An introduction to the categorical p-adic Langlands program
Matthew Emerton, Toby Gee, and Eugen Hellmann

Abstract. We give an introduction to the “categorical” approach to the padic Langlands program, in both the “Banach” and “analytic” settings.

Contents
1. Introduction
1.1. A rapid overview of the Langlands program
1.2. The p-adic perspective
1.3. Proofs of reciprocity
1.4. Unramified geometric Langlands and the Fargues–Scholze conjecture
1.5. Some differences between the `-adic and p-adic cases
1.6. Motivation from Taylor–Wiles patching
1.7. The “Galois to automorphic” direction
1.8. Topics we omit
1.9. A brief guide to the notes
1.10. Acknowledgements
2. Notation
2.1. Fields
2.2. p-adic Hodge theory
2.3. Hodge–Tate weights vs. highest weights
3. Taylor–Wiles patching as a motivation for categorical p-adic local
Langlands
3.1. A very brief introduction to patching
3.2. p-adic Langlands: an overview and history
4. Moduli stacks of (ϕ, Γ)-modules: the Banach case
4.1. An overview of [EG22]
4.2. Fixed determinant variant
5. Moduli stacks of (ϕ, Γ)-modules: the analytic case
5.1. (ϕ, Γ)-modules over the Robba ring
5.2. Stacks of de Rham objects

2
2
3
4
6
8
9
11
11
11
12
12
12
13
13
14
14
18
31
32
34
34
34
47

M.E. was supported in part by the NSF grants DMS-1902307, DMS-1952705, and DMS2201242. T.G. was supported in part by an ERC Advanced grant. This project has received
funding from the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 884596). E.H. was supported by Germany’s Excellence Strategy EXC 2044-390685587 “Mathematics Münster: Dynamics–Geometry–
Structure” and by the CRC 1442 Geometry: Deformations and Rigidity of the DFG.
1

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M. EMERTON, T. GEE, AND E. HELLMANN

5.3. Drinfeld-style compactifications
52
6. Categorical p-adic local Langlands conjectures74
6.1. Expectations: The Banach Case
74
6.2. Expectations: the analytic case83
7. Some known cases of categorical p-adic Langlands
97
7.1. GL1
97
7.2. The Banach case for GL2 (Qp ) — I. The structure of X
102
7.3. The Banach case for GL2 (Qp ) — II. Proof sketches
111
7.4. The Banach case for GL2 (Qp ) — III. Examples
121
7.5. Semiorthogonal decompositions
127
7.6. Preliminaries in the Banach case for GL2 (Qpf )133
7.7. Partial results and examples in the Banach case for GL2 (Qp2 )139
7.8. The adjoint functor
158
7.9. Global functions and Hecke algebras
163
8. Categorical local Langlands in the case ` 6= p
164
8.1. C-groups
164
8.2. Langlands parameters in the local case — ` 6= p
165
8.3. Categorical local Langlands
166
9. Moduli stacks of global Langlands parameters and the cohomology of
Shimura varieties
168
9.1. Global moduli stacks
168
9.2. The global-to-local map
1699.3. Computing the cohomology of Shimura varieties via categorical
Langlands
170
9.4. An example in the global case: (PGL2 )/Q
172
odd
9.5. The structure of XGL
173
2 ,Q,S
9.6. Eigenvarieties and sheaves of overconvergent p-adic automorphic forms 174
Appendix A. Background on category theory
186
Appendix B. Coherent, Ind-Coherent, and Pro-Coherent sheaves on
Ind-algebraic stacks
199
Appendix C. Localization and Morita theory
207
Appendix D. Topological abelian groups, rings and modules
217
Appendix E. Representations of p-adic analytic groups
221
Appendix F. Derived moduli stacks of group representations
228
References
230

1. Introduction
The goal of these notes are to discuss some of the p-adic aspects of the Langlands
program, and especially the emerging “categorical” perspective. We begin withan overview of some history and our goals in these notes; we refer the reader to
Section 2 for any unfamiliar notation.
1.1. A rapid overview of the Langlands program. The Langlands program began with Langlands’s celebrated letter to Weil (reproduced e.g. in [DS15,§6]), the contents of which were elaborated on in several subsequent writings ofLanglands, including his article “Problems in the theory of automorphic forms” [Lan70]

CATEGORICAL p-ADIC LANGLANDS

3and his Yale lectures on Euler products [Lan71]. The article focuses on the functoriality conjecture for automorphic representations and its consequences, while theYale lectures explain his construction of automorphic L-functions, and discuss the
problem of proving their expected analytic properties (analytic continuation andfunctional equation).1
In his Yale lectures Langlands already touches, if only tangentially, on theidea that the automorphic L-functions may include all motivic L-functions. Subsequently, this idea developed into the reciprocity conjecture. (See [Lan79] forone articulation of this conjecture.) Roughly speaking, the reciprocity conjecture
articulates a correspondence between certain automorphic representations (thosewhich are algebraic) of G(AF ) (for a connected reductive algebraic group G overa number field F ), and motives (with coefficients in Q, say) over F whose motivic
Galois group is closely related to to the C-group2 c G of G. Conjectures on Galois
representations (especially the Fontaine–Mazur conjecture [FM95]) suggest thatsuch motives in turn may be identified with certain compatible systems of `-adic
Galois representations GalF → c G(Q` ).
There are many subtleties involved in trying to formulate a precise reciprocity
conjecture. For example, in the case that G is not some GLd , one has to worry about
L-packets; and one should restrict to automorphic representations which are notanomalous in the sense of [Lan79] (or else replace GalF by GalF × SL2 and work in
the framework of Arthur’s conjectures regarding non-tempered endoscopy [Art89]).
We refer to [BG14] for a more precise discussion of the conjecture.
We note that there is a relationship between reciprocity and functoriality:
namely, since functoriality is tautologically true for (compatible systems of) Galois
representations, cases of reciprocity can be used to deduce cases of functoriality.
Also, since L-groups and C-groups involve Galois groups in their definition, representations of Galois groups into algebraic groups can sometimes be related to
L-homomorphisms of L-groups or C-homomorphisms of C-groups, and thus somecases of reciprocity can be subsumed into functoriality. (This is Langlands’s original perspective on the Artin conjecture, as explained in [Lan70].) If one replaces
the Galois group GalF by the hypothetical Langlands group LF , and compatible
systems of GalF -representations by representations of LF , then one can also extend the reciprocity conjecture to non-algebraic automorphic representations, or(using the “LF -form” of the L-group or C-group) entirely subsume reciprocity intofunctoriality. (A version of this last-mentioned perspective was already adopted attimes by Langlands, by using the “Weil group form” of the L-group.)
1.2. The p-adic perspective. Our intention is not so much to focus on reciprocity in the manner described above (the relationship between automorphic representations and motives, or compatible systems of Galois representations), butrather to fix a prime p and consider the relationship between automorphic representations and p-adic Galois representations (for this fixed choice of p). At first, this
1Since the standard L-functions associated to automorphic representations on GL coincidedwith the L-functions constructed by Godement and Tamagawa, which are known to admit analytic
continuations and functional equations, the expected properties would follow from functoriality.
Langlands’s Yale lectures discuss another approach, via constant terms of Eisenstein series.
2The C-group, introduced in [BG14], is a refinement of the L-group introduced by Langlands,
which is better adapted to the problem of relating automorphic forms and Galois representations;
we also recall the definition below.

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doesn’t much change the problem, since (at least for representations valued in GLd )a compatible system of semisimple Galois representations is determined by any oneof its members, and the Fontaine–Mazur conjecture gives a purely Galois-theoretic
condition for a p-adic Galois representation to be motivic.
But focusing on a particular prime p brings to the fore certain aspects of thetheory of automorphic forms and Galois representations which are absent in the
more motivic perspective on reciprocity. For example, since at least the work of
Ramanujan, it has been known that automorphic forms can satisfy interesting congruences modulo powers of p. Since the work of Swinnerton–Dyer [SD73] andSerre [Ser73], it has been understood that these congruences are related to (ormanifestations of, if one prefers) analogous congruences between p-adic Galois representations. Extending the notion of congruences between automorphic forms,
one is led to take various p-adic completions of spaces of automorphic forms to obtain the notion of p-adic automorphic forms, with associated Galois representations
that need not be motivic. Related to this, one has notions of mod p automorphic
forms, to which one might associate mod p Galois representations; and reciprocity
conjectures have been formulated in this context, famously by Serre [Ser87] inthe context of classical modular forms, and more recently in greater generality byothers (e.g. [BDJ10]).
We do not intend at all to survey these developments; rather, our aim is simply
to indicate that the p-adic perspective emerged naturally over a long period of time,and has led to a natural collection of problems and concerns: e.g. how to arrive atsome notion of automorphic form which admits p-adic integral or mod p coefficients,
and which allows for genuinely p-adic objects? And how to phrase reciprocity ina manner which allows for p-adic integral or mod p Galois representations, which
will be associated to the p-adic integral or mod p objects that one introduces on
the automorphic side?
We make one last remark on the p-adic theory for now: once one allows oneself
to p-adically interpolate automorphic forms, one sees that automorphic forms naturally lie in families (e.g. Hida families, Coleman families, . . . ; see [Eme11b] for asurvey). And Galois representations also lie in families (e.g. via Mazur’s deformation theory [Maz89], which he was at least partly motivated to develop in response
to Hida’s theory of ordinary families of p-adic modular forms). These phenomenaof continuous families of objects — on both the automorphic and Galois side — are
not a feature of reciprocity in its more motivic formulation. (Cuspidal automorphic
representations are rigid objects, and so are motives, if one has fixed a particularnumber field as the field of definition.) Before passing to our next topic, we notethat families of Galois representations are a feature of the geometric Langlands
program. We will return to this point below.

1.3. Proofs of reciprocity. Some of the very first results on reciprocity wereproved in the case of representations with solvable image, by combining class field
theory (which implies reciprocity in the abelian context) with techniques more classically automorphic in nature (see in particular the work of Langlands and Tunnell,[Lan80, Tun81]). In 1995, a breakthrough in our understanding of reciprocity
was achieved with the proof by Wiles [Wil95] and Taylor–Wiles [TW95] of themodularity of semistable elliptic curves over Q, a result which was soon improved

CATEGORICAL p-ADIC LANGLANDS

5

to handle all elliptic curves over Q [BCDT01]. Building on these methods, reciprocity has since been proved in many further interesting contexts; see Calegari’s
recent survey [Cal21] for some of the highlights of the last 25 years.
The crux of the method of [Wil95, TW95] and the many subsequent resultsthat build on their ideas is to prove the modularity — or, more generally, automorphy — of a p-adic Galois representation. More precisely, the Taylor–Wiles method
gives a way to deduce from the modularity of a single p-adic Galois representation ρ1
the modularity of another p-adic Galois representation ρ2 which is congruent to ρ1
modulo p. It turns out that there are many such congruences between Galois representations, and modularity can be propagated from a single representation to
many others in this way (either using a fixed prime p, or using several different
primes p). Of course, it is necessary to have an initial supply of representations
whose modularity is already known, such as CM forms, or 2-dimensional Artin representations with solvable image (the latter being used in Wiles’ proof of Fermat’s
Last Theorem [Wil95]).
Modularity lifting theorems are also known as “R = T” theorems, because
they are proved by identifying a Zp -algebra R (a “Galois deformation ring”) parameterizing Galois representations congruent to a fixed representation ρ1 withanother Zp -algebra T, a “Hecke algebra”, the endomorphism algebra of an appropriate space of (p-adic) modular forms. Over the last 25 years it has become
apparent that such theorems should hold in great generality, although finding appropriate definitions of R and T so that we literally have R = T is still somethingof an art. In particular, it has become clear (following in particular Skinner–Wiles[SW99], Wake–Wang-Erickson [WWE17], Newton–Thorne [NT19]) that in general R should be taken to be a so-called pseudodeformation ring, i.e. a deformation
ring for pseudorepresentations, rather than a deformation ring for literal Galois
representations.
Roughly speaking, pseudorepresentations capture the information given bycharacteristic polynomials of representations; for this reason they are also known as
pseudocharacters. They were originally introduced for GL2 by Wiles [Wil88], andwere considerably developed for GLd by many authors (we highlight in particular
Chenevier’s notion of a determinant [Che14], which showed how to define themfor arbitrary primes p). A theory valid for general reductive groups was introduced
by Vincent Lafforgue [Laf18].
From Lafforgue’s point of view, as recently made precise (in the setting of local
Galois representations) by Fargues–Scholze [FS21, §VIII], a pseudodeformation
ring R is the ring of global functions on a moduli stack of L-parameters, so thatthe moduli space of pseudorepresentations is a coarse moduli space for the moduli
stack of L-parameters; and for general reductive groups, the Hecke algebra T is
replaced with a larger algebra, the algebra of “excursion operators” (which agrees
with the usual Hecke algebra for GLd ).
From this perspective, it is natural to wonder whether it is possible to upgrade
the putative equality R = T to a statement on the actual moduli stack of Lparameters, rather than on its coarse moduli space; for example, one could ask
whether spaces of p-adic modular forms admit an interpretation in terms of sheaves
on the stack of L-parameters. The possibility of such statements, and their local
analogues, is the main subject of these notes; but before describing them further, we6

M. EMERTON, T. GEE, AND E. HELLMANN

turn to the geometric Langlands program, which was also a substantial motivation
for Vincent Lafforgue’s work.
1.4. Unramified geometric Langlands and the Fargues–Scholze conjecture. Very roughly, the unramified geometric Langlands correspondence for aconnected reductive group G and a curve X takes the form of an equivalence of
categories∼ QCoh(LocSys b (X)),(1.4.1)D(const. BunG (X)) =G

where the left hand side is a derived category of “constructible sheaves” on the stack
BunG of G-bundles on X, and the right hand side is a derived category of “quasib
coherent sheaves” on a stack of “G-local
systems” on X; see for example [Gai17].
The recent work of Fargues–Scholze [FS21] geometrizes the classical local Langlands correspondence by “taking X to be the Fargues–Fontaine curve”. (One motivation for doing this is that the fundamental group of the Fargues–Fontaine curve
is GalF .) Slightly more precisely (but still only approximately), if F/Qp is a finite
extension, and X is the Fargues–Fontaine curve for F , then for each prime ` 6= p,
one expects an equivalence of the approximate form (1.4.1), where the left hand side
is replaced by an appropriate derived category of (solid) `-adic sheaves on BunG ,and the right hand side is a derived category of (Ind-)coherent sheaves on a modulibstack of G-valued`-adic representations of GalF .
The stack BunG of G-bundles on the Fargues–Fontaine curve admits a stratification into locally closed substacks, with one of the open strata being a quotient
stack [·/G(F )]. The `-adic sheaves on this quotient stack correspond to smooth`-adic representations of G(F ), so by considering an appropriate pushforward of
sheaves from this quotient stack to BunG , an equivalence of the form (1.4.1) implies in particular that there is a fully faithful functor(1.4.2)D(sm. G(F )) → QCoh(LocSysGb )where the left hand side is a derived category of `-adic representations of G(F ),and the right hand side is a derived category of quasicoherent sheaves on the stackbof G-valued`-adic representations of GalF .
Specialising to the case G = GLd , we have the following rough conjecture,
where we now allow the case ` = p.
Conjecture 1.4.3. If F is a finite extension of Qp , if O is the ring of integers
in a finite extension of Q` (p and ` each denoting some fixed prime), and if d isa positive integer, then the category of smooth GLd (F )-representations on locallytorsion O-modules admits a fully faithful embedding into the category of quasicoherent sheaves on X , an appropriate moduli stack parameterizing d-dimensional `-adic
representations of the absolute Galois group GalF .
We also aim for a version of the conjecture where smooth p-power torsion
representations of GLd (F ) are replaced by locally analytic representations (i.e. thekind of p-adic representation theory relevant for the theory of overconvergent padic automorphic forms, Coleman families, . . . ), and the stack X is replaced bya stack of equivariant vector bundles on the Fargues–Fontaine curve. As there is
not even an informal way to incorporate this case into the above rough conjecture(that simultaneously treats the cases ` 6= p and ` = p) we leave the formulation of
the conjecture in the p-adic locally analytic case to the body of the notes. We only

CATEGORICAL p-ADIC LANGLANDS

7

remark that the conjecture in this case should be seen as an overconvergent versionof the p-adic limit of Conjecture 1.4.3 (given by passing from torsion O-modules tocomplete torsion-free O-modules) in the ` = p case.
We have described this conjecture as “rough” for several reasons: one should
be precise about the stack X appearing in the statement of the conjecture — when
` 6= p, it should be understood to be the stack of Weil–Deligne representations introduced in [Zhu20] (see also [DHKM20] and [FS21] for alternative constructionsof the underlying classical stacks), while in the case ` = p it should be one of themoduli stacks constructed in [EG22] (and one should a priori consider these stacks
as derived stacks — although in the ` 6= p context, it is shown in [Zhu20] that theresulting stacks are in fact classical, and we anticipate that the same will be true inthe ` = p context); one should also be precise about what sort of categories are being considered — the envisaged embedding will not be compatible with the natural
abelian category structure on source and target, so one should work with appropriate derived (or, probably better, stable infinity) categories (where “appropriate”
is also doing some work — e.g. one should work with the triangulated/stable ∞category of Ind-coherent complexes on X , and then make an analogous modification
to the triangulated category of representations as well, as in [Zhu20, §4.1]).
Remark 1.4.4. As far as we know, the possibility of a conjecture along thelines of Conjecture 1.4.3 with ` = p was first raised by Michael Harris (see [Har16,
Question 4.7]). We first learned that a p-adic version of the conjecture of Fargues–
Scholze in the form (1.4.1) should hold from Laurent Fargues in the summer of
2016. Peter Scholze explained some related conjectures to T.G. in 2018, and arough formulation of a version of Conjecture 6.1.14 was discussed by M.E., T.G.,
E.H. and Scholze at the Hausdorff school in Bonn in 2019.
Remark 1.4.5. We motivated Conjecture 1.4.3 by the work of Fargues–Scholze.
In fact, in the case ` 6= p various conjectures along the lines of Conjecture 1.4.3 wereindependently proposed by Ben-Zvi–Chen–Helm–Nadler, Fargues–Scholze, E.H.,
and Zhu. The history of such conjectures is discussed in the introductions to thepapers [FS21, Zhu20, Hel20, BZCHN20]. It is natural to guess that a similar
conjecture should hold in the case ` = p, but it is less clear what the precise formulation should be. We propose two precise conjectures in Section 6, one in the“Banach” case (focusing on representations on torsion O-modules and in the limit
on lattices in Banach space representations), and one in the “analytic” setting (focusing on locally analytic representations), and explain some relationships between
them. However, it does not seem that either conjecture implies the other, and we
do not know whether they admit a common refinement.
The two cases correspond to two classical notions of p-adic automorphic forms,
or p-adic modular forms. Serre defined p-adic modular forms as limits of p-adic
modular forms with torsion coefficients (or more precisely: as the p-adic limits of
the reductions modulo powers of p of the q-expansions of modular forms). These
spaces of p-adic modular forms certainly belong to the “Banach case”, whereas thesubspace of overconvergent modular forms, as introduced by Katz, belong to the“analytic case”. In the discussion in these notes the space of p-adic modular forms is
replaced by the completed cohomology of modular curves (or more general towers of
Shimura varieties), as in Sections 3.2.11 and 9.3.1, and the space of overconvergent
modular forms is then replaced by the space of locally analytic vectors underlying
the completed cohomology, which is used to construct eigenvarieties, see Section 9.6.

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Remark 1.4.6. The reader might ask why we are only looking for fully faithful
functors, rather than equivalences of categories as in the work of Fargues–Scholze
described above. It does seem reasonable to hope that there are such equivalencesof categories (even in the global setting, as speculated in [FS22, §7]). One difficulty
is to even define categories of p-adic sheaves on BunG , but this difficulty is likely
to be resolved by the recent work of Lucas Mann [Man22]. This work should thenplay an important role in formulating a precise conjecture.
Nevertheless, at the time of writing there is no such precise conjecture in theliterature (let alone a theorem — and we note that as far as we are aware, it is
not expected that the Fargues–Scholze construction of the spectral action will goover to the p-adic setting in any simple manner). In contrast, we can make precise
conjectures about the fully faithful embeddings of categories of representations ofGLn (F ), and even prove theorems in the case of GL2 (Qp ). Furthermore, at least
for GL2 (Qp ) and GL2 (Qp2 ) (and their inner forms), we produce semiorthogonal
decompositions of the categories of (Ind-)coherent sheaves, which we expect to correspond to a semiorthogonal decomposition of the appropriate category of sheaves
on BunG induced by the closure relations on BunG itself. (See Section 7.5 for someresults in this direction.)
Remark 1.4.7. There is no particular reason to restrict to the group GLd inthe statement of Conjecture 1.4.3, rather than considering dual groups or L-groups
or C-groups of more general reductive groups over F (as in [BZCHN20, Hel20,
Zhu20]). However, since we are interested in the ` = p case, we will for the mostpart focus on the case of GLd , since the relevant “stacks of parameters” have been
constructed in this case. (Since the literature on Taylor–Wiles patching, which sits
in the background as an important source of intuition and motivation, also focuseson this case, this provides another reason for us to focus on it in this paper as well.)
1.5. Some differences between the `-adic and p-adic cases. While it ispossible to formulate `-adic and p-adic versions of Conjecture 1.4.3 in a somewhat
uniform fashion, there are significant differences between the two cases.
That there should be differences is not surprising from the point of view ofrepresentation theory. Indeed, already for GL2 (F ), the classification of irreduciblesmooth mod ` representations is well-understood for ` 6= p, and can be formulated uniformly for any F (these results are due to Vignéras, see [Vig89]). In
contrast, for ` = p, there is only a classification when F = Qp (due to Barthel–
Livné and Breuil [BL94, Bre03a]), and already for F a non-trivial unramified
extension of Qp there are many complications, for example infinite families of admissible irreducible representations (as constructed by Breuil–Paškūnas [BP12]),and (absolutely) irreducible representations which are not admissible (constructedby Le [Le19]). Furthermore, supersingular irreducible admissible representations
are never finitely presented (see Schraen’s paper [Sch15c] and Wu’s [Wu21b]).
These differences and difficulties are mirrored on the Galois side of the correspondence. For ` 6= p the moduli stacks of Galois representations are 0-dimensional
ind-algebraic stacks of a rather mild type: the ind-structure arises from there being
infinitely many connected components (controlled by the conductors of the Galois
representations), and each connected component is a quotient of an affine schemeby a reductive group. Accordingly these stacks admit many global functions, andCATEGORICAL p-ADIC LANGLANDS

9have coarse moduli spaces which can be described in terms of pseudorepresentations (at least away from a small number of bad primes, this has been proved by
Fargues–Scholze [FS21, Thm. VIII.3.6]).
In the Banach case for ` = p, the moduli stacks we consider are those definedin [EG22]. These are also ind-algebraic stacks, but their dimension grows with [F :Qp ], and the ind-structure is much more complicated: indeed, they are naturally
formal algebraic stacks with only finitely many connected components (just comingfrom a condition on central characters). Furthermore, we do not expect them toadmit any interesting global functions, and beyond the case of GL2 (Qp ) (and thecase of tori) we do not expect them to have non-trivial associated moduli spaces.
In the analytic case the moduli stacks are defined in these notes (a more detailed
discussion is going to appear in the forthcoming paper [HHS]) as the moduli stacks
of equivariant vector bundles on the Fargues–Fontaine curve (or equivalently as
moduli stacks of (ϕ, Γ)-modules over the Robba ring). As far as the dimension isconcerned the analytic case behaves similar to the Banach case: the (expected)
dimension grows with [F : Qp ] and should coincide with the dimension of thegeneric fiber of the formal stacks in the Banach case (and in fact conjecturally thisgeneric fiber should be closely related to an open substack of the analytic moduli
stack). However, these stacks now should not have an ind-structure but rather
should be rigid analytic Artin stacks. Though there are a few more interesting
global functions on these stacks we expect that they all arise from the theory of
Hodge–Tate weights and hence the algebra of global functions still loses a lot of
information about the moduli stack.
Remark 1.5.1. Another difference between the cases ` = p and ` 6= p is thedependence on the field F . In the case ` 6= p, the (in general conjectural) functors
admit a qualitatively uniform description over all F , and it does not seem to be
any easier to establish instances of the conjecture for F = Qp or for a general F .
In contrast, if ` = p then we only know how to construct a functor for GL2 (Qp ) in
the Banach case, in which case it shares some qualitative similarities with the case
` 6= p (for example, it is “not very derived”, with the only derived phenomena relating to finite-dimensional representations, and supersingular irreducible admissible
representations roughly correspond to skyscraper sheaves supported on irreducible
Galois representations).
However already for GL2 (Qp2 ) we expect it is essential to consider the functor
on the derived level, and supersingular irreducible admissible representations nolonger correspond to skyscraper sheaves, but instead should correspond to complexes with non-coherent H −1 . (See Section 7.7.)
Remark 1.5.2. The differences between the `-adic and p-adic settings are reminiscent of the differences between the Betti and de Rham versions of geometricLanglands (see for example [BZN18] for the former and [Gai15] for the latter),although for reasons of ignorance we are unsure of how seriously one should takethis analogy.
1.6. Motivation from Taylor–Wiles patching. Fargues has emphasisedthat the conjectures described in Section 1.4 did not arise from attempting to“copy and paste” from the geometric Langlands program to the setting of the
Fargues–Fontaine curve, but rather from a study of p-adic period morphisms, the10

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p-adic geometry of Shimura varieties and Rapoport–Zink spaces, and the classical local Langlands correspondence (in particular the phenomena of L-packets andendoscopy). Similarly, while it may be possible to arrive at the conjectures presented here by an appropriate procedure of “setting ` = p in the Fargues–Scholze
conjecture”, we instead came to them via a roundabout process over many years,
largely motivated by considerations coming from p-adic automorphic forms and the
Taylor–Wiles patching method.
As well as its importance in shaping our ideas, the Taylor–Wiles patching
method has suggested several important properties of the conjectural functors as
in Conjecture 1.4.3, even in the case ` 6= p. For example, the patching methodsuggests that the kernel of such a functor should be a genuine sheaf (rather thana complex thereof), a property that was not at all apparent in the first versionsof the constructions of Ben-Zvi–Chen–Helm–Nadler and Zhu. (There is also astrong connection between the Taylor–Wiles method and the classical p-adic local
Langlands correspondence for GL2 (Qp ), for which see [CEG+ 18].)
In brief, the connection between Conjecture 1.4.3 and the Taylor–Wiles method
is as follows. A typical example of a smooth representation of GLd (F ) is a compact
GL (F )
induction c-IndGLdd (OF ) V, where V is a continuous (i.e. smooth) representation of
GLd (OF ) on a literally finite O-module V . One envisages that the embeddingof Conjecture 1.4.3 will map such representations to genuine coherent sheaves (asopposed to complexes of such), say A(V ), and that these coherent sheaves will be“globalizations” over X of the patched modules M∞ (V ) that arise in the Taylor–Wiles–Kisin patching method, in the sense that if R∞ is one of the rings usuallydenoted this way in the theory of patching, and if Spf R∞ → X is the natural versal
morphism, then the pullback of A(V ) along this map coincides with M∞ (V ). In
particular, the conjecture entails that patched modules should be “purely local”(where now “local” is understood to pertain to the local Galois-theoretic situation,in opposition to the global automorphic/Galois-theoretic choice that was made inorder to perform patching).
Ascertaining the truth or otherwise of this “pure locality” in general is wellknown to be a major open problem in the arithmetic theory of automorphic forms.
It is worth noticing that, building upon [CEG+ 16], the patching strategy has been
extended to the “analytic case” in [BHS17b]. Using roughly the same formulationrig
as above one can associate patching modules M∞(V ) to locally analytic principalG
anseries representations V = IndB (δ) . The corresponding coherent sheaf should
then be the pullback of the evaluation of the conjectural “analytic” functor on V
(and is related to the fiber over δ of the sheaf of overconvergent p-adic automorphic
forms of finite slope on eigenvarieties in a similar way as the classical patchingmodules are related to modules of classical automorphic forms).
In the opposite direction, since the patched modules M∞ (V ) are constructedfrom the cohomology of Shimura varieties (with coefficients in a local system corresponding to V ), it is natural to ask whether the cohomology of Shimura varieties
can in some way be computed in terms of the (conjectural) functors of Conjecture 1.4.3. This should indeed be the case, as we explain in Section 9.2 (and as will
be more thoroughly explained in the paper in preparation of M.E. and Zhu [EZ]).
In parallel to this, Section 9.6 gives a precise computation of the sheaves of overconvergent automorphic forms on eigenvarieties (that can be defined in terms ofCATEGORICAL p-ADIC LANGLANDS

11

completed cohomology of Shimura varieties) in terms of the conjectural functor in
the analytic case.
1.7. The “Galois to automorphic” direction. A variant of the expectation that patched modules are “purely local” is the hope that it should be possible
to canonically associate an admissible (smooth) representation of GLn (F ) to eachn-dimensional p-adic representation of GalF ; and that this association should be
realised in the cohomology of Shimura varieties. This expectation is borne out by
a good deal of evidence in the case of Shimura curves, the most recent example
being the spectacular results on GK-dimensions due to Breuil, Herzig, Hu, Morra,
and Schraen [BHH+ 20, BHH+ 21].
While our focus is mostly on functors from smooth representations to sheaves on
stacks of Galois representations, our conjectures also entail the existence of adjoint
functors from sheaves to representations of p-adic groups. In particular, applying
these to “skyscraper” sheaves (that is, to sheaves of finite length, supported atparticular Galois representations), we are able to show that our conjectures are
consistent with the above-mentioned hope. (See e.g. Remark 7.8.5.)
1.8. Topics we omit. For reasons of time and ignorance, we do not discuss many recent (and even not so recent) developments in the p-adic Langlands
program, which we nonetheless expect have important connections to the ideas discussed here. In particular we say nothing about the cohomology of the Drinfeld
upper half plane and related p-adic locally symmetric spaces, as studied by Colmez,
Dospinescu, Le Bras, Niziol and Pan [CDN20, CDN22, CDN21, DLB17, Pan17](the interested reader could see https://mathoverflow.net/questions/432932/
how-does-the-cohomology-of-the-lubin-tate-drinfeld-tower-fit-into-ca
tegorical-p for an explanation of the expected connections); about Scholze’s
p-adic Jacquet–Langlands functor [Sch15b]; or about the coherent cohomologyof Shimura varieties, and in particular about any relationship to Boxer and Pilloni’s higher Hida and Coleman theories [Pil20, BP20, BP21] or to Diamondand Sasaki’s geometric Serre weight conjectures [DS22].
1.9. A brief guide to the notes. These notes cover many topics, and wehope that it is possible to read many of the sections more or less independently fromeach other, referring back for definitions where necessary. In Section 3 we give an
idiosyncratic introduction to the Taylor–Wiles method, guided by its connections
to the classical p-adic local Langlands correspondence for GL2 (Qp ), and its role inmotivating our conjectures. In Section 4 we very briefly recall the definitions andbasic properties of the stacks of (ϕ, Γ)-modules from [EG22]. The rather longer
Section 5 explores analogues of these constructions in the “analytic” context of(ϕ, Γ)-modules over Robba rings.
Section 6 states our main (local) p-adic Langlands conjectures, in both the
Banach and analytic settings, and explains their relationships to other topics inthe literature, e.g. the Breuil–Mézard conjecture. We then explain what is known
about these conjectures in Section 7; as well as the relatively straightforward caseof GL1 , we explain forthcoming work of Andrea Dotto with M.E. and T.G. which
proves the Banach conjecture for GL2 (Qp ), and we explain some evidence for the
Banach conjecture for GL2 (F ) for F/Qp unramified. Section 8 briefly recalls someof the results and expectations in the case ` 6= p. This material is used in Section 9,

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M. EMERTON, T. GEE, AND E. HELLMANN

which explains conjectures on the (completed) cohomology of Shimura varieties andeigenvarieties.
The appendices recall various results on (infinity-) category theory and therepresentation theory of p-adic analytic groups, and establish others for which we
do not know of a reference.
Remark 1.9.1. The categorical p-adic Langlands correspondence is still in its
infancy, and (as these notes make plain) it is unclear to us what the ultimate formof many of the conjectures we make should be. There is also a considerable amountof technical machinery needed to get off the ground. While we have endeavoured
to make precise and correct statements, our aim has been where possible to explain
and illustrate phenomena rather than give a detailed development of all of thismachinery. In particular, we for the most part treat infinity categories and derived
algebraic geometry as a black box.
It seems extremely likely that the future development of the theory will rely on
condensed mathematics, and in particular the theory of solid rings (and modules,
sheaves. . . ). It is also likely that some of the definitions and constructions would be
streamlined by phrasing them in these terms. However, due to our own limitations,
we have for the most part not attempted to do this.
1.10. Acknowledgements. We would like to thank the organisers (PierreHenri Chaudouard, Wee Teck Gan, Tasho Kaletha, and Yiannis Sakellaridis) andparticipants of the IHES Summer School on the Langlands Program for the opportunity and encouragement to write these notes.
We would like to thank the many colleagues, collaborators and friends withwhom we have discussed the ideas presented in these notes. In particular we would
like to thank Johannes Anschütz, David Ben-Zvi, Roman Bezrukavnikov, George
Boxer, Christophe Breuil, Frank Calegari, Ana Caraiani, Pierre Colmez, Gabriel
Dospinescu, Andrea Dotto, Laurent Fargues, Tony Feng, Dennis Gaitsgory, DavidGeraghty, Ian Grojnowski, David Hansen, Michael Harris, David Helm, Christian
Johansson, Arthur-César Le Bras, Bao Le Hung, Brandon Levin, Jacob Lurie,
Lucas Mann, David Nadler, James Newton, Vytautas Paškūnas, Vincent Pilloni,
Alice Pozzi, Joaquin Rodrigues Jacinto, David Savitt, Peter Scholze, Benjamin
Schraen, Jack Sempliner, Sug Woo Shin, James Timmins, Pol van Hoften, Akshay
Venkatesh, Carl Wang-Erickson, and Xinwen Zhu.
We would also like to thank Pierre Colmez, Fred Diamond, Andrea Dotto,
David Hansen, Claudius Heyer, Vytautas Paškūnas, and David Savitt for theircomments on earlier versions of these notes.
2. Notation
2.1. Fields. We fix an algebraic closure Q of Q, and algebraic closures Qp
of Qp for each p. If M is a number field, we write GalM := Gal(Q/M ) for the absolute Galois group of M . If S is a set of places of M then we write M (S)/M for the
maximal extension unramified outside of S, and write GalM,S := Gal(M (S)/M ).
We fix embeddings Q ,→ Qp for each p and Q ,→ C, so that if v is a place of M ,there is a homomorphism GalMv → GalM .
If F/Qp is a finite extension, write GalF for the absolute Galois group Gal(Qp /F ).
Write IF for the inertia subgroup of GalF , WF for the Weil group, and kF for theCATEGORICAL p-ADIC LANGLANDS

13

residue field of the ring of integers OF of F . We normalise local class field theory
so that a uniformizer corresponds to a geometric Frobenius element.
Let O denote the ring of integers in a fixed finite extension L of Qp , let k be
the residue field of O, and let $ denote a uniformizer of O.
2.2. p-adic Hodge theory. If ρ is a de Rham representation of GalF on an
L-vector space W , then we will write WD(ρ) for the corresponding Weil–Deligne
representation of WF , and if σ : F ,→ L is a continuous embedding of fields thenwe will write HTσ (ρ) for the multiset of Hodge–Tate numbers of ρ with respectb (i))GalF ; forF
to σ, which by definition contains i with multiplicity dim (W ⊗Lσ,Fexample, if ε denotes the p-adic cyclotomic character, then HTσ (ε) = {−1}.
Suppose that L contains the images of all continuous embeddings F ,→ L. By ad-tuple of labeled Hodge–Tate weights λ, we mean a tuple of integers {λσ,i }σ:F ,→L,1≤i≤dwith λσ,i ≥ λσ,i+1 for all σ and all 1 ≤ i ≤ d − 1. We will also refer to λ as a Hodge
type. We say that λ is regular if λσ,i > λσ,i+1 for all σ and all 1 ≤ i ≤ d − 1. We
say that a de Rham representation ρ has Hodge type λ (or labeled Hodge–Tateweights λ) if for each σ : F ,→ L we have HTσ (ρ) = {λσ,i }1≤i≤d .
By an inertial type τ we mean a representation τ : IF → GLd (L) which extends
to a representation of WF with open kernel (so in particular, τ has finite image).
We say that a de Rham representation ρ has inertial type τ if WD(ρ)|IF ∼= τ.
2.3. Hodge–Tate weights vs. highest weights. Under the p-adic local
Langlands correspondence, Hodge–Tate weights on the Galois-theoretic side of thecorrespondence will be related to highest weights on the representation theoretic
side, and it is useful to have notation adapted to either side of the correspondence.
Definition 2.3.1. If λ is a regular Hodge type, then we write ξσ,i = i − 1 −λσ,d+1−i , so that ξσ,1 ≥ · · · ≥ ξσ,d .
Definition 2.3.2. If λ is a regular Hodge type, then 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). We let Wλ be the corresponding Orepresentation of GLd (OF ), defined as follows: for each σ : F ,→ L, we write Wξσfor the algebraic OF -representation of K (more precisely, the dual Weyl module)of highest weight ξσ . Then we define
Wλ := ⊗σ Wξσ ⊗OF ,σ O,
and write L(ξ) = Wλ [1/p], a representation of GLd (F ) (or of the algebraic group
Q
ResF p GLd ).
Remark 2.3.3. The preceding notation may not be optimal, but serves our
purposes. The “W ” stands for (dual) Weyl, and emphasizing the dependence onλ (rather than on ξ) facilitates the comparison with other considerations of padic Hodge theory. The notation L(ξ) is chosen because it is traditional in thetheory of category O. Indeed, for any (not necessarily dominant, not necessarilyQ
integral) weight ξ (of the Lie algebra of ResF p GLd ), we use M (ξ) to denote theVerma module of highest weight ξ, and L(ξ) to denote the unique simple quotientof M (ξ). When ξ is dominant integral, one finds that L(ξ) is the usual highest
weight representation Wλ [1/p], explaining our choice of notation in this case.

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M. EMERTON, T. GEE, AND E. HELLMANN

3. Taylor–Wiles patching as a motivation for categorical p-adic local
Langlands
In this section we review some of the history of the Taylor–Wiles patching
method and of the p-adic local Langlands correspondence for GL2 (Qp ), emphasising where possible the connections between them. We do not pretend to givea thorough overview, and we view everything through the lens of the categorical
Langlands program. The reader may wish to consult Calegari’s survey [Cal21] fora more mainstream account of developments in the Taylor–Wiles method since its
inception.
3.1. A very brief introduction to patching. Patching was first introduced
by Taylor and Wiles as a technique for proving modularity lifting theorems [TW95].
It was then further developed by Kisin, who showed that it provides a mechanism for relating local and global aspects of the theory of p-adic Galois representations [Kis09c, Kis07a, Kis09a, Kis10b, GK14].
We place ourselves in the following simple but illustrative context. Fix a continuous, absolutely irreducible, modular representation r : GalQ,{p,∞} → GL2 (k) withdeterminant ε−1 , and suppose that ρ := r| GalQp and r|GalQ(ζp ) are also absolutely
irreducible.
For any set S of primes containing p and ∞, we may consider the global
deformation space XS (r) of r over O, which parameterizes deformations of r to
representations of GalQ,S over complete Noetherian local O-algebras, with fixed
determinant ε−1 . We may also consider the local deformation space X (ρ), whichparameterizes deformations of ρ to representations of GalQp over complete Noetherian local O-algebras. These are each the Spf of a complete Noetherian local
O-algebra, denoted RS (r) and R(ρ) respectively. Restriction of Galois representations induces a morphism(3.1.1)

XS (r) → X (ρ),

which is a finite morphism (see [EP20] and [AC14]).
The reduced tangent space to the closed point of XS (r) (resp. X (ρ)) is equal
to the global Galois cohomology group H 1 GQ,S , Ad0 (r) (resp. to the local Galois
cohomology group H 1 GQp , Ad0 (ρ) ), and the restriction map between deformation
spaces induces the restriction map
(3.1.2)H 1 GQ,S , Ad0 (r) → H 1 GQp , Ad0 (ρ)
on reduced tangent spaces, which is given by the usual pullback of cohomology
classes.
In order to simplify the situation, we will assume that (3.1.1) is in fact a
closed immersion when S = {p, ∞}; equivalently, we assume that the Selmer
group3 Sel GQ,{p,∞} , Ad0 (r) vanishes. Under this assumption (and the running assumption that r|GalQ(ζp ) is absolutely irreducible), the formula of Greenberg–Wiles([DDT97, Thm. 2.18]) shows that
dimk Sel GQ,{p,∞} , Ad0 (r)(1) = dimk Sel GQ,{p,∞} , Ad0 (r) + 1 = 1.
3This is a “strict” Selmer group, i.e. defined by the condition of a cohomology class vanishing
locally at p.

