists an n0 ≥ 1 such thatfor all n ≥ n0 , M is Nisnevich (and in particular fpqc) locally free as an A((un ))module. In particular we find it useful to regard a projective A((u))-module as alocally free A((un ))-module together with an action of u.
4.2. Scheme-theoretic images. The main result of [EG21] is the following [EG21,
Thm. 5.4.20].
4.2.1. Theorem. Rad is a limit preserving Ind-algebraic stack, whose diagonal isrepresentable by algebraic spaces, affine, and of finite presentation.
The key point is of course the Ind-algebraicity. In fact we show that Rad =limh Rad,h , where Rad,h is the scheme-theoretic image (in the sense of Theorem 2.5.4)−→
aof the morphism Cd,h→ Rad . It is at least intuitively reasonable that we haveaa
Rd = limh Rd,h (this says that étale locally, every étale ϕ-module comes from a−→finite height ϕ-module, which in the free case is immediate by just choosing a basis
and scaling it by a sufficiently large power of F ), and the hard part is to showthat Rad,h is actually an algebraic stack.
The key point in applying Theorem 2.5.4 is to show that Rad,h admits effective
Noetherian versal rings. Apart from the effectivity, this is at least morally quite
straightforward: the versal rings for Rad correspond to (infinite-dimensional) unrestricted Galois deformation rings, and those for Rad,h are closely related to theirfinite height analogues, which are Noetherian. In particular, if we are in the Kummer setting and we take h = 1, then Rad really admits the unrestricted deformation
rings for GK∞ as versal rings, and Rad,1 admits a version of the height 1 deformation
rings, which can be identified with finite flat deformation rings for GK (which arein particular Noetherian).
The real work is in proving the effectivity, i.e. that the versal morphism Spf R →Rad,h can be upgraded to a morphism Spec R → Rad,h . The key point here is to checkthat the composite morphism Spf R → Rad comes from a morphism Spec R → Rad ; itis then relatively straightforward to check that this morphism factors through Rad,h ,MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

21

using arguments which are very similar to those that we will give in Lecture 5.3.
Slightly more explicitly, having a morphism Spf R → Rad means that we have anétale ϕ-module over the mR -adic completion of AR (i.e. where we are allowed tohave infinite Laurent tails, as long as they tend to zero mR -adically), and we needto show that this arises from a projective étale ϕ-module over AR itself.
The reason this is true is that there is a universal (not necessarily projective)
ϕ-module MR over A+
R of F -height at most h, arising as the pushforward of theauniversal ϕ-module over Cd,h. We then set MRh = MRh [1/T ], which gives us a(not necessarily projective) étale ϕ-module over AR , whose mR -adic completion
agrees with the ϕ-module obtained from Spf R → Rad . (This compatibility withthe mR -adic completion is not completely obvious, but ultimately follows from thetheorem on formal functions.) It remains to check that that MR is projective; thisis immediate from the following general theorem ([EG21, Thm. 5.5.20]).
4.2.2. Theorem. Let R be a complete Noetherian local Z/pa -algebra with maximal
ideal m, let M be an étale ϕ-module over AR , and suppose that the m-adic comc is projective, or equivalently, free (over the m-adic completion of AR ).
pletion M
Then M itself is projective (over AR ).
Finally, passing to the limit over a ≥ 1 and using some results of [Eme], we
obtain the following.
4.2.3. Corollary.
(1) Cd,h is a p-adic formal algebraic stack of finite presentation over Spf Zp ,with affine diagonal.
(2) Rd is a limit preserving Ind-algebraic stack, whose diagonal is representableby algebraic spaces, affine, and of finite presentation.
(3) The morphism Cd,h → Rd is representable by algebraic spaces, proper, andof finite presentation.
(4) The diagonal morphism ∆ : Rd → Rd ×Spf Zp Rd is representable by algebraic spaces, affine, and of finite presentation.
Lecture 5. Stacks of (ϕ, Γ)-modules
5.1. Moduli stacks of (ϕ, Γ)-modules. Recall that for simplicity we are always
assuming that K/Qp is unramified when we discuss (ϕ, Γ)-modules.
5.1.1. Definition. We let XK,d denote the moduli stack of projective étale (ϕ, Γ)modules of rank d. More precisely, if A is a p-adically complete Zp -algebra, then we
define XK,d (A) (i.e., the groupoid of morphisms Spf A → XK,d ) to be the groupoidof projective étale (ϕ, Γ)-modules of rank d with A-coefficients, with morphisms
given by isomorphisms. If A → B is a morphism of complete Zp -algebras, and M
is an object of XK,d (A), then the pull-back of M to XK,d (B) is defined to be thetensor product AK,B ⊗AK,A M . It again follows from Theorem 4.1.2 that XK,d isindeed a stack.
Note that by the equivalence between (ϕ, Γ)-modules and GK -representations
over finite Zp -algebras, the Fp -points of XK,d are in bijection with representations
ρ : GK → GLd (Fp ). More generally, fix a point Spec F → XK,d for some finite
field F, giving rise to a continuous representation ρ : GK → GLd (F), and let Rρdenote the universal framed deformation W (F)-algebra for lifts of ρ. Then it is22

M. EMERTON AND T. GEE

easy to check that the natural morphism Spf Rρ → XK,d (which again is defined
via the equivalence between (ϕ, Γ)-modules and GK -representations over finite Zp algebras) is versal.
One of the main results that we will prove is that XK,d is a Noetherian formal
algebraic stack. However, the proof of this is quite involved, and in this lecture
we establish the preliminary result that XK,d is an Ind-algebraic stack, which we
deduce from the results that we have proved for stacks of étale ϕ-modules.
5.1.2. Definition. We let RK,d (frequently abbreviated to Rd ) denote the moduli
stack of rank d projective étale ϕ-modules, taking A to be AK .
We begin by studying Γ-actions on our ϕ-modules. We choose a topological
generator γ of Γ, and let Γdisc := hγi; so Γdisc ∼= Z. Note that since Γdisc is densein Γ, in order to endow M with the structure of an étale (ϕ, Γ)-module, it suffices toequip M with a continuous action of Γdisc (where we equip Γdisc with the topology
induced on it by Γ).
There is a canonical action of Γdisc on Rd (that is, a canonical morphism
γ : Rd → Rd ): if M is an object of Rd (A), then γ(M ) is given by γ ∗ M :=AK,A ⊗γ,AK,A M . Then we set
RΓd disc := Rd

×∆,Rd ×Rd ,Γγ

Rd ,where ∆ is the diagonal of Rd and Γγ is the graph of γ, so that Γγ (x) = (x, γ(x)).
We claim that RΓd disc is nothing other than the moduli stack of projective étale
ϕ-modules of rank d equipped with a semilinear action of Γdisc . This is an exercise
in unwinding the usual construction of the 2-fibre product: RΓd disc consists of tuples∼∼(x, y, α, β), with x, y being objects of Rd , and α : x −→ y and β : γ(x) −→ ybeing isomorphisms. This is equivalent to the category fibred in groupoids given by∼pairs (x, ι) consisting of an object x of Rd and an isomorphism ι : γ(x) −→ x. ThusΓdisc
an object of Rd (A) is a projective étale ϕ-module of rank d with A-coefficients M ,∼together with an isomorphism of ϕ-modules ι : γ ∗ M −→ M ; and this isomorphism
is precisely the data of a semilinear action of Γdisc = hγi on M , as required.
Since Rd is an Ind-algebraic stack, so is RΓd disc ; indeed it follows from Corollary 4.2.3 that RΓd disc is a limit preserving Ind-algebraic stack, whose diagonal isrepresentable by algebraic spaces, affine, and of finite presentation.
Restricting the Γ-action on an étale (ϕ, Γ)-module to Γdisc , we obtain a morphism
XK,d → RΓd disc , which is fully faithful. Thus XK,d may be regarded as a substackof RΓd disc ; in particular, we deduce that its diagonal is representable by algebraic
spaces, affine, and of finite presentation. Although XK,d is a substack of the Indalgebraic stack RΓd disc , it is not a closed substack, but should rather be thoughtof as a certain formal completion (as in the discussion of the rank one case in
Lecture 1.3); in particular, since it is not a closed substack, we cannot immediately
deduce the Ind-algebraicity of XK,d from the Ind-algebraicity of RΓd disc . Instead,we will argue as in the proof that Rd is Ind-algebraic, and exhibit XK,d as thescheme-theoretic image of an Ind-algebraic stack.
From now on we will typically drop K from the notation, simply writing Xd ,Rd and so on. To go further we need to understand the continuity condition in
the definition of Xd more carefully. It is easy to check that we have γ(T ) − T ∈MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

23(p, T )T A+
A , and using this one can check the following special case of [EG22, Lem.
D.26]:
5.1.3. Lemma. Suppose that A is a Z/pa -algebra for some a ≥ 1. Let M be a
finite projective AA -module, equipped with a semilinear action of Γdisc . Then thefollowing are equivalent:(1) The action of Γdisc extends to a continuous action of Γ.
s(2) For any lattice M ⊆ M , there exists s ≥ 0 such that (γ p − 1)(M) ⊆+T M. (Here a lattice is a finitely generated AK,A -submodule M ⊆ M whose
AK,A -span is M .)s(3) For some lattice M ⊆ M and some s ≥ 0, we have (γ p − 1)(M) ⊆ T M.
(4) The action of γ − 1 on M ⊗Z/pa Fp is topologically nilpotent.
It is easy to use this criterion, together with the fact that RΓd disc is limit preserving, to show that Xd is limit preserving; see [EG22, Lem. 3.2.19].
5.2. Weak Wach modules. In this section we introduce the notion of a weak
Wach module of height at most h and level at most s. These will play a purely
technical auxiliary role for us, and will be used only in order to show that Xd is an
Ind-algebraic stack; we won’t use their relation to crystalline representations.
By Lemma 5.1.3, if A is a Z/pa -algebra for some a ≥ 1, and M is a rank dprojective ϕ-module of T -height ≤ h over A, such that M[1/T ] is equipped witha semilinear action of Γdisc , then this action extends to a continuous action of Γs
if and only if for some s ≥ 0 we have (γ p − 1)(M) ⊆ T M. This motivates thefollowing definition.
5.2.1. Definition. A rank d projective weak Wach module of T -height ≤ h andlevel ≤ s is a rank d projective ϕ-module M over A+
K,A , which is of T -height≤ h, such that M[1/T ] is equipped with a semilinear action of Γdisc which satisfiess(γ p − 1)(M) ⊆ T M.
5.2.2. Definition. We let Wd,h denote the moduli stack of rank d projective weak
Wach modules of T -height ≤ h. We let Wd,h,s denote the substack of rank dprojective weak Wach modules of T -height ≤ h and level ≤ s.
We will next show that the stacks Wd,h,s are p-adic formal algebraic stacks offinite presentation over Spf Zp . Since the canonical morphism lims Wd,h,s → Wd,h−→is an isomorphism (by definition), this will show in particular that Wd,h is an Indalgebraic stack; we will also see that the transition maps in this injective limit areclosed immersions.
Recall that we have the p-adic formal algebraic stack Cd,h classifying rank dprojective ϕ-modules over A+
K,A of T -height at most h. We consider the fibre
Γdisc
product Rd
×Rd Cd,h , where the map RΓd disc → Rd is the canonical morphism
given by forgetting the Γdisc action; this is the moduli stack of rank d projective
ϕ-modules M over A+
K,A of T -height at most h, equipped with a semilinear actionof Γdisc on M[1/T ]. It follows from Corollary 4.2.3 that RΓd disc ×Rd Cd,h is a p-adic
formal algebraic stack of finite presentation over Spf Zp .
Restricting the Γ-action on a weak Wach module to Γdisc , we may regard Wd,has a substack of RΓd disc ×Rd Cd,h . The following is [EG22, Prop. 3.3.5].

24

M. EMERTON AND T. GEE

5.2.3. Proposition. For s ≥ 1, the morphism/ RΓdisc ×R Cd,hWd,h,sdd

is a closed immersion of finite presentation. In particular, each of the stacks Wd,h,s
is a p-adic formal algebraic stack of finite presentation over Spf Zp ; and for eachs0 ≥ s, the canonical monomorphism Wd,h,s ,→ Wd,h,s0 is a closed immersion offinite presentation.
The proof of this is fairly straightforward: by definition, we need to show thats
the condition that (γ p − 1)(M) ⊆ T M is a closed condition, and is determined byfinitely many equations. We do this by reducing to the free case and considering
the equations on the level of matrices.
5.3. Xd is an Ind-algebraic stack. By definition, we have a 2-Cartesian diagram(5.3.1)Wd,h/ RΓdisc ×R Cd,h
Xd/ RΓdiscddd

If h0 ≥ h then the closed immersion Cd,h ,→ Cd,h0 is compatible with the morphisms
from each of its source and target to Rd , and so we obtain a closed immersion(5.3.2)

Wd,h ,→ Wd,h0 .

By construction, the morphisms Wd,h → Xd are compatible, as h varies, withthe closed immersions (5.3.2). Thus we also obtain a morphism(5.3.3)lim W → Xd .
−→ d,hh

Roughly speaking, we will prove that Xd is an Ind-algebraic stack by showing that
it is the “scheme-theoretic image” of the morphism limh Wd,h → RΓd disc induced−→by (5.3.3). More precisely, choose s ≥ 0, and consider the composite(5.3.4)Wd,h,s → Wd,h → Xd → RΓd disc .

This admits the alternative factorization
Wd,h,s → Wd,h → RΓd disc ×Rd Cd,h → RΓd disc .
Proposition 5.2.3 shows that the composite of the first two arrows is a closed embedding of finite presentation, while Corollary 4.2.3 shows that the third arrow isrepresentable by algebraic spaces, proper, and of finite presentation. Thus (5.3.4)
is representable by algebraic spaces, proper, and of finite presentation.
a
Fix an integer a ≥ 1, and write Wd,h,s:= Wd,h,s ×Spf Zp Spec Z/pa . Proposition 5.2.3 shows that Wd,h,s is a p-adic formal algebraic stack of finite presentationaover Spf Zp , and so Wd,h,s
is an algebraic stack, and a closed substack of Wd,h,s .
Note that since at this point we don’t know that Xd is Ind-algebraic, we can’tadirectly define a scheme-theoretic image of Wd,h,s
in Xd . It might be possible to do
this using the formalism of [EG21]; we take a slightly different approach.
a5.3.5. Definition. We let Xd,h,s
denote the scheme-theoretic image of the composite(5.3.6)(5.3.4)aWd,h,s,→ Wd,h,s −→ RΓd disc .

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

25

This is a morphism of Ind-algebraic stacks, and the scheme-theoretic image has
the obvious meaning: since RΓd disc is an Ind-algebraic stack, constructed as the
2-colimit of a directed system of algebraic stacks whose transition morphisms areclosed immersions, the morphism (5.3.6), whose domain is a quasi-compact algebraic stack, factors through a closed algebraic substack Z of RΓd disc . We then defineaaaXd,h,sto be the scheme-theoretic image of Wd,h,s
in Z. Then Xd,h,s
is a closed
Γdisc
algebraic substack of Rd , and is independent of the choice of Z.
a
Our next goal is to prove that Xd,h,s
is a (necessarily closed) substack of Xda . Our
argument for this is a little indirect. By definition, it is enough to check that if A isaa finite type Z/pa -algebra, then for any morphism Spec A → Xd,h,s, the compositeΓamorphism Spec A → Xd,h,s→ Rd disc factors through Xd . More concretely, if M
denotes the étale ϕ-module over A, endowed with a Γdisc -action, associated to thegiven point Spec A → RΓd disc , then we must show that the Γdisc -action on M iscontinuous. By Lemma 5.1.3, we need to show that M contains a (not necessarilysprojective) lattice M such that for some s ≥ 0, we have (γ p − 1)(M) ⊆ T M.
To do this, we note that the natural map A → B := limi Ai is injective, where Ai←−
runs over the Artinian quotients of A. It is enough to find such a lattice for MB ,and using that lattices of bounded height are uniformly commensurable, it is infact enough to find a lattice of some fixed height for each MAi . We are therefore
reduced to the following lemma [EG22, Lem. 3.4.8].
5.3.7. Lemma. Suppose that M is a projective étale ϕ-module of rank d over a finite
type Artinian Z/pa -algebra A, and that M is endowed with an action of Γdisc , suchathat the corresponding morphism Spec A → RΓd disc factors through Xd,h,s. Then Mspcontains a ϕ-invariant lattice M of T -height ≤ h, such that (γ − 1)(M) ⊆ T M.
Our proof of this lemma is a little involved. We can immediately reduce to theArtin local case, and then to a problem about the universal framed deformation
rings of a fixed ϕ-module equipped with an action of Γdisc . Writing R for thisuniversal deformation ring, we can consider the subfunctor of deformations which
admit a lattice M of the required type, and it is straightforward to check that thisis representable by a quotient S of R. The statement of the lemma reduces toshowing that Spf S contains the scheme-theoretic image Spf T of the morphism
X := Wd,h,s ×RΓdisc Spf R → Spf R.
d

We prove this using the following criterion [EG22, Lem. A.32].
5.3.8. Lemma. Let R → S be a continuous surjection of pro-Artinian local rings,
and let X → Spf R be a finite type morphism of formal algebraic spaces.
Make the following assumption: if A is any finite-type Artinian local R-algebra
for which the canonical morphism R → A factors through a discrete quotient of R,
and for which the canonical morphism XA → Spec A admits a section, then the
canonical morphism R → A furthermore factors through S.
Then the scheme-theoretic image of X → Spf R is a closed formal subschemeof Spf S.
The result then follows by unwinding the definitions: the point is that admitting
a section to XA in particular gives a morphism Spec A → Wd,h,s , and the corresponding weak Wach module is a lattice of the kind being considered, which gives
the required factorisation through S.

26

M. EMERTON AND T. GEE

Finally, we can prove that Xd is Ind-algebraic.
a5.3.9. Proposition. The canonical morphism lim Xd,h,s→ Xd is an isomorphism.
−→
Thus Xd is an Ind-algebraic stack, and may in fact be written as the inductive
limit of algebraic stacks of finite presentation, with the transition maps being closed
immersions.

Proof. We just need to show that if T = Spec A for a Noetherian Z/pa -algebra A,athen any morphism T → Xd factors through some Xd,h,s, or equivalently, that theclosed immersion(5.3.10)a
Xd,h,s
×Xd T → T

is an isomorphism, for some choice of h and s.
If M denotes the étale (ϕ, Γ)-module corresponding to the morphism Spec A →Xd , then [EG21, Prop. 5.4.7] shows that we may find a scheme-theoretically dominant morphism Spec B → Spec A such that MB is free of rank d. (In outline, theproof of that result is to note that M is free if and only if M/T M is free, so that M
is automatically free over a dense open subset of Spec A, and use Noetherian induction.) If we show that the composite Spec B → Spec A → Xd factors throughaXd,h,sfor some h and s, then we see that the morphism Spec B → Spec A factorsathrough the closed subscheme Xd,h,s
×Xd Spec A of Spec A. Since Spec B → Spec A isscheme-theoretically dominant, this implies that (5.3.10) is indeed an isomorphism,
as required.
Since MB is free, we may choose a ϕ-invariant free lattice M ⊆ MB , of height ≤ hfor some sufficiently large value of h. Since the Γdisc -action on M , and hence on MB ,sis continuous by assumption, Lemma 5.1.3 then shows that (γ p − 1)(M) ⊆ T M forasome sufficiently large value of s. Then M gives rise to a B-valued point of Wd,h,s,Γdiscwhose image in Rd
is equal to the étale ϕ-module MB . Thus the morphismaSpec B → Xd corresponding to MB does indeed factor through Xd,h,s.
Lecture 6. Crystalline and semistable moduli stacks
From now on we allow K/Qp to be arbitrary.
6.1. Breuil–Kisin–Fargues modules admitting all descents. The connection
between Breuil–Kisin modules and crystalline representations was first establishedby Kisin in [Kis06], where he showed that if K∞ is as in Example 3.1.2, andρ : GK → GLd (Zp ) is a lattice in a crystalline representation with Hodge–Tateweights in [0, h], then the étale ϕ-module corresponding to ρ|GK∞ (via the functors explained in Lecture 3.3) arises from a (unique) Breuil–Kisin module of heightat most h. Kisin also showed that on crystalline representations, the restriction
from GK to GK∞ is fully faithful. Unfortunately, when h > 1, Kisin’s descriptionof the Breuil–Kisin modules which arise from lattices in crystalline representations
is transcendental in nature (it is phrased in terms of a certain meromorphic connection, which only exists after an extension of scalars, admitting at worst log poles),and it is unclear (to us) how to define crystalline moduli stacks purely in terms of
Breuil–Kisin modules for K∞ .
Our approach to constructing crystalline and semistable moduli stacks relieson the observation that while most Breuil–Kisin modules do not give crystalline
or semistable representations of GK , this is “mostly” for the simple reason thatthe corresponding GK∞ -representations do not even extend to GK . However, if ρ

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

27

is a GK -representation such that ρ|GK∞ admits a Breuil–Kisin module, then ρ
is “almost” semistable: in particular, ρ is potentially semistable, and becomesmsemistable over K(π 1/p ) for some m depending only on K. Using this result it
is straightforward to deduce that ρ is semistable if and only if ρ|GK∞ admits anBreuil–Kisin module for all possible choices of (π and ) (π 1/p )n≥0 . This is themain idea behind our construction of the stacks.
In fact, this statement (that ρ|GK∞ admits a Breuil–Kisin module only if ρ
is potentially semistable) was only proved after [EG22] was completed, by Hui
Gao [Gao19]. Instead, we use a slightly weaker statement, which is proved by TG
and Tong Liu in [EG22, App. D] (see Theorem 6.1.5 below).
[
For each choice of uniformiser π of K, and each choice π [ ∈ OCof p-powerroots of π, we write Sπ[ for S = W (k)[[u]], regarded as a subring of Ainf vias
u 7→ [π [ ]. For each choice of π [ and each s ≥ 0 we write Kπ[ ,s for K(π 1/p ),and Kπ[ ,∞ for ∪s Kπ[ ,s . Write Eπ (u) for the Eisenstein polynomial for π, and Eπ[
for its image in Ainf . There is a natural ring homomorphism θ : Ainf → OC , which[
satisfies θ([x]) = x] for any x ∈ OC
; and Eπ[ is a generator of ker θ.
6.1.1. Definition. A Breuil–Kisin–Fargues GK -module of height at most h is a
Breuil–Kisin–Fargues module of height at most h which is equipped with a semilinear GK -action which commutes with ϕ.
6.1.2. Remark. Note that if Minf is a Breuil–Kisin–Fargues GK -module, then W (C[ )⊗Ainf
Minf is naturally a (ϕ, GK )-module in the sense of Definition 3.4.2.
6.1.3. Definition. Let Minf be a Breuil–Kisin–Fargues GK -module of height at
most h with a semilinear GK -action. Then we say that Minf admits all descents if
the following conditions hold.
(1) For every choice of π and π [ , there is a Breuil–Kisin module Mπ[ of heightGKat most h with Mπ[ ⊂ (Minf ) π[ ,∞ for which the induced morphism
Ainf ⊗Sπ[ Mπ[ → Minf is an isomorphism.
(2) The W (k)-submodule Mπ[ /[π [ ]Mπ[ of W (k) ⊗Ainf Minf is independent of
the choice of π and π [ .
(3) The OK -submodule ϕ∗ Mπ[ /Eπ[ ϕ∗ Mπ[ of OC ⊗θ,Ainf ϕ∗ Minf is independentof the choice of π and π [ .
6.1.4. Definition. Let Minf be a Breuil–Kisin–Fargues GK -module which admits
all descents. We say that Minf is furthermore crystalline if for each choice of π
and π [ , and each g ∈ GK , we have(g − 1)(Mπ[ ) ⊂ ϕ−1 ([ε] − 1)[π [ ]Minf .
There is an equivalence of categories between the category of (ϕ, GK )-modules
M of rank d and the category of free Zp -modules T of rank d which are equippedwith a continuous action of GK , with the Galois representation corresponding to M
being given by T (M ) = M ϕ=1 . Write V (M ) := T (M ) ⊗Zp Qp .
We deduce the following theorem [EG22, Thm. F.11] from the results of Tong
Liu’s paper [Liu18] and Laurent Fargues’ correspondence between Breuil–Kisin–
Fargues modules and B+
dR -lattices [BMS18, Thm. 4.28].
6.1.5. Theorem (T.G. and Tong Liu). Let M be a (ϕ, GK )-module. Then V (M )is semistable with Hodge–Tate weights in [0, h] if and only if there is a (necessarily

28

M. EMERTON AND T. GEE

unique) Breuil–Kisin–Fargues GK -module Minf which is of height at most h, which
admits all descents, and which satisfies M = W (C[ ) ⊗Ainf Minf .
Furthermore, V (M ) is crystalline if and only if Minf is crystalline.
More recently, Bhargav Bhatt and Peter Scholze [BS21] have proved a classification of lattices in crystalline GK -representations in terms of prismatic F -crystals;
this puts Theorem 6.1.5 (in the crystalline case) in its natural context.
6.2. Definition of the crystalline stacks. For simplicity of exposition, we will
only address the crystalline case from now on; the proofs in the semistable caseare identical (in fact slightly simpler, because the crystalline case has the extra
condition on the Breuil–Kisin–Fargues modules). The actual definition of thesestacks, and the proof of their basic properties, is quite involved, but the basic idea
is quite simple: we will take the scheme-theoretic image in Xd of the moduli stack of
Breuil–Kisin–Fargues GK -modules which admit all descents. There is one wrinkle:
we do not know that the stack of Breuil–Kisin–Fargues GK -modules admitting alldescents is reasonably behaved, and we instead work with a refinement, for which
the GK -action is “canonical”, in the following sense.
We have the following reinterpretation of some of the results and argumentsof [CL11, §2] (see [EG22, §4.3, 4.4]).
6.2.1. Proposition. For any fixed a, h, and any sufficiently large N , there is apositive integer s(a, h, N ) with the property that for any finite type Z/pa -algebra A,
any projective Breuil–Kisin module M of height at most h, and any s ≥ s(a, h, N ),there is a unique continuous action of GKs = GK(π1/ps ) on Minf := Ainf,A ⊗SA M
which commutes with ϕ and is semilinear with respect to the natural action of GKs
on Ainf,A , with the additional property that for all g ∈ GKs we have (g − 1)(M) ⊂uN Minf .
Conversely, for any Breuil–Kisin–Fargues GK -module Minf (with Zp -coefficients)
corresponding as above to a semistable GK representation with Hodge–Tate weights
in [0, h], and for any choice of π [ with corresponding descent Mπ[ , if s ≥ s(a, h, N )then the restriction to GKπ[ ,s of the action of GK on Minf /pa agrees with the actionof the previous paragraph.
Proof. The proof of the first part is quite straightforward; the point is that bya standard argument with the ϕ-structure, one can uniquely upgrade approximate
homomorphisms (i.e. maps which are homomorphisms modulo uN ) between Breuil–Kisin(–Fargues) modules to actual homomorphisms (see [EG22, Lem. 4.3.2]). We
can write down an “approximate action” of GKs on Minf , by just letting it act
trivially on M (after fixing some basis - since the action is semilinear, it doesn’tliterally make sense to ask that it be trivial).
The second part is proved in [CL11].

Note that Definition 6.1.4 carries over in an obvious fashion to the case with coefficients in a topologically of finite type p-adically complete Zp -algebra A (see [EG22,
Defn. 4.2.4]). Using the faithful flatness results of Proposition 3.2.3, one can checkthat the existence of a descent to a Breuil–Kisin module depends only on π, andnot on π [ , and that any such descent is unique ([EG22, Lem. 4.2.7, 4.2.8]).
a6.2.2. Definition. For any h ≥ 0 we let Cd,crys,hdenote the limit preserving cataegory of groupoids over Spec Z/p determined by decreeing, for any finite type

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY29a
Z/pa -algebra A, that Cd,crys,h(A) is the groupoid of Breuil–Kisin–Fargues GK modules with A-coefficients, which are of height at most h, which admit all descents,
whose GK -actions are canonical, and which are crystalline. We let Cd,crys,h :=alima Cd,crys,h.
−→It is easy to check that if A is a p-adically complete Zp -algebra which is topologically of finite type, then Cd,crys,h (A) is the groupoid of Breuil–Kisin–Fargues
GK -modules with A-coefficients, which are of height at most h, which admit alldescents, whose GK -actions are canonical, and which are crystalline. There is anatural morphism
Cd,crys,h → Xd ,which is defined, for finite type Z/pa -algebras, via Minf 7→ W (C[ )A ⊗Ainf,A Minf ,with the target object being regarded as an A-valued point of Xd via the equivalenceof Proposition 3.4.3.
For now we will admit the following theorem [EG22, Thm. 4.5.20], which we
prove below.
6.2.3. Theorem. Cd,crys,h is a p-adic formal algebraic stack of finite presentation
and affine diagonal. The morphism Cd,crys,h → Xd is representable by algebraic
spaces, proper, and of finite presentation.
fldenote the flat part of Cd,crys,h (i.e. the maximal substack which is flat
Let Cd,crys,hover Spf Zp ; see [Eme, Ex. 9.11]). Then we define Xdcrys,h to be the scheme-theoretic
flimage of the morphism Cd,crys,h→ Xd . The following result is essentially [EG22,
Thm. 4.8.12] (for the purposes of exposition, we are ignoring inertial types andHodge–Tate weights for now).