CATEGORICAL p-ADIC LANGLANDS

15

The Taylor–Wiles method then requires us to consider primes q ≡ 1 mod p suchthat the restriction morphism H 1 GQ,{p,∞} , Ad0 (r)(1) → H 1 GalQq , Ad0 (r)(1)
is non-zero (equivalently, injective), and such that r(Frobq ) has distinct eigenvalues;
these are the so-called Taylor–Wiles primes. Another application of the Greenberg–
Wiles formula then shows that
Sel GalQ,{p,q,∞} , Ad0 (r) = 0
for such a Taylor–Wiles prime q, and thus that the restriction morphism of deformation spaces
X{p,q,∞} (r) → X (ρ)(3.1.3)is again a closed immersion.
Write rq := r|GalQq . We now consider the restriction morphism of deformation
spaces
X{p,q,∞} (r) → X (rq ).
Under our hypotheses that q ≡ 1 mod p and that r(Frobq ) has distinct eigenvalues,
it can be shown that X (rq ) = Spf R(rq ) takes a very simple form: namely, R(rq )nis of the form O[[x, y]]/((1 + x)p − 1), where pn is the largest power of p dividing(q − 1). (More precisely, one checks that all deformations of rq are reducible, andthe variables x, y then correspond respectively to a generator of the tame inertia
group and to a Frobenius element.) In particular, we obtain a mapnX{p,q,∞} (r) → Spec O[[x]]/((1 + x)p − 1),(3.1.4)whose fibre over the locus x = 0 is equal to X{p,∞} (r); in this way we may regard
X{p,q,∞} (r) as a kind of thickening4 of X{p,∞} (r). (This interpretation of the situation would be optimal if the morphism (3.1.4) were flat. This is not always thecase, but what comes out of the Taylor–Wiles method is that X{p,q,∞} (r) supportsna faithful module which is flat over O[[x]]/((1 + x)p − 1), namely the completed
e 1 (q) introduced below, so the situation is always fairly close tohomology group H
the optimal one.)Now it turns out that X{p,∞} (r) has relative dimension 2 over O, while X (ρ) has
relative dimension 3 over O. Thus the closed immersion (3.1.1) has codimension 1,
and the closed immersion (3.1.3) then realizes X{p,q,∞} (r) as a thickening (in the“x direction”, if we refer to (3.1.4)) of X{p,∞} (r) along the transverse direction toX{p,∞} (r) in X (ρ).
A key point is that for any given power pn of p, we can find plenty of Taylor–Wiles primes q (ultimately via a Čebotarev argument), such that q ≡ 1 mod pn .
Thus we can thicken up X{p,q,∞} (r) as much as we like inside X (ρ), and the rather
surprising idea at the heart of the patching method is to try to realize X (ρ) as somekind of limit of all these thickenings X{p,q,∞} (r).
To make this precise, we first regard each deformation ring R(rq ) as an S∞ :=O[[x]]-algebra (this gives a uniform meaning to the “x direction” referred to above
as q varies), and then try to glue together the various Spec S∞ -schemes X{p,q,∞} (r)as q varies in a manner compatible with their embeddings
into X (ρ); equivalently,
we try to glue together the various S∞ , R(ρ) -bialgebras R(rq ). As part of the
4Since if we look modulo the uniformizer of O, then we find that Spec k[[x]]/(1 + x)pn − 1) =Spec k[x]/xpn

is a nilpotent thickening of Spec k.

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M. EMERTON, T. GEE, AND E. HELLMANN

patching process, we simultaneously glue to together various (completed) homology
groups.
Recall that we define completed (first) homology of the modular curve as
e 1 := lim H1 Y (pr ), O) ∼(3.1.5)H
lim Hc1 Y (pr ), O .
=
←−
←−
r

r

Here Y (pr ) is the modular curve of level pr (or more precisely a congruence quotientof PGL2 -symmetric spaces, which is a union of connected components of the usual
modular curves for GL2 ).
This is acted on by a Hecke algebra T generated by the prime-to-p Hecke operators, as well as by PGL2 (Qp ) and GalQ,{p,∞} . All of these actions commute with
one another. There is a maximal ideal m of T corresponding to the modular Galois
e 1 at m to get a Tm -module He 1,m . Similarly, forrepresentation r. We can localize Heach q as above we have the completed homology groupe 1 (q) := lim H1 Y (pr q), O) ∼H
H 1 Y (pr q), O),
= lim
←−
←− c
r

r

which has actions of PGL2 (Qp ), of GalQ,{p,q,∞} , of prime-to-pq Hecke operators,
and of certain Hecke operators at q. We can and do localize at m to get S∞ - ande 1,m (q). In fact, since the action of S∞ on H
e 1,m (q) is by definition
R(ρ)-modules H
via the composite of our chosen morphism S∞ → R(rq ) and the natural morphism
R(rq ) → R{p,q,∞} (r), we can use the formal smoothness of S∞ and the closed
immersion (3.1.3) to produce a morphism S∞ → R(ρ), compatible with the actions
e 1,m (q).
of both rings on H
Following [Sch15b, §9], we make an ultraproduct construction to produce anR(ρ)-module M∞ with an action of PGL2 (Qp ), with the property that
e 1,m .
M∞ ⊗R(ρ) R{p,∞} (r) = H
By construction (and local-global compatibility at the primes q), M∞ is a finite
projective S∞ [[K]]-module, where K = PGL2 (Zp ).

In general one cannot assume that the Selmer group Sel GQ,{p,∞} , Ad0 (r)vanishes (equivalently, we cannot assume that (3.1.1) is a closed immersion) andthe Taylor–Wiles method goes as follows. Let g := dimk Sel GQ,{p,∞} , Ad0 (r) + 1.
The Čebotarev argument alluded to above shows that for each n there is a set Qnof Taylor–Wiles primes with the properties that

(3.1.6)
dimk Sel GQ,{p,∞}∪Qn , Ad0 (r) = dimk Sel GQ,{p,∞} , Ad0 (r) ,and for each q ∈ Qn , we have q ≡ 1 mod pn . We write S∞ := O[[x1 , . . . , xg ]]and R∞ := R(ρ)[[y1 , . . . , yg−1 ]] for formal variables xi , yi . By (3.1.6), we may
lift (3.1.1) and (3.1.3) to (non-canonical) closed immersions X{p,∞} (r) → Spf R∞
and X{p,∞}∪Qn (r) → Spf R∞ , and straightforward generalizations of the constructions explained above in the case g = 1 yield a finite projective S∞ [[K]]-module
M∞ with an action of R∞ , and we can again produce a morphism S∞ → R∞
compatible with the actions of each ring on M∞ . By construction, we again havee 1,m .
(3.1.7)
M∞ ⊗R R{p,∞} (r) = H∞

The patched module M∞ can be thought of as being a way to extract the local
at p information from completed cohomology. Its construction depends on many
choices, but the perspective first taken in [CEG+ 16] is that it should be purely
local in an obvious sense, and that the same construction for more general Shimura

CATEGORICAL p-ADIC LANGLANDS

17

varieties should give a candidate for a p-adic local Langlands correspondence beyond
the case of GL2 (Qp ). This expectation has yet to be verified, but it is one thathas been important for us in finding the conjectures presented in these notes, andthe modules M∞ appear in the statements of our conjectures (see in particular
Remark 6.1.30).
3.1.8. Patching at finite level. If V is any finitely generated O-module with acontinuous action of K, then we set(3.1.9)M∞ (V ) := M∞ ⊗O[[K]] V.

Then V 7→ M∞ (V ) is an exact functor, and (3.1.7) implies that
M∞ (V ) ⊗R∞ R{p,∞} (r) = Hc1 (V )m ,where Hc1 (V )m denotes the (localized at m) compactly supported cohomology groupwith coefficients in the local system determined by V . (Since m is non-Eisenstein,
the natural map Hc1 (V )m → H 1 (V )m is an isomorphism, but we have used the Hc1
notation to emphasize that we are working with homology.)
Remark 3.1.10. As we indicate below, the definition just given of M∞ (V ) interms of M∞ is historically backwards. Indeed the original Taylor–Wiles method
fixed a V and constructed an associated patched ring — denoted R∞ in the original literature, but which we might denote R∞ (V ) — which in retrospect can beinterpreted as that quotient of the ring R∞ introduced above which acts faithfully
on M∞ (V ). In the contexts originally considered by Taylor and Wiles, the patched
module M∞ (V ) is cyclic over R∞ , and hence free over R∞ (V ), and so only appears implicitly (in the use of mod p multiplicity one results that show that certain
Hecke modules arising from the cohomology of modular curves are free over the
Hecke algebras that act on them). The patched modules M∞ (V ) themselves wereintroduced (independently) by Diamond [Dia97] and Fujiwara [Fuj06], and it was
the work of Kisin [Kis09a] (as interpreted by [EGS15]) that then recast patchingin terms of the exact functor V 7→ M∞ (V ). That this functor is determined bya single “large” patched module M∞ was observed in [CEG+ 16]. (Note thoughthat the formula (3.1.9) does not literally appear in [CEG+ 16], where instead one
finds essentially equivalent (but visually more involved) formulas involving Homsand duals.)
The reader may wish to compare the formula (3.1.9) for the patching functor
to Conjecture 6.1.14 (2), which suggests the mnemonic “M∞ = M∞ (O[[K]])”. The
jump from considering the functor V 7→ M∞ (V ) to the conjectures of these notes isroughly to move from considering representations of PGL2 (Zp ) to representations
PGL (Q )of PGL2 (Qp ), by replacing V by the compact induction c-IndPGL22 (Zpp) V , and thento move from modules over Galois deformation rings to sheaves on stacks of (ϕ, Γ)modules.
Actually, in the context of patching, the passage from PGL2 (Zp ) to PGL2 (Qp )was one of the main contributions of [CEG+ 16], and so there is natural progression of ideas from [EGS15] (patching as a functor on PGL2 (Zp )-representations)
to [CEG+ 16] (patching as a functor on PGL2 (Qp )-representations) to Expected
Theorem 7.2.1 below (patching is promoted to a functor from PGL2 (Qp )-representations
to (complexes of) sheaves on a stack of (ϕ, Γ)-modules).
As already indicated above, the patching method can be extended from thecase of modular curves to more general Shimura varieties and related congruence

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M. EMERTON, T. GEE, AND E. HELLMANN

quotients. However, the case of PGL2 (Qp ) considered here is particularly nice,because a p-adic local Langlands correspondence has been constructed in this case(in fact for GL2 (Qp )). The idea of a p-adic local Langlands correspondence was
motivated by Taylor–Wiles patching, but the correspondence itself was constructedby quite different techniques. We review some of the history in the rest of thissection.
3.2. p-adic Langlands: an overview and history. Following preliminary
and for the most part unpublished investigations by many researchers (including inapproximate historical order Fontaine–Langlands, Serre, Mazur, Harris, Vignéras,
and Schneider–Teitelbaum), the investigation of a possible p-adic Langlands correspondence, relating p-adic representations of the group GL2 (Qp ) to 2-dimensional
p-adic representations of the local Galois group GalQp , was instigated by Christophe
Breuil [Bre03a, Bre03b]. He was motivated to a considerable extent by various considerations that arose out of the proof of modularity lifting theorems[Wil95, TW95, CDT99, BCDT01]; in particular, he was directly motivatedby considerations arising from the Breuil–Mézard conjecture (see [Bre08, §1.4] foran account of these motivations). We begin our overview by discussing some ideas
and methods related to these theorems, before turning to a discussion of the correspondence itself, its application to the study of the Breuil–Mézard conjecture, andits relationship to completed (co)homology via local-global compatibility.
3.2.1. Taylor–Wiles–Kisin patching. Taylor–Wiles patching, as introduced in[TW95] and briefly reviewed above, is a process that constructs modules over
local deformation rings out of the (co)homology of Shimura varieties (thought of as
Hecke modules).
The patching method was originally developed to prove modularity lifting theorems, showing that certain p-adic Galois representations (in the original applications, those associated to elliptic curves) correspond to modular forms. Originallythe patching arguments required as inputs a number of deep results about congruences between modular forms, including mod p multiplicity one theorems, levelraising theorems, cases of the weight part of Serre’s conjecture, and Serre’s modularity conjecture itself. Subsequently, the situation has reversed, and such results
can now be deduced from patching arguments.
While patching arguments are still used to prove that Galois representations areautomorphic, they are now also used to study the internal properties of Galois representations and of congruences between automorphic representations, and it is thisapplication which interests us. We will not attempt to give a full account of theseshifts, and will instead discuss the contemporary perspective on the patching construction, but in brief we note that the dependence on mod p multiplicity one theorems was removed independently by Diamond [Dia97] and Fujiwara [Fuj06], levelraising was addressed by Taylor’s “Ihara avoidance” argument [Tay08], the weightpart of Serre’s conjecture by T.G. [Gee11], and Serre’s conjecture was proved by
Khare–Wintenberger [KW09] and Kisin [Kis09b]. Many of these developments,and much of what we explain below, rely crucially on Kisin’s improvement of the
Taylor–Wiles method [Kis09c].
We maintain the notation above, but to avoid clutter we write Rp for R(ρ)
from now on. We also go back to considering GL2 rather than PGL2 , so that inparticular the exact functor V 7→ M∞ (V ) is a functor from the category of continuous GL2 (Zp )-representations on finitely generated O-modules to the category

CATEGORICAL p-ADIC LANGLANDS

19

of finitely generated modules over R∞ , where R∞ is a power series ring over Rp .
There are certain choices of GL2 (Zp )-representation that are of particular interest,
namely those that are lattices in a locally algebraic type, and those that are irreducible representations of GL2 (Fp ) over k. Evaluating the patching functor M∞ on
the first kind of representation relates patching to the study of modularity lifting;and this provided the original motivation for the patching construction. Evaluating
the patching functor on the second kind of representation establishes a relationshipbetween patching and the weight part of Serre’s conjecture.
To be more precise, recall that by a locally algebraic type σ, we mean anirreducible Qp -representation of GL2 (Zp ) obtained by tensoring an algebraic representation with a finite-dimensional representation corresponding to an inertial
type τ via Henniart’s inertial local Langlands correspondence [BM02, App. A];
we say that a lift of ρ is of type σ if it has Hodge–Tate weights corresponding to
the highest weight of the algebraic representation, and inertial type corresponding
to τ . Suppose that σ is a locally algebraic type, and let σ ◦ be a GL2 (Zp )-invariant
lattice in σ. If Rp (σ) denotes the quotient of Rp parameterizing lifts of type σ, then(as a consequence of local-global compatibility) the Rp -action on M∞ (σ ◦ ) factors
through Rp (σ), and a modularity lifting theorem for lifts of type σ (i.e. a theorem
proving that lifts of r of type σ are necessarily modular) can be interpreted in termsof patching as saying that the action of the corresponding quotient R∞ (σ) of R∞
on M∞ (σ) should be faithful.
As for the weight part of Serre’s conjecture, this can be rephrased as the statement that M∞ (σ) is non-zero for precisely those Serre weights σ of GL2 (Fp ) thatlie in a specified set W (ρ) (where by definition a Serre weight is an irreducible
k-representation of GL2 (Fp )).
3.2.2. An observation. There is an important observation we can make that
motivated much of the further development of the subject. If V is a GL2 (Zp )representation on a finite rank free O-module (e.g. a lattice in a locally algebraictype), and V := V /$V is the associated residual representation, then M∞ (V ) is
O-torsion free (by exactness), and so M∞ (V ) 6= 0 iff M∞ (V ) 6= 0 iff M∞ (σ) 6= 0 forsome Jordan–Hölder factor σ of V . Thus, if the weight part of Serre’s conjecture
is known, then the question of whether or not M∞ (V ) is non-zero can be answered
via an examination of the set of Jordan–Hölder factors of V .
3.2.3. The Breuil–Mézard conjecture. As already noted, in order to prove a
modularity lifting theorem for Galois representations of locally algebraic type σ,
one must show that R∞ (σ) acts faithfully on M∞ (σ ◦ ), for some (equivalently,any) invariant lattice σ ◦ contained in σ. It follows from the patching constructionthat M∞ (σ ◦ ) is maximal Cohen–Macaulay over R∞ (σ); so if Rp (σ) is a domain,then R∞ (σ) (which is then also a domain) necessarily acts faithfully on M∞ (σ ◦ ).
Perhaps the simplest way to establish that Rp (σ) is a domain is to show that it isformally smooth, and the arguments of [CDT99] and [BCDT01] were devoted toestablishing such a smoothness result (for appropriately chosen σ). In analyzing theconditions introduced in these papers related to ascertaining such results, Breuil andMézard were led to formulate their eponymous conjecture [BM02]. This conjecture
describes the Hilbert–Samuel multiplicity of Rp (σ)/$ in terms of the Serre weightsof ρ that appear as constituents of σ ◦ /$.
A key breakthrough in understanding the Breuil–Mézard conjecture was madeby Kisin (see e.g. [Kis10b]), who strengthened the observation of (3.2.2) to note20

M. EMERTON, T. GEE, AND E. HELLMANN

that it follows from the weight part of Serre’s conjecture, the exactness of thepatching functor, and the maximal Cohen–Macaulay nature of patched modules,
that the formula of the Breuil–Mézard conjecture precisely describes the Hilbert–
Samuel multiplicity of R/$, where R denotes that quotient of R∞ (σ) that actsfaithfully on M∞ (σ ◦ ). Thus the Breuil–Mézard conjecture, when combined withthe weight part of Serre’s conjecture, is seen to be essentially equivalent to thefaithfulness of the action of Rp (σ) on M∞ (σ ◦ ), i.e. to a modularity lifting theorem.
(See [EG14, §5] for a more precise formulation of this equivalence in a more general
setting.)
By combining this observation with an argument using the p-adic Langlands
correspondence, Kisin was in fact able to prove the Breuil–Mézard conjecture (andso also to deduce corresponding modularity lifting theorems) [Kis09a]. We recall
more of the details of this argument below. For now, we merely remark that,even without knowing whether or not Rp (σ) acts faithfully on M∞ (σ ◦ ), there is an
obvious inequality of Hilbert–Samuel multiplicities eRp (σ)/$ ≥ eR/$ , and so Kisin’s
observation always yields a weak form of the Breuil–Mézard conjecture, in whichequality is replaced by an inequality in one direction.
3.2.4. A locally analytic variant. Above we discussed an approach to studying
Serre weights via patching functors. There is a related story in the analytic setup
which is discussed in [Bre15] and [BHS19]. The question of which Serre weights
are associated to a residual Galois representation ρ̄ can be understood as the question of which irreducible GL2 (Zp )-representations (more generally K = GLn (OF )representations) embed into the (conjectural) mod p representation Π(ρ̄). (Notethat here we are adopting the more traditional perspective on p-adic and mod p
local Langlands; the connection between this perspective and the categorical conjecture emphasised in these notes is discussed in Remark 7.8.5). In some analogy
with this question, Breuil [Bre15, Conj. 6.1] has formulated a conjecture about
the irreducible constituents of locally analytic principal series representation thatembed into the p-adic local Langlands correspondence Π(ρ) of a p-adic Galois representation ρ. While the question about Serre weights is motivated by the global
question about the weights of modular forms whose associated Galois representation has fixed reduction modulo p, Breuil’s socle conjecture is motivated by the
global question about (the weights of) non-classical overconvergent p-adic modular
forms with prescribed associated Galois representation.
In [BHS19] many cases of Breuil’s socle conjecture are proven using a similar
strategy (patching functors and a multiplicity formula in the spirit of the BreuilMézard conjecture) as discussed above: Instead of studying V 7→ M∞ (V ) as afunctor on certain locally algebraic K-representations, one studies a related functor
on certain locally analytic representations (or the BGG category O and a functorfrom that category to locally analytic representations). In this case the functor
produces coherent sheaves on the rigid analytic generic fiber of SpfR∞ that aresupported on closed subvarieties defined in terms of trianguline Galois representations (see also Section 3.2.6 below for a discussion of trianguline representations).
For these trianguline deformation spaces there is a locally analytic analogue ofthe Breuil-Mézard conjecture formulated in [BHS19, Conj. 4.3.4]. Roughly thisconjecture predicts that the multiplicities of irreducible constituents in locally analytic representations that are parabolically induced from a character δ match themultiplicities of certain cycles in trianguline deformation spaces whose parameters

CATEGORICAL p-ADIC LANGLANDS

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correspond to δ. In fact this conjecture can be proven in many cases [BHS19,
Theorem 4.3.8] by relating it to multiplicity formulas in geometric representation
theory.
3.2.5. The p-adic local Langlands correspondence for GL2 (Qp ). The p-adic local Langlands correspondence, associating admissible continuous unitary p-adic Banach space representations Π(ρ) of GL2 (Qp ) to two dimensional continuous representations ρ : GalQp → GL2 (L), was first proposed by Breuil for those representations ρ that are potentially semistable with distinct Hodge–Tate weights[Bre03a, Bre03b].
More precisely, Breuil proposed in this context that Π(ρ) should be constructed
as a certain completion of the locally algebraic representation Π(ρ)alg obtained bytensoring together the smooth representation of GL2 (Qp ) attached to WD(ρ) (theWeil–Deligne representation underlying the Dieudonné module of the potentiallysemistable representation ρ) with a finite dimensional algebraic representation of
GL2 (Qp ) encoding the Hodge–Tate weights of ρ. (The representation Π(ρ)alg contains a locally algebraic type σ, and Breuil’s proposal was motivated by the Breuil–Mézard conjecture.) Furthermore, in the case when ρ is crystabelline or semistabelline (i.e. becomes crystalline, or semistable, upon restriction to a finite abelian
extension of Qp ), he made his proposal completely explicit: namely, in the irreducible crystabelline case, he proposed that Π(ρ) should be the universal unitarycompletion of Π(ρ)alg , while in the irreducible genuinely semistabelline (i.e. noncrystabelline) case, he proposed that Π(ρ) should be a certain explicit completionof Π(ρ)alg depending on the L-invariant of ρ.
The motivation for proposing the universal unitary completion as the candidatefor Π(ρ) in the irreducible crystabelline case is that in this case there is no “extra”
p-adic Hodge theoretic data carried by ρ besides that already carried by Π(ρ)alg(that is, there is no L-invariant or the like in this case; more precisely, there is up toisomorphism a unique weakly admissible filtration with fixed Hodge–Tate weights
on Dcris (ρ) such that the corresponding Galois representation is irreducible). The
representation Π(ρ)alg determines (and is determined by) Dcris (ρ) and the Hodge–
Tate weights, and the universal unitary completion is the only evident choice ofcompletion available that doesn’t depend on any additional choices.
The major initial difficulty in investigating Breuil’s conjectured correspondence
was in establishing any non-trivial properties of his proposed completions; it was
not even evident that these completions were non-zero. One exception to this situation was in the case when ρ was the restriction to GalQp of a global two-dimensional
representation r : GalQ → GL2 (L) arising from a Hecke eigenform. In this casethe theory of completed cohomology, together with local-global compatibility forthe classical Langlands correspondence, gives rise to a non-zero unitary completion of Π(ρ)alg . In the semistabelline case, Breuil was furthermore able to show,
using methods arising from the theory of p-adic L-functions (and we remind thereader that the notion of L-invariant had its origin in that same theory) that thiscompletion factored through his proposed completion of Π(ρ)alg [Bre10b].
Inspired by the approach of Kato, Perrin-Riou and others to the theory of padic L-functions via Galois cohomology, explicit reciprocity laws, and p-adic Hodge
theory, and his own reinterpretation and further developments of these ideas interms of (ϕ, Γ)-modules, Colmez was then able to make a major breakthrough,and establish, in the irreducible and genuinely semistabelline case, that Breuil’s

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proposed completion Π(ρ) is non-zero, topologically irreducible, and admissible.
(See [Col10a] for the culmination of Colmez’s investigations in this direction.)
Soon thereafter, using similar techniques, Berger and Breuil were able to prove theanalogous result in the crystabelline case [BB10]. A key consequence of their result
is that, for crystabelline ρ, the universal unitary completion Π(ρ) of Π(ρ)alg is theunique non-zero unitary completion of Π(ρ)alg .
Since the theory of (ϕ, Γ)-modules applies to arbitrary representations ρ, notjust potentially semistable ones, Colmez’s results led him to propose that the padic local Langlands correspondence should exist for arbitrary continuous representations ρ : GalQp → GL2 (L). (In the case when ρ is not potentially semistablewith distinct Hodge–Tate weights, one no longer has a representation Π(ρ)alg available from which to even conjecturally construct Π(ρ) as some kind of completion.)
Reverse-engineering his (ϕ, Γ)-module approach to studying Breuil’s completions,
Colmez discovered the first of his two functors, the functor Π 7→ V (Π) mapping topologically finite length admissible unitary continuous Banach GL2 (Qp )representations (satisfying a finite length condition on their reductions mod $,which was shown by Paškūnas to be superfluous [Paš13]) to continuous representations of GalQp [Col10c, §IV].
In the light of his investigations of the Breuil–Mézard conjecture, Kisin thenmade the crucial suggestion that Colmez’s proposed general p-adic local Langlands
correspondence should be regarded as taking place over local Galois deformation
spaces, and that deformation-theoretic tools could play a role in its construction.
Following this suggestion, Colmez was able to extend his (ϕ, Γ)-module methods
to construct Π(ρ) forgeneral ρ [Col10c]. The functors are related via a natural
isomorphism V Π(ρ) ∼= ρ. Furthermore, he was able to prove that when ρ ispotentially semi-stable with distinct Hodge–Tate weights, the representation Π(ρ)
is indeed a completion of Π(ρ)alg , as Breuil had originally proposed. In the cases
when ρ is crystabelline or semistabelline, this followed from the results alreadymentioned due to him and Berger–Breuil (indeed, the general construction of Π(ρ)
is motivated by the constructions in those cases). However, in general (i.e. in casesthat are not crystabelline, or semistabelline, or, more generally, trianguline), theaction of the entirety of GL2 (Qp ) on Π(ρ) is constructed in a very indirect way(it is only the action of the upper triangular Borel subgroup of GL2 (Qp ) that canbe seen directly in terms of (ϕ, Γ)-modules) by deformation-theoretic methods (or,one could say, by rigid analytic interpolation), and so the nature of the locallyalgebraic vectors in Π(ρ) is not obvious. Colmez’s proof that they contain (and infact coincide with) Π(ρ)alg ultimately rests on a comparison with the global context
of completed cohomology, mediated via the work of one of us (M.E.) on local-global
compatibility.
The theory of the p-adic local Langlands correspondence for GL2 (Qp ) was completed by Paškūnas, who showed that the functor Π 7→ V (Π) induces a bijection
between isomorphism classes of supersingular topologically irreducible admissible
unitary continuous Banach space representations of GL2 (Qp ), and irreducible continuous two-dimensional representations of GalQp [Paš13]. In fact, more was established: essentially, V induces an equivalence between the category of topologicallyfinite length admissible Banach space representations of GL2 (Qp ), and the category
whose objects are deformations of two-dimensional continuous representations of
GalQp . Furthermore, this equivalence can be described Morita-theoretically: there

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are certain injective objects in the category of GL2 (Qp )-representations, whose endomorphism rings may be identified with Galois deformation rings, that mediate
the equivalence. Since these injective objects (or, equivalently, their topologicalduals, which are then projective objects in an appropriate category) play an important role in the work of Andrea Dotto, M.E. and T.G. which we explain in
Sections 7.2–7.4, we will discuss them further below.
3.2.6. The locally analytic p-adic Langlands correspondence. As we will discuss
further below below, the Banach space representations occurring in the p-adic local
Langlands correspondence for GL2 (Qp ) are ultimately constructed by passage tothe limit of constructions modulo pn (and then by inverting p), and in particularthey naturally fit into the “Banach” context in these notes.
On the other hand, an important tool for studying Banach space representations
such as Π(ρ) is to study their space of analytic vectors Π(ρ)an (in the sense of[ST03, §7]), which are locally analytic representations of GL2 (Qp ). This goes in
the opposite direction to the consideration of mod pn representations, since Π(ρ)anforgets about the Banach space (and hence p-adic integral) structure on Π(ρ). On
the other hand, Colmez’s construction shows that the locally analytic representation
Π(ρ)an can be constructed directly from the rigid analytic (ϕ, Γ)-module Drig (ρ)
over the Robba ring R associated to ρ, so that it still makes sense to study the padic local Langlands correspondence in this context. Motivated by this observation
Colmez [Col16] extended the p-adic local Langlands correspondence for GL2 (Qp )to the setup of (possibly non-étale) rank two (ϕ, Γ)-modules over the Robba ring andlocally analytic representations of GL2 (Qp ). This naturally fits into the “analytic”
context in these notes.
We point out that usually p-adic Hodge-theoretic properties of the representation ρ are not so easy to read off from the associated (ϕ, Γ)-module D(ρ), but
it is much easier to read them off from the rigid analytic variant Drig (ρ). Hence
Hodge-theoretic information, e.g. the Hodge filtration on DdR (ρ) (assuming that ρ
is de Rham) should be connected to properties of the locally analytic representation Π(ρ)an (and it should not be easy to understand this directly in terms of the
Banach space representation Π(ρ)). We now elaborate on this point.
One of the original motivations in the p-adic Langlands program was to improve the understanding of local-global compatibility in the cohomology of modular
curves (or more general Shimura varieties). To each system of Hecke eigenvalues
that occurs in the cohomology H 1 (Y (N ), Qp ) of the modular curve Y (N ) there
corresponds a continuous 2-dimensional representation ρ of GalQ on a Qp vector
space. The restriction ρ|GalQ` to decomposition groups at primes ` 6= p is completely determined (at least up to Frobenius semisimplicity, which is expected to
be automatic) by the eigenvalues of the Hecke operators at `. However the actionof the Hecke operators at p only determines the Weil–Deligne representation associated to the de Rham representation ρ|GalQp . The missing piece to determine
ρ|GalQp is the Hodge filtration which is mysteriously determined by the Hecke action away from p but not captured by the Hecke action at p. The idea is thatp-adic local global compatibility involving completed (co-)homology and the p-adic
local Langlands correspondence will make it possible to read of the Hodge filtrationfrom the GL2 (Qp )-representation in the corresponding system of Hecke eigenvale 1 (Y (N ), Qp ) without using the action of Hecke
ues on completed (co-)homology H

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operators away from p. In fact, the Hodge filtration should be determined by the loe 1 (Y (N ), Qp ). A generalization of this philosophy, whichcally analytic vectors in H
goes back to Breuil’s initial ideas about the p-adic local Langlands correspondence,is visible for example in Breuil’s socle conjecture (see Section 9.6.29), where therelative position of the Hodge filtration with respect to Frobenius stable flags isdetermined by the socle of a certain locally analytic representation. We will discuss
the p-adic local global compatibility involving completed (co-)homology of modular
curves in more detail in Section 3.2.11 below.
Passing from a (classical) (ϕ, Γ)-module D to its variant Drig over the Robbaring R has the advantage that the (ϕ, Γ)-module can become reducible, even though
D was not reducible. For example this happens if D comes from a crystalline orsemi-stable (or more generally crystabelline or semistabelline) representation. The(ϕ, Γ)-modules that are completely reducible are called trianguline and were studied
extensively by Colmez [Col08]. In terms of the p-adic local Langlands correspondence a representation ρ is trianguline (i.e. Drig (ρ) is trianguline) if and only if thelocally analytic vectors Π(ρ)an contain (irreducible constituents of) locally analytic
principal series representations. In this case usually Π(ρ)an is an extension of twoparabolically induced representations, and the extension class of these representations is determined by the extension class of (ϕ, Γ)-modules of rank 1 defined by
Drig (D) (and vice versa). This implies in particular that in the case of crystalline
or semi-stable representations ρ the Hodge filtration on the Weil–Deligne representation WD(ρ) is determined by the locally analytic vectors Π(ρ)an , as it should beaccording to the above discussion.
3.2.7. The mod p local Langlands correspondence for GL2 (Qp ). In our discussion so far, we have set the p-adic local Langlands correspondence in the context of
p-adic representations of GL2 (Qp ). However, from the beginning of the theory, modp considerations have played a fundamental role. In his paper [Bre03a] (and building on earlier work of Barthel and Livné [BL94]), Breuil classified the absolutely
irreducible admissible smooth representations of GL2 (Qp ) over a finite field k, andGL (Q )
showed grosso modo that they are of the form c-IndGL22 (Zpp)Z σ ⊗H(σ),x k, where
GL (Q )
Z denotes the centre of GL2 (Qp ), H(σ) = End( c-IndGL22 (Zpp)Z σ is the relevant∼ k[S ±1 , T ], and where
Hecke algebra, which admits an explicit isomorphism H(σ) =x : H(σ) → k is some given character of H(σ). The key contribution of [Bre03a]
is to show that such representations are irreducible in the so-called supersingularcase, i.e. the case when x(T ) = 0 (and it is this statement that is special to thecontext of GL2 (Qp )).
Building on this classification, Breuil was able to define a mod p semisimple
local Langlands correspondence, by attaching to each semisimple two-dimensional
representation ρ : GalQp → GL2 (k) a corresponding semisimple admissible smooth
representation κ(ρ) of GL2 (Qp ) over k. Breuil further conjectured that this correspondence was compatible with the conjectured p-adic local Langlands correspondence, in the sense that if ρ : GalQp → GL2 (L) is continuous, then one should havessΠ(ρ) ∼= κ(ρss ). (Here, on either side of the conjectured isomorphism, the notationss(–) indicates that we reduce an invariant lattice modulo the uniformizer $, andthen semisimplify. Note also that on either side, the reductions are only uniquelydetermined after semi-simplifying, and so it is natural to pass to semi-simplifications
when making such a comparison between the p-adic and mod p settings.)

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Subsequently, Berger [Ber10] was able to obtain a description of Breuil’s mod p
correspondence in terms of (ϕ, Γ)-modules, and using this, was able to verify Breuil’s
conjectured compatibility in the context of the paper [BB10]. The same (ϕ, Γ)module arguments apply to show this compatibility naturally extends to the generalp-adic local Langlands correspondence of [Col10c].
Colmez was subsequently able to define a fully fledged mod p local Langlands correspondence, i.e. to define the correspondence in such a fashion that,in the case when ρ is reducible but indecomposable, it takes into account thecorresponding extension class of characters. More precisely, for each representation ρ : GalQp → GL2 (k), one may define an associated representation κ(ρ) of
GL2 (Qp ), with the property that V κ(ρ) = ρ. If ρ is not a twist of an extensionof the mod p cyclotomic character by the trivial character, then κ(ρ) can in fact becharacterized by this last property, together with the condition that it admits nofinite-dimensional GL2 (Qp )-invariant subobject or quotient. (See e.g. [Eme11a,Thm. 3.3.2].)
In the exceptional case, when ρ is a twist of an extension of the mod p cyclotomic character by the trivial character, then the “correct” definition of κ(ρ) (bywhich we mean the one that is compatible with correspondence that we explain in
Section 7.2) is not compatible with semi-simplification. In fact, even in this case, it
is possible to alter the definition of κ(ρ) so that it is so compatible, and Colmez did
indeed adopt this alternative definition; but this alternative choice of κ(ρ) fails tosatisfy compatibility with deformations, in the form discussed in more detail below.
(See [CEG+ 18, Rem. 6.22] for a more precise description of the difference between
Colmez’s definition and ours.)
3.2.8. p-adic local Langlands over deformation space and proofs of the Breuil–Mézard conjecture. As already remarked, the construction of the p-adic local Langlands correspondence for GL2 (Qp ) is intimately related to describing the variationof the correspondence over deformation spaces of Galois representations, and we
now turn to discussing the various forms this description has taken in the literature. The first key input to all of these descriptions is that Colmez’s functor V iswell-defined on the mod p and p-adic integral level, so that it can be utilized to
study the p-adic local Langlands correspondence in a deformation-theoretic context.
In [Eme11a], a deformation-theoretic description of the correspondence isgiven for deformations of a representation ρ : GalQp → GL2 (k) that is not a twistof an extension of cyclotomic character by the trivial character; in what follows weassume for simplicity that ρ also admits only trivial endomorphisms, so that we canand do take Rp to be the universal deformation O-algebra of ρ with some fixed determinant (rather than having to work with framed deformations). For simplicity of
exposition, we furthermore assume that this fixed determinant is ε−1 ; correspondingly, we will assume that all of the representations of GL2 that we consider havetrivial central character, and accordingly we will regard them as representationsof PGL2 .
In this case, the arguments of Colmez [Col10c] and Kisin [Kis10a] can be
encapsulated as follows: there exists a representation π univ of PGL2 (Qp ) on anmRp -adically complete and orthonormalizable Rp -module, deforming the smooth
representation κ(ρ), with the property that V (π univ ) ∼= ρuniv , the universal deformation of ρ. The PGL2 (Qp )-representation attached by the p-adic local Langlands

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correspondence to any particular deformation of ρ is then obtained by specializing
π univ over this deformation.
In [Paš13], a deformation-theoretic description of the correspondence is giventhat is more flexible and more conceptual than the formulation just recalled. Inorder to make the description as simple as possible, we assume that ρ is not a twistof an extension of the trivial character by the cyclotomic character. (Note that thisis different to the exceptional case considered above; the forbidden extension has
the characters in the opposite order.) We then let Pe denote a projective envelope of
the Pontryagin dual κ(ρ)∨ in a suitable category of representations of PGL2 (Qp ) on
compact O-modules. One of the main results of [Paš13] is that there is a canonicalisomorphism
Rp ∼= End(Pe),(3.2.9)so that Pe may be naturally regarded as “living over” Spf Rp , and that the consequent Rp -structure on V (Ped ) realizes V (Ped ) as the universal deformation runiv of ρ.
If ρ : GalQp → GL2 (L) admits a lattice deforming ρ, corresponding to a morphism
x : Rp → O, then the unitary Banach space representation Π(ρ) is determined via
the isomorphism
Π(ρ) ∼= L ⊗O (Pe ⊗R ,x O)d ,pdwhere the superscript denotes Schikhof duality on pseudocompact O-torsion freeO-modules, i.e. Homcont
O (−, O).
If ρ is not a twist of an extension of the cyclotomic character by the trivial
character (so that we are in neither of the exceptional cases considered above),then it is furthermore proved in [Paš13] that Pe is topologically flat over Rp (inb Rp Pe on pseudo-compact Rp -modules is exact). In
the sense that the functor −⊗this case, the representation π univ associated to ρ as above is related to Pe via theisomorphism (of Rp [PGL2 (Qp )]-modules)e
π univ ∼= Homcont
Rp (P , Rp ).
In [Kis09a], Kisin gives quite a different description of the p-adic local Langlands correspondence, not over all of Spf Rp , but over the locus Spf Rp (σ) parameterizing lifts of a fixed locally algebraic type σ. For deformations ρ of ρ that are
parameterized by Spf Rp (σ), Colmez’s proof of Breuil’s conjecture describing Π(ρ)
as a completion of the locally algebraic representation Π(ρ)alg shows that the restriction to Spf Rp (σ)[1/p] of the p-adic local Langlands correspondence, which we
PGL (Q )denote by π(σ), may be described as a certain completion of c-IndPGL22 (Zpp) σ. This
description of π(σ) allows Kisin to deduce the Breuil–Mézard conjecture for ρ, as
we now recall.
A choice of Jordan–Hölder filtration in the reduction σ of some chosen PGL2 (Zp )invariant lattice σ ◦ in σ determines a corresponding filtration on π(σ) (the reduction
mod $ of some invariant lattice in π(σ)) by completions of representations of the
PGL (Q )form c-IndPGL22 (Zpp) σ i , where σ i runs through the Jordan–Hölder factors of σ. For
each σ i , we let σi be an algebraic L-representation of PGL2 (Qp ) with σi◦ ⊗O k ∼= σi ,and set Rp (σ i ) = Rp (σi ) ⊗O k (which is easily checked to be independent of the
choice of σi ). Now the explicit classification of the irreducible representations of
PGL2 (Qp ), and the fact that Colmez’s functor V is compatible with reduction,

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shows that the only σ i that can contribute a non-zero completion are those that appear as Serre weights of ρ, and that if a given Serre weight σ i contributes a complePGL (Q )tion, this completion must be the mx -adic completion of c-IndPGL22 (Zpp) σ i , where mx