6.2.4. Theorem. The closed substack Xdcrys,h of Xd is a p-adic formal algebraic
stack, which is of finite type and flat over Spf Zp , and is uniquely determined asa Zp -flat closed substack of Xd by the following property: if A◦ is a finite flat Zp algebra, then Xdcrys,h (A◦ ) is precisely the subgroupoid of Xd (A◦ ) consisting of GK representations which are crystalline with Hodge–Tate weights contained in [0, h].
That Xdcrys,h is a p-adic formal algebraic stack and is of finite type and flatover Spf Zp follows from its construction and [EG22, Prop. A.21] (that is, “thescheme-theoretic image of a p-adic formal algebraic stack is a p-adic formal algebraic
stack”). The characterisation of its A◦ -points can be reduced to the local case, andthus to the following important statement about versal rings: fix a point Spec F →Xd (F) for some finite field F, giving rise to a continuous representation ρ : GK →GLd (F). Let Rρcrys,h denote the universal framed deformation W (F)-algebra for liftsof ρ which are crystalline with Hodge–Tate weights in [0, h] (which exists by [Kis08]).
Then there is an induced morphism Spf Rρcrys,h → Xd , and we have [EG22, Prop.
4.8.10]:6.2.5. Proposition. The morphism Spf Rρcrys,h → Xd factors through a versal morphism Spf Rρcrys,h → Xdcrys,h .
Proof. This is proved using Lemma 5.3.8, in a similar way to the way we used it in
the sketch proof of Lemma 5.3.7; the actual argument is slightly more complicatedas we prove an algebraization statement and then work with p inverted.
30

M. EMERTON AND T. GEE

We now prove Theorem 6.2.3. We will deduce it from the corresponding statements for the stacks Cd,h of Breuil–Kisin modules, which we discussed in Lecture 4.
Then for any h ≥ 0 and any choice of π [ , we write Cπ[ ,d,h for the moduli stackof rank d Breuil–Kisin modules for Sπ[ ,A of height at most h, and Rπ[ ,d for thecorresponding stack of étale ϕ-modules. We have a natural morphism XK,d → Rπ[ ,dgiven by Proposition 3.4.3 and restriction from GK to GK∞ ; for each s ≥ 1, we canfactor this as
XK,d → XKπ[ ,s ,d → Rπ[ ,d .
Then for any fixed a, h, and any N and s ≥ s(a, h, N ) as in Proposition 6.2.1, thereais a canonical morphism Cπa[ ,d,h → XK
,d obtained from the canonical action ofπ [ ,s
Proposition 6.2.1, which fits into a commutative triangle
Cπa[ ,d,h
z
a
XK
[ ,d
π ,s


/ Ra [

π ,dWe then define a stack Cπa[ ,s,d,h by the requirement that it fits into a 2-Cartesian
diagram(6.2.6)Cπa[ ,s,d,h/ C a[


a
XK,d

/ Xa

π ,d,hKπ[ ,s ,dThe lower horizontal arrow in this diagram is again defined via Proposition 3.4.3,
and it is not so hard to show (using the properties that we have proved about XK,dand its diagonal) that it is representable by algebraic spaces and of finite presentation. Since Cπa[ ,d,h is an algebraic stack of finite presentation over Z/pa , it follows
that Cπa[ ,s,d,h is also an algebraic stack of finite presentation over Z/pa . It is alsostraightforward to check that the right hand vertical arrow in this diagram is representable by algebraic spaces, proper, and of finite presentation, so that the sameis true of the left hand vertical arrow.
Given all this, the key statement remaining to be proved is the following [EG22,
Prop. 4.5.15].
6.2.7. Proposition. For each π [ and each s as above, there is a natural closedaaimmersion Cd,crys,h→ Cπa[ ,s,d,h . In particular, Cd,crys,his an algebraic stack ofafinite presentation over Z/p , and its diagonal is affine.
aaProof. We have a morphism Cd,crys,h→ XK,d(given by extending scalars to W (C[ )),aand it follows that there is a natural morphism Cd,crys,h→ Cπa[ ,d,h , defined viaaaMinf 7→ Mπ[ . The composite morphisms Cd,crys,h→ Cπa[ ,d,h → XK
,d andπ [ ,saaa
Cd,crys,h → XK,d → XK [ ,d coincide by construction, so we have an induced morπ ,sphism(6.2.8)aCd,crys,h→ Cπa[ ,s,d,hwhich we need to show is a closed immersion.

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

31

To see that (6.2.8) is at least a monomorphism, it is enough to note that if A is a
finite type Z/pa -algebra, and Minf is a Breuil–Kisin–Fargues module over A which
admits all descents, then Minf = Ainf,A ⊗Sπ[ ,A Mπ[ is determined by Mπ[ , and the
GK -action on Minf is determined by the GK -action on W (C[ )A ⊗Ainf,A Minf .
Finally, the proof (6.2.8) is a closed immersion is a bit more involved: we need toshow that the conditions that Minf = Ainf,A ⊗Sπ[ ,A Mπ[ is GK -stable and admits
all descents are closed conditions. We make repeated use of the results of [EG22,
App. B], particularly [EG22, Lem. B.28] to show that the vanishing loci of various
morphisms of Breuil–Kisin–Fargues modules are closed.

6.3. Potentially crystalline moduli stacks with fixed Hodge–Tate weights.
For applications, it is important to be able to fix the Hodge–Tate weights in our
crystalline moduli stacks, and also to consider representations which are only potentially crystalline. To this end, let L/K be a fixed finite Galois extension; thenthere is an obvious notion of a Breuil–Kisin–Fargues GK -module which admits
all descents over L, and an obvious extension of the above results to give stacks
of GK -representations which become crystalline over L with Hodge–Tate weights
L/K,flcontained in [0, h]. We write Cd,crys,h for the corresponding stack of Breuil–Kisin–
Fargues modules.
Since the inertia types and Hodge–Tate weights are discrete invariants, we atleast morally expect these stacks to decompose as a disjoint union of stacks ofpotentially crystalline representations with fixed inertial and Hodge types; so thekey point should be to see how to read this data off from the Breuil–Kisin–Fargues
modules.
Fix a finite extension E/Qp with ring of integers O, uniformizer $, and residue
field F, and assume that E is large enough to contain the images of all embeddings
K ,→ Qp . We will abusively write Xd for the base change of Xd to Spf O, without
further comment. From now on A◦ will denote a p-adically complete flat O-algebra
which is topologically of finite type over O, and we write A = A◦ [1/p].
Let MA◦ be a Breuil–Kisin–Fargues GK -module with A◦ -coefficients which admits all descents over L, and write MA◦ := MA◦ ,π[ /[π [ ]MA◦ ,π[ (for some choiceof π [ , with π now denoting a uniformiser of L). Then MA◦ has a natural W (l)⊗Zp Asemilinear action of Gal(L/K), which is defined as follows: if g ∈ Gal(L/K),
then g(MA◦ ,π[ ) = MA◦ ,g(π[ ) , so the morphism g : MA◦ ,π[ → g(MA◦ ,π[ ) = MA◦ ,g(π[ )induces a morphismg : MA◦ ,π[ /[π [ ]MA◦ ,π[ → MA◦ ,g(π[ ) /[g(π [ )]MA◦ ,g(π[ ) ,and the source and target are both canonically identified with MA◦ .
This action of Gal(L/K) on MA◦ induces an L0 ⊗Qp A-linear action of IL/K on
the projective L0 ⊗Qp A-module MA◦ ⊗A◦ A. Fix a choice of embedding σ : L0 ,→ E,
and let eσ ∈ L0 ⊗Qp E be the corresponding idempotent. Then eσ (MA◦ ⊗A◦ A) is
a projective A-module of rank d, with an A-linear action of IL/K . Up to canonicalisomorphism, this module does not depend on the choice of σ. We write
WD(MinfA◦ ) := eσ (MA◦ ⊗A◦ A),
a projective A-module of rank d with an A-linear action of IL/K . It is easy to checkthat this is compatible with base change (of p-adically complete flat O-algebras
which are topologically of finite type over O).

32

M. EMERTON AND T. GEE

We now turn to Hodge types. Fix some choice of π [ a uniformiser of L, write MA◦ifor Mπ[ ,A◦ , and u for [π [ ]. For each 0 ≤ i ≤ h we define Fili ϕ∗ MA◦ = Φ−1MA◦ (E(u) MA◦ ),and we set Fili ϕ∗ MA◦ = ϕ∗ MA◦ for i < 0. Then for each 0 ≤ i ≤ h,(Fili ϕ∗ MA◦ /E(u) Fili−1 ϕ∗ MA◦ ) ⊗A◦ A
is a finite projective L ⊗Qp A-module, whose formation is compatible with basechange (see [EG22, Prop. 4.7.2]). There is a natural action of Gal(L/K), which
is semilinear with respect to the action of Gal(L/K) on L ⊗Qp A induced by its
action on the first factor. Since L/K is a Galois extension, the tensor product
L ⊗Qp A is an étale Gal(L/K)-extension of K ⊗Qp A, and so étale descent allows
us to descend (ϕ∗ MA◦ /E(u)ϕ∗ MA◦ ) ⊗A◦ A to a filtered module over K ⊗Qp A;
concretely, this descent is achieved by taking Gal(L/K)-invariants. This leads tothe following definition.
6.3.1. Definition. In the preceding situation, we write∗∗
DdR (Minf
A◦ ) := (ϕ MA◦ /E(u)ϕ MA◦ ) ⊗A◦ A

Gal(L/K),

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

Gal(L/K)

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

where each eσ DdR (MinfA◦ ) is a projective A-module of rank d. For each i, we writei
infFili eσ DdR (MinfA◦ ) = eσ Fil DdR (MA◦ ).

Each Fili eσ DdR (MinfA◦ ) is again a projective A-module.
6.3.2. Definition. A Hodge type λ of rank d is by definition a set of tuples ofintegers {λσ,i }σ:K,→Qp ,1≤i≤d with λσ,i ≥ λσ,i+1 for all σ and all 1 ≤ i ≤ d − 1. We
say that it is effective if all λσ,i ≥ 0, and bounded by h if all λσ,i ≤ h.
If D := (Dσ )σ:K,→E is a collection of rank d vector bundles over Spec A, labeled(as indicated) by the embeddings σ : K ,→ E, then we say that D has Hodge typeλ if Fili Dσ has constant rank equal to #{j | λσ|K,j ≥ i}.
As the notation suggests, the Hodge type of V (Minf ) agrees with the Hodge typeof DdR (Minf ). Putting this all together, we have [EG22, Prop. 4.8.2]:
L/K,fl6.3.3. Proposition. Let L/K be a finite Galois extension. Then the stack Cd,crys,hL/K,fl,λ,τ

is a scheme-theoretic union of closed substacks Cd,crys,h , where λ runs over all
effective Hodge types that are bounded by h, and τ runs over all d-dimensional Erepresentations of IL/K . These latter closed substacks are uniquely characterisedby the following property: if A◦ is a finite flat O-algebra, then an A◦ -point of

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY
L/K,fl33

L/K,fl,λ,τCd,crys,h ) is a point of Cd,crys,hif and only if the corresponding Breuil–Kisin–
inf
Fargues module MA◦ has Hodge type λ and inertial type τ .
6.3.4. Remark. It is not obvious (at least to us) from the definition that the Hodge
filtration on DdR (Minf ) is independent of the choice of π [ ; rather, we deduce thisindependence from its compatibility with the Hodge filtration on DdR (V (Minf )).
crys,λ,τWe then define the corresponding stacks Xd,; we can extend the definition
to possibly negative Hodge–Tate weights by twisting by an appropriate power ofthe cyclotomic character. Finally we can deduce [EG22, Thm. 4.8.14]:
crys,λ,τ

6.3.5. Theorem. The algebraic stack Xd
×Spf O Spec F is equidimensional ofdimensionX#{1 ≤ i < j ≤ d|λσ,i > λσ,j }.
σcrys,λ,τIn particular, if λ is regular, then the algebraic stack Xdequidimensional of dimension [K : Qp ]d(d − 1)/2.

×Spf O Spec F isThis follows from Proposition 6.2.5, from which it follows that the versal rings
crys,λ,τcrys,λ,τto Xd
are given by the corresponding deformation rings Rρ
, and thecomputation of the dimensions of these deformation rings in [Kis08] (which comes
down to a computation on the generic fibre, using weakly admissible modules).
Lecture 7. The Herr Complex, and families of extensions
We continue to allow K/Qp to be an arbitrary finite extension.
7.1. Herr Complex. Let M be a projective étale (ϕ, Γ)-module over AK,A , where A
can be any Zp -algebra, but to make nontrivial statements we will often think about
the case where A is of finite type over Z/pa for some a ≥ 1. Then we define theHerr complex(ϕ−1,γ−1)(γ−1)⊕(1−ϕ)

C • (M ) = [M −−−−−−−→ M ⊕ M −−−−−−−−−→ M ]concentrated in degrees 0, 1, 2. Note that this is a complex of A-modules, rather
than AK,A -modules (the terms are AK,A -modules, but the maps are only AK,A semilinear with respect to the ϕ- or γ-actions). The following is [EG22, Thm.
5.1.22].
7.1.1. Theorem. Suppose A is a Noetherian Z/pa -algebra with A/p countable.
Then the Herr complex C • (M ) is a perfect complex of A-modules. If A is of finite
type over Z/pa , and B is a finite type A-algebra, then there is a natural quasi∼•
isomorphism C • (M ) ⊗L
A B −→ C (M ⊗AK,A AK,B ).
Proof. Because A is Noetherian, AK,A is flat over A. Since M is a projective AK,A module, it is therefore also a flat A-module, so the Herr complex is a complex of flat
A-modules (which are however not finite over A, but only over AK,A ). To checkthat C • (M ) is perfect, it therefore suffices to show that the cohomology A-modules
are of finite type over A. By Nakayama’s lemma for complexes, it in fact suffices
to prove the theorem in the case that A is an Fp -algebra, so we assume this fromnow on.