PGL (Q )is the kernel of the character x : H(σ i ) → k for which c-IndPGL22 (Zpp) σ i ⊗H(σ),x k isa constituent of κ(ρ). Furthermore, this completion, when thought of as an Rp (σ)module, in fact is supported on Rp (σ i ), the mx -adic completion of Rp (σ i ) being
explicitly identified with the mx -adic completion of H(σ i ). In this way, one obtains
an inequality of Breuil–Mézard type, but in the opposite direction to the one givenby the argument described in (3.2.3) above. Combining these two inequalities gives
an equality, which is the Breuil–Mézard conjecture. As noted in (3.2.3), this conjecture then implies that each patched module M∞ (σ ◦ ) is faithful over the ringRp (σ), giving rise to a modularity lifting theorem for global deformations of typeσ. This is the approach to the Fontaine–Mazur conjecture taken in [Kis09a].
In the argument that is sketched in the preceding paragraph, the reason that
only an inequality (rather than the precise equality of the Breuil–Mézard conjecture) is obtained is more-or-less because completions (being a form of projective
limit) are not a priori right exact, so that is it not clear a priori that all the posPGL (Q )sible completions of the representations c-IndPGL22 (Zpp) σ i that might occur in π(σ)
actually do so occur.
Issues of exactness of this kind are ultimately of a homological nature, and so
it is perhaps not surprising that having stronger homological information available
allows the argument to be strengthened. This was achieved by Paškūnas in thepaper [Paš15]. Using the construction of Pe as described above, together with
Colmez’s result on locally algebraic vectors, one finds that Rp (σ) is precisely the
support of the Rp -module HomPGL2 (Zp ) (σ ◦ , Pe∨ ) (for any choice of invariant latticeσ ◦ in σ). In fact, it is more convenient to work with the Rp -module(3.2.10)M (σ ◦ ) := Pe ⊗O[[PGL2 (Zp )]] σ ◦ ,since this turns out to be a coherent Rp -module. Since Pe is projective as a
PGL2 (Qp )-representation, it is in particular projective as a PGL2 (Zp )-representation,and so the functor M (–) implicitly defined by the preceding formula is exact. Arguing with this functor one can essentially combine the two prongs of Kisin’s argument
into a single argument that yields the Breuil–Mézard equality directly.
3.2.11. Local-global compatibility. In [Eme11a] a local-global compatibility relating the p-adic local Langlands correspondence to completed cohomology is proved,
giving a p-adic analogue of the classical local-global compatibility [Lan73, Del73,
Car83] relating cohomology at infinite level and the classical local Langlands correspondence. In order to state the result, we fix as above a modular irreducible
residual representation r : GalQ → GL2 (k), setting ρ := r|GalQp , and work at atame level N divisible by the prime-to-p Artin conductor of r.
e 1 := H
e 1 Y1 (N ), O denote
Similar to the completed homology (3.1.5) we let Hét
the p-adically completed étale cohomology of the “modular curves” (again: more
precisely, congruence quotients of PGL2 -symmetric spaces, which are unions of theconnected components of the usual modular curves for GL2 ) of tame level N ; bydefinition we have1e 1 := lim lim Hét
H
(Y1 (N pn ), O/$s ),
←− −→
s

n

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M. EMERTON, T. GEE, AND E. HELLMANN

e 1 is naturally equipped with commuting actions of the groups GalQ andand H
PGL2 (Qp ), and of a Hecke algebra T (generated by the usual Hecke operators Sl , Tl
at primes l - N p). The representation r gives rise to a maximal ideal m(r) of T, ande1e1
we may form the corresponding localisation H
m(r) of H . There is also a universal
pro-modular deformation ρuniv of r over Tm(r) , and if we set ρ := r| GalQp , thenrestriction from GalQ to GalQp induces a morphism Rp → Tm(r) .
Assume now that ρ is not a twist of an extension of the cyclotomic characterby the trivial character; the main result of [Eme11a] then states that there is an
isomorphism (of Tm(r) [GalQ × PGL2 (Qp )]-modules)
univ
e1 ∼
b Tm(r) O,
H⊗Rp π univ ) ⊗m(r) = (ρwhere π univ is the orthonormalizable PGL2 (Qp )-representation over Rp discussed
in Section 3.2.8 above, and where O is a p-torsion free cofinitely generated Tm(r) module5 of “oldforms” related to the chosen tame level.6 If ρ is furthermore nota twist of an extension of the trivial character by the cyclotomic character, thenpassing to continuous Rp -duals, and taking into account the relationship between
π univ and Pe in this case, we may rephrase this as an isomorphisme 1 )m(r) ∼b R (ρuniv )∗ ⊗T(3.2.12)(HO∗ ,= Pe ⊗pm(r)e 1 = HomO (He 1 , O) denotes the completed homology of the modular curves
where Has in (3.1.5) at tame level N , and O∗ (the O-dual of O) is now a finitely generated
Tm(r) -module. The proof of the isomorphism is a combination of a Morita-theoretic
argument, similar to the more sophisticated such arguments that are developedin [Paš13], and an interpolation argument, related to the “capture” argumentsthat appear in [CDP14]. The key input is a combination of classical local-global
compatibility and the result of Berger–Breuil (that Π(ρ) is the unique unitarycompletion of Π(ρ)alg when ρ is crystabelline): classical local-global compatibility
e 1 , and then the result of Berger–Breuil shows
gives rise to copies of Π(ρ)alg inside He 1 are necessarily isomorphic to Π(ρ).
that the closures of these in H
Local-global compatibility gives another approach to the Fontaine–Mazur conjecture. Namely, if x : Tm(r) → L is a homomorphism corresponding to a promodular deformation r of r, with kernel px , and if ρ := r| GalQp , then passing topx -torsion in the local-global compatibility isomorphism (3.2.12) yields an isomorphism
e 1 [px ] ⊗O E ∼(3.2.13)H= r ⊗ Π(ρ).
If ρ is potentially semi-stable with distinct Hodge–Tate weights, so that by Colmez’s
results Π(ρ)alg 6= 0, then one concludes that the system of Hecke eigenvalues givene 1 . The comparison between locally algebraic vectors in completed
by x appears in H
algcohomology and the classical cohomology of local systems on modular curves thenshows that x, and thus r, arises from a classical modular form.
5I.e. the O-dual of a finitely generated T

m(r) -module.
If we work at full level N , and then pass to a colimit over all levels N , the resulting limiting
space of oldforms acquires an action of GL2 (Apf ), and can be described in terms of the local
Langlands correspondence in families of [EH14] and [HM18]. This is related to the ` 6= p caseof categorical local Langlands that we discuss briefly in Section 8, and which plays a key role in
the global conjecture (Conjecture 9.3.2 below). For simplicity of exposition, though, we don’t say
anything more about it here.
6

CATEGORICAL p-ADIC LANGLANDS

29

The Fontaine–Mazur conjecture for deformations of r therefore reduces to the
problem of showing that all such deformations are pro-modular. Under standard
hypotheses on r needed for the Taylor–Wiles method, this can be accomplished byproving a “big R = T” theorem, identifying Tm(r) with the universal deformation
O-algebra for deformations of r which are minimally ramified away from p. Suchtheorems were first proved in [Böc01], using the infinite fern of Gouvêa and Mazur
together with modularity lifting theorems in small weight (where the deformation
rings Rp (σ) can be described explicitly via Fontaine–Laffaille theory). Moreover,
we note that (3.2.13) and the construction of the p-adic Langlands correspondencee 1 [px ] ⊗O L is non-zero for every x : Tm(r) → L. This is in fact a
imply that H
stronger statement than the pro-modularity result of the “big R = T theorem”.
The results of [Eme11a] therefore give an alternative proof of the resultsof [Kis09a] (with slightly different hypotheses); while both proofs rely on Taylor–Wiles–Kisin patching and the p-adic Langlands correspondence, they differ in thatthe approach of [Kis09a] is to use the p-adic Langlands correspondence to proveenough about the ring Rp (σ) to deduce that the patched R∞ (σ)-module M∞ (σ ◦ )is faithful, while the approach of [Eme11a] can be viewed as computing the functor M∞ in terms of the p-adic Langlands correspondence.
The rephrasing (3.2.12) of the main result of [Eme11a] in terms of completed
homology is highly suggestive of two things. Firstly, from the optic of these notes,
where we replace Galois deformation rings with moduli stacks of L-parameters,
it suggests that we can compute the completed cohomology of modular curves
by pulling back some canonical sheaf of PGL2 (Qp )-representations from a stackof local parameters to a stack of global parameters; we pursue this suggestion in
Section 9. Secondly, it suggests the possibility of using the Taylor–Wiles patchinge 1 ; this possibility was realized
method to extract Pe from the completed homology H
in [CEG+ 18]. In particular, one finds that the patching functor M∞ “is” thefunctor M implicitly described by (3.2.10). (Note, though, that it seems hopelessto construct the sheaf L∞ of Section 7.2 in this way, or from any other global
construction.)
3.2.14. Comparison to the proof of classical local Langlands. Our constructionof the p-adic local Langlands correspondence via global means may be comparedwith the construction of the classical local Langlands correspondence, which also
proceeds by realising a local representation globally [HT01, Hen00, Sch13]. However, as discussed in [CEG+ 16, §1.3], there are some significant differences between
our construction and the classical one. Classically, one first uses the classification of
Weil–Deligne representations, and the parallel classification of irreducible smooth
representations of GLn (Qp ) [BZ77, Zel80] to reduce to constructing a bijection
between irreducible n-dimensional Weil–Deligne representations and cuspidal representations of GLn (Qp ). A trace formula argument allows one to realize any cuspidal
representation (up to twist) as the local component of a cuspidal automorphic representation, and the associated Weil–Deligne representation is then constructed in
the cohomology of a Shimura variety (in particular, this construction presumes that
local-global compatibility will hold, just as that of [CEG+ 18] does). In order toverify that the correspondence so constructed is in fact purely local, one ultimately
relies on the prescription for the local Langlands correspondence in terms of L- andε- factors.

30

M. EMERTON, T. GEE, AND E. HELLMANNIn the p-adic context, most of these steps aren’t available (at least at present).
For example, the only known classification of admissible unitary Banach representations of GL2 (Qp ) is that of [Paš13], which is in terms of the p-adic local Langlands
correspondence, rather than being independent of it. Furthermore, we can’t expect
to realize a local p-adic Galois representation globally, even if we require it to bepotentially semistable of some prescribed weights and type: the family of such representations is one-dimensional (thus uncountable), while there are only countably
many modular representations of GalQ . This is why Taylor–Wiles patching is so
crucial to the approach of [CEG+ 16, CEG+ 18]: it provides a mechanism for relating the local and the global contexts in the p-adic setting, in the absence of theBernstein–Zelevinsky classification/trace formula argument that relates the local
and global contexts in the classical setting.
Perhaps the biggest difference between the p-adic and classical contexts is that,in the p-adic context, we don’t have any a priori prescription for the correspondence in terms analogous to the classical prescription in terms of L- and ε- factors;
the closest analogue is the requirement that there should be a compatibility between the p-adic and classical correspondence mediated by the passage to locallyalgebraic vectors (which should parallel the passage from a potentially semistable
Dieudonné module to the underlying Weil–Deligne representation on the Galoisside). However, it is only in the principal series case, for the group GL2 (Qp ),that this requirement has the possibility of providing enough information to determine the p-adic correspondence. This is why the consideration of the (unramified)principal series case is so crucial in all known proofs of a p-adic local Langlands
correspondence for GL2 (Qp ).
3.2.15. Patching and the analytic setup. In a series of papers [BHS17b], [BHS17a]and [BHS19] it was shown that the patching construction can be used not only
to prove results about automorphic forms and Galois representations (for example,
to prove the existence of an automorphic form whose associated Galois representations has prescribed local properties), but also to prove results about overconvergent
p-adic automorphic forms and their associated Galois representations.
A key observation in the theory of overconvergent p-adic automorphic forms isthat the Hecke eigenvalues of overconvergent p-adic automorphic eigenforms nicely
vary in families and form a so-called eigenvariety Y . The p-adic automorphic forms(of which the eigenvariety parameterizes the Hecke eigenvalues) themselves form a
coherent sheaf M on this eigenvariety. In particular we note that in the theory of
eigenvarieties, spaces of (overconvergent finite slope) p-adic automorphic forms canbe described as the (dual of) global sections of a coherent sheaf on a rigid analytic
space, that is closely related to the generic fiber of a deformation space of Galois
representations.
Characterizing the support Y of this coherent sheaf M purely in terms of Galois representations can be regarded as an overconvergent version of the Fontaine–
Mazur conjecture: it amounts to giving a necessary and sufficient condition on a
Galois representation ρ that ensures that ρ is associated to a overconvergent padic automorphic form of finite slope. Kisin [Kis03] constructed a rigid analytic
subvariety (the finite slope space of loc. cit.) of (the product of Gm with) a deformation space of 2-dimensional GalQ -representations that is defined in purely

CATEGORICAL p-ADIC LANGLANDS

31

Galois-theoretic terms (asking that the Galois representations are trianguline7 at p
and unramified almost everywhere) and that conjecturally equals the eigencurve ofColeman–Mazur. This construction in fact inspired the definition of the trianguline variety in [BHS17b] and the following overconvergent version of the Fontaine–
Mazur conjecture (that we state slightly informally, and which was first proposedby Kisin in the case d = 2; see [Kis03, 11.7(2), 11.8]): a p-adic d-dimensional odd
global Galois representation is associated to an overconvergent p-adic automorphic
form of finite slope on GLd if any only it is unramified almost everywhere and trianguline at places dividing p. (In the case d = 2, this conjecture was proved under
auxiliary technical hypotheses in [Eme11a, Thm. 1.2.4]; a slightly weaker version,under slightly different hypotheses, is established in [Kis09a, Cor., p. 642].)
However, these constructions and conjectures only give a characterization of the
support Y of the coherent sheaf M. We refer to Section 9.6 for a precise conjecture
characterizing the coherent sheaf M purely in terms of local Galois representations
and a global-to-local map of stacks of Galois representations. The idea that acharacterization like this should exist originates in the insight that the patching
construction can also be carried out for eigenvarieties.
More precisely, in a similar way as M∞ over R∞ (and M∞ (σ) over R∞ (σ))
is related (via pullback along a local-global map) to a space of automorphic forms
over a global Galois deformation ring, there is (in the setup of a definite unitarygroup G over a CM field F satisfying some assumptions necessary for the patching
construction) a patched version [BHS17b] of the eigenvariety Xp (ρ̄) and a coherent
sheaf M∞ on Xp (ρ̄) such that M and Y may be reconstructed from M∞ andXp (ρ̄) via pullback along a local-global map. The main idea of the construction
is to mimic the construction of eigenvarieties of M.E. [Eme06b] (using a locallyanalytic Jacquet functor), but using a big patched module M∞ (respectively its
dual Π∞ , which is a big Banach space representation of G(Fp )), instead of thespace of p-adic automorphic forms.
The key idea of [BHS17a] and [BHS19] is to use Galois-theoretic tools (i.e. thegeometry of the space XG,tri from Section 5.3) to analyze the local geometry of thespace Xp (ρ̄), which in turn gives information about the behavior of the sheaf M∞ .
4. Moduli stacks of (ϕ, Γ)-modules: the Banach caseWe now review the (sometimes conjectural) constructions of stacks of (ϕ, Γ)modules that we will work with. We work in two different settings: in this section
we cover the “Banach” case treated in [EG22], which (conjecturally) relates tosmooth p-adic representations of G(Qp ). In Section 5 we discuss the “analytic”
case, relating to locally analytic representations of G(Qp ).
In this section we very briefly recall some of the main results of [EG22]. The
reader may find it useful to refer to [EG20] for a more extended introduction toand overview of [EG22], and to Sections 7.2 and 7.6 below for a more explicit
description of the stacks in some simple cases.
The stacks that we consider below are formal algebraic stacks in the senseof [Eme]; colloquially, they are “the formal analogue of Artin stacks”, in the sensethat they admit smooth covers by formal schemes (or more precisely by formal algebraic spaces). We note that they are in general not p-adically formal; equivalently,
7The name trianguline was only invented later by Colmez using a different but equivalentdefinition as the one in Kisin’s paper.

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M. EMERTON, T. GEE, AND E. HELLMANN

their special (i.e. mod p) fibres are genuinely formal, rather than algebraic. We
refer the reader to [EG22, App. A] for a basic overview of formal algebraic stacks
and their properties.
4.1. An overview of [EG22]. We have our fixed finite extension F/Qp , andwe work with coefficient rings which are O-algebras, where O = OL is sufficiently
large. We let Xd denote the moduli stack of projective étale (ϕ, Γ)-modules over Fof rank d, in the sense of [EG22, Defn. 3.2.1].
Remark 4.1.1. There are several different possible descriptions of the objects
that are parameterized by Xd . For example, one can consider them as (ϕ, Γ)modules with respect to either the full cyclotomic extension of GalF , or its Zp
subextension; but it is also sometimes helpful to think of them as étale (GalF , ϕ)modules over W (C[ ), in the sense of [EG22, Defn. 2.7.3]. More generally it would
probably be helpful to give a description in terms of prismatic F -crystals following [BS21, Wu21c, DL21b].
By its very definition (and by Fontaine’s theory of (ϕ, Γ)-modules [Fon90]),
the groupoid of Fp -points of Xd , which coincides with the groupoid of Fp -points of
the underlying reduced substack Xd,red , is naturally equivalent to the groupoid ofcontinuous representations ρ : GalF → GLd (Fp ). More generally, if A is any finite
O-algebra, then the groupoid Xd (A) is canonically equivalent to the groupoid ofcontinuous representations GalF → GLd (A). It follows easily that universal Galois
lifting rings are versal rings to Xd at finite type points; so we can think of Xd as analgebraization of Mazur’s Galois deformation rings.
Remark 4.1.2. If A is not a finite O-algebra (e.g. if A = k[X]), then there
is typically no Galois representation corresponding to an object of Xd (A). Thereis a maximal substack XdGal of Xd over which the universal (ϕ, Γ)-modules comesfrom a GalF -representation; this stack was originally constructed and studied by
Wang-Erickson [WE18].
The inclusion XdGal ⊂ Xd induces a bijection on Fp -points (and is versal atsuch points), and one can roughly imagine that XdGal is the union of the formal
completions of Xd at its closed Fp -points (which correspond to semisimple GalF representations). Thus although the stack XdGal is the more obvious definitionof a moduli stack of “Langlands parameters”, the stack Xd has richer geometricstructure, and although for example the restriction maps from the stacks of global
Galois representations to the stacks Xd factor through the stacks XdGal , we anticipate
that it is the stack Xd (or rather, its derived categories of quasicoherent sheaves)that is the natural setting for the p-adic local Langlands correspondence.
In the p-adic Langlands program it is often important to impose conditions
on the GalF -representations under consideration, such as demanding that they
be crystalline of fixed Hodge–Tate weights, or more generally of being potentiallycrystalline or semistable of a fixed inertial type. For any8 Hodge type λ and inertial
crys,λ,τ
ss,λ,τ
type τ , there is a corresponding substack Xd(and similarly Xd), which arecharacterized as being the unique closed substack of (Xd )O which is flat over O andwhose groupoid of A-valued points, for any finite flat O-algebra A, is equivalent to
8For the following construction, we need to assume that L is “large enough”, e.g. so that τ
is defined over L. We largely suppress this point in what follows.

CATEGORICAL p-ADIC LANGLANDS

33

the groupoid of A-valued GalF -representations satisfying the corresponding p-adic
Hodge theoretic property (of being potentially crystalline or semistable of Hodge
type λ and inertial type τ ). In the below if τ is trivial then we drop it from thenotation.
These substacks are constructed in [EG22] using (among other techniques)
Kisin’s results on Breuil–Kisin modules and crystalline representations [Kis06];
again, they are probably more naturally defined in terms of (log-) prismatic F crystals, following [BS21].
The following theorem summarises the main results of [EG22].
Theorem 4.1.3.
(1) Xd is a Noetherian formal algebraic stack.
ss,λ,τ
crys,λ,τ
and Xdis a p-adic formal algebraic stack.
(2) Each substack Xd(3) The underlying reduced substack Xd,red of Xd (which is an algebraic stack)is of finite type over Fp , and is equidimensional of dimension [F : Qp ]d(d−1)/2.
crys,λ,τ
ss,λ,τ
(4) The special fibre of each of Xd
and Xd(which is an algebraic
stack) is equidimensional of dimension equal to that of the flag variety
determined by λ. In particular, if λ is regular, then each of these stacks
is equidimensional of dimension [F : Qp ]d(d − 1)/2.
(5) The irreducible components of Xd,red admit a natural labelling by Serreweights.
Remark 4.1.4. The notion of a Serre weight is recalled in Section 6.1.37.
The precise meaning of the labelling of the irreducible components of Xd,red bySerre weights is somewhat involved; see [EG22, §5.5]. Roughly, the description
is as follows. Each irreducible component of Xd,red admits a dense open substack,whose Fp -points correspond to GalF -representations which are maximally nonsplitof niveau one; that is, they admit a unique filtration by characters. The orderedtuple of inertial weights of these characters determines the irreducible component,
and these ordered tuples correspond to Serre weights, because Serre weights are bydefinition (isomorphism classes of) irreducible Fp -representations of GLd (k), andas such are indexed by their highest weights. (There is one subtlety in making
this identification, which is that it is necessary to distinguish Serre weights whose
highest weights are congruent modulo p − 1; this manifests itself on the Galois side
as a distinction between peu and très ramifiée representations.)
Remark 4.1.5. The two appearances of [F : Qp ]d(d−1)/2 in Theorem 4.1.3 —as the dimension of the underlying reduced substack Xd,red , and of the dimensionof the special fibres of potentially crystalline/semistable substacks of regular Hodge
type — is not a coincidence. As well as being an important ingredient in the proofof Theorem 4.1.3, it underlies the Breuil–Mézard conjecture (see Section 6.1.37).
Remark 4.1.6. As far as we are aware, there is no general dimension theory
for formal algebraic stacks in the literature. It is, however, reasonable to think thatsuch a theory exists, and that this dimension can be computed from the dimensionsof the versal rings. This being the case, the results of [BIP21] (which in particular
compute the dimensions of unrestricted local Galois lifting rings) imply that Xd
will be equidimensional of dimension 1 + [F : Qp ]d2 . Since its underlying reduced
substack Xd,red is equidimensional of dimension [F : Qp ]d(d − 1)/2, we see that Xd

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M. EMERTON, T. GEE, AND E. HELLMANN

should have 1 + [F : Qp ]d(d + 1)/2 directions which are purely formal (including
the p-adic direction). In particular, it is (provably) never a p-adic formal algebraic
stack.
Remark 4.1.7. Again, as far as we are aware, there is no notion in the literature of what it means for a formal algebraic stack to be lci. We anticipate thatreasonable definitions exist, and that Xd is lci; again, this should follow from theresults of [BIP21] (which show that the unrestricted local Galois lifting rings are
lci).
4.2. Fixed determinant variant. It is often convenient to consider a variantof these stacks where we fix the determinant. If χ : GalF → O× is a character, thenwe let Xdχ (A) be the groupoid of pairs (D, θ) where D is a rank d projective étale(ϕ, Γ)-module with A-coefficients, and θ is an identification of ∧d D with χ. These
stacks (and their potentially crystalline and semistable variants) are studied in an
appendix to [DEGa], where the analogue of Theorem 4.1.3 is established (the only
difference being that in the case of “Steinberg” Serre weights, there can be multiple
corresponding irreducible components, indexed by the dth roots of unity in Fp ).
In particular, the underlying reduced substack of Xdχ is again equidimensional ofdimension [F : Qp ]d(d − 1)/2.
5. Moduli stacks of (ϕ, Γ)-modules: the analytic case
Similarly to Section 4, we want to define and study stacks of (ϕ, Γ)-modules on
the category of rigid analytic spaces. It is sometimes more convenient to think ofthese objects in terms of equivariant vector bundles on the Fargues–Fontaine curve.
Most of the results in this section will appear in the forthcoming paper [HHS]. We
start by recalling the setup and the comparison with (ϕ, Γ)-modules.
5.1. (ϕ, Γ)-modules over the Robba ring. Associated with the field F (orrather with the cyclotomic extension F∞ and the algebraic closure F̄ ) we can define
two versions of the Fargues–Fontaine curve. Let $[ ∈ OF̂ [ denote the choice of a∞pseudo-uniformizer. We set
YF∞ = Spa(W (OF̂ [ ), W (OF̂ [ ))\V (p[$[ ]),∞∞YF̄ = Spa(W (OF̄ˆ [ ), W (OF̄ˆ [ ))\V (p[$[ ]).
These adic spaces over Qp come equipped with an automorphism ϕ and a continuous
action of Γ, respectively GalF , that commutes with ϕ. The Fargues–Fontaine curves
XF∞ and XF̄ are the quotients
XF∞ = YF∞ /ϕZ ,XF̄ = YF̄ /ϕZ ,which inherit a continuous action of Γ, respectively GalF .
Note that the curves XF∞ and XF̄ have a distinguished point ∞ defined by
the kernel of the usual θ maps of Fontaine
θF∞ : W (OF̂ [ ) −→ OF̂∞ ,∞θF̄ : W (OF̄ˆ [ ) −→ OF̄ˆ .
In fact ∞ is the unique fixed point for the Γ, respectively GalF -action.
We define the following categories of (semi-)linear algebra objects:

CATEGORICAL p-ADIC LANGLANDS

35- the category VBΓXF∞ of Γ-equivariant vector bundles on XF∞ ,F- the category VBGal
of GalF -equivariant vector bundles on XF̄ ,XF̄

- the category VBϕ,Γ
YF∞ of (ϕ, Γ)-equivariant vector bundles on YF∞ ,F- the category VBϕ,Galof (ϕ, GalF )-equivariant vector bundles on YF̄ .
YF̄
In fact in all the above categories we ask that the action of Γ resp. GalF on the vector
bundles is continuous in the evident sense (that we will not spell out explicitly).
By the work of Berger, Fargues–Fontaine and Kedlaya, all these various categories
are known to be canonically equivalent.

Theorem 5.1.1.
(i) The projection XF̄ → XF∞ is equivariant with respect to the canonical projection
GalF → Γ and induces via pullback an equivalence of categoriesF
VBΓXF∞ → VBGal
XF̄ .

The corresponding statement for YF̄ → YF∞ also holds true.
(ii) The projection YF∞ → XF∞ is Γ-equivariant and induces, via pullback, anequivalence of categories
VBΓXF∞ → VBϕ,Γ
YF ∞ .
The corresponding statement for YF̄ → XF̄ also holds true.
Remark 5.1.2. In Scholze’s world of diamonds one has XF∞ /Γ = XF̄ / GalF ,giving a more geometric interpretation of the first equivalence.
Furthermore there are obvious compatibilities between these equivalences. Thereis a further description of these categories in terms of the Beauville–Laszlo gluing
lemma [BL95]. This gives rise to Berger’s category of B-pairs [Ber08a]. We set+
BdR= ÔXF̄ ,∞ ,Be = Γ(XF̄ \{∞}, OXF̄ ),+BdR = BdR
[1/t].

Each of these rings is equipped with a continuous GalF -action. Using the Beauville–Fwith the category of triples (M, Λ, ξ)Laszlo gluing lemma, we can identify VBGal
XF̄
+consisting of a finite projective Be -module M and a finite projective BdR-moduleΛ equipped with continuous semi-linear GalF -actions and a GalF -equivariant isomorphism∼ξ : M ⊗Be BdR −→ Λ ⊗B + BdR .
dR

F ∼There is yet another description of VBGal= VBΓXF∞ using (ϕ, Γ)-modules
XF̄over imperfect period rings. We write RF for the Robba ring[r,s]RF = lim
lim BF−→ ←−,r→1 s→1

[r,s]

[r,s]

where BF is the ring of rigid analytic functions on the closed annulus BF∞,0 of
inner radius r and outer radius s in the closed unit disc over F∞,0 , the maximal
unramified subextension of F∞ . For s > r  0 there is a canonical way to define a[r,s]continuous action of Γ on BF and to define a ring homomorphism[r,s]

ϕ : BF[r 1/p ,s1/p ]→ BFcommuting with the Γ-action. These data induce a ring homomorphism ϕ : RF →RF and a continuous Γ-action on RF commuting with ϕ (we refer e.g. to [KPX14,
2.2] for the details of the definition of the ϕ- and Γ-actions, and for the variants of
the ring RF with coefficients in an affinoid algebra below).

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M. EMERTON, T. GEE, AND E. HELLMANN

Definition 5.1.3. The category Modϕ,Γ
RF is the category of finite projective
RF -modules together with continuous Γ-action and an isomorphism ϕ∗ D → D
commuting with the Γ-action.
Remark 5.1.4. We recall the relation of (ϕ, Γ)-modules over RF with the(ϕ, Γ)-modules used in Section 4. The (ϕ, Γ)-modules of Section 4 are definedover a p-adically complete ring AF . This ring contains a subring A†F ⊂ AF of
overconvergent elements that is stable under the action of ϕ and Γ. MoreoverA†F embeds (equivariantly for ϕ and Γ) into the Qp -algebra RF . By a theoremof Cherbonnier–Colmez [CC98] every étale (ϕ, Γ)-module over BF = AF [1/p] isoverconvergent, where by definition the étale (ϕ, Γ)-modules over BF are thosethat come by base change from a (ϕ, Γ)-module over AF . More precisely, thescalar extension from B†F = A†F [1/p] to BF induces an equivalence between thecorresponding categories of étale (ϕ, Γ)-modules. Moreover, the extension of scalars
from B†F to RF induces a fully faithful embedding of the category of étale (ϕ, Γ)modules over B†F to the category of (ϕ, Γ)-modules over RF . By definition the(ϕ, Γ)-modules over RF in the essential image of this functor are called étale.
By Kedlaya’s slope filtration theorem [Ked08] the étale (ϕ, Γ)-modules are
precisely the objects that are semi-stable of slope zero, in the following sense: If D
is a (ϕ, Γ)-module D of rank r, then by theVclassification of the (ϕ, Γ)-modules ofrrank 1 (see Section 7.1 below), we may write
D = RF (δ), where δ is a continuous
character of F × . We then define the slope of D to bevalp δ($F ),slope(D) =rwhere $F is a choice of a uniformizer in F . We furthermore say that D is semistable if slope(D0 ) ≥ slope(D) for all (ϕ, Γ)-stable subobjects D0 ⊆ D.
Theorem 5.1.5. There is an equivalence of categories
ϕ,Γ
Modϕ,Γ
RF −→ VBYF∞ .

Proof. (Sketch.) The rough idea behind this theorem is the following: any[r,s](ϕ, Γ)-module over RF admits a model over some closed annulus BF∞,0 . Using the
choice of the coordinate function [$[ ] we can map a corresponding closed annulus[r,s]in YF∞ to BF∞,0 , equivariantly for the action of Γ (and compatibly with ϕ in an[r,s]evident way). The pullbacks of the ϕ-bundles with Γ-action on the various BF∞,0
to the closed annuli in YF∞ spread out to a Γ-equivariant ϕ-bundle on YF∞ (as ϕis an automorphism of YF∞ that shifts the radii of closed annuli). The hard partof the theorem is the essential surjectivity, sometimes referred to as deperfection.
It is proved using the additional structure on the cyclotomic tower given by Tate’s
normalized traces. A proof of the theorem can be given by combining [FF18,
10.1.2] and [Ber08a, Theorem 2.2.7]. We refer to [BC08, 4.2] for a variant of thedescent argument using Tate’s normalized traces.
We will be interested in families of these (equivalent) objects with coefficientsin rigid analytic spaces. More precisely we want to define a stack Xd whose Sp(Qp )F
valued points are given by the groupoid of rank d objects in VBGal
or any of theXF̄
equivalent categories described above. We briefly describe the variants of the above
categories over an affinoid rigid space Sp(A), which suffices to define our stacks.

CATEGORICAL p-ADIC LANGLANDS

37

Let A be an affinoid algebra topologically of finite type over Qp . By a result of
Kedlaya [Ked16] the Fargues–Fontaine curves XF∞ , XF̄ are strongly Noetherian,and the same applies (locally) to their coverings YF∞ , YF̄ . Hence the fiber products
XF∞ ,A := XF∞ ×Sp(Qp ) Sp(A)etc. are well defined in the category of adic spaces, and come equipped with acanonical continuous action of Γ, respectively GalF .
Similarly to the absolute case considered above we can define- the category VBΓXF∞ ,A of Γ-equivariant vector bundles on XF∞ ,A ,F
- the category VBGal
XF̄ ,A of GalF -equivariant vector bundles on XF̄ ,A ,- the category VBϕ,Γ
YF∞ ,A of (ϕ, Γ)-equivariant vector bundles on YF∞ ,A ,F
- the category VBϕ,Galof (ϕ, GalF )-equivariant vector bundles on YF̄ ,A .
YF̄ ,A
Again we note that the Γ- respectively GalF -action on the vector bundles is supposed to be continuous in the evident way. Moreover, there is a variant withcoefficients of the category of (ϕ, Γ)-modules over the imperfect Robba ring RF .
We define[r,s]b Qp A
RF,A = lim
lim BF ⊗−→ ←−r→1 s→1

which is again equipped with a continuous Γ-action and an endomorphism ϕ commuting with the Γ-action. Then the category Modϕ,Γ
RF,A is defined to be the categoryof finite projective RF,A -modules D together with an isomorphism ϕD : ϕ∗ D → D
and a continuous Γ-action commuting with ϕD .
The conclusions of Theorems 5.1.1 and 5.1.5 still hold true for the categories
with coefficients in A, i.e. there are equivalences of categories
VBΓXF∞ ,A∼/ VBϕ,Γ

YF∞ ,A

∼



F
VBGal
XF̄ ,A∼o∼/ Modϕ,Γ

RF,A∼/ VBϕ,GalF
YF̄ ,A

induced by similar functors as in the case without coefficients. Moreover, given amorphism A → B of affinoid algebras, there are obvious base change functors
b A B : VBΓX(−)⊗−→ VBΓXF∞ ,B
F∞ ,Aetc. and the above equivalences of categories are obviously compatible with the basechange from A to B.
Remark 5.1.6. It is also worth noticing that the description of these equivalent
categories using the Beauville–Laszlo gluing lemma generalizes to families. This
allows us to modify a given family of equivariant vector bundles over XF̄ ,A by agiven family of invariant lattices over the completion of XF̄ ,A along {∞} × Sp A.
5.1.7. Rigid analytic Artin stacks. We introduce basic notions in the framework of Artin stacks on a category of rigid analytic spaces to set the ground forthe definition of the stacks of rigid analytic (ϕ, Γ)-modules. Denote by RigL thecategory of rigid analytic spaces over a fixed base field L that is a finite extensionof Qp . We will equip RigL with the Tate-fpqc topology [CT09, 2.1]. The coverings in this topology are generated by the usual (admissible) Tate coverings and the38

M. EMERTON, T. GEE, AND E. HELLMANN

morphisms Sp(A) → Sp(B) of rigid spaces for faithfully flat maps B → A of affinoid
algebras. With respect to this topology all representable functors are sheaves, andcoherent sheaves satisfy descent [Con06, Theorem 4.2.8]. In the following a stackon RigL will be a category fibered in groupoids over RigL that satisfies descent for
the Tate-fpqc topology. As in the case of Artin stacks on schemes, we have to startwith the definition of an analogue of algebraic spaces.
Definition 5.1.8. A quasi-analytic space is a sheaf F on RigL such that the
diagonal F → F × F is representable and such that there exists an étale surjection
U → F from a representable sheaf U onto F.
Remark 5.1.9.
(i) In fact below we do not need this level of generality: by a result of Conrad
and Temkin [CT09, Theorem 1.2.2] every separated quasi-analytic space is representable by a rigid analytic space. Below we will only meet separated quasi-analytic
spaces.
(ii) The PhD thesis of Evan Warner [War17] also develops a theory of Artin stacks
on adic spaces. The main difference to our set up is that Warner works with allstrongly Noetherian adic spaces (and uses the étale topology), whereas we allow
only rigid analytic spaces (and use the Tate-fpqc topology); moreover Warner uses
a different terminology. We mainly restrict ourselves to rigid analytic spaces as, atleast for the time being, the theory of (ϕ, Γ)-modules with coefficients [KPX14] andin particular the finiteness results about their cohomology, is limited to coefficientsin affinoid algebras of classical rigid analytic geometry.
Definition 5.1.10. A rigid analytic Artin stack is a stack X on RigL such thatthe diagonal ∆X : X → X ×L X is representable by quasi-analytic spaces and suchthat there exists a rigid analytic space U and a smooth surjection U → X.
Given a rigid analytic Artin stack X we can define as usual its category ofcoherent sheaves as
Coh(X) = lim
Coh(Sp(A)),←−
Sp(A)→X

where the limit is taken over all maps of affinoid rigid spaces Sp(A) to X. As usual,
given a smooth surjection U → X from an (affinoid) rigid analytic space U to X
this can be computed as the limit of the diagram→

Coh(U ) ⇒ Coh(U ×X U )→→....
Similarly the stable ∞-category of coherent sheaves on X can be defined as thehomotopy limit of ∞-categories
Dcoh (X) =lim←−
Sp(A)→X

Dcoh (Sp(A)),

and for a given covering it can be computed as the homotopy limit of the diagramof ∞-categories→
Dcoh (U ) ⇒ Dcoh (U ×X U )→→....
For a morphism of rigid analytic spaces f : X → Y, or more generally for amorphism between rigid analytic Artin stacks, the definition implies that we havea canonical derived pullback Lf ∗ : Dcoh (Y) → Dcoh (X). If the map f is proper wealso have (derived) pushforward Rf∗ : Dcoh (X) → Dcoh (Y) which is right adjoint
to Lf ∗ . More precisely, for general f the pushforward Rf∗ can be defined on the

CATEGORICAL p-ADIC LANGLANDS

39

subcategory of Dcoh (X) consisting of sheaves with proper support over Y. For the
rigid analytic Artin stacks that we consider there always exists a dualizing complex
ωX , and the internal duality
DX (−) = RHomOX (−, ωX )allows us to use
f ! = DX ◦ Lf ∗ ◦ DY : Dcoh (Y) → Dcoh (X)as an ad hoc definition of the upper shriek pullback f ! .
Remark 5.1.11. The definitions above are a bit restrictive, as we only defined
categories of coherent and not of quasicoherent sheaves. This is mainly due to thefact that quasicoherent sheaves on rigid analytic spaces are (for topological reasons)a bit delicate. In the world of condensed rings and modules of Clausen–Scholze it
is possible to develop a theory of a (derived) category of quasicoherent sheaves
together with a full six-functor formalism that gives a more uniform approach than
the above ad hoc definition of f ! .
We mention two sources of rigid analytic Artin stacks: Given an Artin stack X
on the category SchL of L-schemes such that X is (locally) of finite type over L (i.e.
X can be covered by a scheme U (locally) of finite type over L), its analytification
can be defined as the sheafification of
Xan (Sp(A)) = X(Spec(A)).
In particular: for a presentation V = U ×X U ⇒ U → X the analytificationof X can be defined as the stack quotient of Van ⇒ Uan . Given X we have an
obvious analytification functor on the (derived) category of coherent sheaves which
commutes with Lf ∗ , f ! , Rf∗ whenever they are defined.
On the other hand, if X is a formal algebraic stack over Spf(OL ), then we candefine its generic fiber as(5.1.11)Xηrig (Sp(A)) = lim
X (Y ),−→
Y

where the limit is taken over all formal models (in the sense of Raynaud) Y of
Sp(A).
Proposition 5.1.12. Let X be a formal algebraic stack. Then Xηrig is a rigid
analytic Artin stack.
5.1.13. Rigid analytic stacks of (ϕ, Γ)-modules. We now return to the main
objective of this subsection and define stacks of (ϕ, Γ)-modules.
Definition 5.1.14. For d ≥ 1 the category fibered in groupoids XGLd = Xd
over RigL is the groupoid that assigns to an affinoid space Sp A the groupoid of(ϕ, Γ)-modules of rank d over RF,A .
Using the fact that a (ϕ, Γ)-module over RF,A admits a model over an annulus
over Sp A, descent for vector bundles on rigid analytic spaces implies:
Proposition 5.1.15. The category fibered in groupoids Xd is a stack.
Remark 5.1.16. For the expected relationship between Xd and the stack Xdof Section 4, see (5.1.27) and the surrounding discussion.

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M. EMERTON, T. GEE, AND E. HELLMANN

Remark 5.1.17. By the above discussion the stack Xd can be described equivalently as the stack of GalF -equivariant vector bundles on the Fargues–Fontaine
curve and in the following we will freely choose whichever description fits betterwith our purposes. Hence there are some obvious similarities with the classical theory of (stacks of) vector bundles on an algebraic curve. We list the most importantsimilarities and differences:(i) There is a cohomology theory
RΓGalF (X, −) := RΓ(GalF , RΓ(Sp A × XF̄ , −))for GalF -equivariant vector bundles on Sp A × XF̄ . In terms of (ϕ, Γ)modules this cohomology theory coincides with the usual (ϕ, Γ) cohomology that is computed by the Herr complex [KPX14, Def. 2.2.3]. Moreprecisely, given a (ϕ, Γ)-module D over RF,A , the corresponding Herrcomplex isRΓϕ,Γ (D) = [D∆(ϕ−1,γ−1)/ D∆ ⊕ D∆ (γ−1)⊕(1−ϕ) / D∆ ],where ∆ ⊂ Γ is the torsion subgroup and where γ ∈ Γ/∆ is a topological
generator. This complex is (quasi-isomorphic to) a perfect complex of
A-modules concentrated in degrees 0, 1, 2, see [KPX14, Theorem 4.4.5].
(ii) As a consequence of (i) an analogue of Grothendieck’s cohomology andbase change theorem for coherent sheaves holds true in the context ofequivariant vector bundles on the Fargues–Fontaine curve, respectively inthe context of (ϕ, Γ)-modules. More precisely, for a given (ϕ, Γ)-module
D over RF,A and a point x ∈ Sp A the base change mapii(D) ⊗A k(x) −→ Hϕ,Γ(D ⊗RF,A RF,k(x) )bcix : Hϕ,Γis surjective if and only if it is an isomorphism; and if bcix is surjective,ithen Hϕ,Γ(D) is a vector bundle in a neighborhood of x if and only if
i−1
bcx is surjective.
(iii) There is no analogue of an ample line bundle: for a given equivariant
vector bundle V the Euler characteristic formula (see op. cit.)Xi(−1)i dim HGal
(XF̄ , V) = −rk V[F : Qp ]Fi≥0implies that there is always a non-vanishing H 1 (unless of course V = 0).
In particular we cannot twist away higher cohomology. (Without taking
the GalF -equivariance into account, there is an analogue of an ample line
bundle on XF̄ . This implies that we can at least twist the vector bundle
V such that its H 2 vanishes.)(iv) Local Tate duality [KPX14, Theorem 4.4.5 (3)] asserts that given a(ϕ, Γ)-module over RF,A there is a canonical isomorphism∼RΓϕ,Γ (D) −→ RHomA (RΓϕ,Γ (D∨ (ε)), A)[−2].
Here D∨ = HomRF,A (D, RF,A ) and ε is the cyclotomic character.
(v) There is a theory of Hodge–Tate–Sen weights, which we explain in Section
5.1.30 below. It introduces substantial differences in the discussion of P structures, for a parabolic subgroup P ⊆ GLd , in Section 5.3 below.

CATEGORICAL p-ADIC LANGLANDS

41

Conjecture 5.1.18. The stack Xd is a rigid analytic Artin stack of dimension
d2 [F : Qp ] = dim ResF/Qp GLd , which is furthermore normal and a local complete
intersection.
We now describe some partial results in the direction of this conjecture.
Theorem 5.1.19.
(i) The diagonal ∆Xd : Xd → Xd × Xd of Xd is representable by rigid spaces.
(ii) Let L0 be a finite extension of L and let x ∈ Xd (L0 ) be an L0 -valued point. Then
there exists an open neighborhood Ux of x in Xd which is a rigid analytic Artinstack. Moreover, if d = 2, then Ux has the expected dimension and is normal anda local complete intersection.
Sketch of Proof: (i) This basically follows from the fact that RΓϕ,Γ (D)is universally computed by a perfect complex. More precisely, given two families
D1 , D2 of (ϕ, Γ)-modules over an affinoid rigid analytic space Sp A, we have to showthat the functorb A B, D2 ⊗b A B)
Isom(D1 , D2 ) : Sp B 7−→ Isomϕ,Γ (D1 ⊗on the category of affinoid spaces over Sp A is representable by a rigid analytic
space.
By looking at the induced map on the top exterior powers it follows that
Isom(D1 , D2 ) is an open subfunctor ofb A B, D2 ⊗b A B)
Hom(D1 , D2 ) = H 0ϕ,Γ (D1∨ ⊗ D2 ) : Sp B 7−→Homϕ,Γ (D1 ⊗0b A B).
=Hϕ,Γ
((D1∨ ⊗ D2 )⊗Let us write D = D1∨ ⊗ D2 . Then there is a complex of vector bundles E • concentrated in degrees [0, 2] together with an isomorphism H 0ϕ,Γ (D) ∼= ker(E 0 → E 1 ),0
where the functor on the right hand side maps Sp B to ker(E ⊗A B → E 1 ⊗A B). As
E 0 and E 1 are vector bundles, this functor is representable by the preimage of thezero section under the induced map on the corresponding geometric vector bundles
over Sp A.
(ii) We sketch the proof in the case d = 2. Obviously we can reduce to the case
L0 = L. Let D be a (ϕ, Γ)-module of rank 2 over RF,L . In order to construct asmooth surjection from a rigid analytic space to a neighborhood of D, we distinguish
two possible cases:(a) There exists a character δ : F × → L× such that D0 = D ⊗ RF,L (δ) isétale.
(b) No such character exists.
Case (a): As D0 is étale there exists a Galois representation ρ0 : GalF → GL2 (L)with Drig (ρ0 ) = D0 . After conjugating ρ0 by some element of GL2 (L) if necessary,
we may assume that ρ0 takes values in GL2 (O) (i.e. we choose a stable lattice in
the Galois representation). Let ρ̄0 : GalF → GL2 (k) denote the reduction of ρ0and let Rρ̄0 denote the universal framed deformation ring of ρ̄0 . Then ρ0 defines apoint of the rigid analytic generic fiber (Spf Rρ̄0 )rig and we are left to show that themorphism(Spf Rρ̄0 )rig −→X2
ρ 7−→Drig (ρ) ⊗ R(δ −1 )

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M. EMERTON, T. GEE, AND E. HELLMANN

that maps ρ0 to D is smooth in a neighborhood of ρ0 . This follows from theinfinitesimal lifting criterion together with the fact that the complete local ringof (Spf Rρ̄0 )rig at ρ0 pro-represents the universal framed deformation functor ofρ0 : GalF → GL2 (L), see [Kis09c, Lemma 2.3.3, Prop. 2.3.5].
Case (b): After again twisting with some rank 1 object we may assume that all
slopes of subobjects of D are non-negative. By Lemma 5.1.21 below there exists a
character δ : F × → L× and a non-split extension
0 −→ D −→ D0 −→ RF,L (δ) −→ 0such that D0 is étale and such that Ext0ϕ,Γ (RF,L (δ), D) = Ext2ϕ,Γ (RF,L (δ), D) = 0,
as well as dim Homϕ,Γ (D0 , RF,L (δ)) = 1. Again we choose an O-lattice ρ0 in the
Galois representation associated to D0 and consider ρ0 as a point of (Spf Rρ̄0 )rig therigid analytic generic fiber of the universal framed deformation ring of the reductionρ̄0 of ρ0 .
We construct a locally closed subset Y ⊆ (Spf Rρ̄0 )rig containing ρ0 and asmooth morphism Y → X2 as follows: Let D̃ denote the universal (ϕ, Γ)-module2(D̃(δ −1 ε)).
over (Spf Rρ̄0 )rig and let Y1 denote the scheme-theoretic support of Hϕ,Γ
As
2
2
Hϕ,Γ
(D̃(δ −1 ε)) ⊗ k(ρ0 ) = Hϕ,Γ(D0 (δ −1 ε)) = (Homϕ,Γ (D0 , RF,L (δ)))∨is of dimension 1, we can find a neighborhood Y2 ⊆ Y1 of ρ0 and a lift f˜ ∈2
Γ(Y2 , Hϕ,Γ(D̃(δ −1 ω))) of the dual basis to a chosen basis vectorf ∨ ∈ Homϕ,Γ (D0 , RF,L (δ)).
After replacing Y2 by a smaller open neighborhood of ρ0 if necessary, we may assume2(D̃(δ −1 ε))|Y2 is free of rank 1 with basis f˜. Localizing further we findthat Hϕ,Γthat there is a neighborhood Y ⊆ Y2 of ρ0 together with a surjectionf˜∨ : D̃|Y −→ RF,Y (δ)
that specializes to the chosen surjection f ∨ : D0 → RF,L (δ) at ρ0 and we moreover
may assume that at all points y ∈ Y we have
(5.1.20)

Ext0 (RF,k(y) (δ), D̃ ⊗ k(y)) = Ext2 (RF,k(y) (δ), D̃ ⊗ k(y)) = 0.

We claim that the map Y → X2 defined by D̃|Y 7→ ker f˜∨ is smooth. Again it isenough to check this on the complete local ring of a point y ∈ Y . The complete
local ring at y pro-represents the functor of deformations of D̃y = D̃ ⊗k(y) together
with a deformation of the map fy∨ : D̃y → RF,k(y) (δ). By our choices of Y thisfunctor may be identified with the functor of deformations of Dy = ker fy∨ together
with a deformation of the extension
0 −→ Dy −→ D̃y −→ RF,L (δ) −→ 0.
By the vanishing of the Ext-groups (5.1.20) this functor is formally smooth overthe deformation functor of Dy .
We are left to prove the statement about the dimension and the claims on thelocal structure. This is Proposition 5.1.25 below.
Lemma 5.1.21. Let D be a (ϕ, Γ)-module of rank 2 over RF,L and assume thatall slopes of subobjects of D are non-negative and that D ∼6 RF,L (δ)⊕2 for any
=
×
×
×
×
character δ : F → L . Let η : F → L be a continuous character such that

CATEGORICAL p-ADIC LANGLANDS

43- for all embeddings τ : F ,→ L one has (wtτ (η) + Z) ∩ wtτ (D) = ∅, where
we refer to Section 5.1.30 below for the definition of the weights.
- the slope of η is slope(η) = −slope(D).
Then there exists a non-split extension(5.1.22)0 −→ D −→ D0 −→ RF,L (η) −→ 0such that D0 is étale and moreover
Ext0ϕ,Γ (RF,L (η), D) = Ext2ϕ,Γ (RF,L (η), D) = 0,dimL Homϕ,Γ (D0 , RF,L (η)) = 1.
Remark 5.1.23. Given D as in the lemma, a character η satisfying the assumptions in the lemma obviously always exists.
Proof. As D(η −1 ) and D∨ (ηω) have non-integral Hodge–Tate–Sen weights,
the vanishing of the Ext-groups is a direct consequence of the identifications
0
Ext0ϕ,Γ (RF,L (η), D) = Hϕ,Γ(D(η −1 ))0(D∨ (ηω)∨ .
Ext2ϕ,Γ (RF,L (η), D) = Hϕ,ΓChoose any non-split extension (5.1.22). The claim on the Hodge–Tate weights
also implies the claim on the dimension of Homϕ,Γ (D0 , RF,L (η)), so we only needto show that D0 can be chosen to be étale. By the condition on the slope of η we
automatically have slope(D0 ) = 0 and by Kedlaya’s theorem we are left to showthat for all (saturated) subobjects D00 ⊆ D we have slope(D00 ) ≥ 0. By assumption
this is true if D00 ⊆ D. It therefore suffices to show that we can choose D0 suchthat all saturated (ϕ, Γ)-stable subobjects of D0 are contained in D. Assume to thecontrary that that there is some D00 such that the map D00 → RF,L (η) is non-zero.
We distinguish the cases rk D00 = 1 and rk D00 = 2.
If rk D00 = 1, then D00 → RF,L (η) is an isomorphism after inverting t, which
implies that that the extension (5.1.22) is split after inverting t. However, theassumption (wtτ (η) + Z) ∩ wtτ (D) = ∅ implies that the canonical map
Ext1ϕ,Γ (RF,L , D) −→ Ext1ϕ,Γ (RF,L [1/t], D[1/t])
is an isomorphism, which is a contradiction, as we chose (5.1.22) to be non-split.
If rk D00 = 2, then E = D00 ∩ D ⊆ D is a (ϕ, Γ)-submodule of rank 1. By
assumption, D has only finitely many saturated subobjects of rank 1. Each suchsubobject E gives rise to a short exact sequence(5.1.24)0 −→ RF,L (δ1 ) −→ D −→ RF,L (δ2 ) −→ 0

and our assumptions on the Hodge–Tate–Sen weights implies the vanishing of certain Ext0 and Ext2 groups and hence a short exact sequence
0 → Ext1ϕ,Γ (RF,L (η), RF,L (δ1 )) → Ext1ϕ,Γ (RF,L (η), D) →→ Ext1ϕ,Γ (RF,L (η), RF,L (δ2 )) → 0.
Moreover, as in the rank 1 case, the condition (wtτ (η) + Z) ∩ wtτ (D) = ∅ implies
that inverting t induces an isomorphism between this exact sequence and the exact
sequence
0 → Ext1ϕ,Γ (RF,L (η)[1/t], RF,L (δ1 )[1/t]) → Ext1ϕ,Γ (RF,L (η)[1/t], D[1/t]) →→ Ext1ϕ,Γ (RF,L (η)[1/t], RF,L (δ2 )[1/t]) → 0.

44

M. EMERTON, T. GEE, AND E. HELLMANNAs D has only finitely many subobjects, we can choose the extension
D0 ∈ Ext1ϕ,Γ (RF,L (η), D)such that for all possibilities to write D as an extension (5.1.24), the extension D0
is not in the image of Ext1ϕ,Γ (RF,L (η), RF,L (δ1 )).
In this case, if D00 ⊆ D0 is a saturated subobject with D00 ∩ D = E = RF,L (δ1 )and such that D00 → RF,L (η) is non-zero, then
RF,L (η)[1/t] ∼= (D00 /E)[1/t] ⊆ (D0 /E)[1/t]
implies that the image (D0 /E)[1/t] of D0 in Ext1ϕ,Γ (RF,L (η)[1/t], RF,L (δ2 )[1/t])
vanishes. This contradicts our choice of D0 above.
The existence of local charts implies the existence of versal rings at rigid analytic
points of Xd .
Proposition 5.1.25. Let d = 2 and x ∈ Xd and let R be a versal ring at x.
Then R is a local complete intersection of dimension d2 [F : Qp ] and is normal.
Sketch of proof. As any two versal rings are smoothly equivalent, we may
use any choice of a versal ring to prove the assertion. We use the notation fromthe proof of Theorem 5.1.19. If x is as in case (a) of that proof, then R can bechosen to be isomorphic to the complete local ring of (Spf Rρ̄0 )rig at the point ρ0 .
The result then follows from [BIP21].
In case (b), where x is not the twist of an étale (ϕ, Γ)-module, standard deformation theory arguments give us a presentation of a versal ring R such that
Spf R → Xd is formally smooth of dimension m and such that every irreducible
component of R has dimension at least m + [F : Qp ]d2 , and in order to showthat R is lci, we need to show that for all irreducible components equality holds.
By the existence of local charts (i.e. Theorem 5.1.19 (2)), we may view R as thecomplete local ring of a point y ∈ Sp A, where Sp A → Xd is smooth. We claimthat each Z be an irreducible component of Sp A that contains y is such that the(ϕ, Γ)-module on Z is generically on Z the twist of an étale (ϕ, Γ)-module. Thenthe argument in case (a) implies that Z has the expected dimension m+[F : Qp ]d2 .
It is enough to show that there are no irreducible components Z of Sp(A) containing y on which the (ϕ, Γ)-module is (generically) reducible (because otherwise it isgenerically irreducible and thus étale up to twist).
To see this, given an extension (5.1.24) we consider the groupoid of deformations
D of this short exact sequence and denote by S a versal ring to D such thatSpf S → D is formally smooth of dimension m. Then it is enough to show that all
irreducible components of S are of dimension strictly less than m + [F : Qp ]d2 . In
the case d = 2 this can easily be checked by a direct computation (compare also
the discussion of the stack of B-bundles in 5.3 below).
For normality we need to check in addition (if F = Qp ) that for a generic choiceof a reducible (ϕ, Γ)-module
0 → RF,L (δ1 ) → D → RF,L (δ2 ) → 0
the stack X2 is smooth at D. Again this follows from a tangent space computation(resp. a computation of Ext-groups) together with the fact that generically on thestack of B-bundles, the ratio δ1 /δ2 is very regular in the sense that
Ext0ϕ,Γ (RF,L (δ2 ), RF,L (δ1 )) = Ext2ϕ,Γ (RF,L (δ2 ), RF,L (δ1 )) = 0.



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Remark 5.1.26. Basically the same strategy (using not just the Borel B but
all parabolic subgroups) should settle the claim on the dimension and the local
structure for general d. Only the computation of the dimension of spaces/stacks
of extensions becomes more involved and is not finished yet. We note that similar
computations of dimensions of Ext-groups also show up in the proof of the main
theorem of [BIP21].
In connection with the integral theory in Section 4, there is a morphism(5.1.27)rig
πd : Xd,η
−→ Xd

from the rigid analytic generic fiber of Xd to the stack Xd that is basically given
by forgetting the étale lattice. This morphism should be formally étale, but not
representable by rigid analytic spaces (nor by quasi-analytic spaces), compare thed = 1 case in Section 7.1. It is not clear from the definitions that such a map exists,
but the existence is a consequence of the following conjecture, which recently has
been proved by Gal Porat [Por22]:
Conjecture 5.1.28. Let A be a p-adically complete Zp -algebra topologically offinite type over Zp . Then every étale (ϕ, Γ)-module over AF,A is overconvergent.
It is worth noticing that the conjecture follows from work of Berger–Colmez [BC08]
if the étale (ϕ, Γ)-module arises from a family of Galois representations over A. The
work of Porat [Por22] also treats the case that the étale (ϕ, Γ)-module is not attached to a Galois representation, which can happen if A is not finite over Qp .
Remark 5.1.29. We point out that the proof of Theorem 5.1.19 only gives
charts locally around rigid analytic points. As a set theoretic cover of a rigid
analytic space by admissible open subsets is not necessarily an admissible cover,this is not enough to give a full proof of Conjecture 5.1.18. Extending the argumentin the above proof to all points of the corresponding adic space (not just thosepoints x such that L0 = k(x) is finite over L) would prove Conjecture 5.1.18. The
given argument would directly generalize if one had an analogue of Kedlaya’s slope
filtration theorem for (ϕ, Γ)-modules over RF,L0 , where L0 is allowed to be the(completed) residue field of any point in an adic space of finite type over Qp and if
moreover one could prove that the map πd (that exists due to [Por22]) is formallyétale. While étaleness is not expected to be difficult to prove, we have no idea
whether the generalization of Kedlaya’s theorem to general coefficient fields is trueor not.
5.1.30. Hodge–Tate–Sen weights and global sections. The theory of Hodge–
Tate–Sen weights for representations of GalF on finite dimensional Qp -vector spaces
generalizes to families of (ϕ, Γ)-modules, respectively to families of GalF -equivariant
vector bundles on XF̄ . Let A denote an affinoid algebra and let DA be a (ϕ, Γ)module DA over RF,A . We write EA = EF∞ (DA ) for the corresponding Γ-equivariant
vector bundle on XF∞ ,A . We then consider the special fiberb Sen (DA ) = EA |Sp A×{∞} = DA ⊗b RF,A (Fb∞ ⊗b Qp A)
D
where the tensor product on the right hand side is induced by a certain (explicit)
map RF → Fb∞ . The map RF → Fb∞ factors through F∞ ⊂ Fb∞ and the moduleb Sen (DA ) has a decompleted version
D
DSen (DA ) = DA ⊗RF,A (A ⊗Qp F∞ )46

M. EMERTON, T. GEE, AND E. HELLMANN

which is a finite projective A⊗Qp F∞ -module that carries a continuous action of the
1-dimensional p-adic Lie group Γ. On the decompleted version DSen (DA ) the Γaction is locally analytic and hence one can derive this action, defining a morphism
Lie Γ −→ Lie(GL(DSen (DA ))) = End(DSen (DA )).
The cyclotomic character induces a trivialization Zp ∼= Lie Γ and hence theimage of 1 ∈ Zp defines an endomorphism
Θ : DSen (DA ) −→ DSen (DA ).
This endomorphism commutes with the Γ-action, so the coefficients of its characteristic polynomial take values in F ⊗Qp A (though we caution the reader that
this does not mean that the pair (DA , Θ) descends to an object overQF ⊗Qp A).
If A is an L-algebra and |Hom(F, L)| = [F : Qp ], then F ⊗Qp A ∼= τ :F ,→L Aτ ,where Aτ = A. For each embedding τ we obtain an endomorphism Θτ of the finite
projective A-module
DSen,τ (DA ) = DSen (DA ) ⊗A⊗Qp F Aτwith characteristic polynomial χΘτ with coefficients in A. The zeros of this polynomial (i.e. the eigenvalues of Θτ ) are called the Hodge–Tate–Sen weights of Dlabeled by τ . If A = L is a field containing all the zeros of χΘτ , then we writewtτ (D) ⊆ L for the set of all these zeros.
We write gln //GLn for the GIT quotient of the Lie algebra gln by the adjoint
action of GLn . Mapping a (ϕ, Γ)-module DA over RF,A to the coefficients of thecharacteristic polynomial χΘ of Θ defines a map
(5.1.31)
ωd : Xd −→ ResF/Qp (gln //GLn ) L = WTd,L .
Q
If |Hom(F, L)| = [F : Qp ], then of course WTd,L = τ :F →L WTτ,d,L and each
WTτ,d,L = (gln //GLn )L is an affine space.
Taking the top exterior power of an equivariant vector bundle defines a map
det : Xd −→ X1 −→ T ,where T is the space of continuous characters of F × (see Section 7.1 below for a
description of X1 and for the definition of the last map). Similarly, the top exterior
power defines a map WTd,L → WT1,L and ω1 induces a map T → WT1,L . We
expect that these morphisms are the source of all global functions on the stack Xd .
Conjecture 5.1.32. The maps ωd and det induce an isomorphismb Γ(WT1,L ,OWT ) Γ(T , OT ).
Γ(Xd , OXd ) ∼= Γ(WTd,L , OWTd,L )⊗1,LCorollary 5.1.33. Assume Conjecture 5.1.32. Then the connected components of Xd are in bijection (via the determinant) with the connected componentsof X1 and hence with the connected components of the space
W = (Spf O[[OF× ]])rigof continuous characters of OF× .
Note that the connected components of W are in turn in bijection with thecharacters µ(F ) → O× . We point out that that a similar result for the componentsof universal deformation rings was proven by Böckle–Iyengar–Paškūnas [BIP21].

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5.2. Stacks of de Rham objects. The stack of (ϕ, Γ)-modules Xd has closed
substacks defined in terms of p-adic Hodge theory, similarly to the closed substacks
ss,λ,τ
Xd⊂ Xd . In the case of (ϕ, Γ)-modules over the Robba ring these closed
substacks can be studied in terms of (filtered) Weil–Deligne representations. In the
conjectural relation of locally analytic representations with sheaves on the stack Xdthese closed substacks of de Rham objects (of given weight and inertial type) willplay a role in the comparison with the smooth categorical Langlands conjectures.
We will now define these “de Rham loci”.
Let F 0 be a finite Galois extension of F with Galois group Gal(F 0 /F ) andwrite F00 respectively F0 for the maximal unramified subextension of F 0 respectively F . Moreover, let σ denote the Frobenius lift on F00 . We consider the stack
Modd,ϕ,N,F 0 /F on the category of L-schemes that maps an L-algebra A to thegroupoid of finite projective A ⊗Qp F00 -modules D of rank d together with∼- an id ⊗ σ-linear automorphism ϕD : D −→ D.
- an A ⊗Qp F00 -linear endomorphism N : D → D satisfying
N ◦ ϕD = pϕD ◦ σ ∗ N.
- an action of Gal(F 0 /F ) on D, commuting with ϕD and N , such thatg((a ⊗ f ) · d) = (a ⊗ g(f )) · g(d)for all a ∈ A, f ∈ F00 , d ∈ D and g ∈ Gal(F 0 /F ).
We further recall the stack of d-dimensional Weil–Deligne representations WDd,Fof F which is an Artin stack on the category of L-schemes, see [DHKM20],[Zhu20, §3]. Recall that for a given L-scheme S the groupoid WDd,F (S) parameterizes vector bundles E on S together with a smooth action ρ of the Weil group
WF and an endomorphism N : E → E satisfying N g = q ||g|| gN . Given F 0 we write
WDd,F 0 /F ⊂ WDd,F for the open and closed substack of WDd,F consisting of those
Weil–Deligne representations (E, ρ, N ) such that the restriction of ρ to the inertia
subgroup IF 0 of F 0 is trivial.
We note that Fontaine’s construction of the Weil–Deligne representation associated to a (ϕ, N, Gal(F 0 /F ))-module [Fon94] (see also [BS07, Prop. 4.1]) implies
that these two stacks become (non-canonically) isomorphic over a large enoughfield L:
Lemma 5.2.1. Assume that L is large enough such that [F00 : Qp ] = |Hom(F00 , L)|and fix an embedding F00 ,→ L. Then there is an isomorphism of stacks∼

WDd : Modd,ϕ,N,F 0 /F −→ WDd,F 0 /F .
We will spell out this isomorphism in detail in Section 5.2.7 below, and also
give explicit descriptions of the stacks involved (at least for F 0 = F ).
In order to link a (ϕ, N, Gal(F 0 /F ))-module D ∈ Modd,ϕ,N,F 0 /F (A) to equivariant vector bundles, we need to introduce a filtration on D. Let us write
G = ResF/Qp GLd and G0 = ResF 0 /Qp GLd for the moment. Moreover, we write
G00 = ResF00 /Qp GLd .
Let T ⊆ G denote the Weil restriction of the diagonal torus of GLd and let
B ⊆ G denote the Weil restriction of the upper triangular matrices. For a choice ofa dominant cocharacter λ ∈ X∗ (TL̄ )+ we write Lλ for the reflex field of λ and Pλ ⊆GLλ for the parabolic subgroup (containing BLλ ) defined by λ. The flag variety

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M. EMERTON, T. GEE, AND E. HELLMANN

GLλ /Pλ then parameterizes filtrations on Ldλ ⊗Qp F of type λ. More precisely, using
the standard identification
YX∗ (T )+ =Zd+
τ :F ,→L̄

we write
λ = (λτ,1 ≥ λτ,2 ≥ · · · ≥ λτ,d )τ :F ,→L̄ .
Then an L̄-valued point of GLλ /Pλ is given by a product of descending, exhaustiveand separated Z-filtrations onYL̄d ⊗Qp F =L̄dττ :F ,→L̄(with L̄τ = L̄) such that (λτ,1 ≥ λτ,2 ≥ · · · ≥ λτ,d ) are precisely the weights(counted with multiplicity) of the filtration on L̄dτ .
Using the canonical map G → G0 we can also view λ as a cocharacter of a torus
in G0 (more precisely again of the Weil restriction of the diagonal torus in GLd )and we obtain a similar parabolic subgroup Pλ0 and a flag variety G0Lλ /Pλ0 .
Note that there is a forgetful map
WDd,F −→ BGLd = ∗/GLd
to the stack of rank d vector bundles. And similarly there is a canonical map
Modd,ϕ,N,F 0 /F −→ ∗/G00
forgetting the action of ϕ, N and Gal(F 0 /F ) on a projective A ⊗Qp F00 -module ofrank d.
The scheme G/Pλ is equipped with a (diagonal) left translation action of GLd
and we can define the fiber productFilλ WDd,F = WDd,F ×∗/GLd GLd \GLλ /Pλthat is called the stack of filtered Weil–Deligne representations (of Hodge–Tateweight λ). Its S-valued points parameterize Weil–Deligne representations on vector
bundles E over S together with a filtration on F ⊗Qp E of type λ.
Similarly, the scheme G0 /Pλ0 is equipped with a (diagonal) left translation actionof G00 and we can consider the closed substackFilλ Modd,ϕ,N,F 0 /F ⊂ Modd,ϕ,N,F 0 /F ×∗/G00 G00 \G0Lλ /Pλ0
parameterizing (for a given object D ∈ Modd,ϕ,N,F 0 /F ) those filtrations (of type λ)on D ⊗F00 F that are stable under the action of Gal(F 0 /F ).
Remark 5.2.2. We point out that Gal(F 0 /F )-stable filtrations can only exist
if the cocharacter λ of (our maximal torus in) G0 comes from (our maximal torusin) G via the map G → G0 .
The following lemma then generalizes Lemma 5.2.1.
Lemma 5.2.3. Assume that L is large enough such that [F 0 : Qp ] = |Hom(F 0 , L)|and fix an embedding F 0 ,→ L. Then there is a canonical isomorphism∼Filλ Modd,ϕ,N,F 0 /F −→ Filλ WDd,F 0 /F
lying above the isomorphism WDd of Lemma 5.2.1.

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49

The stack Filλ Modd,ϕ,N,F 0 /F plays an important role in the characterization ofde Rham (ϕ, Γ)-modules and via the above isomorphism we can hence link filteredWeil–Deligne representations to (ϕ, Γ)-modules, at least if the base field is large
enough. More precisely we have the following theorem:
Theorem 5.2.4. For fixed λ ∈ X∗ (TL̄ )+ there is a canonical closed embedding(Filλ Modd,ϕ,N,F 0 /F )an −→ Xd,Lλ .
Moreover, the image of this morphism coincides with the closed substack of de
Rham (ϕ, Γ)-modules of Hodge–Tate–Sen weight λ that become semi-stable after
base change to F 0 .
The construction of a family of (ϕ, Γ)-modules of rank d over the rigid analytic
stack (Filλ Modd,ϕ,N,F 0 /F )an is the generalization of the work of Berger [Ber08b]
to families of the objects involved.
To make sense of the “moreover” part of Theorem 5.2.4 we need to define thenotion of a family of (potentially) semi-stable or of de Rham (ϕ, Γ)-modules over
RF,A . Given a (ϕ, Γ)-module DA over RF,A we can consider the base change
Ddif (DA ) = DA ⊗RF,A F∞ ((t))
which is endowed with a derivation∂ : Ddif (DA ) −→ Ddif (DA )dthat is defined by the derivative of the Γ-action. Then DA is defined toabove t dt
be de Rham if Ddif (DA )∂=0 is a projective A ⊗Qp F∞ -module of rank d and if thecanonical map
Ddif (DA )∂=0 ⊗F∞ F∞ ((t)) −→ Ddif (DA )is an isomorphism. The proof of the “moreover” part of Theorem 5.2.4 then follows
closely the proof of [BC08, Thms. 5.3.1, 5.3.2]
Remark 5.2.5. Recall that the (ϕ, Γ)-modules in the analytic set up are notrequired to be étale. Hence the stacks of de Rham (ϕ, Γ)-modules can be describedby filtered (ϕ, N )-modules (with descent data) as done in Theorem 5.2.4 without
putting an additional “weak admissibility” condition on the filtration. It is however
possible to pass to an open substack of (Filλ Modd,ϕ,N,F 0 /F )an where the (ϕ, Γ)modules associated to the filtered (ϕ, N, GalF )-modules are étale, see [HH20] and[Hel13] for that construction in the case of a trivial GalF -action. This étale locus
then agrees with the (union over appropriate τ of the) image of the generic fibers
ss,λ,τ
of the stacks Xd
under the map (5.1.27).
Remark 5.2.6. We point out that it is important to fix the cocharacter λ in thetheorem. If λ is allowed to vary arbitrarily, then the union of the corresponding substacks (Filλ Modd,ϕ,N,F 0 /F )an is not closed in Xd but expected to be Zariski-dense.
This should be seen as an analogue of the density of crystalline representations inlocal deformation rings, see [Che13, Nak14, BIP22].
5.2.7. The role of the embedding F00 ,→ L. In Lemma 5.2.1 above, we haveidentified the stacks Modd,ϕ,N,F 0 /F and WDd,F 0 /F over a field L containing F00 using
the choice of an embedding σ0 : F00 ,→ L. We elaborate a bit further about the
choice of this embedding. For simplicity (and in order not to overload the notation)
we limit ourselves to the case F 0 = F . We refer to [BG19, §2.6] and [HH20, §3]
for the explicit description of the stacks that we are using in the following.

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M. EMERTON, T. GEE, AND E. HELLMANN

Assume that L contains F00 = F0 (the equality holding because we have assumed
that F 0 = F ); then for an L-algebra A we have a decomposition
YYA ⊗Qp F0 = A ⊗L (L ⊗Qp F0 ) = A ⊗L
Lς =
Aς ,
ς:F0 ,→L

ς:F0 ,→L

where Lς = L and Aς = A for all ς. The action of Gal(F0 /Qp ) of course justpermutes the factors in the product. For the remainder of this section we fix an
Qfembedding ς0 and rewrite A ⊗Qp F0 = i=1 Ai with Ai = Aς0i = A. We deducethat we have isomorphisms(ResF0 /Qp GLd )L =

f
Y

GLd

i=1

(ResF0 /Qp Matd×d )L =f
Y

Matd×d .

i=1Note that WDd,F/F = Xd /GLd , where
Xd = {(ϕ, N ) ∈ GLd × Matd×d |N ϕ = qϕN }/GLd ,and where GLd acts on GLd and Matd×d by conjugation. On the other hand
Modd,ϕ,N,F/F = Yd /ResF0 /Qp GLd , where
Yd = {(ϕ, N ) ∈ ResF0 /Qp GLd × ResF0 /Qp Matd×d | N ϕ = pϕσ ∗ N }/ResF0 /Qp GLd ,and where σ ∗ (N1 , . . . , Nf ) = (Nf , N1 , . . . , Nf −1 ) and ResF0 /Qp GLd acts by conjugation on ResF0 /Qp Matd×d and acts on ResF0 /Qp GLd by σ-conjugation, i.e. g ·(ϕ, N ) = (g −1 ϕσ(g), g −1 N g). In other words(g1 , . . . gf ) · ((ϕ1 , . . . , ϕf ), (N1 , . . . , Nf ))=((g1−1 ϕ1 gf , g2−1 ϕ2 g1 , . . . , gf−1 ϕf gf −1 ), (g1−1 N1 g1 , . . . , gf−1 Nf gf )).
Then there is a canonical morphism Xd → Yd given by(ϕ, N ) 7−→ ((ϕ, 1, . . . , 1), (N1 , p−1 N1 , . . . , p−(f −1) N1 )which is equivariant for the map of group schemes GLd → ResF0 /Qp GLd given byg 7−→ (g, g, . . . , g)and which induces the isomorphism of stacks∼

fς0 : WDd,F/F = Xd /GLd −→ Modd,ϕ,N,F/F = Yd /ResF0 /Qp GLdin Lemma 5.2.1. The inverse of this isomorphism is induced by the map Yd → Xd
given by((ϕ1 , . . . , ϕf ), (N1 , . . . , Nf )) 7−→ (ϕf · · · ϕ2 ϕ1 , N1 ).
These formulas show that the isomorphism fς0 depends on the choice of ς0 : for adifferent choice ς00 the induced automorphism fς00 ◦fς−1
is not the identity. Moreover,0
we easily see that the isomorphism fς0 does not descend to a subfield L0 ⊂ L not
containing F0 : if τ : L → L is a field automorphism of L, then it commutes withfς0 if and only if its restriction to F0 is the identity.
On the other hand we note that the map
WDd,F/F → GLd //GLd ∼= Ad−1 × Gm

CATEGORICAL p-ADIC LANGLANDS

51

mapping (ϕ, N ) to the coefficients of its characteristic polynomial induces an isomorphism on global sections, see e.g. [Hel20, Remark 3.4]. We deduce that fσ0
induces an isomorphism on the (invariant) global sections(5.2.7)

Γ(Modd,ϕ,N,F/F , OModd,ϕ,N,F /F ) −→ Γ(WDd,F/F , OWDd,F /F )that is independent of the choice of ς0 and that, moreover, this isomorphism descends to any subfield L0 ⊆ L. We now make this descent explicit in the case
L = F0 and L0 = Qp . Then the action σYd of the generator σ ∈ Gal(F0 /Qp ) on theϕ-part in Yd is given by(ϕ1 , . . . , ϕf ) 7−→ (σ ∗ ϕf , σ ∗ ϕ1 , . . . , σ ∗ ϕf −1 ),whereas the action σXd of σ on the ϕ-part in Xd is clearly given by ϕ 7→ σ ∗ ϕ. We
deduce that (spelling out only the ϕ-part of the formulas)(fς0 ◦ σYd )(ϕ1 , . . . , ϕf ) = (σ ∗ ϕf )−1 σXd ◦ fς0 ((ϕ1 , . . . , ϕf )) (σ ∗ ϕf )and hence fς0 does not descend to Qp . On the other hand the characteristic polynomials of the ϕ-parts of fς0 ◦ σYd and σXd ◦ fς0 agree, and hence the isomorphism
between the global sections (5.2.7) descends to Qp .
In fact this observation generalizes to WDd,F 0 /F as follows:
Proposition 5.2.8. Let L be any finite extension of Qp and let F 0 /F be a
finite extension. For a field extension M of L containing F00 and any choice of anembedding ς0 : F00 ,→ M the isomorphismΓ((Modd,ϕ,N,F 0 /F )M , O(Modd,ϕ,N,F 0 /F )M ) −→ Γ((WDd,F 0 /F )M , O(WDd,F 0 /F )M )induced by fς0 is independent of the choice of ς0 and descends to a canonical isomorphism
Γ(Modd,ϕ,N,F 0 /F , OModd,ϕ,N,F 0 /F ) −→ Γ(WDd,F 0 /F , OWDd,F 0 /F ).
5.2.9. Invariant functions and the Bernstein center. Recall that the local Langlands correspondence for GLd (F ) defines a bijection τ 7→ Ωτ between the ddimensional inertial types τ (defined over L) of IF and the Bernstein blocks Ω(defined over L) of the category Repsm GLd (F ) (we view our representations as representations on L̄-vector spaces for an algebraic closure L̄ of L). More precisely,
if (r, N ) is the Weil–Deligne representation associated to an irreducible smooth
representation π of GLd (F ), then r|IF ∼= τ if and only if π lies in the Bernsteinblock Ωτ .
Moreover, note that over an algebraic closure L̄ of L there is a bijection
τ 7→ WDd,F,τ between the d-dimensional inertial types τ (defined over L̄) andthe connected components of the stack of Weil–Deligne representations, such thata Weil–Deligne representation (r, N ) lies in WDd,F,τ if and only if r|IF ∼= τ , see
[BZCHN20, §5.2]. As we restrict ourselves to inertial types that become trivial on
IF 0 we may choose L to be large enough such that all inertial types τ over L̄ (withτ |IF 0 being trivial) are defined over L and such that all the geometric connectedcomponents WDd,F,τ are defined over L, and (by slight abuse of notation) we will
still write WDd,F,τ for this stack on L-schemes.
The following proposition (which is a consequence of [BZCHN20, Theorem
5.13], see [HM18] for an integral version) relates the Bernstein center ZΩτ of Ωτ(that we view as a block of the category of smooth representations on L-vector
spaces, so in particular ZΩτ is an L-algebra) to the invariant functions on WDd,F,τ .

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M. EMERTON, T. GEE, AND E. HELLMANN

Proposition 5.2.10. Let τ be an inertial type defined over L. Then there is anatural isomorphism∼ZΩτ −→ Γ(WDd,F,τ , OWDd,F,τ ).
Remark 5.2.11. Note that there are two possibilities to normalize the isomorphism of Proposition 5.2.10, corresponding to two possible normalizations of thelocal Langlands correspondence. In the usual (unitary) normalization the central
character of a smooth representation π matches the determinant of the corresponding Weil–Deligne representation (ρ, N ). Sometimes it is more convenient to choosed−1

a normalization such that (ρ, N ) corresponds to π ⊗ |det|− 2 . This normalization
is better suitable for local-global compatibility, and has the advantage that it does
not depend on a choice of q 1/2 (and hence has better properties when consideredover non-algebraically closed fields, compare [BS07]). In these notes we will fromnow on always use the non-unitary normalization.
For a fixed inertial type τ we can consider the rigid analytic generic fiber
ss,λ,τ
ss,λ,τ rig)η of the formal stack Xd
defined in Section 4. By the very construction(Xd
ss,λ,τ
of the stacks Xd
this generic fiber admits a map(5.2.12)ss,λ,τ

πdss,λ,τ rig)η −→ (Filλ Modd,ϕ,N,τ )an ⊂ Xd ,: (Xdthat is a generalization of the period map [PR09, (5.37)] of Pappas–Rapoport inthe case τ = 1 and λ minuscule. Moreover, it fits into a commutative diagram/ X rig

ss,λ,τ rig
)η

(Xd

d,η

ss,λ,τπdπd

(Filλ Modd,ϕ,N,τ )an/ Xdwith the map πd from (5.1.27). In particular the isomorphism∼ZΩτ −→ Γ(Modd,ϕ,N,τ , OModd,ϕ,N,τ )(5.2.13)

obtained by combining Propositions 5.2.8 and 5.2.10, induces a map
ss,λ,τ rig)η , O(X ss,λ,τ )rig)ηZΩτ −→ Γ((Xd(5.2.14)dthat is defined over any chosen base field L over which τ is defined (not necessarily
containing the field F00 over which we can identify the stack of (ϕ, N, Gal(F 0 /F ))modules with a substack of the stack of Weil–Deligne representations).
Remark 5.2.15. After composing with the map to the functions on the genericfiber of a potentially semi-stable deformation ring this map coincides with themorphism of [CEG+ 16, Theorem 4.1, Prop. 4.2]. In fact the result of [CEG+ 16]
is stronger: it asserts that the map takes values in the ring of bounded functions
ss,λ,τ

Γ(Xdss,λ,τ rig)η , O(X ss,λ,τ )rig).
η

, OX ss,λ,τ )[ p1 ] ⊂ Γ((Xddd

5.3. Drinfeld-style compactifications. We discuss variants of the stack Xdparameterizing flags on a family of equivariant vector bundles of rank d. These
stacks are naturally associated with a choice of a parabolic subgroup P ⊆ G = GLd .
Given a parabolic subgroup P with Levi quotient M ∼= GLd1 × · · · × GLdr we defineCATEGORICAL p-ADIC LANGLANDS

53

XP to be the stack of GalF -equivariant P -bundles on the Fargues–Fontaine curve
XF̄ , which comes with two projections
XP
βP

Xd

}

αP

!