34

M. EMERTON AND T. GEE

If A is in fact a finite-dimensional Fp -vector space, Herr [Her98] showed thatC • (M ) computes H • (GK , ρ), where ρ is the associated GK -representation corresponding to M ; so the statement we want to prove is analogous to proving a version of the finiteness of Galois cohomology in the presence of coefficients. Our proof
largely follows that of Herr.
We don’t have an equivalence between (ϕ, Γ)-modules and Galois representations
for general coefficient rings, but we have similar behavior, as in the following lemma.
7.1.2. Lemma. H i (C • (M1∨ ⊗ M2 )) = Exti(ϕ,Γ)/AK,A (M1 , M2 ) for i = 0, 1. In addition, H 2 (C • (ad M )) controls the obstructions to infinitesimal deformations of M .
It may well be the case that H 2 (C • (M1∨ ⊗ M2 )) = Ext2(ϕ,Γ)/AK,A (M1 , M2 ), but
we have not attempted to verify this.
There is an A-linear map ψ : AK,A  AK,A , which is given by ψ = p1 tr ϕ, andan induced map ψ : M  M , which is defined so that for a ∈ A and m ∈ M wehave ψ(ϕ(a)m) = aψ(m) and ψ(aϕ(m)) = ψ(a)m.
Since ϕ commutes with Γ, so does ψ, and we have an induced morphism (1 − γ) :M ψ=0 → M ψ=0 . In fact, this morphism is an isomorphism. The proof of this takessome work, but at least morally it has the following explanation in the simplest case,e and (A0 )+ =when A = Fp , K = Qp and M = AQp . Working instead with Γ
Qp
e
]]
(as
an
F
[
Γ]-module)=F[∆]⊗
Zp [[Zp ]], we have Fp [[Zp ]]ψ=0 = Fp [[Z×pp
Fp Fp [[x]],pwhere x = γ − 1. Thus we see that (γ − 1) acts on Fp [[x]] by multiplication
by x, which is injective. After passing to the fraction field of Fp [[Zp ]], i.e. after0
passing to A0Qp , i.e. after inverting TQ, one finds, upon passing to ψ-invariants,pthat this corresponds to inverting x; i.e. that (A0Qp )ψ=0 = Fp [[∆] × Fp ((x)). Thus
multiplication by γ − 1 = x now acts invertibly.
Using the ψ operator, we can give an alternative description of the Herr complex
as follows. Let Cψ• (M ) be the complex in degrees 0, 1, 2 given by0(ψ−1,γ−1)/M/ M ⊕ M (γ−1)⊕(1−ψ) / M/ 0.

We have a morphism of complexes C • (M ) → Cψ• (M ) given by
0

/M

0


/M

(ϕ−1,γ−1)/ M ⊕ M (γ−1)⊕(1−ϕ) / M(−ψ,1)1(ψ−1,γ−1)

/0

−ψ



/ M ⊕ M (γ−1)⊕(1−ψ) / M/0That this is a morphism of complexes follows from the facts that ψ ◦ ϕ = id, andthat ψ commutes with γ; and since (1 − γ) : M ψ=0 → M ψ=0 is an isomorphism,
we see that is in fact a quasi-isomorphism.
With some more work, we can show that this morphism induces topological isomorphisms of the cohomology groups. We deduce the required finiteness from this:the basic idea is that a Frobenius truncation argument shows that the cohomology
groups of C • (M ) are subquotients of finitely generated A((T ))/A[[T ]]-modules, andthus have the discrete topology, while the cohomology groups of Cψ• (M ) are subquotients of finitely generated A[[T ]]-modules (so the slogan is: “discrete and compact
implies finite”). It is in these arguments that we use the countability assumption
on A, in order to use the theory of Polish groups.

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

35∼

Finally, to see that the obvious map of complexes C • (M ) ⊗A B −→ C • (M ⊗AK,A
AK,B ) induces a quasi-isomorphism, we use an argument of [KPX14] to formallyreduce to the case that B is a finite A-module, in which case the claim follows easily
as AK,B = AK,A ⊗A B.

7.2. Constructing families of extensions. Our next goal is to cover the underlying reduced substack Xd,red by certain families of étale (ϕ, Γ)-modules, having
rather explicit parameterizations, with the goal of showing that Xd,red is of finite
presentation over Fp , and of bounding its dimension from above,
We begin with the case of K = Qp and d = 1, and to this end return briefly
to the discussions of Lecture 1.4. By local class field theory (see also Lecture 1.3),
×
the Fp -valued characters of GK are all of the form ω i urα , for i = 0, . . . p − 2 and×α ∈ Fp . Going over to the setting of (ϕ, Γ)-modules, we can take the trivial (ϕ, Γ)module over AQp ,Fp [α,α−1 ] and twist ϕ by α and Γ by ω i , to get a Gm worth of×

representations (one for each α ∈ Fp ; here i is being kept fixed), but also a Gm
worth of scalar automorphisms. As we run over i we obtain a map
p−2
G[Gm /Gm ] → X1,red

i=0

which with a little work can be shown to be an isomorphism.
Now let d = 2. Any irreducible representation has the form
GQ

urα IndGQp ω2i
p22(for some 0 < i < p − 1 with (p + 1) - i). This gives us a point [Gm /Gm ] ,→ X2,red(again the α gives a Gm and then we quotient by scalars). But we know (looking
ahead to Theorem 9.1.1) that dim X2,red = 1 and that it’s equidimensional, so theirreducible representations lie in a substack of positive codimension. We can now
consider families of reducible representations, so we want to look at representations
coming from Ext1 (urα ω i , urβ ω j ). For generic choices of α, β there’s a unique nontrivial extension, and the intuition is that as α, β vary, the extension class varies
with them, and then you mod out by scalar endomorphisms to get the expected
one-dimensional stack (two dimensions coming from the choice of α, β).
In fact, we want to apply this same technique in more generality. The idea
is to begin with some representation, and then to iteratively build its spaces ofextensions by some given irreducible representations. More generally, assume that
we’re given a family ρT → T of rank d étale (ϕ, Γ)-modules, where T is a reduced
finite type variety over Fp . (This is our starting representation — of course inpractice we work with a (ϕ, Γ)-module, but we think of it as being a family of
Galois representations.) Then take σ an irreducible Galois representation (or, more
properly, the associated étale (ϕ, Γ)-module). As a technical hypothesis, assume
furthermore that the spaces Ext2GK (σ, ρt ) are of constant dimension when we vary t
over the closed points of T .
The base change property for C • (M ) in Theorem 7.1.1 involves a derived tensor
product, so there’s a spectral sequence relating C • (M ) and C • (M ⊗A B), but sinceH 2 is the highest degree, it satisfies naive base change. Therefore, our assumption
on the Ext2 having constant degree implies that H 2 (C • (σ ∨ ⊗ ρT )) is actually avector bundle over T . This implies that τ≤1 C • (σ ∨ ⊗ ρT ) is still perfect, and can

36

M. EMERTON AND T. GEE

thus be modeled as a two term complex [C 0 → Z 1 ] concentrated in degrees 0, 1,
whose terms are finite locally free over T . Note that we have a surjection Z 1 
H 1 (C • (σ ∨ ⊗ ρT )) = Ext1 (σ, ρT ).
Now let V be the vector bundle over T corresponding to Z 1 . Let ρV be thepullback of ρT to V . By analogy with the 2-dimensional case, we want to study a“universal extension” EV of ρV by σ, which should fit in the short exact sequence
0 → ρV → EV → σ → 0.
To construct this thing, we consider
Ext1 (σ, ρT ⊗ (Z 1 )∨ ) = Ext1 (σ, ρT ) ⊗ (Z 1 )∨ ,which is a quotient of
Z 1 ⊗ (Z 1 )∨ .
This tensor product has a “trace element”, whose image in Ext1 (σ, ρT ⊗ (Z 1 )∨ ) isan extension class corresponding to a short exact sequence
0 → ρT ⊗ (Z 1 )∨ → extension → σ → 0.
If we extend scalars from Fp to OV , and then pushforward under the map
ρT ⊗OT (Z 1 )∨ ⊗Fp OV → ρT ⊗OT OV
induced by the inclusion (Z 1 )∨ ,→ Sym• (Z 1 )∨ = OV together with the multiplicative structure of OV , we obtain the desired extension
0 → ρT ⊗OT OV → EV → σ ⊗Fp OV → 0.
Here EV is now an étale (ϕ, Γ)-module over V , and so is classified by a map
V → Xd,red (where d = dim ρT + dim σ). The object EV will admit various automorphisms, and if we compute them precisely, we can hope to get an embeddingof a quotient stack of V into Xd,red . Iterating this construction, we obtain familiesof étale (ϕ, Γ)-modules parameterized by quotient stacks of the total spaces of thevarious vector bundles V appearing via applications of the preceding construction,
which will ultimately cover the various Xd,red . (Since any Galois representation
over Fp can be written as an iterated extension of irreducible representations.)
7.3. Explicit families and labelling by Serre weights. The iterative construction that we just explained ultimately allows us to get a description of the underlying reduced stacks Xd,red for general d. Before stating our general result, we returnto the case of K = Qp and d = 2. Again, while the stack X2 is really a stack of(ϕ, Γ)-modules, for the purposes of gaining intuition, it is reasonable to imagine
our families of extensions as being extensions of Galois representations, and we will
describe things from this perspective without further comment for the rest of thissection. For simplicity we assume that p > 2 throughout this discussion.
Take T = Gm = Spec Fp [α, α−1 ] and ρT = urα ω i . For the moment let σ = 1(this will suffice to describe all the geometric phenomena; twisting by a 1-dimensionalfamily urβ ω j for some 0 ≤ j < p − 1 gives the remaining cases). Fix some 0 ≤ i <p − 1. Then
Ext2 (1, urα ω i ) = H 2 (GQp , urα ω i )which (for example by Tate local duality) vanishes unless i = 1 and α = 1, in whichcase it’s 1-dimensional.

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

37

We begin with the case that 2 ≤ i ≤ p − 2. Then
Ext2 (1, urα ω i ) = 0,
and
Ext0 (1, urα ω i ) = (urα ω i )GQp = 0
so actually (by Tate’s local Euler characteristic formula) Ext1 (1, urα ω i ) has constant rank 1, so Ext1 (1, ρT ) is a line bundle. Its total space V is then a copy of
A1 × Gm that parameterizes a family of extensions
urα ω i ∗.
01
We can add one dimension to this family by forming unramified twists — thus
obtaining a family parameterized by V × Gm = A1 × Gm × Gm , parameterizing
extensions of the formurαβ ω i∗.
0
urβThere is an action of (a different copy of) Gm × Gm on this family, arising fromthe action of Gm by automorphisms on each of the characters urαβ ω i , and urβ .
If we let (a, b) denote the coordinates on this copy Gm × Gm that is acting, andlet x denote the coordinate on A1 (thus x provides a coordinate on each of the
1-dimensional spaces Ext1 (urβ , urαβ ω i )) — so that the coordinates on V = A1 ×
Gm × Gm are (x, α, β) — then this action is given by the formula (a, b) · (x, α, β) =(ab−1 x, α, β). (The fact that the stabilizer of any particular point (x, α, β) is the
diagonal copy of Gm if x 6= 0, and the entirety of Gm × Gm when x = 0, isa manifestation of the fact that the only automorphisms of a non-split extensionof distinct characters are the scalar matrices, while the automorphisms of a split
extension (i.e. a direct sum) of distinct characters are the diagonal matrices.) Thus
we get an embedding[(A1 × Gm × Gm )/(Gm × Gm )] ,→ X2,red ,whose source visibly has dimension 3 − 2 = 1.
We next turn to the case i = 0, where Tate local duality gives us that
Ext2 (1, urα ) = H 2 (GQp , urα ) = H 0 (GQp , urα ⊗ −1 ) = 0
×for all α ∈ Fp . However, we now run into the issue that Ext0 is not a vector bundle,
because it has rank 0 everywhere except for when α = 1, and at this point the rank
jumps from 0 to 1; correspondingly, the rank of Ext1 (1, urα ) jumps from 1 to 2. In
the framework of Lecture 7.2, this means that we can’t choose the complex C 0 → Z 1
so that C 0 = 0 and Z 1 = Ext1 ; we have to instead allow C 0 to have rank 1, andZ 1 to have rank 2. If we go through the construction of that subsection, we end up
with a family parameterized by the total space V of a rank 2 vector bundle on Gm .
We can add in the unramified twists, to obtain a family parameterized by V × Gm ,and we obtain a map(7.3.1)

V × Gm → X2,redclassifying the family of Galois representations that V × Gm parameterizes. Thismap won’t be an embedding, of course, because various different points on V × Gm
will give rise to isomorphic Galois representations. Just as in the case 2 ≤ i ≤ p − 2
considered above, there is a scaling action of Gm × Gm on the various Ext1 spaces.