XM

induced by the maps P ,→ GLd and P  M , where XM ∼= Xd1 × · · · × Xdr denotes
the stack of GalF -equivariant M -bundles on XF̄ . It will be convenient to write
XG = Xd in the following.
In the special case of P = B the (ϕ, Γ)-modules in the image of the morphism
XB → Xd are, by definition, precisely the (ϕ, Γ)-modules that are trianguline (inthe sense of Colmez, see [Col08, Col14] for the definition and discussion of two
dimensional trianguline representations and their role in the p-adic Langlands correspondence). Beside the fact that trianguline (ϕ, Γ)-modules are very natural objects
to consider (as a triangulation is nothing but a B-structure), their definition was
inspired by the fact that the Galois representations associated to overconvergent
p-adic modular forms of finite slope (i.e. those p-adic modular forms whose eigensystems are parameterized by the Coleman-Mazur eigencurve) are trianguline at p, as
discovered by Kisin [Kis03] (before the name “trianguline” was invented and using
an equivalent description). This was later extended to more general eigenvarieties(in particular to eigenvarieties for unitary groups, see e.g. [BC09]). We will returnto this point in Section 9.6 when we discuss the relation of eigenvarieties and p-adic
automorphic forms with sheaves on Xd defined using XB and its compactification.
Remark 5.3.1. There are of course variants, where GLd is replaced by anarbitrary reductive group, but for simplicity of the exposition we limit ourselves tothe case of GLd here. We define the stacks in a way such that it is clear what thedefinitions will look like in general.
Following Drinfeld, Braverman and Gaitsgory [BG02] define two types of socalled Drinfeld compactifications of the stack of P -bundles on an algebraic curve,e P in the context of GalF -equivariant
and we will define their analogues XP and X
vector bundles, respectively in the context of (ϕ, Γ)-modules, below. Both of thesecompactifications have their advantages and their disadvantages: the compactification XP has the advantage that it is functorial with respect to inclusions P 0 ⊆ P ofparabolic subgroups, but has the disadvantage that the map αP : XP → XM doese P has the advantage that αP
not extend to XP . Conversely the compactification Xeeextends to αeP : XP → XM , but P 7→ XP is not functorial with respect to inclusions
P 0 ⊆ P of parabolic subgroups.
Before giving the formal definitions let briefly explain the motivation in thecase P = B. A B-structure on a (ϕ, Γ)-module DA is a complete (ϕ, Γ)-stable flag
0 = Fil0 ⊆ Fil1 ⊆ · · · ⊆ Fild = DAsuch that Fili /Fili−1 is a projective RF,A -module of rank 1. Using Plücker coordinates the datum of such a filtration is the same as the datum of the (ϕ, Γ)-stablerank 1 subobjects
^i
^i
Li =
Fili ⊆DA

54

M. EMERTON, T. GEE, AND E. HELLMANN

Vi
such that the quotient ( DA )/Li is a projective RF,A -module. The lines Li satisfy
some relations (called Plücker relations) that we do not spell out explicitly, but werefer to the more abstract description below that allows a more uniform descriptionof these relations (but has to pay the price that we consider the evaluation of DA
Vi
on all algebraic representations of GLd instead of just the exterior powers
DA ,compare also Remark 5.3.5 below).
The following example illustrates that there are families DA over rigid spaces
Sp A that admit a B-structure on a Zariski-open and dense subspace, but this Bstructure does not extend to all of Sp A. Hence we aim to somehow compactify theVisituation. The idea is to still parameterize lines Li ⊆DA that satisfy the Plücker
Virelations, but to drop the condition that ( DA )/Li is projective over RF,A . Thisobviously fixes the issue in the following example.
Example 5.3.2. Let F = Qp and consider the following family of filtered ϕmodules over the projective line P1 . Let
M = OP1 e1 ⊕ OP1 e2 and ϕM = diag(ϕ1 , ϕ2 )•for fixed ϕ1 6= ϕ2 ∈ Q×
p . Moreover we consider the Z-filtration Fil on M given byM i ≤ −1i
Fil M = L i = 0


0
i≥1

where L ⊂ M is the universal line over P1 . We consider the submodule M 0 =OP1 e1 ⊂ M . Then it is not possible to define a filtration Fili M 0 on M 0 (in thesense that Fili M 0 ⊂ M 0 is locally on P1 a direct summand) such that for all x ∈ P1
the subobject Fili M 0 ⊗ k(x) ⊂ M 0 ⊗ k(x) is the intersection of Fili M ⊗ k(x) withM 0 ⊗ k(x). If we insist that Fili M 0 ⊂ M 0 is (locally) a direct summand, then we
can only arrange this on A1 ⊂ P1 by setting(
M 0 i ≤ −1i0.
Fil M =
0i≥0However at x = ∞ the specialization Fili M 0 ⊗ k(x) is a proper subset of theintersection Fili M ⊗ k(x) ∩ M 0 ⊗ k(x).
Using the construction of (ϕ, Γ)-modules associated to filtered ϕ-modules (seeTheorem 5.2.4 and the discussion below it), this translates to the following situation.
Let
−1 ⊕2
R⊕2
RF,P1
F,P1 ⊂ D ⊂ t
be the (ϕ, Γ)-module over P1 associated to M and let L be the (ϕ, Γ)-stable line
defined by the sub-filtered ϕ-module M 0 ⊂ M . Then L ⊂ D is a B-structure on
D|A1 , but after specializing to ∞ ∈ P1 the cokernel of this inclusion has t-torsion.
In order to define these compactifications in general we introduce the following
notation, compare [BG02]. We fix a Borel subgroup B ⊆ G = GLd and write
T = B/U , where U is the unipotent radical of B. As usual we write h·, ·i forthe canonical pairing between X∗ (T ) and X ∗ (T ). Given a parabolic subgroup
B ⊆ P ⊆ G with unipotent radical UP and Levi quotient M = P/UP , we write+∗X ∗ (T )+M ⊃ X (T )G for the semi-groups of dominant weights for M respectively
for G. The set of simple coroots {αi ∈ X∗ (T ), i ∈ I} of G is indexed by the set

CATEGORICAL p-ADIC LANGLANDS

55

I = IG = {1, . . . , d − 1} of vertices of the Dynkin diagram of G. Given M as above,
the Dynkin diagram for M gives rise to a subset IM ⊆ I and we set
X ∗ (T )G,P = {λ̌ ∈ X ∗ (T ) | hαi , λ̌i = 0 for i ∈ IM }.
Moreover, we write SM = M/[M, M ] for the maximal torus quotient of M .
Given a (family of) GalF -equivariant vector bundle(s) E on XF̄ (or a family of(ϕ, Γ)-modules D over RF ) and an algebraic representation V of G we write V (E)(respectively V (D)) for the GalF -equivariant GL(V )-bundle (respectively a (ϕ, Γ)module with GL(V )-structure) given by the push-forward of E along G → GL(V ).
We view V (E) again as a vector bundle instead of a GL(V )-bundle, and similarly
λ̌
we view V (D) again as a (ϕ, Γ)-module. For a character λ̌ ∈ X ∗ (T )+G we write Vλ̌
for the irreducible G-representation of highest weight λ̌ and L for the canonicalrepresentation λ̌ : T → Gm of T .
naive
e naive below areas well as the definition of X
The following definition of XPP
the analogues of the corresponding compactifications of BunP defined in [BG02].
naive

Definition 5.3.3. The stack XP
is the category fibered in groupoids over
RigL parameterizing triples)(
EG ∈ XG (Sp A), ESM ∈ XSM (Sp A)naive
XP (Sp A) =(κλ̌ : Lλ̌ (ESM ) → V λ̌ (EG ))λ̌∈X ∗ (T )+ ∩X ∗ (T )G,P
G

such that the restriction of κλ̌ to fibers over Sp A is injective, the cokernels of thevarious κλ̌ are flat over Sp A and such that the following Plücker relations are
satisfied:- for λ̌ = 0 the map κ0 is the identity.
∗- for λ̌, µ̌ ∈ X ∗ (T )+G ∩ X (T )G,P the composition
κλ̌+µ̌

Lλ̌ (ESM ) ⊗ Lµ̌ (ESM ) = Lλ̌+µ̌ (ESM ) −−−→V λ̌+µ̌ (EG )→ (V λ̌ ⊗ V µ̌ )(EG ) =V λ̌ (EG ) ⊗ V µ̌ (EG )coincides with the map κλ̌ ⊗ κµ̌ .
The stack XP of equivariant P -bundles on XF̄ can be identified with the open
naive
substack of XP , where the cokernels of the maps κλ̌ are flat over Sp A × XF̄
(not just over Sp A) for all λ̌. It is obvious from the definition that the maps
Xnaive→ XG and Xnaive→ XM → XSM extend to morphisms
PP
β P : XP −→ XG
αP : XP −→ XSM .
naive
Moreover P 7→ XP
is functorial with respect to inclusions P 0 ⊆ Pof parabolic

subgroups (compatible with the projection maps α and β).
e naive is the category fibered in groupoids overDefinition 5.3.4. The stack X
P
RigL parameterizing triplesEG ∈ XG (Sp A), EM ∈ XM (Sp A)e naive (Sp A) =X
P
(κV : V UP (EM ) → V (EG ))V
where the maps κV are indexed by the finite dimensional G-representations, suchthat the restrictions of κV to the fibers over Sp A are injective, the cokernels of the

56

M. EMERTON, T. GEE, AND E. HELLMANN

various κV are flat over Sp A, and such that the following version of the Plücker
relations are satisfied:- if V is the trivial representation, then the map κV is the identity.
- for any G-representations V1 and V2 the following diagram commutes:
κV1 ⊗κV2(V1UP ⊗ V2UP )(EM )/ (V1 ⊗ V2 )(EG )=(V1 ⊗ V2 )UP (EM )κV ⊗V / (V1 ⊗ V2 )(EG )12- for a morphism of G-representations V1 → V2 the following diagram commutes:
κV1/ V1 (EG )V UP (EM )
1



κV2
/ V2 (EG ).
V2UP (EM )e naive where the cokernels
Again XP can be identified with the open substack of X
P
of the maps κV are flat (over Sp A × XF̄ , not just A-flat). Again, it is obvious fromthe definition that the maps XP → XG and XP → XM extend to morphisms
e naive −→ XG
βeP : XP

e naive −→ XM .
αeP : XP
e naive → Xnaive , inducing the identity on
Moreover, there is a forgetful map X
PP
XP , that is given by only remembering the maps κV for V = V λ̌ with λ̌ ∈ X ∗ (T )+
G∩
λ̌ UP
∗
factors through SM .
X (T )G,P . In this case the M -representation (V )naive
e naive coincide in the case P = B, but
Remark 5.3.5. The stacks XP
and X
P
are distinct otherwise. In our setting with G = GLd and in the case P = B it isenough to remember the maps κλ̌ for the fundamental weights of GLd . That is,an Sp A-valued point of XB can be described by tuples (E, L1 , . . . , Ld , ν1 , . . . , νd ),where E is an equivariant vector bundle of rank d, the Li are equivariant line bundles
and the νi are morphisms^iνi : Li −→Ethat are fiberwise (over Sp A) injective with A-flat cokernel and such that the νisatisfy some Plücker relations (that we do not spell out explicitly).
It turns out the the equivariance condition on vector bundles adds a new phenomenon which makes the above definitions too naive (hence the name): the stacknaive
e P . Roughly the reason for this behavior
XP is not dense in XP
respectively in X
is the existence of Hodge–Tate–Sen weights. The following example makes thisphenomenon more explicit:
Example 5.3.6. Let d = 2 and D ∈ X2 (L). Then a B-structure on D is given
by a subobject L = RF,L (δ1 ) ⊂ D such that the quotient D/RF,L (δ1 ) is a (ϕ, Γ)module of rank 1, say RF,L (δ2 ). It follows that wtτ (D) = {wtτ (δ1 ), wtτ (δ2 )}. As
the labeled Hodge–Tate weights of both D and the subobject L vary continuously

CATEGORICAL p-ADIC LANGLANDS

57

naive

on XB , it follows that on the closure of XB the weight of the subobject L must
be one of the weights of D. On the other hand, by definition of XB all the objects(D, tm L) for m ≥ 0 lie in XB .
In order to define good compactifications of XP we would like to take its closure
inside the naive compactifications in a reasonable way. In the case P = B this is
the content of the following theorem that will be proven in [HHS].
Theorem 5.3.7.
naive
e naive are rigid analytic Artin stacks.
=X(i) The stacks XB and XB
Bnaive(ii) There is a well defined “scheme-theoretic” image XB of XB in XB .
Sketch of proof: We sketch the proof of the fact that XB is an Artin stackin the 2-dimensional case. The proof can be divided into two steps: in a first step weconstruct the restriction Xwreg
of XB to a weakly regular subset Xwreg⊂ XT . ThisB
T
weakly regular subset ensures that all Ext2 -terms vanish which makes it possible toconstruct explicit charts. In a second step we use the fact that we can twist away2
and use the Beauville–Laszlo gluing lemma to reduce
the second cohomology Hϕ,Γthe general case to the first step.
Step 1 : Let us write Xwreg⊂ XT for the open subset of characters (δ1 , δ2 ) suchT2
(R(δ1 /δ2 )) = 0. If the base field L contains all Galois conjugates of F ,that Hϕ,Γ
then Xreg
T may be described as the set of all (δ1 , δ2 ) such thatY
δ1 /δ2 ∈/ {εz λ | λ ∈Z+ }.
τ :F ,→L.
Sp(A) be an affinoid open subset of Xwreg
T

We will write (δ1 , δ2 ) for
Let U =
the restriction of the universal characters to U , and use the same notation for the
2(RA,F (δ1 /δ2 )) vanishes
pullback of δi along maps from rigid spaces to U . As Hϕ,Γpointwise on U , there is a quasi-isomorphismdf••[E 0 −→ E 1 ] −→ (CA
, ∂•)
•
from
from a perfect complex concentrated in degrees [0, 1] to the Herr-complex CA•
Remark 5.1.17 (i) computing Hϕ,Γ (RA,F (δ1 /δ2 )). Moreover, this quasi-isomorphism
1stays a quasi-isomorphism after any pullback. The map f 1 : E 1 → ker ∂ 1 ⊂ CA
gives rise to a universal extension

0 −→ RY,F (δ1 ) −→ DY −→ RY,F (δ2 ) −→ 0
over the geometric vector bundle Y → U associated to E 1 . This extension induces
a morphism
g : Y −→ XB |U .
We claim that g is a smooth surjection. Indeed, g is surjective by construction, andrelatively representable as the diagonal of XB is representable. It is left to showthat g satisfies the infinitesimal lifting criterion. Let B  B̄ be a surjection ofaffinoid A-algebras and consider a commutative diagram
Sp(B̄)

h̄

/Y
g


Sp(B)

h



/ XB .

58

M. EMERTON, T. GEE, AND E. HELLMANN

We need to find a lift Sp(B) → Y making the diagram commute. The morphism h̄11
defines an element sB̄ ∈ Γ(Sp(B̄), EB̄) (where we write EB̄
for the pullback of E 1 toSp(B̄), and similarly for the other objects involved) and its image f 1 (sB̄ ) defines
an extension
0 −→ RB̄,F (δ1 ) −→ DB̄ −→ RB̄,F (δ2 ) −→ 0.
The morphism h gives a lift of the extension DB̄ to an extension DB of RB,F (δ2 )by RB,F (δ1 ) and (after fixing appropriate bases) we may view DB as a cocycle11
c1B ∈ CB
lifting f 1 (sB̄ ) ∈ CB̄
. To find a lift Sp(B) → Y we need to find an element
1sB ∈ Γ(Sp(B), EB ) such that the extension defined by f 1 (sB ) is isomorphic to theextension DB via an isomorphism lifting the canonical identification of DB̄ with0
the extension defined by f 1 (sB̄ ). In other words we need to find sB and c0B ∈ CB
0
1
1
0 0
reducing to 0 in CB̄ such that cB − f (sB ) = ∂ (cB ).
1
•
1
As EB
→ H 1 (EB) = Hϕ,Γ
(RB,F (δ1 /δ2 )) is surjective, there exists an element1
lifting sB that represents the cohomology class [c1B ] defined by the extens0B ∈ EB
0 00
1 0
1
0
sion DB . Hence there exists some element c00
B ∈ CB such that cB −f (sB ) = ∂ (cB ).
00
does
not
necessarily
reduce
to
0
over
,
as
a
prioric
We need to modify s0B and c00
B
B
B̄. However, it follows from the construction that its reduction maps to zero under∂ 0 , i.e.
0
0
1
0
1
c00
B̄ ∈ ker(∂ : CB̄ → CB̄ ) = ker(EB̄ → EB̄ ).
00
be a lift of
In particular we may regard c00
as an element e0B̄ of EB̄
. Let e0B ∈ EB
B̄00
cB̄ and set
sB = s0B + d(e0B ),0 0
c0B = c00
B − f (eB ).
10
0 0
Then c0B̄ = c00− f 0 (e0B̄ ) = 0 and c1B − f 1 (sB ) = ∂ 0 (c00
B ) − f (d(eB )) = ∂ (cB ) asB̄
desired.
Step 2 : Let U ⊂ XT be an open neighborhood of a point (δ1 , δ2 ) that is not
weakly regular and that admits a smooth surjection from an affinoid space Sp(A).
2Then there exists some N  0 such that Hϕ,Γ(RA,F (z N δ1 /δ2 )) = 0, compare
Remark 5.1.17 (iii). We consider the neighborhood U 0 = {(η1 , z N η2 ) | (η1 , η2 ) ∈ U }of (δ10 , δ20 ) = (δ1 , z N δ2 ) ∈ Xwreg
and the morphism a : U → U 0 given by (η1 , η2 ) 7→TN(η1 , z η2 ). This morphism fits in a diagram

V0

b


/ XB |U 0a/ U 0,XB |U

U

where V 0 → XB |U 0 is a smooth surjection from a rigid analytic space (that exists
by Step 1) and where b is the morphism defined by the pullback0/ R(η1 )(5.3.8)/DOι

0/ R(η1 )?
/ D0/ R(η2 )O/0?
/ z N R(η2 )/ 0.

CATEGORICAL p-ADIC LANGLANDS59

We define V → XB |U as the projection to the first factor of the fiber product
V = V 0 ×XB |U 0 XB |U −→ XB |U .
As V 0 → XB |U 0 is a smooth surjection we are left to show that b is relatively
representable. This is the content of Lemma 5.3.9 below. This finishes the proof
that XB is an Artin stack.
We refer to Remark 5.3.10 below for some comments on how to adapt thenaivestrategy to prove the corresponding claims for XB and XB .
Lemma 5.3.9. The morphism b : XB → XB defined by mapping an extension
D to the pullback D0 defined by (5.3.8) is relatively representable.
Sketch of proof: Let V 0 = Sp(A) → XB be a morphism defining the extension D0 . For any D as in (5.3.8) the morphism ι induces an isomorphism∼
D0 [1/t] −→ D[1/t]. Let us write D̂ for the scalar extension D̂ = D ⊗RF F∞ [[t]] andsimilarly D̂0 . By the Beauville–Laszlo gluing lemma, the (ϕ, Γ)-module D may bereconstructed from D0 and the datum of the Γ-stable (A ⊗Qp F∞ )[[t]]-lattice in thefree (A ⊗Qp F∞ )((t))-module D̂0 [1/t]. We may assume that D0 is free and choosea basis of D0 such that the first basis vector spans the subobject RA,F (η1 ). Using
this basis we may view D̂ as a Γ-stable A-valued point of the affine Grassmannian
Grass∞ = L(ResF∞ /Qp GL2 )/L+ (ResF∞ /Qp GL2 ).
The lattice D̂0 is then identified with the base point e0 ∈ Grass∞ . Moreover thefact that D is an extension as in (5.3.8) translates to the fact that the lattice D̂lies in the closed subspacetν L<0 U∞ t−ν e0 ⊂ Grass∞ ,where ν = (1, −N ) and where U∞ ⊂ ResF∞ /Qp GL2 is the Weil restriction of theunipotent upper triangular matrices.
Conversely, the Γ-stable lattices in tν L<0 U∞ t−ν e0 precisely describe the pullbackof V 0 → XB along b.
There are two technical difficulties in this proof:(a) As F∞ is not finite over Qp , the Weil restriction ResF∞ /Qp GL2 is not
representable by a rigid analytic space.
(b) The subspace tν L<0 U∞ t−ν e0 is an ind-scheme (or ind-rigid analytic space)rather than a scheme (or a rigid analytic space).
By working a bit more carefully, we can overcome (a) by descending to some finite subextension Fn = F (µpn ) ⊂ F∞ . To overcome (b), we need to show thatthe Γ-invariant lattices in fact lie inside some closed rigid analytic subspace oftν L<0 U∞ t−ν e0 . In the case F = Qp the main ingredients of this argument are as
follows (for general F a variant of this argument works as well): on L-valued points
either D̂ is uniquely determined by D̂0 and the diagram (5.3.8), or D0 is de Rhamup to twist. In the latter case both D̂0 and D̂ lie in certain affine Schubert cells withrespect to a third base point D̂00 in the affine Grassmannian (in the de Rham casethe lattice D̂00 can be chosen such that it defines a (ϕ, Γ)-module with Hodge–Tateweights (0, 0) and the Schubert cells containing D̂0 respectively D̂ depend on theirrespective Hodge–Tate weights).
60

M. EMERTON, T. GEE, AND E. HELLMANN

Remark 5.3.10.
naive(i) In order to prove that XB
and XB are rigid analytic Artin stacks we follow
the same idea as in the above sketch. In order to construct charts for XB we haveto consider the closure of tν L<0 U∞ t−ν e0 in the affine Grassmannian. To construct
naive
charts for XB
we have to consider the union of certain tν L<0 U∞ t−ν e0 .
(ii) The construction of smooth surjections from a rigid analytic space to XB (renaive
spectively XB , XB ) crucially uses the fact that XT is an Artin stack. For a moregeneral parabolic subgroup P we can prove the same result for XP and its come P , if we assume that XM is an Artin stack, i.e. if we canpactifications XP and X
prove Conjecture 5.1.18 for M .
e P and will be(iii) The maps αP , β P , αeP , βeP of course restrict to XP , respectively X
denoted by the same letters in the following.
One important difference with the classical case of vector bundles on algebraic
curves is that the map
β B : XB −→ XG
is neither proper nor surjective (and a similar remark applies to more general parabolic subgroups as well). The reason that the map is not surjective is simply
that not every Galois representation (or more generally not every (ϕ, Γ)-module)is trianguline. The easiest way to see this is to compare the dimensions of thestacks Xd and XB (or of some related rigid analytic spaces of Galois representations). The failure of properness is explained in Remark 5.3.14 below. We state thefollowing results/expectations that should clarify differences and similarities withthe algebraic situation.
Theorem 5.3.11. The morphisms(5.3.12)XP −→ XG × XSM ande P −→ XG × XM
X

are representable by rigid analytic spaces and proper.
This result is proven using results on triangulations in families by Liu [Liu15]
and Kedlaya–Pottharst–Xiao [KPX14]. The following conjecture relates the stack
XB (or rather its image in Xd × XT ) to the trianguline variety and hence to eigenvarieties and p-adic automorphic forms. Conjectures 9.6.8 and 9.6.27 will make therelation to spaces of p-adic automorphic forms more precise.
Conjecture 5.3.13.
(i) The scheme-theoretic images of the morphisms (5.3.12) are equidimensional ofdimension dim ResF/Qp P .
(ii) Let P = B and let Rρ̄ denote the framed deformation ring of a fixed residual
representation ρ̄ : GalF → GLd (k). Then the pullback of the scheme theoretic imageof (5.3.12) along the smooth map(Spf Rρ̄ )rig × T d −→ XG × XT
coincides with the trianguline variety of [BHS17b].
Here scheme-theoretic image should of course be understood as an analogue ofthe usual notion of scheme-theoretic images in the context of rigid analytic Artinstacks.

CATEGORICAL p-ADIC LANGLANDS

61

Remark 5.3.14. In fact one expects that the image of XB in XG is Zariskidense: it should be seen as an analogue of the infinite fern of Gouvea–Mazur[GM98] (and its generalization by Chenevier [Che13] and Nakamura [Nak14]).
The infinite fern argument implies directly that the image β B (XB ) is dense in theunion of all irreducible components of XG that contain a crystalline point in theirinterior. We expect that all components of XG contain a crystalline point in theirinterior (which in fact would be a consequence of the conjectural description of theconnected components of XG as a consequence of Conjecture 5.1.32). In light of therelation with eigenvarieties the failure of properness can also be seen as consequenceof the fact that the construction of eigenvarieties involves Fredholm hypersurfacesof infinite degree over the weight space.
Remark 5.3.15. From a purely deformation theoretic viewpoint the expecteddimension of the stack XP of equivariant P -bundles is dim ResF/Qp P . In fact
we expect that XP is equidimensional of this dimension and is a local complete
intersection. The p-adic situation seems to be rather different to the situation withthe spaces of L-parameters [Hel20, Remark 2.2], [Zhu20, Remark 2.3.8] where
the stacks of L-parameters for parabolic subgroups can fail to be equidimensional(whereas the stacks of L-parameters for reductive groups are always equidimensionaland complete intersections). In the situation with stacks of L-parameters one ishence forced to work with derived stacks due to this phenomenon. In the p-adic
case we expect that the stacks XP (and similarly their compactifications XP ande P ) do not have any additional non-trivial derived structure. In fact for fixed dX
one can show that XB is equidimensional if [F : Qp ] is large (compared with d).
For the time being we do not have an argument for all F , but it seems reasonable
to expect equidimensionality independent of the degree [F : Qp ].
Motivated by Conjecture 5.3.13 we write XG,tri ⊂ XG × XT for the scheme
theoretic image of XB under the proper map β B . The following theorem gives apartial description of the preimages of the morphism XG,tri → XG that will turn out
to be important in the construction of companion points in eigenvarieties (i.e. in
the construction of overconvergent p-adic automorphic forms of finite slope withprescribed associated Galois representation) in Section 9.6. Before describing thepreimages we need some preparation.
Let D ∈ XG (L) be a crystalline (ϕ, Γ)-module and assume that the eigenvalues ϕ1 , . . . , ϕd of the Frobenius on the associated Weil–Deligne representation
WD(D) lie in L and satisfy ϕi /ϕj 6= 1, q for all i 6= j. Moreover, we write
λ = (λτ,1 ≥ · · · ≥ λτ,d )τ :F ,→L for the Hodge–Tate weights of D and assume that λis regular, i.e. we assume λτ,i 6= λτ,j for all τ and i 6= j. We fix an ordering of the
Frobenius eigenvalues ϕ1 , . . . , ϕd or equivalently a complete Frobenius stable flagF• on WD(D). For each embedding τ : F ,→ L we define the Weyl group element
wF ,τ ∈ Sn as the relative position of the τ -part of the Hodge filtrationFil•τ ⊆ WD(D) = WD(D)τ ⊆Y
τ 0 :F ,→L

WD(D)τ 0 = WD(D) ⊗Qp L

with respect to F• (note that by the regularity assumption on λ the filtration Fil•τ
is indeed a full flag).

62

M. EMERTON, T. GEE, AND E. HELLMANN

We associate to the flag F• (equivalently, to the ordering of the Frobenius
eigenvalues ϕ1 , . . . , ϕd ) the unramified character δF• : T = (F × )n → L× given by(x1 , . . . , xn ) 7−→(5.3.16)dY

unrϕi (xi ).

i=1

Moreover, associated to a tuple (µτ )τ ∈ ZHom(F,L) we have the characterY(5.3.17)z µ : x 7−→
τ (x)µτ .
τMore generally for µ = (µ1 , . . . , µd ) with µi ∈ ZHom(F,L) we write
z µ : (x1 , . . . , xd ) 7−→

d
Y
i=1

z µi (xi ) =d YYτ (xi )µi,τ .

i=1 τThe following characterization of the points in XG,tri above D is [BHS19, Theorem
4.2.3].
Theorem 5.3.18. Let D ∈ XG (L) be a crystalline (ϕ, Γ)-module of regular
Hodge–Tate weight λ and such that the eigenvalues ϕ1 , . . . , ϕd of the Frobenius on
WD(D) lie in L and satisfy ϕi /ϕj 6= 1, q for i 6= j. Then{δ ∈ XT | (D, δ) ∈ XG,tri }[= {δF• z wλ |w = (wτ )τ ∈ W such that wF ,τ  wτ w0 for all τ },F•

where the union runs over all Frobenius stable flags F of WD(D) and where w0 ∈ Sd
is the longest element.
Remark 5.3.19.
(i) Following Hansen [Han17, Conjecture 6.2.3], there is a general conjectural description of the fibers of XG,tri → XG without assuming that the (ϕ, Γ)-module D
is crystalline (and without the regularity assumptions in the theorem).
(ii) In the case of crystabelline (ϕ, Γ)-modules this description is more or less thesame as in the crystalline case. In the semi-stable case (and more generally in thesemistabelline case) we only take those flags F• that are stable under the monodromy on the Weil–Deligne representation.
(iii) It is reasonable to ask for a generalization from semistabelline to all de Rham(ϕ, Γ)-modules D. In this case the image of XB in XG × XT should be replacedby the image of XPQin XG × XSM , where P is the parabolic subgroup containingr
B with Levi M = i=1 GLdi such that the Weil–Deligne representation WD(D)associated to D is a direct sum of indecomposable Weil–Deligne representations
D1 , . . . , Dr of respective dimension di (compare [BD21]).
The following expectation is motivated by the relation of the trianguline variety
with eigenvarieties and with the patched version of the eigenvariety [BHS17b] (andby Conjecture 6.2.4(2) respectively Conjecture 9.6.16).
Conjecture 5.3.20. Let us write πB : XB → XG × XT for the canonical(proper) projection.
(i) There are isomorphisms∼∼
RπB,∗ O, RπB,∗ ω
.
= OX
= ωX
XB

G,tri

XB

G,tri

CATEGORICAL p-ADIC LANGLANDS

63(ii) The direct image RπB,∗ (OXB ([F : Qp ]ρ0 )) is concentrated in degree 0 and isa Cohen–Macaulay module (i.e. its dual RπB,∗ (ωXB (−[F : Qp ]ρ0 )) is also concentrated in degree 0).
Here ρ0 is the algebraic character (0, −1, . . . , −d + 1) of T = Gdm that we viewas a line bundle on ∗/Gdm and pull it back to XB along the canonical map
XB → XT = Xd1 = (T /Gm )d → ∗/Gdm ,see Section 7.1 for the identification of X1 .
Moreover, it makes sense to ask whether the trianguline variety XG,tri is normal
and Cohen–Macaulay (note that Cohen–Macaulayness would be a consequence of(i) of the conjecture, if XB is Cohen–Macaulay). We point out that the sheafin (ii) of the conjecture should be regarded as a variant of the coherent Springer
sheaf from [BZCHN20] (respectively [Zhu20, 4.4] or [Hel20, 2.3]) and hence (ii)should be seen as a variant of [BZCHN20, Conjecture 4.15] (respectively [Zhu20,
Conjecture 4.4.2] or [Hel20, Conjecture 2.17]).
In the case F = Qp and GL2 we can prove the above conjecture and computethe singularities as well as the fiber dimensions of the Cohen–Macaulay modules
in (ii) (these fiber dimensions have some interpretation in terms of the conjectural
correspondence of locally analytic representations with coherent sheaves on Xd ,Conjecture 6.2.4).
Theorem 5.3.21. Let F = Qp and d = 2, i.e. G = GL2 .
(i) The stack XB is smooth.
(ii) The stack XG,tri is a rigid analytic Artin stack9 and normal and Cohen-Macaulay.
(iii) The morphism πB : XB → XG,tri is birational with geometrically connected
fibers and the canonical map
OXG,tri → RπB,∗ OX B
is an isomorphism. In particular we also have an isomorphism
RπB,∗ ωX B ∼= ωXG,tri .
(iv) Let F = RπB,∗ (OXB (ρ0 )) and G = RπB,∗ (ωXB (−ρ0 )). Then F and G areconcentrated in degree zero and in particular are Cohen-Macaulay modules.
(v) For x = (D, δ1 , δ2 ) ∈ XG,tri the fibers F ⊗ k(x), G ⊗ k(x) and ωXG,tri ⊗ k(x) are
one-dimensional unless D = R(δ) ⊕ t−n R(δ) with n ≥ 0 and δ1 = δ. In this casedim F ⊗ k(x) = 2(
2
dim G ⊗ k(x) =1(3
dim ωXG,tri ⊗ k(x) =
2

n=0
n≥1
n=0
n ≥ 1.

9As we do not know whether X = X is a rigid analytic Artin stack this should be interpreted2
G
as: there is a rigid analytic Artin stack XG.tri that embeds as a closed substack into XG × XT .
The map XB → XG × XT factors through XG,tri and is scheme-theoretically dense.

64

M. EMERTON, T. GEE, AND E. HELLMANN

5.3.22. Local models. Assume for the remainder of this section that L contains
all Galois conjugates of F (inside F̄ ). We discuss the relation of stacks of (ϕ, Γ)modules over RF with stacks of (semi-linear) Γ-representations on F∞ [[t]]. We
don’t know whether it can be expected that the latter stacks are Artin stacks (itseems that they can be rather ill-behaved), but their completions at a fixed Hodge–
Tate weight turn out to be well-behaved. At certain “nice” points the morphismfrom a stack of (ϕ, Γ)-modules to these stacks is formally smooth and hence weobtain rather explicit local models for stacks like XB , generalizing the local models
for the trianguline variety of [BHS19].
We define the groupoid XdR,+
over RigL as follows. For an affinoid algebra A
dwe setcontinuous, semi-linear Γ-representations
XddR,+ (Sp A) =b ∞ [[t]]-modules of rank d ,on finite projective A⊗Fb ∞ [[t]] = (A ⊗Qp F∞ )[[t]].
where (by slight abuse of notation) we write A⊗F
Remark 5.3.23.
(i) In a similar way we can define a groupoid XdR
d by mapping Sp A to the groupoidb ∞ ((t))-modules of rank d with a semi-linear Γ-action thatof finite projective A⊗Fb ∞ [[t]]-lattice.
(locally on Sp A) admit an A⊗F(ii) The motivation to consider these stacks (and the explanation for their name) is
Fontaine’s theorem [Fon04] that the category of semi-linear Γ-representations on
finite free F∞ [[t]]-modules (respectively on finite dimensional F∞ ((t))-vector spaces)+
is equivalent to the category of semi-linear GalF -representations on finite free BdR
modules (respectively on finite dimensional BdR -vector spaces). We expect that asimilar equivalence holds true in families over rigid analytic spaces, but, as we do
not have a proof, we restrict our attention to the semi-linear Γ-representations, as
they seem to be easier to handle.
Expectation 5.3.24.
(i)The groupoids XdR,+
and XdR
d are stacks.
ddR,+(ii) The diagonal of Xd
is representable.
(iii) The morphism XddR,+ → XdR
d is representable.
Though we do not know whether one should expect that XdR,+
and XdR
d are
dArtin stacks, both stacks admit versal rings at rigid analytic points, and moregenerally their completions along a fixed Hodge–Tate–Sen weight are formal rigid
analytic Artin stacks (a notion that we will not formally introduce here). In orderto make sense of this statement, we note that, similarly to the definition of theweight map ωd in (5.1.31), we have a weight map
YωddR : XdR,+−→ WTd,F ∼Ad /Sd=dF ,→Lb ∞ [[t]]-module DA to the characteristicthat maps, on A-valued points, an A⊗F
polynomial of the derivative of the Γ-action on DA /tDA (recall that we have fixed
L large enough such that [F : Qp ] = |Hom(F, L)|).
Remark 5.3.25. The Hodge–Tate–Sen weights of an object DA ∈ XdR
d (Sp A)are only well defined mod Z. More precisely, under the identification of A ⊗Qp F =CATEGORICAL p-ADIC LANGLANDS

Q

65

F ,→L A we can define the weight of DA as an element ofd
A/Z /Sd .

Y
F ,→L

Fontaine’s theory of almost de Rham representations gives a description of thedR,+stacks XdR
after completion at a fixed integral Hodge–Tate weight, andd and Xd
similar results hold true for the completion at any fixed Hodge–Tate–Sen weight.
We briefly recall Fontaine’s classification [Fon04] of BdR -representations (or equivF
alently F∞ ((t))-representations). Let us write ModGal
BdR for the category of continuous semi-linear GalF -representations on finite dimensional BdR vector spaces, andF
similarly we define ModGal, ModΓF∞ ((t)) and ModΓF∞ [[t]] in the obvious way. For a
B+
dR

F ,a
Hodge–Tate–Sen weight a ∈ F we denote by ModGal
the full subcategory of all
BdR
objects whose Hodge–Tate–Sen weights are congruent to a modulo Z. ThenM
F ,aF
ModGalModGal
BdR =
BdR

a∈F/Z
F
is a block decomposition of ModGal
BdR , i.e. the higher Ext-groups between objects indifferent summands vanish (this is a consequence of [Fon04, Thm. 3.19]).
GalF ,0The block ModB
is called the block of almost de Rham representationsdR(or presque de Rham representation in French). It has the following description interms of linear algebra objects (see [BHS19, Prop 3.1.1]):

Theorem 5.3.26 (Fontaine). There is an equivalence of categories
F ,0−→ FilRep(Ga )DpdR : ModGal
B+dR+
from the category of almost de Rham representations over BdR
to the categoryof finite dimensional F -vector spaces equipped with an algebraic representation of
the additive group Ga and with a separated and exhaustive Z-filtration Fil• stable
F ,0
F ,0
to ModGal
under the Ga -action. Under this equivalence, passing from ModGal
BdR
B+
dR

corresponds to forgetting the filtration.
We recall that (over fields of characteristic 0) a Ga -representation is nothing
but the action of a nilpotent operator. Moreover, an almost de Rham representation+
V over BdR(or over BdR ) is de Rham if and only if the nilpotent operator N :DpdR (V ) → DpdR (V ) (defined by the Ga -action) is zero.
There is a definition of the functor DpdR in the spirit of period rings [BHS19,
F
3.1] that we do not recall here. On the equivalent category ModΓF∞ [[t]] ∼= ModGal
B+dR

one can characterize almost de Rham representations and the functor DpdR as
follows: Given an F∞ [[t]]-module Λ with continuous Γ-action, we can consider(similarly to the definition of Hodge–Tate–Sen weights) the derivation of the Γaction at 1 which gives a derivation∂Λ : Λ −→ Λdabove the derivation t dt
on F∞ [[t]]. We can then define[
DpdR,∞ (Λ) =ker(∂ΛN : Λ[1/t] → Λ[1/t])
N ≥1

66

M. EMERTON, T. GEE, AND E. HELLMANN

as the sub-F∞ vector space of Λ[1/t] on which ∂Λ is nilpotent. In particular ∂Λ
induces an F∞ -linear, nilpotent endomorphism N∞ of DpdR,∞ (Λ). The representation Λ is almost de Rham if and only if the canonical map
DpdR,∞ (Λ) ⊗F∞ F∞ ((t)) −→ Λ[1/t]is an isomorphism. In that case we write Λ0 = DpdR,∞ (Λ) ⊗F∞ F∞ [[t]] and referto Λ0 as the standard lattice. Then the lattice Λ ⊂ Λ0 [1/t] defines a separated andexhaustive Z-filtration Fil•∞ on DpdR∞ (Λ) = Λ0 /tΛ0 by sub-F∞ vector spaces thatare stable under the action of N∞ , as Λ is stable under ∂Λ . Again in a similar fashion
to the definition of Hodge–Tate–Sen weights, the triple (DpdR,∞ (Λ), N∞ , Fil•∞ ) is
equipped with a Γ-action which allows us to descend10 (DpdR,∞ (Λ), N∞ , Fil•∞ ) from
F∞ to the desired triple (DpdR (Λ), N, Fil• ) over F .
The equivalence in Fontaine’s theorem lies at the heart of the following de+scription of the completions of the stacks of BdR-representations. Note that for an
L-algebra A we haveYb Qp A =b F A,
F∞ [[t]]⊗F∞ [[t]]⊗τ :F ,→L

and hence we can apply Fontaine’s description for each embedding τ and for eachlabeled Hodge–Tate–Sen weight separately. Let us write G = ResF/Qp GLd andg = Lie G for the remainder of this subsection. Moreover, we fix an integral labeled
Hodge–Tate weightY(Zd )/Sd ⊆ WTd,F (L)λ∈τ :F ,→L

We denote by [λ] its class modulo Z (which of course only remembers that theweight was integral but does not distinguish between various λ).
b
We describe the formal completions (XdR,+)b[λ] and (XdRd )[λ] of the groupoidsd
XdR.+
and XdR
along the fixed weight λ (respectively along the locus where theddHodge–Tate–Sen weight is congruent to λ modulo Z). This result is very similar to[BHS19, Cor. 3.1.6, Cor. 3.1.9]. LetegPλ = {(A, F • ) ∈ g × G/Pλ | AF • ⊂ F • } ⊆ g × G/Pλbe the variant of Grothendieck’s simultaneous resolution of singularities that parameterizes pairs (A, F • ) consisting of an endomorphism A and an A-stable flag F •eof
Q type Pλ (compare Section 5.2 for the notation). Recall that gPλ is a subvariety of
gl
×GL
/P
and
again
we
think
of
the
filtration
on
G/P
d
λτ
λ as an [F : Qp ]-tupledτ
of Z-filtrations that jump in degrees λ.
Proposition 5.3.27. The functor DpdR,∞ induces isomorphisms
∼

b
b
(XdR
d )[λ] −→ g0 /G∼(XdR,+)bλ −→ (egPλ )b0 /G,
dwhere on the right hand side we complete at the closed subspace of all nilpotent endomorphisms (i.e. at the nilpotent cone, respectively its partial Springer resolution).
10We caution the reader that this descent is not completely formal and requires some work.

Recall that we pointed out in the discussion of Hodge–Tate weights that (DSen (DA ), Θ) does notdescend to an object over F ⊗Qp A. However, this can be done after completion at a fixed integral
Hodge–Tate weight. The issue here is a similar one.