38

M. EMERTON AND T. GEE

Also, any two points in of V that lie over the same point α ∈ Gm (Fp ), and whichlie in the same fibre of the surjection(7.3.2)Zα1 → Ext1 (1, urα )(here we write Zα1 to denote the fibre of Z 1 over the point α) parameterize thesame extension of 1 by urα . And there are even more isomorphisms that canoccur between the representations parameterized by this family: for example, if we
restrict to the family of split extensions urαβ ⊕ urβ parameterized by Gm × Gm(the “zero section” of V × Gm , which we also refer to as the “split locus” of thefamily), then the points (α, β) and (α−1 , αβ) both parameterize the direct sum
urαβ ⊕ urβ . Since these “extra isomorphisms” occur only on a proper subvariety
of V × Gm , though, they don’t play a role in determining the dimension of theimage of (7.3.1). Indeed, since the kernel of the surjection (7.3.2) generically (moreprecisely, whenever α 6= 1) has dimension 1, we find that the image of (7.3.1) is
an irreducible constructible substack of X2,red of dimension 4 − 2 − 1 = 1. The
structure of this substack along the image of the split locus, as well as along theimage locus where α = 1, is rather complicated; for example, along the split locus
we get a fold singularity (corresponding to the identification of pairs of points (α, β)and (α−1 , αβ)), degenerating to some kind of cusp singularities along the points
where furthermore α = 1. (See [San14] for some computations of the versal ringsat these points.)
Finally, in the case i = 1, Tate local duality gives
(
1 α=1
2
2
0
dim Ext (1, urα ω) = dim H (GQp , urα ω) = dim H (GQp , urα ) =0 α 6= 1
so Ext2 is no longer a vector bundle. On the other hand, we have
H 0 (GQp , urα ω) = 0.
Following the strategy of Lecture 7.2, we thus break Gm up into two pieces: T0 :=Gm \ {α = 1}, and T1 := {α = 1}. Over T0 , we see that Ext2 (1, urα ω) has constantrank 0, and we can construct a family analogously to the case 2 ≤ i ≤ p − 1,parameterized by the total space of a line bundle over Gm \ {α = 1}. Adding in
the unramified twists, this leads to an embedding[(A1 × (Gm \ {α = 1}) × Gm )/(Gm × Gm )] ,→ X2,red .
Passing now to T1 (which is just a single point! — over which Ext2 (1, urα ω) hasconstant rank 1), we obtain a family parameterized by the 2-dimensional vector
space Ext1 (1, ω). Again adding in unramified twists, we obtain an embedding[(A2 × Gm )/(Gm × Gm )] ,→ X2,red .
Each of the finite number of families that we have just described (along withthe families obtained by taking a twist of one of these families by ω j for somej = 1, . . . , p − 2) gives rise to a 1-dimensional irreducible constructible substackof X2,red . By construction, every Fp -point of X2,red that corresponds to a reducible
Galois representation lies exactly one of these subsets.
We have already seen that the Fp -points corresponding to irreducible Galois representations lie in a finite union of images of the 0-dimensional stack [Gm /Gm ].
We thus deduce that X2,red is 1-dimensional, and that the closure of each of theMODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

39

1-dimensional and mutually disjoint constructible substacks that we’ve just constructed must be an irreducible component of X2,red . We now explain how we canlabel these components by so-called Serre weights.
Temporarily return to the case of general K, d, and let k be the residue fieldof K. Then by definition, a Serre weight is an (isomorphism class of) irreducible Fp representations of GLn (k). These are determined by their highest weights, which
are tuples of integers {kσ,i }σ:k,→Fp ,1≤i≤d with the properties that• p − 1 ≥ kσ,i − kσ,i+1 ≥ 0 for each 1 ≤ i ≤ d − 1, and• p − 1 ≥ kσ,d ≥ 0, and not every kσ,d is equal to p − 1.
We refer the reader to Lecture 9 for some explanations and motivation for theappearance of the representation theory of GLn (k) in the geometry of our stacks.
Returning to the case that K = Qp and d = 2, the Serre weights are the2representations detj Symi−1 Fp with 0 ≤ j < p − 1, 1 ≤ i ≤ p; in the notation
above, these have highest weight (k1 , k2 ) = (j+i−1, j). Our labelling of components
by Serre weights is then as follows.4 An irreducible component which genericallyparameterizes representations of the formurαβ ω i+j∗0
urβ ω jis labelled by the highest weight (i + j − 1, j) (that is, by the representation2
detj Symi−1 Fp ). This is unambiguous except for the possibilities i = 1, p, which
we distinguish as follows. In the discussion above, we saw that if i = 1 there is one
irreducible component for which α = 1 at a dense set of points, and we label this2
component by detj Symp−1 Fp ; and there is one component where α 6= 1 at a dense2

set of points, which we label by detj Sym0 Fp .
We can inductively extend the the construction technique that we have justdescribed for d = 2, so as to prove the following general theorem. The notion of
being “generically maximally nonsplit of niveau one” means that for an open set ofpoints, the corresponding GK -representation is a successive extension of charactersin a unique way; to such a representation one can associate a Serre weight k, via
a straightforward extension of the labelling we just explained in the case K = Qp
and d = 2.
7.3.3. Theorem.
(1) The Ind-algebraic stack Xd,red is an algebraic stack, of finite presentation
over F.
(2) We can write (Xd,red )Fp as a union of closed algebraic substacks of finite
presentation over Fp
[ k
small
(Xd,red )Fp = Xd,red,F∪
Xd,red,F ,ppk4For accuracy, in case the reader tries to carefully compare our present discussion with thecorresponding discussion in [EG22], we note that our labelling here differs from the one given inthat reference; the component that here we label by the Serre weight σ will there be labelled by
the Serre weight σ ∨ ⊗det−1 . We have chosen our present convention just because it is notationallyeasier to work with extensions of 1 by powers of ω rather than with extensions of powers of ω −1
by 1. If you like, we have applied the “Cartier duality” involution — given by ρ 7→ ρ∨ ⊗ ω — toX2,red .

40

M. EMERTON AND T. GEE

where:small• Xd,red,F
is empty if d = 1, and otherwise is non-empty of dimensionpstrictly less than [K : Qp ]d(d − 1)/2.
k• each Xd,red,F is a closed irreducible substack of dimension [K : Qp ]d(d−p1)/2, and is generically maximally nonsplit of niveau one and weight k.
(3) If we fix an irreducible representation α : GK → GLa (Fp ) (for some a ≥1), then the locus of ρ in Xd,red (Fp ) for which dim Ext2GK (α, ρ) ≥ r is ofdimension at most[K : Qp ]d(d − 1)/2 − r.
7.3.4. Remark. Note that in part (3), the locus of points in question corresponds
to a closed substack of (Xd,red )Fp , by upper-semicontinuity of fibre dimension.
k

7.3.5. Remark. Since the Xd,red,F are irreducible, have dimension equal to thatpof (Xd,red )Fp , and have pairwise disjoint open substacks (corresponding to maximally nonsplit representations of niveau 1 and weight k), they are in fact distinct
irreducible components of (Xd,red )Fp .
7.3.6. Remark. We can, and will, be much more precise about the structure of Xd,red .
Namely, in Lecture 9 we combine Theorem 7.3.3 with the results of Lecture 8 toshow that Xd,red is equidimensional of dimension [K : Qp ]d(d−1)/2; accordingly, thekirreducible components of (Xd,red )Fp are precisely the Xd,red,F , and in particularp
are in bijection with the Serre weights k.
7.3.7. Remark. The upper bound of Theorem 7.3.3 (3) is quite crude when [K :Qp ] > 1, although it is reasonably sharp in the case K = Qp . However, it suffices forour purposes. (In fact, in the inductive proof of Theorem 7.3.3 we use a slightly more
complicated upper bound to control the dimensions of various extension groups, but
the stated bound is all that we will need going forward.)
Lecture 8. Crystalline Lifts
8.1. Crystalline lifts. The following theorem is the main result of this lecture.
8.1.1. Theorem. If ρ : GK → GLd (Fp ) is continuous, then there exists ρ◦ :GK → GLd (Zp ) lifting ρ such that the corresponding p-adic Galois representation
ρ : GK → GLd (Qp ) is crystalline with regular Hodge-Tate weights.
8.1.2. Remark. In fact, the lift ρ◦ in Theorem 8.1.1 can be chosen to be potentiallydiagonalizable in the sense of [BLGGT14], which implies in turn that if p - 2d, thenρ can be globalized to come from an automorphic form (see [EG22, Thm. 1.2.3]).
While the statement of Theorem 8.1.1 is purely local (in the sense that the
representation ρ is fixed), and our proof is for the most part via a local argument,
we will also make crucial use of Theorem 7.3.3.
We begin by explaining the “obvious” strategy to prove Theorem 8.1.1. Firstly
consider the case that ρ : GK → GLd (Fp ) is irreducible. Then it is easy to showthat ρ is induced from a character of the unramified extension K 0 /K of degree d,and furthermore it is easy to show that this character can be lifted to a crystalline
character, and because K 0 /K is unramified, the induction of a crystalline character

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

41

of GK 0 to GK gives a crystalline representation. Furthermore, one has considerable
control over the Hodge–Tate weights of such a lift.
An immediate consequence is that all semisimple ρ have crystalline lifts. However, not all mod p representations are semisimple, so the remaining problem is
to show that we can lift extensions of representations. To get a feeling for this,consider the two-dimensional case, and so suppose that ρ : GK → GL2 (Fp ) is of
the form


χ1 ∗
.
0 χ2
×

×

Choose, as we may, crystalline lifts χ1 : GK → Zp , χ2 : GK → Zp of χ1 , χ2
respectively, and suppose that the labelled Hodge–Tate weights of χ1 are slightly
greater than those of χ2 , in the sense that for each embedding K ,→ Qp thecorresponding Hodge–Tate weight of χ1 is greater than that of χ2 , and for at least
one embedding, the gap is greater than 1. Then any extension of χ2 by χ1 isautomatically crystalline (for example by the formulae in [Nek93, Prop. 1.24]).
Since extensions are classified by H 1 , it is therefore enough to show that given any
class in H 1 (GK , χ1 χ−12 ), we can choose χ1 , χ2 in such a way that this class can belifted to H 1 (GK , χ1 χ−1
2 ).
As we will soon see, this can always be arranged, and in “most” cases the natural−1
1
map H 1 (GK , χ1 χ−12 ) → H (GK , χ1 χ2 ) is actually surjective. It is then naturalto imagine that one could inductively prove the existence of lifts in all dimensions
by induction on the number of Jordan–Hölder factors of ρ. Quite a few people
have tried to do this, but no-one succeeded, as far as we know. The basic problem
seems to be that it is in general not possible to fix the lifts of each Jordan–Hölder
factor when arguing inductively; rather, one keeps having to go back and adjust
the previous choices in light of the next step.
For concreteness, here are some examples (due to Tong Liu) which are difficult
to do by hand, and give a sense of why it’s hard to work inductively. For the rest of
the lecture we write ε for the cyclotomic character, and ε for the mod p cyclotomic
character. Firstly, consider(8.1.3)


ε

W =


∗
ε


∗ ∗
0 ∗

ε ∗
1

where ∗ could be zero or not. It is quite easy to convince yourself that there areenough “degrees of freedom” (by choosing for example different unramified twists ofthe characters that you lift to, and lifting to different extension classes) to manage
to make the lift, but quite tricky to come up with an actual argument. If that one
is too easy, you can try something like
 2
ε




∗
ε2

∗
0
ε2


∗
∗
.
∗
W

42

M. EMERTON AND T. GEE

The key to our approach is the following theorem. Note that when we use thistheorem inductively, we are not guaranteeing that you can fix lifts of each Jordan–Hölder factor once and for all before considering the various possible extension
classes, and this increased flexibility is important in our argument.
8.1.4. Theorem. Suppose given a representation ρd : GK → GLd (Fp ) that admits
a lift ρ◦d : GK → GLd (Zp ) which is crystalline with labelled Hodge–Tate weights λ.
Let 0 → ρd → ρd+a → α → 0 be any extension of GK -representations over Fp ,with α : GK → GLa (Fp ) irreducible, and let α◦ : GK → GLa (Zp ) be any crystalline
lifting of α with labelled Hodge–Tate weights λ0 , which we assume to be slightly lessthan λ.
Then we may find a lifting of the given extension to an extension◦
0 → θd◦ → θd+a
→ α◦ → 0of GK -representations over Zp , where θd◦ : GK → GLd (Zp ) again has the property
that the associated p-adic representation θd : GK → GLd (Qp ) is crystalline with◦labelled Hodge–Tate weights λ. Furthermore, θd+a
is crystalline, and we may choose
λ,crys◦θd to lie on the same irreducible component of Spec Rρdthat ρ◦d does.
Given Theorem 8.1.4, it is easy to prove Theorem 8.1.1 by induction on d. In
fact, without much additional difficulty one can prove the following stronger version,
which is useful in applications.
8.1.5. Theorem. Let K/Qp be a finite extension, and let ρ : GK → GLd (Fp ) be acontinuous representation. Then ρ admits a lift to a crystalline representation ρ◦ :GK → GLd (Zp ) of some regular labelled Hodge–Tate weights λ. Furthermore:(1) ρ◦ can be taken to be potentially diagonalizable.
(2) If every Jordan–Hölder factor of ρ is one-dimensional, then ρ◦ can be taken
to be ordinary.
(3) ρ◦ can be taken to be potentially diagonalizable, and λ can be taken to be alift of a Serre weight.
(4) ρ◦ can be taken to be potentially diagonalizable, and λ can be taken to havearbitrarily spread-out Hodge–Tate weights: that is, for any C > 0, we canchoose ρ◦ such that for each σ : K ,→ Qp , we have λσ,i − λσ,i+1 ≥ C foreach 1 ≤ i ≤ n − 1.
We now discuss the proof of Theorem 8.1.4. Return to the two-dimensional case,−1
1
and write χ = χ1 χ−1
2 , χ = χ1 χ2 , so that we want to lift classes in H (GK , χ)1
to H (GK , χ). Fix a sufficiently large finite extension E/Qp with ring of integers O,
uniformizer $, and residue field F, so that in particular χ takes values in O× .
Taking cohomology of$

0 → O(χ) → O(χ) → F(χ) → 0,
we see that we have an exact sequence
H 1 (GK , χ) ⊗O F → H 1 (GK , χ) → H 2 (GK , χ)[$].
By Tate local duality, H 2 (GK , χ) is dual to H 0 (GK , (E/O)(χ−1 )(1)), with the (1)denoting a Tate twist, so we see in particular that if χ 6= ε, then there is nothing
to prove. However, if χ = ε, then there is some work to do.