CATEGORICAL p-ADIC LANGLANDS

67

Remark 5.3.28.
(a) The proposition implies that the completion of XdR,+at an integral Hodge–Tatedweight is a formal rigid analytic Artin stack (with an appropriate definition of suchan object).
(b) In fact, applying Fontaine’s theory for arbitrary (i.e. not necessarily integral)
respecweights λ one can obtain a similar description of the completion of XdR,+dtively XdRatλ.
d
In the neighborhood of certain “nice” points, XdR,+
can be thought of as a local
dmodel of the stack Xd . A sample result for this is the following theorem, compare[HMS18, Theorem 3.1.1]. We will discuss the related situation for B-structures(resp. P -structures) and their Drinfeld compactifications below.
Theorem 5.3.29. Let x ∈ Xd be a crystalline point with regular Hodge–Tateweight and Frobenius eigenvalues ϕ1 , . . . , ϕd that satisfy ϕi /ϕj ∈/ {1, q}. Then themorphism Xd → XdR,+
isformallysmoothatx.
dGiven a parabolic subgroup P ⊆ GLd we can consider P -structures (i.e. partial
flags) on objects in XdR
and XdR,+, which allows us to define stacks XdR
P and
d
d
dR,+
XP . Parallel to the above discussion of Drinfeld compactifications, we woulddR,+
e dR,+ of the stack XdR,+ that fit intoand X
like to define “compactifications” XPPP
the diagram
e dR,+
X
P



dR,+
XP

 {

#

XdR,+
M

# 
XdR,+
SM .

XddR,+

We will define such compactifications, and prove finiteness statements about
them after formal completion at an integral weight λ.
As above we fix an integral labeled Hodge–Tate weight λ = ((λτ,1 , . . . , λτ,d )τ :F ,→L .
We restrict ourselves to the easiest case in which λ is regular. As λ is only welldefined up to permutation, we choose its dominant representative, i.e. we assume
λτ,1 > λτ,2 > · · · > λτ,d for all τ . In particular the parabolic subgroup Pλ equals
the Weil restriction of the Borel subgroup (ResF/Qp B)L . We write e
g = e
gPλ for
simplicity.
We now give a description similar to Proposition 5.3.27 of the completion of
XdR,+
and use this description to define “compactifications” of the formal compleP
tion. Consider the space[(5.3.30)
XB,P = eg ×g egP =Vw ⊆ g × G/B × G/P,w∈WP

Q

where W = τ Sd is the Weil group of ResF/Qp GLd and WP ⊆ W is the set ofminimal length representatives of elements in W/WM (WM ⊆ W being the Weylgroup of the Levi ResF/Qp M of ResF/Qp P ). For w = (wτ )τ ∈ Wp , the space
Vw is defined to be the preimage of the Bruhat cell G(1, w) ⊆ G/ResF/Qp B ×

68

M. EMERTON, T. GEE, AND E. HELLMANN

G/ResF/Qp P (recall that we write G = ResF/Qp GLd ). Over Vw , the intersectionof the universal flag on G/ResF/Qp B and over G/ResF/Qp P defines a well definedflag in ResF/Qp M/ResF/Qp BM . Given w, we write wλ for the permutation of λ byw. Moreover, we can view wλ as a weight of M . In particular for each w ∈ W , weobtain a diagram(5.3.31)

Vw /G
|

&
e
m/Res
F/Qp M,eg/G

e for the Lie algebra of ResF/Qp M .
where by abuse of notation we write m
Proposition 5.3.32. The functor DpdR induces an isomorphism of the diagram(XdR,+)bww0 λPx'
)bww0 λ(XdR,+M(XdR,+)bλdwith the completion of (5.3.31) at the closed subspace of nilpotent endomorphisms.
Here (XdR,+)bww0 λ is the completion of XdR,+
along the preimage ofPP)bww0 λ ⊆ (XdR,+)(XdR,+M
M
dR,+
under the canonical map XP→ XdR,+.
M

In the diagram (5.3.31) there are two ways to complete Vw /G into a stack proper
over eg/G, and again only one of these compactifications will admit a morphism toem/M. The easiest way is to consider the Zariski-closure X w of Vw inside the space
XB,P = eg ×g egP defined in (5.3.30). In this case the intersection of the two flags over
G/ResF/Qp B and G/ResF/Qp P does not induce a well defined family of filtrations
on the graded pieces (with respect to the P -filtration), but only on their top exterior
powers. We obtain a diagram
X w /G{eg/G&
Lie SM /SM .

Note that the formation of X w is functorial with respect to inclusions P 0 ⊆ P as
can be deduced from the identification
XB,P /G = q −1 (p)/ResF/Qp P,
where q : e
g → g is the canonical projection and p ⊆ g is the Lie algebra of0
ResF/Qp P ⊆ G. The stack X w /G is canonically isomorphic to X w /ResF/Qp P ,0

where X w ⊆ q −1 (p) is the closure of the locally closed subset Vw0 ⊆ q −1 (p) suchthat Vw /G = Vw0 /ResF/Qp P .
Note that for fixed w the intersection of the universal flag on G/ResF/Qp B withthe partial standard flag (of type P ) gives rise to a well defined flag in the graded

CATEGORICAL p-ADIC LANGLANDS

69

ew to
e and we can define X
pieces of the standard flag. This induces a map Vw0 → m
0−1
e Then we obtain a diagram
be the closure of Vw inside q (p) × m.
ew /ResF/Q P
X
p

e
g/G

(
e
m/Res
F/Qp M,y

as well as a canonical projection
ew /ResF/Q P −→ X w /G
πP : Xp
compatible with the projections to eg.
Proposition 5.3.33.
ew is normal and Cohen–Macaulay.
(i) If P = B, then the scheme X w = X(ii) For a general parabolic subgroup P the scheme X w is unibranched.
These results about the local geometry of X w are proven in [BHS19, Prop.
2.3.3, Thm. 2.3.6] respectively in [Wu21a, Thm. 2.12].
Expectation 5.3.34.
ew are normal and Cohen–Macaulay.
(i) The schemes X w and X(ii) The morphism
ew /ResF/Q P −→ X w /G
πP : Xpsatisfies RπP,∗ O = O and RπP,∗ ω = ω.
We can use these objects to define compactifications of the completion XdR,+P
by settingdR,+ be dR,+ )b = (Xew /P )b .
(X
) = (X w /G)b and (X
P

wλ

0

P

wλ

0

Remark 5.3.35. Of course, as the notation suggests, we would like to be abledR,+
e dR,+ before completion.
to define stacks XP
and X
P
By construction these compactifications fit into a diagram
e dR,+ )b
(X
wλ
P


dR,+ b
)wλ

(XP
 y

(XddR,+ )bλ'(XdR,+
)bwλ
M
'



b
(XdR,+SM )det(wλ) ,where we write det(wλ) for the weight of SM that is given by the top exterior poweron each GLm -block of the Levi M .
Finally the following result reconnects these stacks, and their explicit local
description (respectively: their definition), to the Drinfeld compactifications of XP .
Part (ii) of the theorem is a generalization of the local model for the trianguline
variety in [BHS19].

70

M. EMERTON, T. GEE, AND E. HELLMANN

Theorem 5.3.36.
(i) The canonical map(XP )bwλ −→ (XdR,+)bwλP
induced by XP → XdR,+on completions extends to morphisms
PdR,+ b)wλ(XP )bwλ −→ (XPe P )b −→ (Xe dR,+ )b .
(X
wλ
wλ
P
e P and write x for its image in XP . Assume that the image xM =(ii) Let xe∈X(D1 , . . . , Dr ) of x
e in XM is regular in the sense thatdR,+
- the map XM → XM
is formally smooth at xM .
- for i 6= j the Ext-groups Ext0 (Di , Dj ) = 0 = Ext2 (Di , Dj ) = 0 vanish.

Then the morphismdR,+ b)wλ(XP )bwλ −→ (XPis formally smooth at x.

Conjecture 5.3.37. In the situation of (ii) of the theorem the morphisme P )b −→ (Xe dR,+ )b
(X
wλ
wλ
P
is formally smooth at xe.
Of course in combination with the results on the local structure of the varieties X w this gives an explicit description of the local geometry of the Drinfeld
compactifications.
In the case P = B the following theorem makes the situation even a bit more
precise: Let ϕ1 , . . . , ϕd ∈ L× such that ϕi /ϕj 6= 1, q for i 6= j. We write δsm for theunramified character (t1 , . . . , td ) 7→ unrϕ1 (t1 ) · · · unrϕd (td ). For w ∈ W let us write
XB,zwλ δsm ⊆ XB
for the preimage of z wλ δsm ∈ XT and let
XB,zwλ δsm ⊆ XBbdenote the formal completion of XB along this closed substack. We write[
Xd,(λ,δsm )−tri =β B (XB,zwλ δsm ) ⊆ Xdw∈W

for the union of the images of XB,zwλ δsm in Xd . Note that the assumptions on δsm
imply that the restriction of β B to (the formal completion of) XB,zwλ δsm is a closed
embedding.
By definition this is the closed substack of Xd consisting of all (ϕ, Γ)-modules
D such that D admits a filtration Fil• with graded pieces
RF (unrϕi z (λwτ (i) )τ )(see 5.3.17 for the notation) for some w ∈ W . Over the closed substack Xd,(λ,δsm )−tri
this filtration does not glue to a filtration of the universal (ϕ, Γ)-module D̃ (asthe Weyl group element w might vary). However, this turns out to be true afterCATEGORICAL p-ADIC LANGLANDS

71

inverting t, and we write M• ⊆ D̃[1/t] for this filtration. With this notation wecan consider the formal substackXbd,(λ,δsm )−tri ⊂ Xd
parameterizing deformations D̃0 of D to infinitesimal neighborhoods of Xd,(λ,δsm )−tri
such that M• lifts to a (by our choice of δsm : necessarily unique) filtration M0• ⊂D̃0 [1/t] stable under ϕ and Γ. The following theorem is basically proven in [BHS19].
Theorem 5.3.38.
(i) The morphismdR,+ b)wλ ∼= (X ww0 /G)b0

XB,zwλ δsm −→ (XBb⊗[F :Qp ]is formally smooth with relative dualizing sheaf L2ρ
(ii) There is a formally smooth morphism.

fλ,δsm : Xbd,(λ,δsm )−tri −→ (g̃ ×g g̃)b0⊗[F :Qp ]with relative dualizing sheaf L2ρ
by the morphism in (i).

, whose restriction to β B (XB,zwλ δsm ) is givenb

We define the sheaf L2ρ that occurs in the statement of the theorem: The stack
has a canonical map to ∗/Gdm (as it sits inside XB which maps to XT
and then further projects to ∗/Gdm ). Hence it makes sense to pull back the line
bundle that is associated to the character given by the sum of the positive roots 2ρ.
This defines the line bundle in (i). To make sense of this line bundle in (ii) we notethat we can write L2ρ as a pullback of a G-equivariant line bundle on X ww0 thatis in fact defined on all of g̃ ×g g̃ ⊂ g × G/B × G/B, as it is given by the pullback
of a line bundle (the line bundle corresponding to 2ρ) on one of the G/B-factors.

b
XB,zwλ δsm

Example 5.3.39. Let us spell out the above result in a bit more detail in thecase of GL2 (Qp ).
(i) Let δ1 = unrϕ1 z λ1 and δ2 = unrϕ2 z λ2 be two locally algebraic characters with
unramified smooth parts, which we assume satisfy ϕ1 /ϕ2 ∈/ {1, p±1 } and λ1 6= λ2 .
We consider a (ϕ, Γ)-module that is an extension(5.3.39)

0 → RL (δ1 ) → D → RL (δ2 ) → 0.

There are basically two cases to consider: either D is crystalline, or the extension is
non-split and λ1 < λ2 . In the crystalline case we also need to distinguish between
λ1 < λ2 and λ1 > λ2 .
(a) The extension is non-split and λ1 < λ2 . In this case D is Hodge–Tate
but not de Rham and (5.3.39) is the only way to write D as an extensionof two rank one (ϕ, Γ)-modules. In particular (δ1 , δ2 ) is the only tuple ofcharacters such that (D, δ1 , δ2 ) ∈ XB . But if we set δ10 = unrϕ1 z λ2 andδ20 = unrϕ2 z λ1 , then (D, δ10 , δ20 ) ∈ XB . The points
x1 = (D, δ1 , δ2 ) and xs = (D, δ1 , δ2 )are the only points in XB above D. Consequently, locally at D the space
Xbd,(λ,δsm )−tri has two components: one component is the image of a neighborhood (inside XB ) of x1 , and the other is the image of a neighborhood(inside XB ) of xs under the projection XB → Xd .

72

M. EMERTON, T. GEE, AND E. HELLMANN(b) The extension D is crystalline. In this case there are again two cases todistinguish:(b1) We have λ1 > λ2 . In this case the extension is crystalline no matter
whether it is split or not. Moreover, the space Xbd,(λ,δsm )−tri is smoothat D.
If the extension D is non-split then there are exactly two points in
XB above D, and both actually belong to XB . The second point does
not arise as in (a) above, but from the fact that D can also be written
as an extension
0 → RL (η1 ) → D → RL (η2 ) → 0,with η1 = unrϕ2 z λ1 and η2 = unrϕ1 z λ2 . That is: we have changed
the refinement, i.e. the order of the ϕ1 and ϕ2 (whereas in (a) we havechanged the ordering of the weights λ1 , λ2 ), i.e. in Theorem 5.3.38
these points would be addressed by a different choice of the smoothcharacter δsm .
If the extension is split, then there are three points in XB above D.
Only one of them corresponds to the refinement given by the ordering
ϕ1 , ϕ2 . The other two points correspond to the different ordering andare hence covered by case (b2) below.
(b2) We have λ1 < λ2 . Then there are three points in XB , two of whichcorrespond to the ordering (ϕ1 , ϕ2 ), i.e. belong to Xbd,(λ,δsm )−tri . Their
description is exactly as in (a). The only difference with (a) is that
there is a third point in XB mapping to D ∈ X2 corresponding to
the refinement given by the ordering (ϕ2 , ϕ1 ). Again this point is ofcourse not covered by Theorem 5.3.38.
(ii) Let us describe the local models in the case d = 2. The local model g̃ ×g g̃ ofTheorem 5.3.38 has two irreducible components both of which are non-singular. To
see this, and in order to give explicit equations for the local models, we note thatg̃ ×g g̃ is smoothly equivalent to q −1 (b), where q : g̃ → g is the canonical projection.
More precisely(g̃ ×g g̃)/G = (q −1 (b))/B
as stacks. Obviously
X1 = b × {[1 : 0]} ⊂ q −1 (b) ⊂ b × P1
is one of the irreducible components. The other component Xs is the closure of
 



a b
Vs =
, [x : 1] (a − c)x + b = 0
0 c
where s denotes the non-trivial element of the Weyl group S2 of GL2 . Note that Vs
is smooth, open in Xs and disjoint from X1 . The intersection X1 ∩Xs is contained in
the preimage of the origin 0 = [1 : 0] ∈ P1 and in the standard affine neighborhoodof this point q −1 (b) is described by
 



a b
, [1 : y] y((a − c) + by) = 0
0 c
whereas Xs is given by the equations
 



a b
, [1 : y] (a − c) + by = 0 .
0 c

CATEGORICAL p-ADIC LANGLANDS73

Let us writeZ = X1 ∩ Xs =

a
0



b
, [1 : 0]afor the intersection of the two components. Under the formally smooth morphisms
in Theorem 5.3.38, the points of type (a) and (b2) in the first part of the example
are mapped to Z, whereas the points of type (b1) are mapped to



a 0
, [x : 1]⊂ Vs
0 aMore precisely the points of type (a) are mapped to


a b
0 a
with b 6= 0, whereas the points of type (b2) are mapped to the diagonal matrix
diag(a, a).
(iii) The image Xd,tri of XB in Xd × XT should be seen as a local Galois-theoretic
counterpart of an eigenvariety. We refer to [BHS17b] for a mathematically meaningful formulation of this slogan. In the case of the eigencurve we briefly indicate
how the points discussed in (i) arise. Let f be a classical modular eigenform of
weight k ≥ 2 level Γ1 (N ) for some N not divisible by p. Then the Galois representation ρ attached to f is crystalline at p. At level Γ1 (N ) ∩ Γ0 (p) there are twostabilizations fα and fβ of f which are eigenforms for the same Hecke eigenvalues
away from p but with different Up -eigenvalues, say α and β. These two different
Up -eigenvalues correspond to the two possible choices of a refinement above. There
are at least two branches of the eigencurve that meet at ρ inside the generic fiber ofa deformation space of Galois representations: the branch containing the classicalmodular form fα and the branch containing the classical form fβ .
In the case that ρ is non-split at p these are the only branches containing the
Galois representation ρ. This corresponds to the non-split extensions in case (b1)above (note that both fα and fβ are of dominant algebraic weight). But if ρ is split
crystalline at p there is third branch of the eigencurve containing a non-classicalp-adic modular form f 0 (of weight 1 − k). This is precisely the case (b2) discussed
above. Let us mention explicitly that the two branches of Xbd,(λ,δsm )−tri in Theorem
5.3.38 meeting at a given (ϕ, Γ)-module D as in (b2) are not the branches containing
the classical forms fα and fβ , but one of these branches should be thought of as
containing a classical modular form, whereas the other branch contains the nonclassical form f 0 (the points on that branch being of non-classical weight). Finally
we note that the points in (a) correspond to non-classical p-adic modular forms(that may or may not have classical weight).
Remark 5.3.40. In [BD21] Breuil and Ding define a variant of the trianguline
variety for paraboline (ϕ, Γ)-modules. In the stacky context discussed above theirconstruction can be described as follows (the notations differ from the ones of
Breuil–Ding): Instead of XM we consider its closed substack of X0M of objects thatare de Rham of prescribed weight and inertial type, up to a twist with an arbitrary0
rank 1 object. Then we can define the preimage X0P of X0M in XP and its closure XP
in the compactification XP . Then, in our language, the analogue of the paraboline
0variety of Breuil–Ding is the image of XP inside Xd × XSM . Breuil and Ding also

74

M. EMERTON, T. GEE, AND E. HELLMANN

construct local models for their spaces. These local models are given by the Zariskiclosure (inside the space XB,P of (5.3.30)) of the closed subspace in Vw of elements
whose image in m is a central element of the Lie algebra.
6. Categorical p-adic local Langlands conjectures
In this section we state our main conjectures, and explain some motivation forthem (in particular, we explain some motivations coming from the Taylor–Wiles
method).
6.1. Expectations: The Banach Case. We now make precise our expectations in the Banach case. In order to do so, we introduce some notation. Fix as usuala finite extension F/Qp and a coefficient ring O. Set G = GLd (F ), K = GLd (OF ),Z = Z(G).
Our goal, as we’ve already explained, is to make a conjecture relating (the derived category of) of smooth representations to coherent sheaves, or, more generally,
Ind-coherent complexes, on the formal algebraic stack Xd . In order to make precise
statements, we require some preliminary discussion of the relevant categories of
G-representations.
6.1.1. Preliminaries on categories of representations. We begin by recallingsome material from Appendix E. We let sm. G denote the abelian category ofsmooth representations of G on O-modules, where we define an O-linear G-representation
to be smooth if any element of the representation is fixed by an open subgroup of G,
and annihilated by some power of p. (See Definition E.1.2.) We let D(sm. G) denotethe derived stable ∞-category of sm. G.
Any smooth G-representation admits a canonical structure of O[[G]]-module,
where O[[G]] is the ring defined in Definition E.1.1, and it’s frequently useful tostudy smooth G-representations from the perspective of their O[[G]]-module structures. To this end, Proposition E.2.2 yields a fully faithful t-exact functor(6.1.2)D(sm. G) ,→ D(O[[G]])(the target being the stable ∞-category of complexes of O[[G]]-modules) with essential image equal to Dsm (O[[G]]) (the full subcategory of objects with smooth
cohomologies).
Since there are many O[[G]]-modules that are not smooth (e.g. O[[G]] itself),
the functor (6.1.2) is far from being an equivalence. On the other hand, via aconsideration of pro-objects, we can construct a functor which is close to being−
left-inverse to (6.1.2). Indeed, if we let Dc.g.
(O[[G]]) denote the full subcategoryof D(O[[G]]) consisting of complexes whose cohomologies are countably generated
and vanish in sufficiently high degree, then (E.4.4) provides a functor−
Dc.g.
(O[[G]]) → Pro Db (sm. G).

(6.1.3)n(For example, O[[G]] itself maps to the formal pro object “ limH,n ”c-IndG
H O/p of←−Pro sm. G; here H runs over all compact open subgroups of G and n over all positive
integers, and O/pn is given the trivial H-action.) Furthermore, the composite(E.4.5)∼(6.1.3)

b
b
−
Dc.g.
(sm. G) −→ Dc.g.sm(O[[G]]) ,→ Dc.g.
(O[[G]]) −→ Pro Db (sm. G)bis simply the canonical fully faithful functor Dc.g.
(sm. G) ,→ Pro Db (sm. G). (Thisprovides the sense in which we regard (6.1.3) as close to being left-inverse to (6.1.2).)

CATEGORICAL p-ADIC LANGLANDS

75

In order to properly state our categorical Langlands conjecture, we make apreliminary conjecture regarding the abelian category sm. G.
Conjecture 6.1.4. The abelian category sm. G of smooth O-linear G-representations
is locally coherent.
Remark 6.1.5. Recall that this means that sm. G is generated by its compact
objects, and that these compact objects form an abelian subcategory of sm. G. We
have seen in Proposition E.3.5 that the compact objects in sm. G are precisely those
representations that are of finite presentation, in the sense of Definition E.3.3, andthat sm. G is generated by these objects. Thus the content of Conjecture 6.1.4 isthat the full subcategory of sm. G consisting of representations of finite presentation
is closed under the formation of kernels in sm. G. (It is obviously closed under theformation of cokernels.)
Remark 6.1.6. Since the smooth representations of finite presentation are precisely those that are finitely presented as O[[G]]-modules (Lemma E.3.4), Conjecture 6.1.4 certainly holds if O[[G]] − Mod is locally coherent, i.e. if O[[G]] is acoherent ring, i.e. if the category of finitely presented O[[G]]-modules is stable under the formation of kernels.
In the case d = 1 the ring O[[G]] is even Noetherian, while if d = 2 thering O[[SL2 (F )]] is coherent by a result of Shotton [Sho20, Cor. 4.4], and we
expect that a variation of his argument (considering the amalgamated product of
two copies of KZ over IZ) will show that O[[GL2 (F )]] is itself coherent. In eithercase it thus follows that the category of finitely presented smooth O[[G]]-modules isabelian (at least after making an additional local Z-finiteness hypothesis if d = 2,
to deal with the SL2 / GL2 distinction, which could be removed if O[[GL2 (F )]] isindeed coherent).
No other cases of Conjecture 6.1.4 are known. It is known that if d > 2, then
O[[G]] is not coherent, by a result of Timmins [Tim21]. Our main reason for
believing Conjecture 6.1.4 is that we believe that something like Conjecture 6.1.14
must hold, and it is difficult to formulate such a conjecture consistently unlessConjecture 6.1.4 holds.
bSuppose now that Conjecture 6.1.4 holds, and let Df.p.
(sm. G) denote the fullsubcategory of D(sm. G) consisting of (cohomologically) bounded complexes whose
cohomologies are finitely presented O[[G]]-modules. As noted in the discussionbfollowing Proposition E.3.5, Df.p.
(sm. G) is then precisely the full subcategory ofD(sm. G) consisting of coherent objects.
b
We have a duality theory for Df.p.
(sm. G) defined as follows. We identify
O[[G]]op with O[[G]] via g 7→ g −1 . In this way, we may canonically convert
right O[[G]]-modules into left G-modules. Then we define an antiequivalencebb
D : Df.p.
(sm. G) → Df.p.
(sm. G)

via
D(–) := RHomO[[G]] (—, O[[G]])[d2 [F : Qp ] + 1].
We anticipate that this is of amplitude [0, d]. It is equipped with a natural isomor∼
phism D ◦ D −→ Id, and has the property that
(6.1.7)G∨
D(c-IndG
H V ) = c-IndH V

76

M. EMERTON, T. GEE, AND E. HELLMANN

for any finite length smooth representation V of a compact open subgroup H of G;
here V ∨ = HomO (V, E/O) denotes the Pontryagin dual of V , equipped with its
contragredient H-action.
6.1.8. Serre duality on Xd . On the Galois side, we have the derived categoriesof (Ind-)coherent sheaves Dcoh (Xd ) and IndCoh(Xd ) (as defined in Appendix B.2).
There is an involution ı of Xd given in moduli-theoretic terms by ρ 7→ ρ∨ . We
let DXd denote the antiequivalence of Dcoh (Xd ) given by composing Grothendieck–
Serre duality with ı∗ (i.e. ı∗ ◦ RHomOXd (–, ωXd ) where ωXd is the dualizing sheafof Xd , thought of as pro-coherent sheaf on X (as discussed in Example B.3.2) andplaced in degree 0; recall that we anticipate that Xd is lci).
6.1.9. Hodge types, inertial types, components of the Bernstein centre, andHecke algebras. By results of Schneider–Zink [SZ99] (the “inertial local Langlands
correspondence”), we may associate to any inertial type τ : IF → GLd (L) a finitedimensional smooth irreducible K-representation σ crys (τ ) as in [EG22, Thm. 8.2.1].
We let σ crys,◦ (τ ) denote an arbitrary choice of K-stable O-lattice in σ crys (τ ). As
is explained in [CEG+ 16], the type τ determines a Bernstein component Ωτ , andthere is a canonical isomorphism∼
crys(6.1.10)ZΩτ −→ EndG c-IndG(τ ) .
Kσ
If λ is a regular Hodge type, then we have the algebraic representation Wλ of K
(see Definition 2.3.2). Since Wλ [1/p] is naturally a G-representation, “push-pull”
for compact induction shows that

crys
crys
c-IndG(τ )◦ [1/p] = c-IndG(τ )K Wλ ⊗O σ
K Wλ [1/p] ⊗E σ
crys
= Wλ [1/p] ⊗E c-IndG
(τ ).
Kσ

As Wλ [1/p] is furthermore an irreducible algebraic representation of G, while thecompact induction is a smooth G-representation, we find that
(6.1.11)
EndG






crys
crys
(τ )
c-IndG(τ )◦ [1/p] = EndG Wλ [1/p] ⊗E c-IndG
K Wλ ⊗O σ
Kσ
crys
(τ )
= EndG c-IndG
Kσ (6.1.10)= ZΩτ .

As recalled in Section 5.2.9, the Bernstein centre ZΩτ can also be identified withthe global functions on a stack of Weil–Deligne representations, and as in (5.2.14),
there is a morphism(6.1.12)crys,λ,τ rig)η , O(X crys,λ,τ )rig).
ηZΩτ −→ Γ((Xdd6.1.13. Statement of the conjecture. We are now ready to state our categorical
Langlands conjecture.
Conjecture 6.1.14. Assume Conjecture 6.1.4. Then there exists an O-linearbb
exact fully faithful functor A : Df.p.
(sm. G) → Dcoh
(Xd ) satisfying the following
properties:(1) The functor A has bounded cohomological amplitude.
(2) (L∞ is a sheaf.) L∞ := A(O[[G]]) is a pro-coherent sheaf, concentrated indegree 0. Furthermore it is flat over O[[K]].
(3) (Compatibility with duality.)There is a natural equivalence(6.1.15)∼A ◦ D −→ (DXd ◦ A)[[F : Qp ]d(d + 1)/2 + 1]

CATEGORICAL p-ADIC LANGLANDS

77bof contravariant functors from Df.p.
(sm. G) to Dcoh (Xd ).
(4) (Locally algebraic vectors.) For any regular Hodge type λ and inertialb 
crys,◦⊗σ(τ)
type τ , the scheme-theoretic support of A c-IndG
WO
λK
crys,λ,τ. (The b here denotes p-adic completion.)
Furis equal to Xd
Gthermore, the action of the Bernstein centre ZΩτ on A c-IndK Wλ ⊗Ob 
σ crys,◦ (τ )[1/p] induced by the identification (6.1.11) and the functori-

ality of A coincides with the action given by (6.1.12).
Remark 6.1.16. The notation A for the functor of Conjecture 6.1.14 is drawn
from the paper [Zhu20], where the same notation is used in the ` 6= p case.
Remark 6.1.17. Assuming that it exists, the functor A has natural Ind and
Pro extensions whose considerations are important (and indeed necessary to makesense of the conjecture as stated). We begin with the Ind-extension of A; this is acocontinuous functorb
Ind Df.p.
(sm. G) → Ind Coh(X ),which we again denote by A.
b
The category Ind Df.p.
(sm. G) is the analog, in our context, of the renormalized categories of smooth representations considered in [Zhu20, §4.1]. It admits acanonical functor(6.1.18)b
Ind Df.p.
(sm. G) → D(sm. G),

which becomes an equivalence if we restrict to bounded below complexes on eachside. Indeed, assuming Conjecture 6.1.4, the t-structure on D(sm. G) is coherent inbthe sense of Definition A.6.9, and Df.p.
(sm. G) is then precisely the subcategory ofcoherent objects in D(sm. G), so that the functor is particular instance of (A.6.10).
b
In particular, the heart of Ind Df.p.
(sm. G) coincides with the heart of D(sm. G),
i.e. with the abelian category sm. G. Thus, extending A to all of Ind Df.p. (sm. G)
allows us in particular to evaluate A on all smooth G-representations, not justfinitely presented ones.
As an aside, we note that in general the t-structure on D(sm. G) need not be
regular (in the sense of Definition A.6.11), and so (6.1.18) need not be an equivalencein general. Indeed, we expect that the t-structure on D(sm. G) is regular, sothat (6.1.18) is an equivalence, precisely when I1 (the pro-p-Iwahori subgroup of K)
is torsion-free.
b
Since Db (sm. G) is a full subcategory of Ind Df.p.
(sm. G), we may restrictb(the Ind-extension of) A to D (sm. G) so as to obtain a functor Db (sm. G) →Ind Coh(X ). (In fact, taking into account Remark 6.1.27 below, we see that thisbfunctor should take values in Dcoh
(X ).) We may Pro-extend this functor to obtaina functor
Pro Db (sm. G) → Pro Ind Coh(X ).
Composing this with the functor (6.1.3), we find that it makes sense to evaluate−
A on objects of Dc.g.
(O[[G]]). The discussion of Section 6.1.1 shows that the re∼bbstriction of this functor to Df.p.
(sm. G) −→ Df.p.sm(O[[G]]) (here the target hasbthe obvious meaning, namely it is the full subcategory of Dsm(O[[G]]) consistingof complexes whose cohomologies are finitely presented; the indicated equivalence

78

M. EMERTON, T. GEE, AND E. HELLMANN

follows directly from Proposition E.2.2 by restricting the equivalence stated there
to the full subcategories of its source and target consisting of complexes whose
cohomologies are finitely generated) coincides with the original functor A.
Remark 6.1.19. To make sense of Conjecture 6.1.14 (2), note that, as discussed
in Remark 6.1.17, we may evaluate A on O[[G]] (thought of as a module over itself).
A priori this gives rise to an object of Pro Ind(Xd ), but part (2) asserts that in fact
we obtain an object of Pro Coh(Xd ).
Remark 6.1.20. As was already mentioned in Section 1.4, it seems likely that
Conjecture 6.1.14 can be viewed as a consequence of a stronger conjecture, to theeffect that there should an equivalence of (derived) categories as in the conjecture of
Fargues–Scholze [FS21, Conj. I.10.2], but with p-adic rather than `-adic coefficients.
However it does not seem to be clear at the time of writing precisely which categoryof sheaves on BunG should be considered in order to obtain such an equivalence.
Remark 6.1.21. The full faithfulness of A implies in particular that there
should be identifications of E1 -rings between p-adic derived Hecke algebras andcertain endomorphism rings of coherent sheaves on Xd (e.g. p-adic analogues ofFeng’s spectral Hecke algebra [Fen19]); see [Zhu20, Conj. 4.3.1] for an analogous
conjecture in the case ` 6= p.
Remark 6.1.22. We could equally well formulate Conjecture 6.1.14 (4) for
potentially semistable representations, but for simplicity of exposition we have restricted to the potentially crystalline case.
Remark 6.1.23. The functor of Conjecture 6.1.14 is necessarily of the form
considered in Appendix C.2. That is, it is of the form
π 7→ L∞ ⊗L
O[[G]] π,
where, as in Conjecture 6.1.14 (2), L∞ := A(O[[G]]) is a pro-coherent sheaf on themoduli stack Xd , and is in addition a right O[[G]]-module (because the endomorphism algebra of O[[G]] is O[[G]]). (In the case d = 1, we see below that L∞ is the
structure sheaf on X1 , by (7.1.2), and in the case of GL2 (Qp ) we give a constructionof L∞ , but in general we do not have a conjectural candidate for L∞ .)Indeed, assuming Conjecture 6.1.4, we may resolve any finitely presented smoothO[[G]]-module π by a (possibly infinite) complex of direct sums of copies of O[[G]]· · · → O[[G]]⊕mn · · · → O[[G]]⊕m0 → π → 0,
and we find that A(π) is computed by· · · → A(O[[G]])⊕mn · · · → A(O[[G]])⊕m0
and thus by
L∞ ⊗O[[G]] (· · · → O[[G]]⊕mn · · · → O[[G]]⊕m0 ) = L∞ ⊗L
O[[G]] π.
(Assumption (1), i.e. the boundedness of the amplitude of A, justifies the manipb
ulations with these possibly unbounded below complexes.) Since Ind Df.p.
(sm. G)is generated under shifts and colimits by the representations of finite presentation,
we have(6.1.24)in general.

A(π) = L∞ ⊗L
O[[G]] πCATEGORICAL p-ADIC LANGLANDS

79

Remark 6.1.25. We note that the only reason for making an a priori assumption of bounded amplitude for A (i.e. the only reason for imposing condition (1) inthe conjecture) is to justify the manipulations that lead to formula (6.1.24), andwe should emphasize that the existence of such a formula for A is basic to our own
point of view on Conjecture 6.1.14. If we grant the validity of this formula, then theboundedness of the amplitude of A should follow, as we explain in Remark 6.1.27
below.
Remark 6.1.26. If V is a smooth O[[K]]-module, of finite type as an O-module(so that it is p-adically complete), then we we may regard the p-adically completedbcompactly induced representation(c-IndG
K V ) as an object of Pro sm. G, and henceGbform A (c-IndK V ) . (And this gives the sense in which we formb 
crys,◦⊗σ(τ)
A c-IndGWOλK
in property (4) of Conjecture 6.1.14.)b
It then follows from Conjecture 6.1.14 (2) that A (c-IndG
is concentrated
KV)
in degree 0. Indeed, by Remark 6.1.23, we haveGbLbL
A(c-IndG
K V ) = L∞ ⊗O[[G]] (c-IndK V ) = L∞ ⊗O[[K]] V

b indicates the p-adically completed tensor product) and the claim follows(here ⊗
from the hypothesized flatness of L∞ over O[[K]].
b 
crys,◦
In particular, the sheaves A c-IndG
W
⊗
σ
(τ
)
considered in propλOK
erty (4) are concentrated in degree 0.
Remark 6.1.27. We can be much more precise about the bounded amplitudeconjectured in statement (1) of the conjecture. Namely, we anticipate that thefunctor A has amplitude [1 − d, 0]. Indeed, in the case d = 2 and p > 2 thisfollows from Remark 6.1.26 and the resolution explained in Remark 7.3.11 (at least
if we ignore the action of the centre, which we do not expect to contribute to theamplitude), and we anticipate that analogous resolutions exist for all d.
bRemark 6.1.28. Since (by construction) the compact objects in Ind Df.p.
(sm. G)b
are the objects of Df.p. (sm. G), and the compact objects in Ind Coh(Xd ) are given bybb
Dcoh(Xd ), it follows from Lemma A.3.2 (3) that if π is an object of Ind Df.p.
(sm. G),bbthen π actually belongs to Df.p. (sm. G) if and only if A(π) is an object of Dcoh
(Xd ).
In particular, if π is concentrated in degree zero, then we see that π is of finite
presentation if and only if A(π) is a bounded complex with coherent cohomology
sheaves. (In fact, we expect that A(π) is automatically bounded, by Remark 6.1.27.)
If π is concentrated in degree zero, and is finitely generated but not of finite
presentation, then A(π) is not coherent. However, we claim that H 0 (A(π)) is coherent. Indeed, since π is finitely generated, it admits a surjection c-IndG
K U → π forsome finite length K-representation U . Since A is right t-exact (see Remark 6.1.27),
G0
we have a surjection H 0 (A(c-IndG
K U )) → H (A(π)); and A(c-IndK U ) is coherentG(because c-IndK U is of finite presentation).

Remark 6.1.29. We have tried to formulate Conjecture 6.1.14 with a minimal
set of useful properties; we explain some motivation for these properties in the following remarks. It is unclear to what extent these properties uniquely determine A,
but in practice they seem to seriously constrain it.

80

M. EMERTON, T. GEE, AND E. HELLMANN

Remark 6.1.30. We can think of L∞ as a “universal”, purely local versionof completed cohomology. More precisely, under the assumption that p - 2d,for any point x ∈ Xd (Fp ) the paper [CEG+ 16] constructs a versal morphism
f : Spf R∞ → Xd at x, together with an R∞ -module M∞ with a commuting action of G. Here R∞ is a power series ring over a universal lifting ring for the
GalF -representation corresponding to x, with the power series variables being the“patching variables” occurring in the Taylor–Wiles–Kisin method, and M∞ is obtained by applying the patching method to the completed cohomology of certain
unitary groups. (See Section 3 for a very similar construction for modular curves.)
The construction of R∞ and M∞ is global and depends on many choices, but
we expect that M∞ = f ∗ L∞ (see also Expected Theorem 9.4.2). This would
confirm the conjecture in [CEG+ 16, §6] that M∞ should be purely local; indeed
the hypothetical L∞ in loc. cit. is just the pullback of our L∞ to a local deformation
ring. It also explains some of our expectations for L∞ ; for example, M∞ is flat overO[[K]] by construction, whence our expectation that L∞ is flat over O[[K]].
Remark 6.1.31. Property (2) of Conjecture 6.1.14 is an analogue of the expectation [BZCHN20, Conj. 4.15] that the coherent Springer sheaf is concentrated indegree 0. In our case it is motivated by the expectation of Remark 6.1.30 (which
implies that the pullback of L∞ to any versal ring is concentrated in degree 0).
Remarkfor any regular Hodge type λ and inertial 6.1.32. By property (4),b crys,λ,τcrys,◦
type τ , A c-IndG
W
⊗
σ
(τ
)
is supported on Xd. We have theλOKcrys,λ,τclosed immersion i : Xd,→ Xd , which we expect to be pure of codimension[F : Qp ]d(d+1)/2 (we say “expect” because we are not aware of a dimension theory
for formal algebraic stacks in the literature; but note that by [Kis10b, BIP21] weknow the corresponding claim on versal rings, i.e. on Galois deformation rings).
Granting this expectation, we have i! ωXd = ωX crys,λ,τ [−[F : Qp ]d(d + 1)/2], wheredwe abusively write ωX crys,λ,τ for the dualizing complex shifted by the dimensiondcrys,λ,τof Xd

crys,λ,τ(so that if Xd

is Cohen–Macaulay then ωX crys,λ,τ is a sheaf indcrys,λ,τdegree zero). Thus for any F ∈ DCoh (Xd) we havei∗ RHomO crys,λ,τ (F, ωX crys,λ,τ ) = RHomOXd (i∗ F, ωXd )[[F : Qp ]d(d + 1)/2].
X
d

d

Write
crys,λ,τDO crys,λ,τ : DCoh (XdX

crys,−λ,τ ∨) → DCoh (Xd)

dfor the composite of Grothendieck–Serre duality and ı∗ , where
crys,−λ,τ ∨ı : Xd∼crys,λ,τ−→ Xd

is induced by theinvolution ı of Xd .
b 
crys,◦Take F = A c-IndG(τ ). Note that it follows from (6.1.7)
K W λ ⊗O σthatb  
 Gcrys,◦∗
crys,◦ ∨ b
D c-IndG
W
⊗
σ
(τ
)
=
c-Ind
W
⊗
σ
(τ
)
[1],
λ O
K
K
−λ O
∗
where we abusively write W−λfor the (not dual!) Weyl module, and choose
crys,◦ ∨σ(τ ) to be a lattice in σ crys (τ ∨ ) that is dual to our chosen lattice σ crys,◦ (τ ).

CATEGORICAL p-ADIC LANGLANDS

81

Bearing in mind (6.1.15), we obtain ∗crys,◦ ∨ b(6.1.33) A c-IndG
W
⊗
σ
(τ
)
O
K
−λ
 
b 
∼
crys,◦−→ DO crys,λ,τ A c-IndG(τ ).
K W λ ⊗O σ
XdRemark 6.1.34. Property (3) of Conjecture 6.1.14 can be thought of explaining
the fact that the patched modules (constructed by Taylor–Wiles–Kisin patching at
finite level) are maximal Cohen–Macaulay over their supports. In particular we
expect that we can strengthen
property (4) to say thatfor any regular Hodge type λb Gcrys,◦
is concentrated in degree 0, is
and inertial type τ , A c-IndK Wλ ⊗O σ(τ )b rig
crys,λ,τcrys,◦
ismaximal Cohen–Macaulay over Xd(τ ), and A c-IndG
K W λ ⊗O σηcrys,λ,τ rig)η . We now sketch an explanation for this.
b 
crys,◦
Since we are assuming that A c-IndG
W
⊗
σ
(τ
)
is concentrated in
λ OKlocally free of rank one over (Xdcrys,λ,τ∗degree 0 and is supported on all of Xd, and similarly for A c-IndG
K W−λ ⊗O


b 
b 11
crys,◦(τ )σ crys,◦ (τ ∨ )is, it follows from (6.1.33) that A c-IndG
K W λ ⊗O σ
crys,λ,τ(see [Sta, Tag 0B5A]).
maximal Cohen–Macaulay over Xdb rig
G
It follows that A c-IndK Wλ ⊗O σ crys,◦ (τ )is maximal Cohen–Macaulayηcrys,λ,τ rig
crys,λ,τ rig)η . Since (Xd)η is smooth, this implies that A c-IndG
over (XdK W λ ⊗Orig
b
σ crys,◦ (τ )is locally free, and we anticipate that the full faithfulness of A, toηgether with [CEG+ 16, Cor. 3.12], implies that it is furthermore locally free of rankone.
The Cohen–Macaulay property of patched modules is usually established bycompletely different means, using the action of the ring S∞ of diamond operators;see [Dia97].
Remark 6.1.35. As already noted when defining our modified version DXd ofGrothendieck–Serre duality on Xd , we have the “main involution” ı on Xd definedb(Xd ). We also have theas ρ 7→ ρ∨ . This induces an auto-involution ı∗ on Dcoh
−1 t
usual main involution g 7→ (g ) of GLd (F ), and precomposing representationsbof GLd (F ) with this involution induces an auto-involution ı0 of Df.p.
(sm. G). We
0
expect that A intertwines the involutions ı and ı.
For example, in the statement of Property (3) of Conjecture 6.1.14, rather
than replacing Grothendieck–Serre duality D on Xd by its twist DXd := ı∗ D, web
could replace the duality D on Df.p.
(sm. G) by its twist D0 := ı0 ◦ D0 . Then
Property (3) could be rephrased as saying that A interchanges D0 and D (up to aG
shift). Now (6.1.7) implies (ignoringpossible shifts) that D0 (c-IndG
K V ) = c-IndK V
GG(respectively that D0 (c-IndK V ◦ )∧ = (c-IndK V ◦ )∧ ) if V is an irreducible modp representation of K (respectively if V ◦ is a K-invariant lattice in an irreducible
locally algebraic representation V of K). Thus we expect that the corresponding
11While we have not made an explicit conjecture involving W ∗ , this statement follows from−λthe version for W−λ , since these representations are isomorphic after inverting p.