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

43

One thing that we could do in this case is to take χ = ε, in which case H 2 (GK , χ)[$]vanishes, and we see that we can lift to semistable (not necessarily crystalline) extensions. One might wonder if this means that in general we should just relax
things and try to only produce semistable lifts, rather than crystalline ones, but inpractice this doesn’t seem to be the case - the inductive arguments still run into
trouble on examples like (8.1.3) above. Indeed, let us try to handle (8.1.3) by lifting inductively, allowing ourselves to use semistable lifts. The bottom right 3 × 3
matrix is a sum of two extensions with χ = ε, so we’ve just seen that we can lift it
to a representation U of the form


ε 0 ∗
U =  ε ∗ .
1
Then


W =

ε

∗
U


,

so we could try to consider lifts of the form


ε ∗
W =
.
U
However, if the classes giving the two extensions of ε by ε aren’t twists by ε ofthe unramified extension of the trivial character by itself, then these lifts won’t besemistable (because the only semistable extensions of the trivial character by itself
are unramified). So we are forced instead to consider something like
 p

ε
∗
W =
,
U
and then we run into difficulties with the existence of torsion in H 2 .
So let’s return to the case of χ = ε, and consider the problem of finding crystalline
lifts. Rather than taking χ = ε, we take χ to be an unramified twist of εp , sothat all extensions are crystalline. In this case one finds that H 2 (GK , χ)[$] is 1dimensional, so there is a possible obstruction to lifting; and indeed H 1 (GK , ε) is
2-dimensional, while H 1 (GK , χ)⊗O F is 1-dimensional, so no fixed χ can provide all
the lifts that we need. It is possible to analyse the situation directly using Kummer
theory, and one finds that given any class in H 1 (GK , ε), one can choose χ so thata lift exists.
It is then natural to consider the moduli space of all such χ, and the universal
extension over this space. The universal χ is the character χt : GK → O[[t]]×
given by χt = εp λ1+t , where λx is the unramified character sending a (geometric)
Frobenius element to x. Then by Tate local duality and local class field theory, it
is easy to see that H 2 (GK , χt ) = O[[t]]/(p, t) = O/p. One checks that H 1 (GK , χt )is a free O[[t]]-module of rank 1, but that the corresponding “universal family” ofextensions over Spec O[[t]] is not in fact universal; more precisely, the fibre over any
given x : O[[t]] → O, t 7→ a sits in a short exact sequence(8.1.6)0 → H 1 (GK , χt ) ⊗O[[t]],x O → H 1 (GK , εp λ1+a ) → O[[t]]/(p, t) → 0.

So we see would like to have a way of “rescaling” H 1 (GK , χt ) by dividing by (p, t),so as to have a universal family that sees every extension; and then we could try tocheck that this family surjects onto H 1 (GK , ε). To do this, we blow up Spec O[[t]]44

M. EMERTON AND T. GEE

along (p, t), giving a scheme X, which is flat over Zp , and for which (p, t) pulls backto an ideal sheaf I. Over X, we do have a universal family H 1 (GK , χX ), and one
can check that its base change to any point x
e : Spec O → X gives the full H 1 .
0More explicitly, since H plays no role, we may assume that the Herr complex
is computed by a complex C 1 → C 2 of locally free modules over O[[t]]. Since
H 2 = O[[t]]/(p, t), we can imagine that this complex looks like(p,t)

O[[t]]2 −→ O[[t]].

(8.1.7)(Note that this is consistent with (8.1.6).) In particular, the image of the coboundary map is the ideal I = (p, t). When we blow up Spec O[[t]] at (p, t) to obtain thescheme X, the complex (8.1.7) pulls back over X to a complex
e 1 = O⊕2 → OX = C
e2 ,
C
X

(8.1.8)and the image of the coboundary map is now locally free, equal to the ideal sheaf
I of the exceptional divisor. The kerrnel Ze1 is then a locally free sheaf (being thekernel of a surjection between locally free sheaves), and gives the desired “universal
family” of extensions.
To see this, consider a class in H 1 (GK , ε) arising from a cocycle c, and choosesome c : O[[t]] → C 1 which specializes at ($, t) = 0 (the closed point of Spec O[[t]])e 1 , which composed with the coboundary
to c. This pulls back to a map ec : OX → C
gives a map δec : OX → I. Since I is a non-trivial invertible sheaf, this map (thoughtof as a section of the line bundle associated to I) has a zero at some closed pointx ∈ X. We choose an affine open neighbourhood U of x in X, and consider thee 2 obtained by restricting (8.1.8) to U (or, equivalently, by pulling
e1 → C
complex CU
Uback (8.1.7) to U ). Now (letting IU denote the restriction of I to U ) since IU is
e 1 → IU splits, and we choose a
locally free, and since U is affine, the surjection C
U
1
1
e = Ze ⊕ IU .
splitting CUUe1 ⊕ IU whose projection to the
The restriction ecU of ec to U is then a section of Z
Usecond factor vanishes at x. Hence, if we let zeU denote the projection of ec to thefirst factor, then zeU and ecU coincide at x; thus zeU is a cocycle over U which lifts
our original cocycle c. (Projecting ec to the first factor has achieved the desiered“division by (p, t)”.) If we specialize this cocycle over any O-valued point of U
passing through x (and since U is flat over Zp , such a point always exists after
extending O if necessary), then we obtain the desired lift of c.
The proof of Theorem 8.1.4 is via a generalisation of this construction, withthe following theorem being the general analogue that we need of the fact used
above that H 2 (GK , χt ) = O[[t]]/(p, t). (To see the relationship with the previous
argument, which used non-triviality of the line bundle associated to I, note thatif δec induced a surjection OX → I, then we would see that c induces a surjection
O[[t]] → I, hence I would be principal, and H 2 = O[[t]]/I would be supported incodimension one, whereas in fact it is supported only in codimension two.)
8.1.9. Theorem. For any regular tuple of labelled Hodge–Tate weights λ, andany fixed irreducible representation α : GK → GLa (Fp ), the locus of points x ∈Spec Rλ,crys /$ for whichdimκ(x) κ(x) ⊗R Ext2GK (α, ρuniv ) ≥ rρ

has codimension at least r.

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

45
λ,crys

Proof. See [EG22, Thm. 6.1.1]. The basic idea is that since the stack Xd
is ap-adic formal algebraic stack, its special fibre is an algebraic stack, and the ringRλ,crys /$ is an effective versal ring at some finite type point. It therefore suffices
to prove a corresponding codimension bound on the support of Ext2GK (α, −) in thespecial fibre of Xd , and this is provided by Theorem 7.3.3 (3).
Lecture 9. Geometric Breuil–Mézard
As in the previous lectures, we will concentrate on the potentially crystalline casewhen formulating the Breuil–Mézard conjecture; most of what we say goes overunchanged to the potentially semistable version, and we refer the reader to [EG22,§8] for the details.
9.1. The irreducible components of Xd,red . We can now complete the analysisof the irreducible components of Xd,red . Recall that by Theorem 7.3.3, Xd,red isan algebraic stack of finite presentation over F, and has dimension [K : Qp ]d(d −1)/2. Furthermore, for each Serre weight k, there is a corresponding irreduciblekcomponent Xd,red,F of (Xd,red )Fp , and the components for different weights k arepdistinct. Using our results on crystalline lifts, we can now show that these are theonly irreducible components. The following is [EG22, Thm. 6.5.1].
9.1.1. Theorem. Xd,red is equidimensional of dimension [K : Qp ]d(d−1)/2, and thekirreducible components of (Xd,red )Fp are precisely the various closed substacks Xd,red,F ;pkin particular, (Xd,red )Fp is maximally nonsplit of niveau 1. Furthermore each Xd,red,Fpk

can be defined over F, i.e. is the base change of an irreducible component Xd,redof Xd,red .
k

Proof. To see that the Xd,red,F may all be defined over F, we need to show thatpthe action of Gal(Fp /F) on the irreducible components of (Xd,red )Fp is trivial.
This follows immediately by considering its action on the maximally nonsplit representations of niveau 1 (since the action of Gal(Fp /F) preserves the property of
being maximally nonsplit of niveau 1 and weight k). It is therefore enough toprove that each irreducible component of (Xd,red )Fp is of dimension of at least[K : Qp ]d(d − 1)/2; this follows by choosing a closed point not contained in any
other irreducible component, and noting that (by Theorem 8.1.1) it is contained incrys,λthe special fibre of some Xd
with λ regular.

We can also prove [EG22, Prop. 6.5.2]:
9.1.2. Proposition. Xd is not a p-adic formal algebraic stack.
Proof. Assume that Xd is a p-adic formal algebraic stack, so that its special fibre X d := Xd ×O F is an algebraic stack, which is furthermore of finite type over F.
Since the underlying reduced substack of X d is Xd,red , which is equidimensional ofdimension [K : Qp ]d(d−1)/2, we see that X d also has dimension [K : Qp ]d(d−1)/2.
Computing with versal rings, we see that for every ρ : GK → GLd (F) the
unrestricted framed deformation ring Rρ /$ must have dimension d2 +[K : Qp ]d(d−1)/2. However, it is known that there are representations ρ for which Rρ /$ isformally smooth of dimension d2 + [K : Qp ]d2 (see for example [All19, Lem. 3.3.1]).
Thus we must have d2 = d(d − 1)/2, a contradiction.
46

M. EMERTON AND T. GEE

9.2. The qualitative geometric Breuil–Mézard conjecture. If λ is a regularcrys,λ,τHodge type, and τ is any inertial type, then the stack Xd
is a finite type p-adic
formal algebraic stack over O, which is O-flat and equidimensional of dimension 1+
crys,λ,τis an algebraic stack[K : Qp ]d(d − 1)/2. It follows that its special fibre X dcrys,λ,τ
over F which is equidimensional of dimension [K : Qp ]d(d − 1)/2. Since Xdcrys,λ,τis a closed substack of Xd , its special fibre X d
is a closed substack of thespecial fibre X d , and its irreducible components (with the induced reduced substack
structure) are therefore closed substacks of the algebraic stack X d,red .
Since X d,red is equidimensional of dimension [K : Qp ]d(d − 1)/2, it follows thatcrys,λ,τthe irreducible components of X d

are irreducible components of X d,red , andk
are therefore of the form X d,red for some Serre weight k.
crys,λ,τk
For each k, we write µk (X d) for the multiplicity of X d,red as a component
crys,λ,τcrys,λ,τ. We write Zcrys,λ,τ = Z(X d) for the corresponding cycle, i.e. forof X dthe formal sum(9.2.1)

Zcrys,λ,τ =X

crys,λ,τµk (X dk

) · X d,kwhich we regard as an element of the finitely generated free abelian group Z[Xd,red ]kwhose generators are the irreducible components X d .
Fix some representation ρ : GK → GLd (F), corresponding to a point x :Spec F → Xd . For each regular Hodge type λ and inertial type τ , we have ancrys,λ,τcrys,λ,τeffective versal morphism Spec Rρ
/$ → X. For each k we setk

Ck (ρ) := Spf Rρ ×Xd Xd ,which we regard as a cycle of dimension d2 + [K : Qp ]d(d − 1)/2 in Spec Rρ /$k(note that since Xd is algebraic, it has effective versal rings, so we really get a
subscheme of Spec Rρ /$, rather than of Spf Rρ /$). The following theorem gives
a qualitative version of the refined Breuil–Mézard conjecture [EG14, Conj. 4.2.1].
While its statement is purely local, we do not know how to prove it without making
use of the stack Xd .
9.2.2. Theorem. Let ρ : GK → GLd (F) be a continuous representation. Then
there are finitely many cycles of dimension d2 + [K : Qp ]d(d − 1)/2 in Spec Rρ /$such that for any regular Hodge type λ and any inertial type τ , each of the special
crys,λ,τfibres Spec Rρ
/$ is set-theoretically supported on some union of these cycles.
crys,λ,τcrys,λ,τProof. We have Spec Rρ
/$ = Spf Rρ ×Xd X. It follows from (9.2.1),
together with the definition of Ck (ρ), that we may write the underlying cycle asXcrys,λ,τcrys,λ,τ
(9.2.3)Z(Spec Rρ/$) =µk (X d) · Ck (ρ).
k

The theorem follows immediately (taking our finite set of cycles to be the Ck (ρ)). We can regard this theorem as isolating the “refined” part of [EG14, Conj.
4.2.1]; that is, we have taken the original numerical Breuil–Mézard conjecture,formulated a geometric refinement of it, and then removed the numerical part

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

47

of the conjecture. The numerical part of the conjecture consists of relating thecrys,λ,τmultiplicities µk (X d) to the representation theory of GLn (k), as we now recall.
The following theorem is essentially due to Schneider–Zink [SZ99].
9.2.4. Theorem. Let τ : IK → GLd (Qp ) be an inertial type. Then there is a finitedimensional smooth irreducible Qp -representation σ crys (τ ) of GLd (OK ) with theproperty that if π is an irreducible smooth Qp -representation of GLd (K), then the
Qp -vector space HomGLd (OK ) (σ crys (τ ), π) has dimension at most 1, and is nonzero
precisely if recp (π)|IF ∼= τ , and N = 0 on recp (π).
For each regular Hodge type λ we let W (λ) be the corresponding representationof GLd (OK ), defined as follows: For each σ : K ,→ Qp , we write ξσ,i = λσ,i −(d−i),so that ξσ,1 ≥ · · · ≥ ξσ,d . We view each ξσ := (ξσ,1 , . . . , ξσ,d ) as a dominant weight of
the algebraic group GLd (with respect to the upper triangular Borel subgroup), andwe write Mξσ for the algebraic
OK -representation of GLd (OK ) of highest weight ξσ .
N
Then we define Lλ := σ Mξσ ⊗OK ,σ O.
For each τ we let σ crys,◦ (τ ) denote a choice of GLd (OK )-stable O-lattice in σ crys (τ ),write σ crys (λ, τ ) := Lλ ⊗O σ crys,◦ (τ ), and write σ crys (λ, τ ) for the semisimplification of the F-representation of GLd (k) given by σ crys (λ, τ ) ⊗O F. For each Serreweight k, we write Fk for the corresponding irreducible F-representation of GLd (k).
Then there are unique integers ncrysk (λ, τ ) such that
crys

⊕nk
σ crys (λ, τ ) ∼= ⊕ k Fk(λ,τ ).