82

M. EMERTON, T. GEE, AND E. HELLMANN

G◦ ∧coherent sheaves A(c-IndG
K V ) (respectively A (c-IndK V )Serre self-dual (up to a shift).

are Grothendieck–

Remark 6.1.36. We also expect a compatibility with parabolic induction, justas in the ` 6= p and analytic cases (see Section 6.2.13 for the latter).
6.1.37. The geometric Breuil–Mézard conjecture. The geometric Breuil–Mézard
conjecture for the stack Xd was formulated in [EG22]; it was motivated by (andindeed is equivalent to) the earlier geometric Breuil–Mézard conjectures for Galois
deformation rings of [BM14, EG14], and the reader who is unfamiliar with theBreuil–Mézard conjecture is referred to these papers (and the references therein)
for the original motivations for the conjecture, which come from automorphy lifting
theorems. In this section we explain how Conjecture 6.1.14 gives a further refinement of the Breuil–Mézard conjecture (lifting the statement from an equality ofcycles to an equality in a Grothendieck group of coherent sheaves).
By a Serre weight k we mean a tuple of integers {kσ,i }σ:k,→Fp ,1≤i≤d with theproperties that• p − 1 ≥ kσ,i − kσ,i+1 ≥ 0 for each 1 ≤ i ≤ d − 1, and• p − 1 ≥ kσ,d ≥ 0, and not every kσ,d is equal to p − 1.
The set of Serre weights is in bijection with the set of irreducible Fp -representationsof GLd (kF ), via passage to highest weight vectors (see for example the appendix
to [Her09]); for each Serre weight k, we write Fk for the corresponding irreducible
k-representation of GLd (kF ).
We write σ crys (λ, τ ) for the semisimplification of the k-representation of GLd (kF )given by Wλ ⊗O σ crys,◦ (τ ) ⊗O k. There are unique integers ncrysk (λ, τ ) such thatss⊕nk (λ,τ )σ crys (λ, τ ) ∼.
= ⊕k Fk

Write Zcrys,λ,τ for the cycle (i.e. an element of the free abelian group Z[Xd,red ] on
the irreducible components of Xd,red ) corresponding to the special fibre of Xdcrys,λ,τ .
The following is [EG22, Conj. 8.2.2].
Conjecture 6.1.38. 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 .
Assume Conjecture 6.1.14. For each regular
Hodge type λ and each inertialtype τ we make a choice of lattice σ crys,◦ (τ ) 12 and setb 
crys,◦
Fλ,τ = A c-IndG
W
⊗
σ
(τ
)
,
λ O
K
which by Remark 6.1.26 is a coherent sheaf (concentrated in degree zero) on Xd ,and for each Serre weight k we set
Fk := A(c-IndG
K Fk ).
An easy induction using [EG14, Lem. 4.1.1] shows that Fk is a coherent sheaf(again, concentrated in degree zero) on the special fibre of Xd .
Since A is a functor, it follows from the definitions that we have an equality in
K0 (Coh(Xd ))X crys
(6.1.39)[Fλ,τ ⊗O k] =nk (λ, τ ) · [Fk ].
k12Since we are passing to underlying cycles, the precise choice of lattice is irrelevant.

CATEGORICAL p-ADIC LANGLANDS

83

As explained in Remark 6.1.34, Fλ,τ is maximally Cohen–Macaulay over its support
Xdλ,τ , so that the support of Fλ,τ ⊗O k is equal to Zcrys,λ,τ . Again, an easy induction
using [EG14, Lem. 4.1.1] shows that the support of each Fk is a cycle Zk in
Z[Xd,red ], so that Conjecture 6.1.38 is an immediate consequence of (6.1.39).

6.2. Expectations: the analytic case. Let G = GLd (F ) considered asa Qp -analytic group. Moreover, we assume in this section that the fixed finite
extension L of Qp contains the normal closure of F . We want to formulate aconjecture parallel to Conjecture 6.1.14 involving locally analytic representations
and sheaves on Xd . In order to do so we first have to discuss derived categories in
the context of locally analytic representations.
6.2.1. Derived categories of locally analytic representations. We write D(G)for the distribution algebra of G with coefficients in L. This distribution algebra
is defined as the dual space of the space of locally analytic functions G → L, see[ST02] for the precise definition (in our case the field of coefficients will always be
understood and we suppress it from the notation).
By a theorem of Schneider–Teitelbaum [ST02, Corollary 3.3] the category of
locally analytic representations of G on L-vector spaces of compact type with continuous G-morphisms is anti-equivalent to the category of separately continuous
D(G)-modules on nuclear Fréchet spaces with continuous D(G)-module homomorphisms. This anti-equivalence is given by mapping an L-vector space V of compact
type to its strong dual V 0 = Vb0 . Under this anti-equivalence of categories theadmissible locally analytic G-representations (by definition) correspond to coadmissible D(G)-modules, i.e. to those D(G)-modules M that are coadmissible overthe distribution algebra D(K) of the maximal compact subgroup K = GLn (OF ) of
G. More precisely this means that M is an inverse limit of finite type modules overcertain Banach completions Dr (K) of D(K), compare [ST03, §3, 4]. In the following we will write Repan
L G for the category of locally analytic G-representations on
L-vector spaces of compact type.
We note moreover that the G-action on a locally analytic representation π
automatically extends to a separately continuous action of the distribution algebra D(G), i.e. the representation π can be regarded as a D(G)-module [ST02,
Prop. 3.2]. Similarly to the category D(sm. G) we would like to consider thederived category D(an. G) of complexes of locally analytic representations. The
definition of this category (and of derived functors from this category to other derived categories) meets some difficulties, as we are working with topological objects
in a homological algebra context. These problems are best dealt with by using condensed structures and solid modules introduced by Clausen–Scholze. A treatmentof locally analytic representations in the framework of solid modules was developedby Rodrigues Jacinto and Rodrı́guez Camargo in [JC21]. We will not go too muchinto detail concerning these definitions and rather use D(an. G) as an informal object that will contain all the locally analytic representations that we are interested
in. Still we try to give some details as to how a formal definition of D(an. G) might
look.

84

M. EMERTON, T. GEE, AND E. HELLMANNIn terms of [JC21]13 the most natural candidate for D(an. G) is to construct
D(an. G) as a full subcategory of the derived category D(L [G]) of solid L [G]modules. If C is an object in D(L [G]), then C is called derived locally analytic if
the natural map C Rla → C from the derived locally analytic vectors in C to C isa quasi-isomorphism (compare [JC21, Definition 4.40]). In this case (by loc. cit.)
C is a (complex of) D(G)-module(s) extending the L[G]-module structure in aunique way. We define the candidate D(an. G) for our derived category as the fullsubcategory of D(L [G]) consisting of derived locally analytic objects. At least
conjecturally this category contains all locally analytic G-representations [JC21,
Proposition 4.43] as well as bounded complexes of those.
b
In particular we can regard the subcategory Df.p.
(Repsm G) of bounded complexes of smooth G-representations on L vector spaces with finitely presented cohomology as a subcategory of D(an. G). We note that this is not a full subcategory,as the computation of derived homomorphisms involves the computation of Liealgebra cohomology, compare [JC21, Theorem 5.5].
Remark 6.2.2. We would like to have a full subcategory of D(an. G) whose objects satisfy a finiteness condition that will ensure that the hypothetical functor ArigG
in Conjecture 6.2.4 below will produce (bounded complexes of) coherent sheaves.
b
As our use of D(an. G) is in any case rather informal we will write Df.p.
(an. G) for
this full subcategory. In analogy with the Banach case one might speculate that thiscategory is the full subcategory of bounded complexes whose cohomology satisfies
some finiteness condition over the distribution algebra D(G). A precise (and still
conjectural) description would involve talking in depth about the condensed set-up
used in [JC21]. As in these notes the category D(an. G) is not properly defined web
use Df.p.
(an. G) is a black box for a derived category of locally analytic representab
tions satisfying appropriate finiteness conditions. In any case Df.p.
(an. G) shouldsmbcontain the derived category Df.p. (Rep G).
6.2.3. Formulation of the conjecture. Recall that the center Z(gL ) of the enveloping algebra U (gL ) of the Lie algebra gL of the Qp -analytic group G embeds
into D(G), [ST02, Prop.3.7]. This center can be identified with the global sections Γ(WTd,L , OWTd,L ) of the space of Hodge–Tate–Sen weights of Xd using theHarish–Chandra isomorphism.14
Further recall [BZCHN20, Theorem 5.13], [Zhu20, Conjecture 4.5.1], [Hel20,
Conjecture 3.6] (see also Section 8.3) that for smooth representations we have afunctorsmbb
Asm
G : Df.p. (Rep G) −→ Dcoh (WDd,F ).
(For a general reductive group this functor is of course conjectural but for G =GLd (F ) the functor can be constructed from the case of the Iwahori–Hecke algebrausing type theory, see [BZCHN20].)
13We caution the reader that in [JC21] the group G usually is a uniform pro-p-group. As our
reductive groups G are locally uniform pro-p groups and as the condition of being locally analytic
only depends on small neighborhoods of 1, we can easily adapt the notions and results from op.
cit.
14Recall that in this section the group G is GL and hence we can easily compare the centerdof the enveloping algebra with the space of functions on the space of Hodge–Tate–Sen weights.
The case of an arbitrary reductive group is harder and would rely on similar constructions as
in [DPS20, 4.6]. But anyway, we did not define spaces of (ϕ, Γ)-modules with G-structures forarbitrary reductive groups.

CATEGORICAL p-ADIC LANGLANDS

85

We formulate a conjecture about locally analytic representations and coherent
sheaves on Xd that roughly parallels Conjecture 6.1.14 in the case of smooth modp representations and sheaves on Xd .
Conjecture 6.2.4. There exists an exact Z(gL )-linear functorbb
Arig
G : Df.p. (an. G) → Dcoh (Xd )satisfying the following conditions:(1) (Compatibility with the smooth case.) Let ξ be a dominant algebraic character associated to a regular Hodge–Tate weight λ, and write Wλ [1/p]for the corresponding irreducible algebraic representation of G of highest
weight ξ as in Definition 2.3.2. Let RepsmΩτ G be the Bernstein block associated to an inertial type τ such that τ |IF 0 is trivial, for some finite extension
F 0 of F . After base change to L0 ⊃ F 0 there is a natural isomorphism
∗
sm
0
∼
Arig
G (Wλ [1/p] ⊗ –) = (prWD (AG (−))((−ρ )σ:F ,→L ),(6.2.5)
where

prWD : Filλ Modd,ϕ,N,τ ∼= Filλ WDd,F,τ −→ WDd,F,τ
is the canonical projections.
(2) (Compatibility with parabolic induction.) Let P ⊆ G be a parabolic subgroup with Levi quotient M and write P̂ respectively M̂ for the corresponding dual groups. Then locally analytic parabolic inductionan
IndG: D(an. M ) → D(an.G),P̄ (–)

where P̄ ⊂ G is the opposite parabolic, satisfies the compatibility(6.2.6)Gan ∼ e
∗
0
eP̂
(Arig
Arig
M (–)([F : Qp ]ρM ))G (IndP̄ (–) ) = βP̂ ,∗ ◦ αwhenever this formula makes sense, see Section 6.2.13 for a more precise
formulation.
Remark 6.2.7. Let us comment on the twists by −ρ0 respectively by ρ0M that
occur in (6.2.5) respectively (6.2.6).
(i) Let us, as usual, write ρ for the half sum of the positive roots in the dual torusT̂ . To fix notations let us fix T̂ to be the diagonal torus in GLd and we canonically
identify T̂ = Gdm . Thenρ=(d−1 d−3−d + 3 −d + 1
,
,...,
,
)
2
2
2
2

in general is not a character of T̂ , but only lies in X ∗ (T̂ ) ⊗ Q. Instead we consider
the character ρ0 = (0, −1, . . . , −(d − 1)) ∈ X ∗ (T̂ ). Note that ρ0 is a shift of ρ byan element of X ∗ (T̂ ) ⊗ Q that factors through the determinant and agrees with ρ
when evaluated on the coroots of T̂ .
If M is a Levi subgroup of G, then we write ρM for the half sum of the rootsthat occur in the T̂ representation on Lie Û , where Û ⊂ P̂ is the unipotent
Qrradical.
0Then again we can apply a shift to ρM to obtain a character
ρ
of
M̂
=
i=1 GLri
Qr M
that factors through the maximal torus quotient SM̂ = i=1 Gm such that ρ0M is
trivial on the first Gm factor of SM̂ .

86

M. EMERTON, T. GEE, AND E. HELLMANN(ii) Let us now describe the twist in (6.2.5). The space Filλ WDd,F,τ has a canonicalprojection toY
GLd \(ResF/Qp GLd )L /Pλ ∼GLd /B̂ .
= GLd \
σ:F ,→L

Here Pλ denotes the parabolic defined in Section 5.2 that agrees with the Weil
restriction of the Borel, as λ is regular. Then we can twist a sheaf on Filλ WDd,F,τ
by the pullback of a line bundle on the product of flag varieties that we choose
to be the line bundle on GLd /B̂ associated with the character −ρ0 of T̂ in eachfactor. We point out that twisting with (−ρ0 )σ is compatible with duality (in thespirit of the compatibility in Remark 6.1.32). Again we have an auto-duality DXdbon Dcoh(Xd ) induced by composing Grothendieck–Serre duality with the pullback
of a map Xd → Xd induced by the main involution on GLd . If F denotes a complex
of sheaves on WDd,F,τ , thenDXd (pr∗WD (F)(−ρ0 )σ ) = pr∗WD (DWDd,F,τ (F))(−ρ0 )σ ,where DWDd,F,τ is defined the same way as DXd .
(iii) Finally let us describe the twist in (6.2.6). The determinant induces a canonicalmorphism Xd → X1 → ∗/Gm (see (7.1.2) for a description of X1 and the canonicalprojection to ∗/Gm ). In particular we obtain a projection
XM̂ → ∗/SM̂
and hence it makes sense to twist sheaves on XM̂ with a character of SM̂ (that weview as a line bundle on ∗/SM̂ and pull back to XM̂ ).
There should be further compatibilities, namely a compatibility with the Banach case discussed in Section 6.2.10 below and a local-global compatibility that
we will talk about in Section 9.6. We now comment a bit on the compatibilities in
the conjecture.
Remark 6.2.8.
(a) Property (1) prescribes the functor Arig
G on locally algebraic representations.
This means that in some sense we expect that the extension of the functor Asm
Gin smooth representations to locally algebraic representations (which takes values
in coherent sheaves supported on de Rham loci) can be interpolated to a functor
that takes values in coherent sheaves on Xd (without restrictions on the support).
In some sense this parallels the point of view that (Hecke eigenvalues on) p-adic
automorphic forms interpolate (Hecke eigenvalues on) classical automorphic forms.
(b) The natural isomorphism in Property (1) depends on the choice of the isomorphismFilλ Modd,ϕ,N,τ ∼= Filλ WDd,F,τ
in Lemma 5.2.3, i.e. on the choice of an embedding F 0 ,→ L0 . In fact we expectthat there is a variant of the functor Asm
G that is a functor
[

b
b
Df.p.
(Repsm G) −→ Dcoh
Modd,ϕ,N,F 0 /F
F 0 /F

which agrees with Asm
G after appropriate scalar extensions, see Remark 8.3.3. Thisversion of Asm
wouldallow for a smoother formulation of Property (1) (i.e. a version
G
without an auxiliary scalar extension to a field containing F 0 ).

CATEGORICAL p-ADIC LANGLANDS

87(c) We cannot hope that the functor Arig
G is fully faithful without putting any adb
ditional conditions on the source Df.p.
(an. G). In the GL1 -case this is discussed
in Remark 7.1.8 below. But the phenomenon is also visible by looking at locallyalgebraic representations. The functor Asm
G is (conjecturally) fully faithful. Given
two smooth representations π1 and π2 (concentrated in degree zero) the module ofhomomorphisms HomG (π1 , π2 ) is the same whether we compute it in the categoryof smooth representations or in the category of locally analytic representations (ofcourse this is not true for higher Ext-groups). By full faithfulness this module cosm
incides with Hom(AsmG (π1 ), AG (π2 )), but in general the space of homomorphisms
between the analytifications of these sheaves is strictly larger. It seems that thisissue will not occur if we either add a finite slope condition, or ask that the representations are admissible; compare the discussion in the GL1 -case. Unfortunately,
simultaneously imposing the assumptions of being finitely presentated over D(G)and being admissible is rather restrictive, and would rule out a lot of interesting
representations.
(d) We expect that the functor Arig
G can be defined on a larger category like D(an. G)binstead of Df.p.
(an. G) if we allow more general (complexes of) sheaves than justcoherent sheaves. Similarly to the case of p-power torsion representations we expectthat in this case admissibility of a representation π will not ensure that Arig
G (π) isa coherent sheaf without assuming some additional finiteness assumption on thelocally analytic representation regarded as a module over D(G). In the p-torsion
case this is discussed in Section 7.7.59, and one knows [Paš10, Thm. 1.1] that,when G = GL2 (F ), there are unitary Banach completions of locally algebraic Grepresentations whose reduction contains any given irreducible admissible mod p
G-representation, so that we expect the mod p phenomena of that discussion tohave analogues in characteristic zero.
Remark 6.2.9. We expect that, similarly to Remark 6.1.23, the functor Arig
Gof Conjecture 6.2.4 should be given byLb D(G) π
π 7−→ L∞ ⊗b L − (most likely: defined in somefor a certain derived completed tensor product −⊗category of solid modules) and a family L∞ of D(G)-modules over the stack Xd suchthat, using the map πd from (5.1.27), its pullback to the generic fiber of Xd should
be given byπd∗ L∞ ∼= L∞ ⊗OXd [[G]] D(G, OXd )for a relative version D(G, OXd ) of the distribution algebra defined in similar terms
as in [BHS17b, 3.1] or [GRJ18, 1.3].
In the case of GL2 (Qp ) we expect that for a point x ∈ X2 (L), given by a(ϕ, Γ)-module D, the D(G)-module L∞ ⊗ k(x) is the dual of the locally analytic
representation associated to D by Colmez’s extension of the p-adic Langlands correspondence to the case of (not necessarily étale) (ϕ, Γ)-modules over RF , [Col16].
More precisely: the family L∞ should interpolate Colmez’s p-adic Langlands correspondence for (ϕ, Γ)-modules over RL . Using the object L∞ from Remark 6.1.23
we can construct L∞ (in the case of GL2 (Qp )) on the open substack of X2 , wherethe (ϕ, Γ)-modules are étale up to twist. The complement of this open subset consists only of trianguline (ϕ, Γ)-modules. On the other hand Gaisin and Rodrigues
Jacquinto [GRJ18] construct an analogue of L∞ over (the regular part of) the stack88

M. EMERTON, T. GEE, AND E. HELLMANN

XB parameterizing trianguline (ϕ, Γ)-modules (together with a triangulation). At
least for the time being it is not clear how to glue L∞ from these two cases.
6.2.10. Compatibility with the Banach case. We expect a strong link between
the conjecture in the Banach case (Conjecture 6.1.14) and the conjecture in theanalytic case (Conjecture 6.2.4). On the representation theoretic side, this compatibility should involve passage from a lattice Λ in a Banach space representationb
V = Λ[1/p] (that we view as a pro-object in Df.p.
(sm. G)) to the locally analytic
vectors (cf. [ST03, §7]) in V . On the side of (coherent) sheaves on stacks of (ϕ, Γ)b
modules, we consider the generic fiber Fη of a pro-object in Dcoh
(Xd ). This (atleast heuristically, but see the discussion following Lemma 7.1.7 for a discussion inrig
the case of GL1 ) gives rise to a complex of sheaves on Xd,η, and we compare thiscomplex of sheaves to the pullback along (5.1.27) of a complex of sheaves on Xd .
bMore precisely, for a pro-object Λ ∈ Df.p.
(sm. G) with V = Λ[1/p], the comparison
should look like
(6.2.11)∼ ∗ rig an ).
A(Λ)rig
η = πd AG (VThis formula meets with some difficulties due to the finiteness conditions that
we have imposed. We expect that these difficulties can be dealt with by using solid
modules on both sides (i.e. solid locally analytic representations and solid quasicoherent sheaves) but we will not pursue this direction here. Instead we discuss
some of the difficulties that one has to deal with.
- We have restricted ourselves to coherent sheaves on rigid analytic spaces(and rigid analytic Artin stacks), though there should be a more flexible
ambient category of sheaves on these spaces (defined in terms of solid
modules), compare Remark 5.1.11. Though it is possible to put conditions
on pro-coherent sheaves F on Xd that will assure that F has a well-definedrig
generic fiber Fηrig that is a coherent sheaf in Xd,η, it is less clear which
finiteness conditions on the pro-object Λ of D(sm. G) would assure thatrig
A(Λ)rig
η is an object of Dcoh (Xd,η ). Given a single representation Λ (sayin cohomological degree zero) the finiteness assumption should at least
involve finiteness over O[[G]].
- In our context of derived categories the passage to the locally analytic
vectors V 7→ V an should be derived. It turns out that this is not necessary on the category of admissible Banach spaces representations. Moreprecisely, if V is an admissible Banach space representation (i.e. its dual isfinitely generated over L[[K]]), then V an is an admissible locally analytic
representation and its dual may be described as follows: By [ST03, §4]
there is a canonical flat map O[[K]] → D(K) from the completed groupring O[[K]] to the distribution algebra D(K). Given a finitely generated
O[[K]]-module π we can consider the extension of scalars π ⊗O[[K]] D(K).
If π is equipped with a G-action (compatible with the O[[K]]-module
structure), then π ⊗O[[K]] D(K) naturally becomes a coadmissible D(G)module. As O[[K]] → D(K) is flat, the functor (−)an is exact on the
category of admissible Banach space representations, and one can in fact
deduce that V an is even derived locally analytic, see [JC21, Prop. 4.48],[Pan22b, Theorem 2.2.3]. In general, i.e. without invoking admissibility,

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89

we have to consider a derived variant of V 7→ V an (that is the derived locally analytic vectors of [JC21]) that can only be defined in the frameworkof solid modules.
- As discussed in Remark 6.2.8, the admissibility condition is not a goodfiniteness condition when dealing with the functors A and Arig
G , i.e. weshould not expect that admissible representations are mapped to coherent
sheaves in general. On the other hand imposing both finiteness conditions,admissibility and finiteness over O[[G]], respectively D(G), is rather restrictive and excludes a lot of interesting representations. Hence it seems
that a general formulation of the compatibility (6.2.11) can only be formulated if we enlarge our ambient categories relying on the theory of solid
modules.
One instance of the compatibility (6.2.11) of the Banach and the analytic caseshould be the following. Given a regular Hodge–Tate weight λ and an inertial
type τ , Conjecture 6.1.14 (4) predicts thatb
crys,◦⊗σ(τ)
F = A c-IndGWO
λK
is a pro-coherent sheaf concentrated in degree 0 and supported on the p-adic stackcrys,λ,τcrys,λ,τXd. We expect that its pullback along any morphism Spf A → Xdfora p-adically complete O-algebra topologically of finite type over O is given by afinitely generated A-module. In this case there is a well defined generic fiber Fηcrys,λ,τ rig
of F which is a coherent sheaf on (Xd)η . We then expect that there is an
isomorphism


rig
crys
(6.2.12)Fηrig ∼σ(τ).
= πd∗ AG Wλ [1/p] ⊗ c-IndGK
We note that (6.2.12) is not quite of the same form as (6.2.11). However, wenote that since Wλ [1/p] is a G-representation, there is an isomorphism
crys
crys
(τ ),
(τ ) ∼
c-IndG= Wλ [1/p] ⊗ c-IndGKσ
K Wλ [1/p] ⊗O σ

and it seems to be reasonable to expect that the locally analytic vectors inbcrys,◦
c-IndG(τ ) ⊗O L
K W λ ⊗O σ
crys
are given by c-IndG(τ ).
K Wλ [1/p] ⊗O σ
Finally let us point out that Colmez [Col19, Rem. 0.2] has described the
Jordan–Hölder factors of the locally analytic vectors Π(D)an in the Banach space
representation Π associated to an étale (ϕ, Γ)-module D of rank 2 under the p-adic
local Langlands correspondence for GL2 (Qp ). It would be interesting to predict
where the representations in [Col19, Corollaire 0.4] are mapped to under the functor Arig
G . In the trianguline case this is basically covered by the expected compatibilities with the smooth case and with parabolic induction. But for the remaining
representations we do not know how to (conjecturally) describe the image.
6.2.13. Compatibility with parabolic induction. We fix a Borel subgroup B ⊆ Gwith split maximal torus T . Further let P ⊇ B be a parabolic subgroup with Levi
M . Let (π, V ) be a locally analytic representation of M on an L-vector space V ofcompact type. The locally analytic parabolic induction of π is defined byan(IndG= {f : G → V locally analytic | f (gp) = π(p)−1 f (g) for all p ∈ P, g ∈ G}.
P π)

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M. EMERTON, T. GEE, AND E. HELLMANN

In particular we obtain functors
an
an
IndG
: RepanP (−)
L M −→ RepL G,an
IndG: D(an. M ) −→ D(an. G),
P (−)

where by abuse of notation we use the same notation for the functor on the (informally defined) derived categories. In order to make the discussion less involved we
focus on the case P = B in the following.
Expectation 6.2.14. There are isomorphisms


G
an ∼
∗
0
Arig
Arig
= β̃B̂,∗ α̃B̂G IndB (π)
T (ι(π)([F : Qp ]ρ ))G(π)an lies
functorial in π, whenever whenever π is a representation such that IndB
rigb∗
in Df.p. (an. G) and such that the support of α̃B̂ (AT (π)) is proper over Xd .

Here ι : D(an. T ) → D(an. T ) twists a T -representation with the character(6.2.15)

δB · (1, . . . , (ε ◦ rec)i−1 , . . . , (ε ◦ rec)d−1 ).

and ε is the cyclotomic character and rec is the isomorphism of local class field
theory.
Remark 6.2.16. The appearance of the twist with the character (6.2.15) isdue to the chosen normalizations. In fact the normalizations we choose dependon the normalization of the functor Asm
G , or (which is more or less the same), thenormalization of the isomorphism in Proposition 5.2.10, compare Remark 5.2.11.
In fact here we use a different normalization than in [Hel20], where compatibility
with parabolic induction is stated without involving a twist (but using normalized
parabolic induction). The normalization in [Hel20] corresponds to the unitary
normalization of the local Langlands correspondence, rather than the non-unitary
normalization that we use here. The price we have to pay is the appearance of thetwist in Expectation 6.2.14.
Let us stress that locally analytic parabolic induction should not be expectedb
to preserve the finiteness conditions, i.e for π ∈ Df.p.
(an. T ) we do not expect thatGanb
IndP (π) lies in Df.p. (an. G). However this should be true if π is of bounded slope.
We expect that the phenomenon is directly related to the question whether the∗
support of αB̂(Arig
T (π)) is proper over Xd or not. As already remarked above we

expect that there is a variant of Arig
G defined on D(an. G) that takes values in alarger category of sheaves on Xd (most probably: solid quasi-coherent sheaves). In
this context one should have a full 6-functor formalism for these sheaves at hand
and the compatibility with parabolic induction 6.2.14 should rather be formulatedwith β̃B̂,! instead of β̃B̂,∗ (of course we don’t see the difference if we assume thatthe support of the corresponding sheaf is proper).
We mention that Orlik and Strauch [OS15] have described the (topologically)
irreducible subquotients of some locally analytic principal series representations.
anMore precisely, they describe the irreducible subquotients of (IndG
for locallyB δ)×
algebraic characters δ : T → L . The main tool of their study is the constructionof certain bi-functors FPG (−, −) that map a pair (M, π) consisting of a Lie-algebra
representation M in the BGG category O (such that the action of Lie P on M canbe integrated to a P -action) and a smooth representation π of the Levi of P to a

CATEGORICAL p-ADIC LANGLANDS

91

locally analytic representation of G. We recall part of their construction. In fact itmakes sense to speculate that the construction of Orlik–Strauch extends to functors
an
FPG : Repan
L M −→ RepL G,

FPG : D(an. M ) −→ D(an. G),
that behave similarly to parabolic induction, see Expectation 6.2.21 below.
Again we focus on the case P = B. In the following we write U (gL ) for theuniversal enveloping algebra of gL = (Lie G) ⊗Qp L (note that here we
Q consider G
as a Qp -analytic group and hence gL is isomorphic to the product τ :F ,→L gld ).
Similarly, we write tL and bL for the Lie algebras over L of the Qp -analytic groups T
respectively B. Let us write O(U (gL )) for the BGG category O of finitely generated
U (gL )-modules on which the Cartan tL acts semi-simply and which are locally finite
as U (bL )-modules. We write O(U (gL ))alg for the subcategory of objects M on which
the action of tL is algebraic. On this subcategory one can integrate the bL -action to
obtain a B-action on M . Then every object M of O(U (gL ))alg can be regarded asa module over the sub-L-algebra D(gL , B) ⊆ D(G) generated by U (gL ) and D(B).
We obtain an exact contravariant functor
FBG (−) : O(U (gL ))alg −→ Repan G
given by
V 7−→ (D(G) ⊗D(g,B) V )0 ;compare [OS15, 3.4].
Remark 6.2.17. On the dual side, in terms of D(G)-modules, the locally analytic parabolic induction is given byan 0= D(G) ⊗D(B) (Lδ )0b ,
(IndG
B δ)
b
where Lδ denotes the 1-dimensional locally analytic representation of T on L via
the character δ, see [OS15, Lemma 2.3]. Now let ξ be an algebraic character of T .
The Verma module M (ξ) = U (gL ) ⊗U (bL ) Lξ is an object of O(U (gL ))alg and wefind that
D(G) ⊗D(gL ,B) M (ξ) = D(G) ⊗D(B) Lξ .
In particular, for an algebraic character ξ we findξ an
FBG (M (−ξ)) = IndG
B (z ) .

In fact Orlik and Strauch [OS15, 4.4] show that the functor FBG (−) can insome sense be twisted by a smooth character of T : for any choice δsm of a smoothcharacter of T there is an exact contravariant functor
FBG (−, δsm ) : O(U (gL ))alg −→ Repan G
such that for an algebraic character ξ the representation FBG (M (−ξ), δsm ) is theξlalocally analytic induced representation (IndG
B z δsm ) .
We point out the relation with locally algebraic representations whose smoothpart is parabolically induced. Fix an integral, dominant and regular Hodge–Tateweight λ and write ξ for the corresponding character of tL (via Definition 2.3.1),
and O(U (gL ))(ξ) ⊆ O(U (gL )) for the block of the category O containing the Vermamodule M (ξ). Then O(U (g))(ξ) ⊆ O(U (g))alg . As λ is regular the character ξis automatically dot-regular (i.e. the Weyl group orbit W · ξ of ξ under the dotaction w · ξ = w(ξ + ρ) − ρ consists of |W | elements). The simple objects of92

M. EMERTON, T. GEE, AND E. HELLMANNO(U (gL ))(ξ) are the simple quotients L(w · ξ) of the Verma modules M (w · ξ).
Moreover, L(ξ) = Wλ [1/p] is the irreducible algebraic representation of G definedin Definition 2.3.2.
Lemma 6.2.18. There is a canonical isomorphism
F G (L(−ξ), δsm ) ∼= L(ξ) ⊗L (IndG δsm )smBBwith the locally algebraic representation given by the tensor product of the irreducible
algebraic representation L(ξ) and the smooth parabolic induction of the smooth character δsm .
Recall that the BGG category has an internal notion of duality M 7→ M ∨(usually referred to as BGG duality) that is roughly given by mapping M to thedirect sum of the duals of its (finite dimensional) t-eigenspaces and using the main
involution on g (in order to pass from left to right modules and to obtain a U (gL )module that is locally U (bL )-finite instead of U (bL )-finite, where b denotes the
Lie-algebra of the opposite Borel B). For a simple module M in O(U (g)L ) wehave M ∨ ∼= M , but as (−)∨ is contravariant the functor exchanges subobjects andquotients. Let δ = z ξ δsm be a locally algebraic character with smooth part δsm andalgebraic part ξ. We then define the representation(6.2.19)GFB(δ) = FBG (M (−ξ)∨ , δsm ).

anG(δ) have theRemark 6.2.20. It follows from [OS15] that IndG
and FBB (δ)
same irreducible subquotients (even with the same multiplicities). Moreover, notethat Lemma 6.2.18 together with the fact that there is a canonical surjectionG ξ
M (−ξ)∨ → L(−ξ) implies that for dominant ξ the representation FB(z δsm ) hasGsm
the locally algebraic representation L(ξ)⊗L (IndB δsm ) as a quotient, whereas thisξ
anlocally algebraic representation appears as a subrepresentation in IndG
B (z δsm ) .

We expect that the Orlik–Strauch functors can be used to define a variant ofparabolic induction (whose derived versions of course will depend on the definitionof the categories D(an. T ) and D(an. G)).
Expectation 6.2.21. There exist natural functors
anG
FB: Repan
L T −→ RepL G,
GFB: D(an. T ) −→ D(an. G),
Gsuch that for a locally algebraic character δ the representation FB(δ) is given by(6.2.19).

Remark 6.2.22. It seems reasonable to speculate that there are internal dualities DG resp. DT of D(an. G) resp. D(an. T ) (similar to the dualities discussed
in Section 6.1) such that (up to shift)Gan
FB(−) = DG ◦ IndG
◦ DT .
B (−)

The restriction of DG to smooth representations should be given by the Zelevinsky
involution, and the above expectation suggests that DG should be related to Serreduality under the hypothetical functor Arig
G .
GUsing the conjectural functor FB
we can formulate a variant of Expectation
6.2.14. This expectation is motivated by the properties of Bezrukavnikov’s functor
in Theorem 6.2.26 below.

CATEGORICAL p-ADIC LANGLANDS

93

Expectation 6.2.23. There are isomorphisms


rig
G
!
0
∼
Arig
G FB (π) = β̃B̂,∗ α̃B̂ AT (ι(π)(−[F : Qp ]ρ ))G
functorial in π, whenever whenever π is a representation such that FB(π) lies inrigb!
Df.p. (an. G) and such that the support of α̃B̂ (AT (π)) is proper over Xd .

Remark 6.2.24. Let us consider the case F = Qp and d = 2 and recall from
Remark 6.2.9 that we expect Arig
G to be of the formbL
π 7→ π ⊗D(G) L∞ ,where L∞ is a family of D(G)-modules over X2 . For a point x ∈ X2 (L) represented
by a (ϕ, Γ)-module D we expect thatL∞ ⊗ k(x) = (Π(D)an )0 .
Here if D is étale, Π(D)an denotes the locally analytic vectors in the representation corresponding to D via the usual p-adic local Langlands correspondence forGL2 (Qp ), and if D is not étale we use Colmez’s extension [Col16] of the correspondence. Given a character δ : T → L, we are lead to expect that


G
G
an 0
∼
H 0 Arig
G (IndB̄ (δ)) ⊗ k(x) = HomG (IndB̄ (δ), Π(D) ) ,


G
G
an 0
∼
H 0 Arig
G (FB̄ (δ)) ⊗ k(x) = HomG (FB̄ (δ), Π(D) ) .
The dimensions of the vector spaces on the right hand side can be determined
explicitly, as Colmez has given a precise description of the locally analytic representations Π(D)an (see [Col14, Col16]), and in particular they are zero unless
D is trianguline. Using Expectations 6.2.14 and 6.2.23 the computation of thesefiber dimensions precisely matches the computation of fiber dimensions in Theorem
5.3.21 (v).
6.2.25. Bezrukavnikov’s functor on the BGG category. We elaborate a bit further on the Orlik–Strauch functor and compatibility with parabolic induction. As
on the Galois side parabolic induction is defined in terms of the compactification
XB of XB and as we can explicitly control the geometry of XB at certain regularpoints (Theorem 5.3.38), we can conjecturally link the evaluation of the functor
Arig
G on certain Orlik–Strauch representations to a functor on the BGG category
constructed by Bezrukavnikov [Bez16].
Fix λ and ξ as above. As every object in O(U (gL ))(ξ) is an extension of objects
that are (sub-)quotients of Verma modules M (w · ξ) for w ∈ W , the expectations
on the functor Arig
G imply that given δsm , there should be a functor
G
∨
b
Arig
G ◦ FB ((−) , δsm ) : O(U (gL ))(ξ) −→ Dcoh (Xd )such that the sheaves in the image of this functor are supported on the closed substack Xd,(λ,δsm )−tri defined just before Theorem 5.3.38. We use an additional duality(−)∨ in this functor to make the functor covariant. Using the local description of
the stack Xbd,(λ,δsm )−tri in Theorem 5.3.38 we can construct this functor, relying on
work of Bezrukavnikov.
Let us writeY
H = (ResF/Qp GLd )L =GLdσ:F ,→L94

M. EMERTON, T. GEE, AND E. HELLMANN

for the remainder of this section. We note that gL ∼= LieH = h as Lie algebras
over L. In particular O(U (gL )) and O(U (h)) are isomorphic. Recall from Section 5.3.22 (where we write G instead of H) that we write
Xw ⊆ eh ×h ehfor the irreducible component that dominates the closure of the Schubert cellH(1, w) ⊆ H/BH × H/BH , where BH ⊆ H is the Weil restriction of the upper
triangular matrices in GLd .
Moreover, we note that there is a canonical projectionpr1
ω : h̃ ×h h̃ −−→
h̃ −→ bt,

(where bt is the dual Lie algebra of tL ) and we write X w,0 for the intersection of X w
with the preimage of 0. Finally note that the zero section gives rise to a copy
(ResF/Qp GLd /Pλ )L × (ResF/Qp GLd /Pλ )L = H/BH × H/BH
⊆ X w0 ,0 ⊆ X w0 ⊆ h̃ ×h h̃.
Here Pλ denotes the parabolic subgroup associated to λ as in Section 5.2 which
agrees with the Borel B as we assume λ to be regular. The following functor
has been constructed by Bezrukavnikov in [Bez16, 11.4], see also [BK, Theorem
13.0.1].
Theorem 6.2.26 (Bezrukavnikov). There is an exact functor
Fξ : O(U (gL ))(ξ) −→ CohH ((h̃ ×h h̃)b0 )such that(i) For w ∈ W the image of the of the Verma module M (ww0 · ξ) is the dualizing sheaf of X̄w :Fξ (M (ww0 · ξ)) = ωX̄w .
(ii) For w ∈ W the image of the of the dual Verma module M (ww0 · ξ)∨ is the
structure sheaf of X̄w :Fξ (M (ww0 · ξ)∨ ) = OX̄w .
(iii) The image of the irreducible algebraic representation L(ξ) is the line bundleon(ResF/Qp GLd /Pλ )L × (ResF/Qp GLd /Pλ )L = H/BH × H/BH ⊆ X w0 ,0associated to the character ((−ρ0 )σ , (−ρ0 )σ ) (compare Remark 6.2.7 for the definition of ρ0 ).
We expect that, using the local description of the stack Xbd,(λ,δsm )−tri via themap
fλ,δsm : Xbd,(λ,δsm )−tri −→ (h̃ ×h h̃)b0of Theorem 5.3.38, the pullback of Bezrukavnikov’s functor gives rise to the restriction of the conjectural functor Arig
G via the functor of Orlik–Strauch:
Expectation 6.2.27. Let ϕ1 , . . . , ϕd ∈ L× such that ϕi /ϕj 6= 1, q for i 6= j
and let δsm be as in 5.3.16. Then the functorGbArig
G ◦ FB (−, δsm ) : O(U (gL ))(ξ) −→ Dcoh (Xd )CATEGORICAL p-ADIC LANGLANDS

95

factors through Xbd,(λ,δsm )−tri and is given by the composition∗(fλ,δ◦ Fξ (−))([F : Qp ]ρ0 ).
smRemark 6.2.28. Here the twist by ρ0 makes sense on Xbd,(λ,δsm )−tri , compare
the discussion following Theorem 5.3.38.
Note that using Lemma 6.2.18 and the fact that fλ,δsm is formally smoothwith relative dualizing sheaf the line bundle associated to −2ρ this expectation iscompatible with the twist by (−ρ0 )σ in (6.2.5) and with the twists in Expectation
6.2.14 and 6.2.23.
6.2.29. The trianguline Breuil–Mézard conjecture. We describe a shadow of thecompatibility with parabolic induction and of the compatibility with smooth categorical Langlands in Conjecture 6.2.4. This shadow can be regarded as a versionof the Breuil–Mézard conjecture (Conjecture 6.1.38) for locally analytic principal
series representations that involves Grothendieck groups rather than groups of cycles (compare the discussion in Section 6.1.37). In certain regular cases, a versionof this conjecture involving cycles was proven in [BHS17b], see Theorem 6.2.35
below.
As above we write G = GLd (F ) and T ⊆ B ⊂ G for the diagonal torus
respectively the upper triangular Borel. We keep the assumption that L is large
enough. Note that we may regard G as the group of Qp -valued points of H =ResF/Qp GLd and similarly T and B are the Qp -valued points of TH and TB the
Weil restrictions of the diagonal torus respectively of the upper triangular Borel.
We fix a regular Hodge–Tate weight λ and as usual write ξ for the corresponding
shifted character (which is a character of TH defined over F̄ ). We regard ξ as an(algebraic) character z ξ : T → L× (compare (5.3.17)) and fix a smooth characterδsm : T → L× .
Q
Let us write W ∼= Sn ⊆= K,→L Sn for the absolute Weyl group of H and W0 ∼W for the relative Weyl group of H. Then for every w ∈ W and w0 ∈ W0 we obtainwell defined characters z w·ξ and w0 δsm .
We write K0 (λ, δsm ) for the free abelian group generated by the irreduciblew·ξ 0
constituents of the locally analytically induced representations IndGw δsm ),B (z0for w ∈ W and w ∈ W0 .
Remark 6.2.30. The notation K0 should remind the reader of the fact that
this free abelian group is the Grothendieck group of the category of finite length
locally analytic representations whose irreducible subquotients occur in the above
locally analytic principal series representations.
On the Galois side, recall the definition of the stack Xd,(λ,δsm )−tri defined justbefore Theorem 5.3.38. We consider the union[
Xd,(λ,w0 δsm )−tri
Xd,[λ,δsm ]−triw0 ∈W0and the Grothendieck group K0 (Coh(Xd,[λ,δsm ]−tri )) of coherent sheaves on thisstack. Note that Xd,[λ,δsm ]−tri is precisely the set of (ϕ, Γ)-modules that admit a
filtration with graded pieces RF (z wλ w0 δsm ) for some w ∈ W and w0 ∈ W0 . The
smooth character δsm defines a Bernstein component Ω = [T, δsm ] of the category ofsmooth representations of G and we write τ for the inertial type such that Ω = Ωτ .