Our “universal” geometric Breuil–Mézard conjecture is as follows (see Lecture 9.4
below for some motivation for this conjecture).
9.2.5. Conjecture. There are cycles Zk with the property
P that for each regular
Hodge type λ and each inertial type τ , we have Zcrys,λ,τ = k ncrysk (λ, τ ) · Zk .
9.3. The relationship between the numerical, refined and geometric Breuil–Mézard conjectures. In brief (see [EG22, §8.3] for the details), the relationship is
as follows: Conjecture 9.2.5 implies (by pulling back to versal rings) the geometricconjecture of [EG14], which in turn implies the numerical conjecture.
Conversely, the numerical conjecture implies Conjecture 9.2.5; in fact, it isenough to know the numerical conjecture for a single sufficiently generic ρ on each
irreducible component of Xd,red . To see this, recall that the numerical conjecturefor ρ is that there are integers µk (ρ) such that for all k, we have
X cryscrys,λ,τ
(9.3.1)e(Spec Rρ/$) =nk (λ, τ )µk (ρ).
k

For each k we choose a point xk : Spec F → X d,red which is contained in Xkk0

and not in any X for k 0 6= k. We furthermore demand that xk is a smooth
point of X d,red . (Since X d,red is reduced and of finite type over F, there is a densekset of points of X satisfying these conditions.) Write ρk : GK → GLd (F) for
the representation corresponding to Xk , and assume that the numerical conjectureholds for each ρk . Then if we set
X
k0(9.3.2)Zk :=µk (ρk0 ) · X ,k0

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M. EMERTON AND T. GEE

it is easy to check that Conjecture 9.2.5 holds.
The numerical Breuil–Mézard conjecture (and consequently Conjecture 9.2.5)
holds if K = Qp and d = 2. Most cases are proved in Kisin’s paper [Kis09] andPaškūnas’ paper [Paš15], and the remaining cases not handled by these papers areproved in the papers [HT15, San14, Tun21a, Tun21b]. The conjecture also holds
if d = 2, p > 2, λ = (0, 1), and K and τ are arbitrary, by the main result of [GK14].
For some recent progress for d > 2 in suitably generic situations, we refer the reader
to [LHLM20].
In the case that d = 2, p > 2, and K is arbitrary, we can make the cycles Zk
completely explicit: we say that a Serre weight k for GL2 is “Steinberg” if for each σ
we have kσ,1 − kσ,2 = p − 1. If k is Steinberg then we define k̃ by k̃σ,1 = k̃σ,2 = kσ,2 .
k

k

k̃

Then if k is not Steinberg, we have Zk = X2 , while if k is Steinberg, Zk = X2 + X2 .
This explicit description follows from the results of [CEGS19]; for the details,see [EG22, Thm. 8.6.2]. (Roughly speaking, the point is that it’s easy to computethe tamely potentially Barsotti–Tate deformation rings for generic extensions ofgeneric characters, and they’re either zero or formally smooth.)9.4. The weight part of Serre’s conjecture. Finally, we very briefly explainsome motivation for Conjecture 9.2.5, and its relationship to the weight part of
Serre’s conjecture. This connection was first explained in the context of the numerical Breuil–Mézard conjecture in [GK14]. For more details, see for example [GHS18](particularly Sections 3 and 4). We will further expand on this discussion in thecontext of a hypothetical p-adic local Langlands correspondence involving sheaves
on Xd in Lecture 10.4.
We expect that the cycles Zk will be effective, in the sense that they are comk
binations of the Xd with non-negative coefficients. Indeed, we expect that after
pulling back to the special fibre of the universal deformation ring at some ρ, thecycle Zk will be precisely the support of the Taylor–Wiles–Kisin patched modulesof mod p automorphic forms of weight k, and the support of a sheaf is an effective
cycle by definition. (Again, see for example [GHS18, §3, 4] for more details; we areslightly simplifying the situation by supposing that the Taylor–Wiles–Kisin method
does not introduce any patching variables, but these are essentially irrelevant forour discussion in any case.)
The weight part of Serre’s conjecture is a prediction that the possible Serreweights of mod p automorphic forms giving rise to a fixed global Galois representation only depends on the restrictions of this representation to decomposition groupsof places above p. If this is the case, there is a way to associate a corresponding
set of Serre weights to each ρ : GK → GLd (Fp ). Such a description was originallygiven by Serre in the case K = Qp and d = 2; while Serre’s recipe is completely
explicit, in general it seems to be unreasonable to hope for such a description.
However, if we admit the standard expectation (which is closely related to the
Fontaine–Mazur conjecture) that all irreducible components of the generic fibresof crystalline deformation rings are witnessed by automorphic forms, it follows formally from the Taylor–Wiles–Kisin method that the cycle Zk is indeed (as conjectured above) the support of the corresponding patched module of weight k modular
forms, and from this one easily deduces that representation ρ admits k as a Serreweight if and only if Zk is supported at ρ.

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

49

Equivalently, we can rephrase the weight part of Serre’s conjecture in the following way: to each irreducible component of Xd,red , we assign the set of weights kwith the property that Zk is supported on this component. Then for each ρ, thecorresponding set of Serre weights is simply the union of the sets of weights for theirreducible components of Xd,red which contain ρ.
k
We expect that in particular the irreducible component Xd,red is assigned theSerre weight k, but that the list of Serre weights associated to this irreducible
component can be longer; indeed, it follows from the description given above of thecycles Zk in the case d = 2 that if kσ,1 − kσ,2 = 0 for all σ, then there are two Serrekweights associated to Xd,red , namely k and the corresponding Steinberg weight.
Lecture 10. Bernstein Centers, Moduli Spaces, and the Categoricalp-adic Langlands program
In this final lecture we discuss some of the general theory of moduli spaces associated to algebraic stacks, and explain how it applies to the stacks we have constructedin the previous lectures. We will see that there is a surprising connection between
these ideas and the p-adic local Langlands correspondence.
10.1. Moduli spaces. An algebraic stack X has an underlying “Zariski” topological space, formed by taking a smooth cover by a scheme, taking its topologicalspace, and then taking a quotient topological space.
10.1.1. Example. If X = [A1 /Gm ] (over some field k, with t ∈ Gm acting on x ∈ A1
via multiplication, i.e. t · x = tx), then the associated topological space consistsof two points, one of which is open and specializes to the other, which is closed.
These two points correspond to the two orbits of Gm acting on A1 — the open
orbit and the closed orbit. The closure relation between the points corresponds tothe closure relation between these orbits.
The topological space underlying X is just a topological space, though; it doesn’tcome equipped with a structure sheaf making it into a scheme. So we can ask: isthere a morphism f : X → X to X a scheme, or, more generally, to an algebraic
space, which is the “best possible approximation to the stack X ”? Somewhat more
precisely, one might call X a moduli space associated to X if f is initial in the spaceof maps to algebraic spaces. Or one could say that f is an associated moduli spacemorphism.
Unfortunately, this doesn’t really help you access X or say anything concrete
about it, or even determine if such an X exists. In [Alp14, Prop. 7.1.1], Jared
Alper gives properties that ensure that f is initial for morphisms to locally separated
algebraic spaces. The properties are:(1) If k is algebraically closed, the induced map X (k) → X(k) identifies thetarget with the quotient of the source by the equivalence relation generated
by x ∼ y if {x} ∩ {y} =6 ∅.
(2) f is a universal submersion.5(3) f∗ OX = OX .
As Alper notes, the first property says that X has the right points, the second
that it has the right topology, and the third that it has the right functions.
5A submersion is a morphism which is surjective and which induces a quotient map on theunderlying topological spaces.

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M. EMERTON AND T. GEE

10.1.2. Example. In the case of X = [A1 /Gm ] over the algebraically closed field k,there are two k-points: the origin, and the open point (corresponding to the open
orbit of Gm on A1 ). Since the open point specializes to the closed point, theassociated moduli space is just the single point Spec k.
This still leaves open the question of when such an X exists. The theoremof Keel–Mori [KM97] (plus various technical improvements) implies that if X is
Deligne–Mumford (see Definition 2.5.1) then there exists X which is a “coarse
moduli space” (the map X → X induces a bijection on k-points for algebraicallyclosed fields k). Unfortunately, this result doesn’t apply in our context: firstly, ourmoduli stacks of étale (ϕ, Γ)-modules are formal algebraic stacks, rather than being
actually algebraic; but, more significantly, they are not Deligne–Mumford: theirpoints have infinite automorphism groups (every Galois representation ρ admits atleast the scalar matrices as automorphisms).
In the context of more general (i.e. not-necessarily Deligne–Mumford) algebraic
stacks, Alper [Alp13, Alp14] has developed a theory of “good” and “adequate”
moduli spaces. His theory is related to earlier ideas in Geometric Invariant Theory(GIT) — the difference between “good” and “adequate” (the latter is the moregeneral notion) is related to the difference of behaviour of the representation theoryof reductive linear algebraic groups in positive characteristic versus in characteristic
zero. One feature of this theory is that if f : X → X is good or adequate, then it
is universally closed. Also, if X admits a good or adequate moduli space, then all
stabilizers at closed k-points are reductive. Further, the relation x ∼ y if {x}∩{y} =6∅ is in fact an equivalence relation.
10.1.3. Example. If G is reductive over k and A is a finite type k-algebra with aG-action, then [Spec A/G] → Spec AG is adequate (or good in characteristic 0).
People often refer to Spec AG as the GIT quotient of Spec A by G.
10.1.4. Example. Take P1 , with Gm acting via scaling. This is similar to theexample of Gm acting on A1 considered above, but now there are two closed orbits(each of the points 0 and ∞), and the open orbit specializes to both of them. Thus
the relation x ∼ y if {x} ∩ {y} =6 ∅ is not an equivalence relation in this example,
and so [P1 /Gm ] does not admit an adequate moduli space.
10.2. Closed points and stacks of Galois representations. As we have justseen, the notion of an underlying moduli space for X involves understanding the
topology on its underlying set of Fp -points, which we know biject with isomorphismclasses of representations ρ : GK → GLd (Fp ).
The following is part of [EG22, Thm. 6.6.3] (which also completely describes theclosure relations between points in terms of “virtual partial semi-simplification”).
10.2.1. Theorem.
(1) The finite type points (equivalently, Fp -valued points) of (Xred )Fp are innatural bijection with the isomorphism classes of continuous representations
ρ : GK → GLd (Fp ).
(2) A finite type point of |(Xred )Fp | is closed if and only if the associated Galois
representation ρ is semi-simple.
(3) If x ∈ |(Xred )Fp | is a finite type point, corresponding to the Galois representation ρ, then the closure {x} contains a unique closed point, whose
corresponding Galois representation is the semi-simplification ρss of ρ.

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

51

The proof of Theorem 10.2.1 involves showing that closure relations in X are
actually realized by representations of GK (rather than just by (ϕ, GK )-modules).
A closely related result is [EG22, Thm. 6.7.2], which makes precise the notion of
the largest substack of X which genuinely parameterizes GK -representations. To
explain this, we recall first that in [WE18] Wang-Erickson constructed a formal
algebraic stack X Gal characterised by the following property: if A is a Zp -algebrain which p is nilpotent, then X Gal (A) is the groupoid of continuous morphisms
ρ : GK → GLd (A)(where A has the discrete topology, and GK its natural profinite topology).
Just as was the case with the stacks of Weil–Deligne representations discussed
in Lecture 1, the geometry of X Gal is quite different from that of X ; in particular,
the members of any connected family of GK -representations over Fp have constantsemisimplification, and so we can write
a
Gal
XD
,
(X Gal )W (Fp ) =D

where D runs over the isomorphism classes of d-dimensional semisimple Fp -representations
Gal(A) is the groupoid of those ρ the semisimplification of whose reof GK , and XD
ductions modulo p is D.
We then have the following result relating Wang-Erickson’s stack to ours.
10.2.2. Theorem. There is a natural monomorphism X Gal → X , which inducesa bijection on Fp -points, and is furthermore versal at these points. For any D as
above, the induced monomorphism
Gal
XD,→ XW (Fp )induces a closed immersion on underlying topological spaces (or, equivalently, on
underlying reduced substacks).
Proof. Other than the claim regarding closed immersions, this is a restatementof [EG22, Thm. 6.7.2], which itself is a straightforward consequence of our almost
Galois descent results described in Lecture 3.4.
To prove the claim about closed immersions, we note that it follows from thedescription of specialization of Fp -points given in Theorem 10.2.1 that the image
Galof the monomorphism XD→ XW (Fp ) is closed under specialization of finite type
Gal
points. It then suffices (since both (XD)red and (Xred )Fp are finitely presented
algebraic stacks over Fp ) to show that this image is constructible. This follows
Galfrom Chevalley’s constructibility theorem, once we write (XD)red as the union ofa finite number of families of extensions, using the ideas of Lecture 7.
In Galois deformation theory, the passage from Galois representations to pseudorepresentations is related to the ideas that we’re discussing (and indeed WangErickson’s arguments use the theory of pseudorepresentations). For example, one
subtlety in the theory of pseudorepresentations is that traces don’t know about extension classes, so the passage to pseudorepresentations factors through semisimplification; indeed, two representations ρ1 , ρ2 : GK → GLd (Fp ) have the same underlyss
ing pseudorepresentation precisely if ρss
1 = ρ2 . The d-dimensional pseudorepresentations of GK over Fp are then in bijection with the semisimple ρ : GK → GLd (Fp ).
These phenomena mirror corresponding phenomena related to the points of Xd,red .

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M. EMERTON AND T. GEE

Namely, as we have just discussed, the closed points of Xd,red (Fp ) correspond tosemisimple representations, and specialization to a closed point corresponds tosemisimplification. That is, if ρ is any element of Xd,red (Fp ), then the closure{ρ} has a unique closed point, which is ρss .
Thus if we could find a moduli space map X → X (whatever its precise meaning
in the context of formal algebraic stacks), we might expect X to be a moduli space ofpseudorepresentations. This is one of our main motivations for studying associatedmoduli spaces in our context.
The fact that any point of Xd,red (Fp ) specializes to a unique closed point implies
that the relation x ∼ y considered above is an equivalence relation. Since closed
points of Xd,red correspond to semisimple representations, we also see that thestabilizers of closed points are reductive. These are two of the properties that arenecessary to admit an adequate moduli space. Nevertheless, we will see that Xd,red
does not admit an adequate moduli space when d > 1, and so we seem to be outside
the scope of any generally developed theory.
10.2.3. Example. Consider [(A2 \{0})/Gm ], with the Gm -action given via t·(x, y) =(x, ty). Then the open orbits in each of the vertical lines satisfying y 6= 0 specialize
to the intersection of the line with the horizontal axis, away from the origin. On theother hand, the vertical line y = 0 is a single closed orbit. There’s an obvious map[(A2 \ {0})/Gm ] → A1 , given by (x, y) 7→ x, which is the associated moduli spacemap (in that it satisfies Alper’s properties (1), (2), (3) above; in fact, it is eveninitial for morphisms to arbitrary algebraic spaces). This map is not adequate, forinstance because it’s not closed.
10.2.4. Remark. The preceding example illustrates that in general, the quotient ofa quasi-affine scheme by a reductive group can be much nastier than the quotientof an affine scheme by a reductive group.
As we saw in Lecture 1, stacks similar to that in the previous example appear
as moduli stacks of two-dimensional Fontaine–Laffaille modules, and so also appear
as irreducible components in X2,red . Thus X2,red (and, more generally, Xd,red ford > 1) will not admit an adequate moduli space.
Nevertheless, we anticipate the following result.
10.2.5. Expected Theorem ([DEG]). Assume that p ≥ 5. Let K = Qp , and fix a×character ψ : GQp → Zp . Then X2det=ψ admits an associated formal algebraic moduli space X (in an appropriately understood sense), with Xred being a certain chainof P1 s. The points of X(Fp ) correspond to 2-dimensional pseudorepresentations of
GQp over Fp with determinant ψ (equivalently, to semi-simple ρ : GQp → GL2 (Fp )with det ρ = ψ). The complete local ring of X at one of its closed points is naturally
identified with the corresponding pseudodeformation ring.
10.2.6. Example. The preceding expected theorem fits nicely with the deformation
theory of 2-dimensional crystalline representations of GQp . The Zp -points of one of
the crystalline stacks we have constructed correspond to Galois-invariant lattices incrystalline representations; but when we pass to pseudodeformations, the different
possible lattices all share a common image.
In particular if we fix the Hodge–Tate weights to be 0, k − 1 with k ≥ 2, theimages of these points in the moduli space X will be a family of crystalline representations Vk,ap , parameterised by the crystalline Frobenius ap with |ap | ≤ 1, i.e.

MODULI STACKS OF (ϕ, Γ)-MODULES: A SURVEY

53

parameterized by the rigid analytic closed unit disk. (That there is just a single parameter ap is a simple calculation with weakly admissible modules.) The image ofthe crystalline moduli stack itself will then be a certain formal model of the closed
unit disk, whose underlying reduced scheme embeds into the chain of P1 s given
c1 Z , whose
by Xred . Now the simplest formal model of the closed unit disk is Ap
special fibre A1 certainly embeds into P1 . More general formal models are obtained
c1 Z , and the special fibre then contains additional P1 s.
by blowing up points in Ap
In the case that k ≤ 2p + 1, this can be seen explicitly by looking at the formulae
for the reductions of crystalline representations in [Ber11, Thm. 5.2.1].
In fact, the results of [Ber11, Thm. 5.2.1] were an important clue that our stacks
should exist. Indeed, Mark Kisin suggested in around 2004 that some kind of nonformal moduli (i.e. with ρ varying) spaces of local crystalline Galois representations
should exist, motivated by the conjectures of [Bre03] and the results of [BB05].
10.3. Bernstein Centers. When we investigate the relationship between automorphic forms and Galois representations,6 the basic link between the two concepts
is that the Hecke eigenvalues of an automorphic eigenform at primes not dividing
the level should match with traces of Frobenius at primes that are unramified in
the associated Galois representation.
At primes that divide the level, there is a more subtle way to extract eigenvalues
from automorphic forms/automorphic representations. Namely, if K is (as always)
a finite extension of Qp , then the abelian category of smooth representations of
GLd (K) on C-vector spaces admits a commutative ring of endomorphisms, called
the Bernstein centre for GLd (K). On any irreducible representation, it will act via
scalars (because of Schur’s Lemma). On unramified representations we recover theusual Hecke eigenvalues.
If f is an automorphic Hecke eigenform, for GLd over some number field F ,generating an automorphic representation π, and if K arises as the completion of F
at some prime v above p, then π has a local factor πv at v, which is a representationof GLd (K). This gives rise to a system of eigenvalues for the Bernstein centre. If v
does not divide the level of f , this is just the usual collection of Hecke eigenvaluesof f .
On the other hand, if ρ : GF → GLd (Q` ) is the global `-adic Galois representation (provably in some cases, conjecturally in others) associated to f , with `chosen so that ` 6= p, then we can consider the restriction of ρ to a decompositiongroup at v, giving rise to a representation ρv : GK → GLd (Q` ). We know in somecases, and anticipate in general, that ρv and πv are related via the local Langlands∼
correspondence (once we fix an isomorphism ı : C −→ Q` ).
On the other hand, the pseudorepresentation associated to ρv encodes the tracesof ρv on all the elements of GK , and so we have two collections of numbers associatedto the local-at-v aspects of our situation: the eigenvalues of the Bernstein centre
acting on πv , and the pseudocharacter of ρv . It turns out (as a kind of numericalshadow of the local Langlands correspondence) that these numbers also determine
one another.
In fact, there is even an integral version of this statement. Namely, we can consider smooth representations of GLd (K) on Z` -modules, and form the corresponding
6This is a huge topic, of course, and we introduce it here only to provide motivation, and in
the most abbreviated way possible!

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M. EMERTON AND T. GEE

Z` -Bernstein centre. And we can consider the moduli stacks VQ = [Spec AQ / GLd ]parameterizing local-at-v Weil–Deligne representations over Z` -algebras, describedin Lecture 1. This stack is the quotient of an affine scheme by a reductive group,dand so the associated moduli space is just Spec AGL
Q .
The following theorem of Helm–Moss encapsulates the manner in which numerical data extracted from Galois representations matches with eigenvalues of theBernstein centre.
dis the Z` -Bernstein center for GLd (K).
10.3.1. Theorem ([HM18]). limQ AGL
Q←−10.3.2. Example. If g is an element of the Weil–Deligne group WDK , then “trace ofdρv (g)” gives an element of each Spec AGL
Q , and thus an element hσ of the Bernsteincentre. The value of this function at a particular ρv — which is just the trace ofthe particular matrix ρv (g) — will then correspond to the eigenvalue of hσ on the
representation πv associated to ρv via local Langlands.

We can then ask if there is an ` = p analogue of this result. Even in the caseof GL2 (Qp ), Expected Theorem 10.2.5 shows the associated formal moduli space
to the stack (X2 )det=ψ is not formally affine, so there is no obvious ring appearing
on the Galois/(ϕ, Γ)-module side to compare with a Bernstein centre. However, wehave the following result.
10.3.3. Expected Theorem ([DEG]). Assume that p ≥ 5. If A is the abeliancategory of smooth GL2 (Qp )-representations on Zp -modules which are locally ppower torsion, with central character equal to ψε, then A localizes to a stack of
categories over the formal scheme X of Expected Theorem 10.2.5.
Given this, we can form a sheaf of Bernstein centres of A over X. Of course, we
also have the structure sheaf OX . Our expectation then is that these two sheaves
can be identified. Just as in the ` 6= p case, this identification will be mediated via
the local Langlands correspondence — but now we will have to use the p-adic local
Langlands correspondence for GL2 (Qp ).
Indeed, if D → X2det χ is the universal (ϕ, Γ)-module, then (D  P1 )/(D\ 
P1 ) (as defined by Colmez [Col10]) should be a quasi-coherent sheaf of GL2 (Qp )representations over X2 . Then the identification between OX and the sheaf ofBernstein centres of A will be determined by saying that the two sheaves of rings
act in the same way on this family.
This is all strongly related to, and uses, Paškūnas’ work [Paš13].
10.4. A conjectural p-adic local Langlands correspondence. We return tothe case of general K and d, and end our lectures with a vague and speculative
discussion of a possible p-adic local Langlands correspondence, analogous to theconjectures made in the ` 6= p setting in [BZCHN20, Hel20, Zhu20]. We also
explain briefly how such a conjecture could explain the geometric Breuil–Mézard
conjecture of the previous lecture.
Each of the papers [BZCHN20, Hel20, Zhu20] proposes a certain conjecturalenhancement of Theorem 10.3.1. We state a rough form of this conjecture here.
10.4.1. Rough Conjecture. The category of GLd (K)-representations on Z` -modules
admits a fully faithful embedding into the category of quasi-coherent sheaves on the
Ind-algebraic stack limQ VQ , compatibly with the identification of Theorem 10.3.1.
−→

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55

We have omitted all technical details and caveats from this statement (in particular, it should be stated at a derived level). The key intuition, though, is that
we are conjecturing that we can localize GLd (K)-representations in some fashion
as sheaves of O-modules over the moduli stack of rank d Weil–Deligne representations. And although the action of the Bernstein centre is not sufficient to separate
all irreducible representations (e.g. any two such representations that admit a nontrivial extension, such as the trivial and Steinberg representations of GL2 (K), will
have the same system of eigenvalues under the Bernstein centre), this more refined
localization process should be able to distinguish non-isomorphic irreducibles.
There is another important facet of the (conjectural) fully faithful embedding ofConjecture 10.4.1 which we want to explain, if only in somewhat general and vague
terms. The essential point is that there is another method of obtaining coherent
sheaves out of GLd (K)-representations, not on the stacks VQ themselves, but on
their versal rings at closed points, which is to say, on formal deformation rings ofGalois representations. This is the method of Taylor–Wiles–Kisin patching (which
already made an appearance in the previous lecture).
Briefly, if we choose some global context related to GLd (K) (e.g. a modular curveor Shimura curve for GL2 , or a certain unitary Shimura variety more generally),
and an irreducible7 automorphic global Galois representation r : GF → GLd (Fp )(here F is a totally real or CM field of which K is a certain completion, and which isrelated to the Shimura variety at hand), and an `-adic representation σ of GLd (OK ),then we are able to “patch” appropriate Hecke localizations of the cohomology ofwhichever Shimura variety is in play so as to obtain a coherent sheaf, traditionallydenoted by M∞ (σ), which lives over the formal neighbourhood8 of r|GK .
The relationship between patching and Conjecture 10.4.1 can be stated very
roughly as follows:9 In sufficiently good situations,10 M∞ (σ) should be equal to
GL (K)
the formal completion at r|GK of the coherent sheaf associated to c−IndGLdd (OK ) σ
by Conjecture 10.4.1. In general, the patched module M∞ (σ) will be related tothis latter coherent sheaf, but we will have to take into account level structure
at completions of F other than K, the most subtle being the level structure atthe primes above `; and this latter case amounts to studying patching for p-adic
GLn (OK )-representations.
The patching described above, in the ` 6= p case (see e.g. [Sho18, Man21, MS21]),is less commonly studied than the case of ` = p, and we now return to the lattercase. In this context, the appropriate stacks over which to state an analogue ofConjecture 10.4.1 are our moduli stacks Xd parameterizing étale (ϕ, Γ)-modules, inplace of the moduli stacks of Weil–Deligne representations.
Our hope, then, is that there is a canonically defined quasi-coherent sheaf Πof GLd (K)-representations on Xd which in particular enjoys the properties that
7In fact r should satisfy a stronger condition than irreducibility, but we will suppress all suchtechnical details.
8We are here ignoring the “patching variables” that might have been introduced to effect thepatching process; these should be ignorable, although this is not known in general.
9For more precise statements, the reader can look at [Zhu20, Conj. 4.6.3], which treats thefunction field analogue of the situation we are describing here — although the connection topatching is not made precise there. The case of number fields should be discussed carefully (justat the level of conjecture, to be sure!) in forthcoming work of Xinwen Zhu and M.E.
10E.g. if p is unramified in F , and if we choose the level and coefficients of our cohomology
carefully enough.

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M. EMERTON AND T. GEE

it satisfies local-global compatibility for completed cohomology of locally symmetric spaces (in the sense of being compatible with Taylor–Wiles–Kisin patching, as
carried out in e.g. [CEG+ 16]), and explains the Breuil–Mézard conjecture and theweight part of Serre’s conjecture in a sense that we will explain shortly.
The analogy with Conjecture 10.4.1 comes by considering the functor π 7→
RHomGLd (K) (π, Π) from the derived category of smooth representations of GLd (K)on Zp -algebras to the derived category of quasi-coherent sheaves on Xd ; this functor
should be fully faithful. This being the case, one could in particular localize theformer derived category over Xd , generalizing the case d = 2, K = Qp explained
above. (However, in this more general context, there does not seem to be any obvious analogue of the moduli space for Xd that we considered above, and it seems
likely that one really has to consider this localization as taking place over the stack.)
In the case that d = 2 and K = Qp , we explicitly construct a candidate quasicoherent sheaf Π in [DEG], by generalising Colmez’s construction from [Col10] of therepresentation he denotes (DP1 )/(D\ P1 ). Note that after pulling back to versal
rings, we know that this sheaf realises the p-adic local Langlands correspondence
for GL2 (Qp ) by results of Colmez and Paškūnas [Paš13] (see also [CEG+ 18] for aperspective related to patching). In particular, a construction of Π in general would
generalize the p-adic local Langlands correspondence from the case of GL2 (Qp ) to
the general case of GLd (K).
If σ is a representation of GLn (OK ) on a finitely generated Zp -module, thenGL (K)
we can apply the functor RHomGLd (K) (–, Π) to π := c−IndGLdd (OK ) σ ∨ . (The Pontrjagin dual is included just so as to make the functor covariant.) We anticipate
that this RHom should be supported in degree 0, and that the resulting sheafM(σ) := HomGLd (Zp ) (σ ∨ , Π) will be coherent. (Just as in the ` 6= p case discussed
above, the formal completions of M(σ) at Fp -points of Xd,red should give the usual
patched modules M∞ (σ).) These (conjectural) coherent sheaves allow us to connect
the present discussion to the Breuil–Mézard and Serre weight conjectures.
Indeed, we expect that if k is a Serre weight (regarded as usual as a representationof GLd (OK ) via inflation from GLd (k)), then the cycle Zk considered in Lecture 9
is simply the support of M(Fk ). Furthermore, we expect that if λ is a regular
Hodge type, τ is an inertial type, and σ ◦ (λ, τ ) is the representation considered incrys,λ,τLecture 9, then the support of M(σ ◦ (λ, τ )) is exactly Xd. This being the case,
Conjecture 9.2.5 is an immediate consequence of the additivity of supports, while
the expectation that the cycles Zk encode the weight part of Serre’s conjecture
should be a consequence of a local-global compatibility for completed cohomologyof Shimura varieties (or more generally locally symmetric spaces), generalising theresults of [Eme11, CEG+ 18] for the modular curve.
This may all seem a speculation too far, but we note that while there is in general
no candidate for Π, the construction of [CEG+ 16] does produce such a candidate
after pulling back to the versal deformation rings; and the conjectural explanation
above for the Breuil–Mézard conjecture describes exactly how the results of [Kis09,
GK14] are proved (in the setting of the numerical conjecture, over a deformationring).
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Email address: emerton@math.uchicago.edu

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59

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