96

M. EMERTON, T. GEE, AND E. HELLMANN

As recalled in Section 5.2.9, the component Ωτ defines connected components
WDd,F,τ ⊂ WDd,F ,Filλ WDd,F,τ ⊂ Filλ WDd,Fof the stacks of rank d Weil–Deligne representations, respectively of rank d filteredWeil–Deligne representations of Hodge–Tate weight λ. The connected component
WDd,F,τ admits a canonical map
WDd,F,τ −→ Spec ZΩτ ,(6.2.31)see (5.2.13), and δsm defines a point [δsm ] ∈ Spec ZΩτ . We write
WDd,F (δsm ) ⊆ WDd,F,τ
for the preimage of [δsm ] under (6.2.31).
Remark 6.2.32. In terms of (smooth) categorical local Langlands this closed
substack can be characterized as the union of the supports of the coherent sheaves
G
Asm
G (π) attached to the irreducible smooth representations π that occur in ιB (δsm ),G
where ιB denotes normalized parabolic induction. Note that we do not need to take
0
G
0
0
G
all the possible ιG
B (w δsm ) for w ∈ W0 into account as ιB (δ)sm and ιB (w δ)sm havethe same Jordan–Hölder factors.
Lemma 6.2.33. The isomorphismFilλ WDd,F,τ ∼= Filλ Modd,ϕ,N,τ
of Lemma 5.2.3 induces a closed embeddingFilλ WDd,F,τ (δsm )an ⊂ Xd,(λ,δsm )−tri .
Proof. This is straightforward from the fact that the right hand side is thestack of all (ϕ, Γ)-modules that are trianguline with parameters z wλ w0 δsm .

The map in the following conjecture should be thought of as the map induced
by Arig
G on Grothendieck groups.
Conjecture 6.2.34. There is a unique injective group homomorphismaλ,δsm : K0 (λ, δ sm ) −→ K0 (Coh(Xd,[λ,δsm ]−tri ))such that(i) For w ∈ W and w0 ∈ W0 one hasaλ,δ [IndG (z ww0 ·ξ w0 δsm )an ] = [βe

∗
0
eB̂
(Arig
M (ι(−)([F : Qp ]ρ )))].
B̂,∗ ◦ α
For an irreducible smooth representation π in the block RepsmΩτ G that is aconstituent of ιG
δ
we
have
B sm
sm

(ii)

B

0
aλ,δsm ([π ⊗ L(ξ)]) = [(prWD ∗ (Asm
G (π))(−ρ )σ ].

Of course the formulas (i) and (ii) should be compared to Properties (1) and (2)in Conjecture 6.2.4. A weaker version of this conjecture is proved in [BHS17b,
Theorem 4.3.8] under more restrictive hypothesis. In fact the proof of loc. cit. globalizes from spectra of deformation rings to the stack Xd,[λ,δsm )]−tri ). More precisely,
let CH0 (Xd,([λ,δsm ]−tri ) denote the free abelian group on the irreducible componentsof Xd,[λ,δsm ]−tri . Mapping a coherent sheaf to its support (and forgetting components of non-maximal dimension) defines a morphism
supp : K0 (Coh(Xd,[λ,δsm ]−tri )) −→ CH0 (Xd,[λ,δsm ]−tri ).

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Then the methods of proof of [BHS17b, Theorem 4.3.8] can be used to prove thefollowing theorem.
Theorem 6.2.35. Let δsm = (unrϕ1 , . . . , unrϕn ) be an unramified characterwith ϕi /ϕj ∈/ {1, q} for i 6= j. Then there is a unique homomorphisma0λ,δsm : K0 (λ, δsm ) −→ CH0 (Xd,[λ,δsm ]−tri )such that the conditions in (i) and (ii) hold after replacing aλ,δsm by a0λ,δsm and after
composing the right hand side of the equalities with the support map supp.
Remark 6.2.36.
(a) In fact, in Conjecture 6.2.34, we could ask for more compatibilities, using parabolic induction from more general Levi subgroups than T . However, this would
make the formulation of the conjecture a bit more involved.
(b) While Conjecture 6.2.4 is rather hard to attack as there is no candidate for afunctor satisfying the constraints, Conjecture 6.2.34 is much more explicit: as in the
Breuil–Mézard conjecture, the group homomorphism aλ,δsm is determined by theconditions (i) and (ii) in the conjecture and the conjecture is rather about checking
relations than about the construction of the map (though in this case, contrary
to the Breuil–Mézard conjecture, there are only finitely many relations to check).
For example, under the additional assumptions in Theorem 6.2.35, Bezrukavnikov’s
functor 6.2.26 and Expectation 6.2.27 already determine aλ,δsm (once one can verify
the corresponding relations).
(c) The version of Conjecture 6.2.34 involving Grothendieck groups is much finer
than its version involving cycles. For example, in the case d = 2, if π is the(parabolically induced) smooth representation that is the non-split extension of thetrivial representation 1 and the (smooth) Steinberg representation St, then the cycle underlying aλ,δsm (L(ξ) ⊗ π) is trivial, whereas the class aλ,δsm (L(ξ) ⊗ π) in theGrothendieck group of coherent sheaves is non-trivial (compare e.g. [Hel20, Remark 4.43]). More precisely, the functor Asm
G maps St to the structure sheaf OX1 (indegree 0) of the irreducible component X1 ⊂ WD2,F,1 (the stack of (ϕ, N )-modulesof rank 2) where the monodromy N is generically non-trivial, whereas 1 is mapped
to a line bundle L on X1 shifted to cohomological degree −1. The line bundle L
and the structure sheaf OX1 are non-isomorphic as coherent sheaves on the stack
X1 , but they have of course the same underlying cycle. In the formula for the cycle
defined by Asm
G (π) these two cycles cancel, as they live in cohomological degrees
with distinct parities. On the other hand the class of Asm
G (π) in the Grothendieckgroup of coherent sheaves is still non-trivial.
(d) The construction of a0λ,δsm lies at the heart of the construction of companion
points on eigenvarieties in Section 9.6.29. In a sloppy way this can be comparedto the Breuil–Mézard conjecture as follows: proving the Breuil–Mézard conjecture
implies that one can (in some nice situations) construct automorphic forms with aprescribed Galois representation. Similarly, the locally analytic Breuil–Mézard multiplicity formula of Theorem 6.2.35 implies that one can construct (overconvergent
finite slope) p-adic automorphic forms with prescribed Galois representation.
7. Some known cases of categorical p-adic Langlands
7.1. GL1 . Unsurprisingly, the GL1 case of Conjecture 6.1.14 is a consequenceof local class field theory. Furthermore in this case we can describe the essential

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image of the functor, and we do not need to pass to the derived level, because thefunctor comes from an equivalence of abelian categories. The key point is that by∼
local class field theory we have an isomorphism WFab −→ F × = GL1 (F ), whileby [EG22, Rem. 7.2.18], the stack X1 admits a description as a moduli stack of
1-dimensional continuous representations of WF , and thus of representations of∼WFab −→ GL1 (F ).
More precisely, if we choose a uniformizer $F ∈ F then we can write F × =×b O O[X, X −1 ]∧ , where O[X, X −1 ]∧ is the
OF ×$FZ , and so write O[[F × ]] = O[[OF× ]]⊗−1
p-adic completion of O[X, X ]. Then by [EG22, Prop. 7.2.17] (see also [Pha22,
Cor. 1.2] for an alternative proof by Dat Pham, which is similar to the approach of
Kedlaya–Pottharst–Xiao mentioned below), there is an isomorphismhi∼b m −→(7.1.1)
Spf O[[F × ]] /G
X1
b m = Spf O[X, X −1 ]∧ is the p-adic completion of Gm , and in the formation(where Gb m -action is taken to be trivial).
of the quotient stack, the G
It follows immediately from (7.1.1) that there is an equivalence of categories
between the (abelian) category O[[F × ]] − Mod and the category of quasicoherentb m -action is trivial. Another way to formulate this equivasheaves on X1 whose G
lence is that the structure sheaf OX1 is naturally an O[[F × ]]-module, and that ourfunctor is given by(7.1.2)

π 7→ OX1 ⊗O[[F × ]] π.

In other words in the case d = 1, the sheaf L∞ of Remark 6.1.23 is OX1 .
There is a parallel description of the stack X1 and an interpretation of coherent
sheaves thereon in terms of locally analytic representations of F × . By a resultof Kedlaya–Pottharst–Xiao [KPX14, Thm. 6.2.14] every rank one (ϕ, Γ)-module
over RF,A , for an affinoid algebra A, is of the form RF,A (δ) ⊗ L for a continuous
character δ : F × → A× and a line bundle L on Sp A. It follows that
(7.1.2)
X1 = T /Gm = (Spf O[[OF× ]])rig × Gm /Gm ,where T = Homcont (F × , Gm (−)) is the rigid analytic space of continuous charactersof F × equipped with the trivial action of the rigid analytic multiplicative group Gm .
As above the choice of a uniformizer allows us to write T = W × Gm , where
W = Homcont (OF× , Gm (−)) = (Spf O[[OF× ]])rig
is the space of continuous characters of OF× . In particular we see that we have acanonical map
rig
b rig )/Gb rig −→ (W × Gm )/Gm = X1 .
X1,η= (W × Gmm

The image of this map is an open substack of X1 , and we note that this map is
not representable. Its non-trivial fibers are zero-dimensional and given by the stackb rig .
quotient Gm /Gm
There is an isomorphism
D(OF× ) ∼= Γ(W, OW )of the locally analytic distribution algebra of OF× with the global sections of thestructure sheaf on the space W of characters of OF× . We note that D(F × ) =CATEGORICAL p-ADIC LANGLANDS

99

D(OF× )[T ±1 ] is a Laurent polynomial ring over D(OF× ), i.e. the ring of functions of
the algebraic15 Gm over D(OF× ) whereasb L Γ(Gm , OG )
Γ(X1 , OX ) = Γ(T , OT ) ∼= Γ(W, OW )⊗1

mis the space of functions on the rigid analytic Gm over D(OF× ).
As in (7.1.2) we hence want to define a functor(7.1.3)Lb D(F × ) π,
π 7→ OX1 ⊗bL
where OX1 ⊗D(F × ) π should be a derived completed tensor product. Unlike in thecase of p-power torsion representations it turns out that the functor (7.1.2) is notdefined on the level of abelian categories, and hence the categorical p-adic Langlands correspondence for locally analytic representations is only a derived statement
already for GL1 ! In order to make this more precise it is better to consider the adjoint functor of (7.1.2). To make the discussion easier we consider the case F = Qp
×and even restrict to the case of Z×
p instead of Qp . The discussion of this case can∼again easily be reduced to the case of (Zp , +) = (1 + pZp , ·).
Recall that the distribution algebra D(Zp ) is isomorphic to the ring of analytic functions Γ(U, OU ) on the open unit disc U over Qp . Here we identify(Spf Zp [[Zp ]])rig , the space of continuous characters of Zp , with the open unit discU = (Spf Zp [[T ]])rig using an identification Zp [[Zp ]] ∼= Zp [[T ]].
The rigid analytic space U is a smooth Stein space. For coherent sheaves onsuch spaces Chiarelotto [Chi90] has constructed cohomology with compact supportRΓc (U, −) as well as a Serre duality(7.1.4)H 0 (U, F)0 ∼= H 1 (U, D(F)),
cwhere (−)0 denotes the strong dual (as above) and D(F) = DOU (F) denotes theSerre dual of the coherent sheaf F. By functoriality Hc1 (U, D(F)) (or more generally RΓc (U, F • )) is a module over D(Zp ) (or a complex of such modules). In fact
the induced Zp -action on the Hc1 is locally analytic (and admissible), as by Serreduality, it is the strong dual of a coadmissible module over D(Zp ). The functor(7.1.2) now should be defined as the right adjoint functor to the functor
F 7−→ RΓc (U, F)
from the derived category of coherent sheaves Dbcoh (U) to a certain derived category of D(Zp )-modules in solid Qp -vector spaces (in the sense of Clausen–Scholze).
We note that the functor RΓc (U, −) does not preserve the cohomological degree of
coherent sheaves: if F is a coherent sheaf concentrated in degree 0 that has finite
support, then RΓc (U, F) agrees with the global sections RΓ(U, F) and is concentrated in degree 0 (and is a finite dimensional Qp vector space). On the other handRΓc (U, OU ) is concentrated in degree 1 where it is given by the space of locallyanalytic functions C an (Zp , Qp ) on Zp (as C an (Zp , Qp ) is identified with the strong
dual of D(Zp ) = RΓ(U, OU )).
bL
Conversely the adjoint functor −⊗D(Zp ) OU takes finite dimensional locally analytic representations to coherent sheaves in degree 0 while it takes C an (Zp , Qp ) toOU [1], the structure sheaf shifted to degree −1.
15Note that we usually write Gm for the rigid analytic space associated with the scheme
Gm . This should not cause confusion as we will always explicitly mention when Gm should beseen as a scheme.

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Remark 7.1.5. Using solid condensed modules and analytic rings as introducedby Clausen–Scholze [Sch] (and in the context of rigid analytic spaces studied further by Andreychev [And21]) it is indeed possible to make the above precise. In
order not to get too involved with condensed structures we will not spell this out
here. However we note that one should compare the above situation to the compactification of the affine line
ι : Spa(Z[x], Z[x]) → Spa(Z[x], Z)
in the context of discrete adic spaces, see [Sch, Theorem 8.1]. In this context thederived completionι∗ = − ⊗ L(Z[x],Z) (Z[x], Z[x]) : D (Z[x], Z) → D (Z[x], Z[x])has a fully faithful left adjoint ι! (the cohomology with compact support) suchthat ι∗ ◦ ι! is the identity, see [Sch, Observation 8.11]. Moreover ι! (Z[x]) =(Z((x−1 ))/Z[x])[−1] lives in cohomological degree 1 (which implies that the derived completion ι∗ maps Z((x−1 ))/Z[x] to the structure sheaf shifted to degree−1), see [Sch, Observation 8.12].
Remark 7.1.6. The above remark suggests that a possible definition of thederived category of locally analytic Zp -representations could be the essential imageof the functor RΓc (U, −) from a category of solid quasicoherent OU -modules to acategory of D(Zp )-modules in the derived category of solid Qp -vector spaces.
It may seem surprising that in the Banach case the functor (7.1.2) is (the derived
functor of) a functor between abelian categories, while in the locally analytic setup
the functor (7.1.3) only makes sense on the derived level. In particular one ofthese functors cannot shift the cohomological degree, while the other one can16.
Still, these two functors are compatible in the sense of Conjecture 6.2.4, as we now
explain.
Consider the pro-system of ind-coherent sheaves
(n)
(n)
Fm
= Fm
⊗Zp [[T ]] OSpfZp [[T ]]m(n)on Spf(Z/pn [[T ]]) = colimm Spec((Z/pn )[T ]/T p ), where Fm = (Z/pn )[Zp /pm Zp ]denotes the representation of Zp on the space of Z/pn -valued functions on Zp thatare constant on pm Zp -cosets (here we of course use the isomorphism Zp [[Zp ]] ∼=Zp [[T ]]). On the level of representations this pro-ind system gives rise (after taking
the limits lim←,n lim→,m ) to the representation of Zp on the space of continuous
maps C cont (Zp , Zp ) and hence, after inverting p and passing to locally analytic
vectors we obtain C an (Zp , Qp ). The compatibility of the Banach and the analytic
context is hence settled by the following lemma.
(n)

(n)

Lemma 7.1.7. The generic fiber (Fm )η of the pro-ind-system (Fm ) (concentrated in degree 0) is the structure sheaf on the open unit disc OU [1] shifted todegree −1.
16One heuristic explanation for this phenomenon is the following: in the torsion case we consider smooth representations, and hence F × is regarded as a locally pro-finite group, in particular
it should be considered as a 0-dimensional object. In the analytic context F × is regarded as ap-adic Lie group which has positive dimension.

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Before sketching the proof, we have to define what we mean by the generic(n)(n)
fiber (Fm )η of a pro-ind system (Fm ). Recall that the open unit disc has anadmissible cover
[
U=
Uk
k≥1

by the closed discs Uk of radius p−1/k and each of the closed discs has a nice affine
formal model Uk . Let jk : Uk ,→ U denote the open embedding and let us write(n)πk : Uk → Spf Zp [[T ]] for the canonical map. Then jk∗ ((Fm )η ) is by definition thegeneric fiber of the coherent sheaf on Uk given by(n)
lim lim πk∗ Fm,←,n →,m(n)

and we define (Fm )η by gluing these sheaves on the cover U =S

k≥1 Uk .

Sketch of proof of Lemma 7.1.7. We sketch the case k = 1 in which case
U1 = Spf Zp hXi with X = p−1 T . For simplicity we only prove that(1)lim π̄1∗ Fm→,mis the structure sheaf on U¯1 = Spec Fp [X] shifted to degree −1 and leave the generalcase as an exercise . Hereπ̄1 : Spec Fp [X] → Spf Fp [[T ]]is the mod p fiber of π1 , i.e. the morphism given by T 7→ 0. One can compute thatm(1)
Fm is the coherent sheaf corresponding to the module Fp [T ]/T p with transition(1)
maps induced by T 7→ T p (a quick way to see this is to note that lim→,m Fm ismthe Pontryagin-dual of Fp [[Zp ]] and lim→,m Fp [T ]/T p is the Pontryagin dual of
Fp [[T ]], but we have fixed an identification Fp [[Zp ]] ∼= Fp [[T ]]).
Computing the derived pullback π̄1∗ using the resolution·T

Fp [[T ]][X] −→ Fp [[T ]][X](1)of Fp [X], we find that π̄1∗ Fm is a two-term complex with cohomology Fp [X] =mm
T p −1 (Fp [T ]/T p )[X] in degree −1 and Fp [X] = (Fp [T ]/T )[X] in degree 0. However the transition maps T 7→ T p induce isomorphisms in degree −1 whereas they(1)
induce the zero maps in degree 0. It follows that lim→,m π̄1∗ Fm is Fp [X] concentrated in degree −1 as claimed.
Remark 7.1.8. We note however that the functor (7.1.3) cannot be fully faithb
ful on the category Df.p.
(an. F × ): Roughly the functor involves analytification fromthe algebraic Gm to the analytic Gm . To give an explicit example, let π be theuniversal unramified representation
F × → F × /O× ∼= Z → Γ(Galg , O× alg ) = L[T, T −1 ]×
F

m

Gm

mapping 1 to T , where we write Galg
m for the scheme Gm . Then (7.1.3) maps π tothe structure sheaf on the quotient of the rigid analytic space Gm = {1} × Gm ⊂W × Gm = T , where 1 ∈ W denotes the trivial character of OF× , by the trivial
Gm -action . It follows that EndF × (π, π) = L[T ±1 ], whereas the endomorphismbLring of OX1 ⊗D(F × ) π is the ring of functions on the rigid analytic Gm . In fact thefunctor (7.1.3) can be seen as some kind of localization of a representation π on X1 .

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We expect that the functor is fully faithful on those representations π such that the
canonical map
bL
π → RΓc (X1 , OX1 ⊗D(F × ) π)
is an isomorphism, or equivalently those representations π on which the D(F × )action extends to Γ(T , OT ). This is the case if π has bounded slope, for example
if π is an admissible representation.
7.2. The Banach case for GL2 (Qp ) — I. The structure of X . In this section and the following two, we summarise the results of the work in progress [DEGa]
of Andrea Dotto with M.E. and T.G., which establishes a version of Conjecture 6.1.14 for GL2 (Qp ). We caution that since this is work in progress, there
may be imprecisions in what follows, and the sketched proofs may not reflect thosein the final version of [DEGa].
For simplicity of exposition, we work throughout these two sections with representations having trivial central character. (The papers [DEGa, DEGb] allow an arbitrary fixed central character, and the arguments are essentially thesame, with some notational overhead.) Accordingly we set G := PGL2 (Qp ),and K = PGL2 (Zp ), so that sm. G is the abelian category of smooth GL2 (Qp )representations with trivial central character.
On the geometric side, we work with the moduli stack X of rank 2 (ϕ, Γ)modules for GL2 /Qp with determinant ε−1 . As explained following Remark 4.1.6,
this is a formal algebraic stack over Spf O, whose underlying reduced substack Xredis equidimensional of dimension 1. We make the geometry of X much more explicitin the discussion below.
The following expected theorem verifies Conjecture 6.1.14 in the case of GL2 (Qp ).
Expected Theorem 7.2.1 (Dotto–Emerton–Gee). Suppose that p ≥ 5, and
let G = PGL2 (Qp ). Then there is a pro-coherent sheaf (concentrated in degree 0)of O[[G]]-modules L∞ over X , such that
A(π) := L∞ ⊗L
O[[G]] πbbdefines an O-linear exact fully faithful functor A : Df.p.
(sm. G) → Dcoh
(X ),bwhich then extends to a cocontinuous fully faithful functor A : Ind Df.p. (sm. G) →Ind Coh(X ).

Remark 7.2.2. We also expect to prove the various properties conjecturedin
GConjecture 6.1.14. In particular, the statement about the support of A c-IndK Wλ ⊗Opb
σ crys,◦ (τ )should follow easily from the construction and the classification of
locally algebraic vectors in the existing p-adic local Langlands correspondence
for GL2 (Qp ).
The expected compatibility with duality is discussed in Section 7.3.22.
The following theorem is an ingredient in the proof of Expected Theorem 7.2.1.
It is essentially the main result of [DEGb] (together with the construction of themorphism Xred → X, which will appear in [DEGa] and is explained below). It
is special to the case of GL2 (Qp ), in the strong sense that we do not expect it togeneralise in any obvious way, even to GL2 (Qp2 ).
Theorem 7.2.3 (Dotto–Emerton–Gee). Assume that p ≥ 5 and G = PGL2 (Qp ).
There is a reduced scheme X given by an explicit chain of P1 s over k with ordinary

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double points, and a morphism Xred → X inducing a bijection on closed Fp -points.
For each open subset U of X, there is a certain localization (sm. G)U of sm. G suchthat (sm. G)X = sm. G, and the collection {(sm. G)U } forms a stack (of abeliancategories) over the Zariski site of X.
Remark 7.2.4. We anticipate that Theorem 7.2.3 can be refined to give much
more precise information about the compatibility of localization with the functor A,
and about the Bernstein centres of the categories (sm. G)U . Indeed, we expect thatb with underlying reduced scheme X, which is equipped
there is a formal scheme X
b and which simultaneously realises X
b as a moduli spacewith a morphism X → X,b should be initial for maps from X to formal
for X (in the sense that X → X
b will satisfy the (formal
algebraic spaces; indeed we expect that the map X → X
algebraic analogues) of the conditions of [Alp14, Prop. 7.1.1]), and whose ring offunctions over each U is identified with the Bernstein centre of (sm. G)U . We hopeto prove such results in a sequel to [DEGa].
Remark 7.2.5. We expect that the statements of Expected Theorem 7.2.1 andTheorem 7.2.3 continue to hold without the assumption that p ≥ 5, and that withsome effort the proofs sketched below can be extended to this case. The difficultywhen p = 2 or 3 is that there are many more special cases to consider (or rather,
the existing special cases overlap in more complicated ways); on the Galois side,
this is because the mod p cyclotomic character is quadratic (if p = 3) and trivial(if p = 2), and on the representation theory side, there are no very generic Serreweights in the sense of Definition 7.2.10 below (and indeed no generic Serre weights
if p = 2).
Remark 7.2.6. Under mild hypotheses, the expected compatibility with Taylor–Wiles–Kisin patching explained in Remark 6.1.30 is a consequence of the construction of our functor below, and local-global compatibility for p-adic local Langlands
for GL2 (Qp ) of [Eme11a] (see also [CEG+ 18] for a more direct connection between
the patching method for modular curves and p-adic local Langlands for GL2 (Qp ),under slightly different hypotheses). See also Expected Theorem 9.4.2.
7.2.7. Motivation. One reason that we are able to prove Conjecture 6.1.14 forGL2 (Qp ) (but not in any other cases) is that we are able to build upon the resultsof Colmez [Col10c] and Paškūnas [Paš13]. Roughly speaking, these results canbe reinterpreted as proving Conjecture 6.1.14 “pointwise”; that is, after completing
X at a closed Fp -point on the spectral side, and restricting to a block of locallyadmissible representations on the automorphic side.
There does not seem to be any hope of directly passing from these results toExpected Theorem 7.2.1, and morally our strategy is to carry out constructions
over X which are analogues of some of the constructions of Colmez and Paškūnas,
and where possible to check properties of these constructions by passing to completions and using their results. In particular, Xred generically consists of reducible
representations, which (for GL2 (Qp ), but not more generally) correspond via themod p local Langlands correspondence for GL2 (Qp ) to (extensions of) principal
series representations, and we are able to interpolate various explicit calculations
in [Paš13] over the reducible locus.
In order to carry out this strategy we need to have an analogue on the automorphic side of familiar operations on sheaves such as restriction to open subsets

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and passage to completion at closed points. In [DEGb] we develop such a localization theory for sm. G, proving (for the most part by purely representation-theoretic
arguments, together with some use of the p-adic local Langlands correspondence
for GL2 (Qp )) that it localizes over the projective scheme X of Theorem 7.2.3.
We then generalize Colmez’s construction in [Col10c] of the GL2 (Qp )-representation
he calls D\ P1 , to give a direct construction of our candidate for L∞ . We therefore
have an explicit functor A, whose behaviour after completion at closed points canbe understood using the results of [Col10c, Paš13].
Our arguments rely on having a good understanding of the geometry of thestack X , as well as an interpretation of this geometry in Galois representationtheoretic terms, and in the remainder of this section we describe this geometry. In
the following section we explain the construction of L∞ and sketch the main points
in the proof of Expected Theorem 7.2.1.
7.2.8. The geometry of Xred . We begin by describing the irreducible components of Xred and their intersections. These are described by the weight part of
Serre’s conjecture, and we begin by recalling this.
××
For any x ∈ Fp , we let λx : GalQp → Fp be the unramified character taking ageometric Frobenius element to x. We write ω = ε for the mod p cyclotomic char×acter, so that every character GalQp → Fp is of the form λx ω i for some uniquely×

determined x ∈ Fp and 0 ≤ i < p − 1.
×

We write ω2 : IQp → Fp for a choice of fundamental character of niveau 2;
writing Qp2 for the quadratic unramified extension of Qp , we can extend ω2 to×a character GalQp2 → Fp in such a way that ω2p+1 = ω|GalQ 2 . Then the 2p
dimensional absolutely irreducible representations GalQp → GL2 (k) are precisely
GalQ

×

those of the form λx ⊗ IndGalQp ω2n for some x ∈ Fp and n ∈ Z/(p2 − 1)Z withp2

(p + 1) - n. In this case x is uniquely determined up to multiplication by −1,
and n is uniquely determined up to multiplication by p. The determinant of thisrepresentation is an unramified twist of ω n .
Serre weights for GL2 (Fp ) are the (isomorphism classes of) irreducible Fp representations of GL2 (Fp ), which are explicitly given by σa,b = deta Symb F2p ,where 0 ≤ a < p − 1 and 0 ≤ b ≤ p − 1. It is sometimes convenient to view a as anelement of Z/pZ, and we will do so without further comment.
Following Serre [Ser87] (and as reinterpreted by Buzzard–Diamond–Jarvis[BDJ10]) one can associate to each representation ρ : GalQp → GL2 (Fp ) a setof Serre weights σ. One way of defining this is to say that σa,b is a Serre weightfor ρ if and only if ρ admits a crystalline lift with Hodge–Tate weights 1−a, −(a+b).
This description can be made completely explicit as follows.
GalQ
If ρ = λx ⊗ IndGalQp ω2n is irreducible, then σa,b is a Serre weight for ρ if andp2

only if n ≡ (p + 1)a + b − p (mod p2 − 1) or n ≡ (p + 1)a + pb − 1 (mod p2 − 1).
If b 6= 0, then σa,b is a Serre weight for a reducible representation ρ if and only
if if and only if
λx ω a+b∗(7.2.9)ρ∼=
0
λy ω a−1
×for some x, y ∈ Fp . If b = 0, then we further demand that ρ is finite flat. If x 6= ythen this is automatic, while if x = y it is equivalent to asking that the extension

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be peu ramifiée (which by definition is equivalent to the corresponding Kummer
class being integral).
Note that if det ρ = ω −1 then any Serre weight for ρ has trivial central character.
We assume that all Serre weights have trivial central character from now on; thatis, we only consider Serre weights σa,b with 2a + b ≡ 0 (mod p − 1). In particular,b is always even.
Then the irreducible components of Xred are as follows: for each Serre weight σ =σa,b with b 6= p − 1, there is a unique irreducible component X (σ), whose Fp -points
are precisely those ρ : GalQp → GL2 (Fp ) admitting σ as a Serre weight. Furthermore if σ = σa,p−1 for a = 0, (p − 1)/2, then there are two corresponding irreducible
components X (σ)± , which differ by a twist by the unramified quadratic character λ−1 . In this case, the representations which admit σa,p−1 as a Serre weightare precisely the union of the Fp -points of the stacks X (σa,p−1 )+ , X (σa,p−1 )− , andX (σa,0 ).
We now describe the stacks X (σ) more explicitly.

Definition 7.2.10. A Serre weight σa,b is generic if 0 ≤ b ≤ p−3 (equivalently,
if b 6= p − 1). It is very generic if furthermore b 6= 0.

The following result is essentially a consequence of Fontaine–Laffaille theory(see Section 7.6.4 for slightly more explanation).

Proposition 7.2.11. If σ = σa,b is generic, then we have an isomorphism∼
X (σ) −→ [(A2 \ {0})/Gm ], where if we write (t, x) for the coordinates on A2 , thenthe Gm -action is given byu · (t, x) = (t, u2 x).

The reducible locus is the locus t 6= 0, and the split locus is x = 0. More precisely,
the locus t 6= 0 is the universal extension

λt ω a+b0

∗.
λt−1 ω a−1

The variable x parameterizes the (unique non-zero) extension class.

106

M. EMERTON, T. GEE, AND E. HELLMANNIn the non-generic case, we have the following description of the stacks X (σa,p−1 )± ,which can be proved by constructing universal families of extensions of rank 1 (ϕ, Γ)modules.
∼

Proposition 7.2.12. If σ = σa,p−1 , then we have isomorphisms [A2 /Gm ] −→X (σ)± , where if we write (x, y) for the coordinates on A2 , then the Gm -action is
given byu · (x, y) = (ux, uy).
±More precisely, X (σ) is the universal extension
λ±1 ω −a∗.
0
λ±1 ω −(a+1)
The variables x, y parameterize the extension class in the 2-dimensional k-vector
space H 1 (GalQp , ω).
Definition 7.2.13. The companion of a generic Serre weight σ = σa,b is theSerre weight σ co := σa+b+1,p−3−b (which is again generic). We refer to the unordered pair of σ, σ co as a companion pair.
(We do not define a companion for a Serre weight of the form σa,p−1 .)We say that a companion pair σ, σ co is very generic if σ and σ co are very genericthey not of the form σa,0 , σa+1,p−3 .
Remark 7.2.14. The companion pairs σ, σ co are precisely the sets of Jordan–Hölder constituents of the reductions modulo p of tame cuspidal types (with trivial
central character).
The following decomposition of Xred will be useful below.
Definition 7.2.15. If σ = σa,b is a generic Serre weight, then we define aclosed substack X (σ/σ co ) of Xred as follows: if b 6= 0, p − 3 then we set
X (σ/σ co ) = X (σ) ∪ X (σ co ),CATEGORICAL p-ADIC LANGLANDS

107

if b = 0 then we set
X (σ/σ co ) = X (σa,0 ) ∪ X (σa,p−1 )+ ∪ X (σa,p−1 )− ∪ X (σa+1,p−3 ),and if b = p − 3 then we set
X (σ/σ co ) = X (σa−1,0 ) ∪ X (σa−1,p−1 )+ ∪ X (σa−1,p−1 )− ∪ X (σa,p−3 ).
Note in particular that X (σ/σ co ) depends only on the unordered pair {σ, σ co }, andthat
Xred =[X (σ/σ co ),σ/σ cowhere σ/σ co runs over all companion pairs of Serre weights.

We let X denote a chain of P1 ’s over k with ordinary double points, of length(p − 1)/2. We choose a coordinate T on each irreducible component of X in sucha way that each singular point corresponds to 0 on one intersecting component
and ∞ on the other. We will refer to the points 0 and ∞ as marked points.
We label the components of X by pairs σ, σ co of companion Serre weights, anddenote the corresponding P1 by X(σ/σ co ). We order the components by demanding that if 0 < b < p − 1, then the component X(σa,b /σa+b+1,p−3−b ) meets thecomponent X(σa+b,p−1−b /σa+1,b−2 ). It is easy to check that there are precisely
two such labellings of the components (the choice corresponding to a choice of endof the chain).

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The point of this construction is that using our explicit descriptions of thestacks X (σ) we can define a morphism π : Xred → X. Again we content ourselveswith this description in the case of a very generic pair σ, σ co . We define a morphism
πσ : X (σ/σ co ) → X(σ/σ co ) as follows. We identify X (σ) with [(A2 \ {0})/Gm ] asin Proposition 7.2.11, and with the coordinates (t, x) there, we send (t, x) 7→ t. We
also identify X (σ co ) with [(A2 \ {0})/Gm ], and send (t, x) 7→ t−1 . Note that bydefinition we have πσco (x, t) = πσ (x, t)−1 .
For each companion pair σ, σ co of Serre weights, we make a choice of one ofπσ : X (σ/σ co ) → X(σ/σ co ) or πσco : X (σ/σ co ) → X(σ/σ co ), in such a way thatthe two maps agree at the intersection points of adjacent components of X. (There
are two possible such choices of maps, corresponding to the two choices of labellingof the components of X; one can convert between the two possible labellings by
twisting by det(p−1)/2 .)Remark 7.2.16. If σ/σ co is very generic, thenX (σ/σ co ) ∼
=



1
Pt × Spec k[x, y]/(xy) /(Gm )u \ (0 × [{x = 0}/Gm ] ∪ ∞ × [{y = 0}/Gm ]),where Gm acts via u · (x, y) = (u2 x, u−2 y). The closed substack X (σ) is cut out by
the equation y = 0 (with x, t, and u then corresponding to the variables with thesame names in the description of X (σ) given by Proposition 7.2.11), while X (σ co )is cut out by x = 0 (with y, t−1 , and u−1 then corresponding to the variables x, t,
and u appearing in the description of X (σ co ) given by Proposition 7.2.11). In termsof this explicit description, the map X (σ/σ co ) → X(σ/σ co ) is simply projection to
the P1 factor.
Remark 7.2.17. It can be checked that if σa,b is a Serre weight with b 6= p − 1,then we have an isomorphism
X (σa,b ) ∪ X (σa+b,p−1−b ) ∼=Spec k[t1 , t2 ]/(t1 t2 ) × [P1x /(Gm )u ] \ ({t1 = 0} × [0/Gm ] ∪ {t2 = 0} × [∞/Gm ]),where Gm acts on P1 via u · x = u2 x.
Here we identify the locus t2 = 0 (resp. t1 = 0) with X (σa,b ) (resp. X (σa+b,p−1−b ))via Proposition 7.2.11, with t1 corresponding to t and x corresponding to x (resp. t2
corresponding to t and x corresponding to x−1 ).

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From this, it is easy to see that the above prescriptions for the morphisms πσand πσco glue to give a morphism Xred → X.
Remark 7.2.18. The motivation for the definition of X and of the morphism π :Xred → X is as follows. Either by general principles, or by our explicit descriptionsof the stacks X (σ), one easily sees that the closed Fp -points of X are precisely
the semisimple ρ : GalQp → GL2 (Fp ) (with determinant ε−1 ). It follows fromthe definition of π that it induces a bijection on closed Fp -points, and we thinkof X as a “coarse” moduli space17 for Xred . On Fp -points, we can think of themorphism π as sending a representation to its semisimplification; more explicitly,
the marked points of the component X(σ/σ co ) correspond to irreducible ρ, and
the Gm obtained by deleting the marked points parameterizes sums of characterswith fixed product, and with the Frobenius eigenvalue determined by the Gm .
Remark 7.2.19. For any choice of F , it is easy to construct associated moduli
space morphisms(7.2.20)(XF,1 )red → Ysatisfying satisfying the conditions of [Alp14, Prop. 7.1.1] Indeed, since (XF,1 )red
is a global quotient stack of the form [Spec A/Gm ], where Gm acts trivially, one
sets Y = Spec A, and the resulting morphism (7.2.20) is even a good moduli spacemorphism, in the sense of [Alp13].
b m ] for some adicb G
Furthermore, since XF,1 can be written in the form [Spf A/b mbaffine formal scheme Spf A which thickens Spec A, endowed with the trivial GbGb m , we obtain a morphism XF,1 → Yb which
action, if we then set Yb := Spf A
satisfies the formal algebraic analogue of the conditions of [Alp14, Prop. 7.1.1],
and realizes Yb as the formal moduli space associated to XF,1 itself.
17We put “coarse” in quotes because the morphism X
red → X is not a coarse moduli spacemorphism in the technical sense of e.g. [Alp14], or even an adequate moduli space morphism(for example, because it is not a closed map on underlying topological spaces). It does, however,satisfy the conditions of [Alp14, Prop. 7.1.1], and in particular is initial for morphisms from Xredto locally separated algebraic spaces.

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Returning to the case of GL2 (Qp ), as already mentioned, we anticipate thatbthe morphism Xred → X constructed above also admits a formal thickening X → X
satisfying the formal algebraic analogue of the conditions of [Alp14, Prop. 7.1.1].
On the other hand, we do not expect there to exist interesting moduli space morphisms XF,d → Yb , or (XF,d )red → Y , if F 6= Qp and d ≥ 2, or even if F = Qp whend > 2. More precisely, in these cases we expect that any morphism (XF,d )red → Y ,with Y a locally separated algebraic space, maps each connected component of its
domain to a single point.
7.2.21. Some open substacks. Fix a very generic companion pair of Serre weights
σ, σ co , and write U (σ/σ co ) to denote the open subset of |X(σ/σ co )| obtained bydeleting the marked points 0, ∞. Then U (σ/σ co ) is also open in |X|, and we writeU(σ/σ co ), or sometimes just U, to denote the preimage of U (σ/σ co ), thought ofan open substack of X . Then Ured = U(σ/σ co )red is the (dense) open substackof X (σ/σ co ) obtained by deleting the locus (consisting of two closed points) corresponding to irreducible ρ. More precisely, Ured is the union of its two closed
substacks U ∩ X (σ) and U ∩ X (σ co ).
coIn terms
7.2.16, U(σ/σ co )red is of the description of X (σ/σ ) given in Remark±1given by Gm × Spec k[x, y]/(xy) /Gm = [Spec k[t , x, y]/(xy)/Gm ] with Gm action given byu · (x, y) = (u2 x, u−2 y).

By a deformation theory argument building on this description of Ured , we can
explicitly describe U as
(7.2.22)b m ],
U(σ/σ co ) = [Spf O[s, t±1 , x, y]∧ /Gb m -action is given by
where the hat denotes (p, s, xy)-adic completion, and the Gu · (s, t, x, y) = (s, t, u2 x, u−2 y).
Here U ∩X (σ) is the locus $ = s = y = 0, and U ∩X (σ co ) is given by $ = s = x = 0(with the identification with the description in Proposition 7.2.11 being given by
y 7→ x−1 , t 7→ t−1 ).

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7.2.23. More open substacks. As in Remark 7.2.17, consider a Serre weight σa,bwith b 6= p − 1. The union X (σa,b ) ∪ X (σa+b,p−1−b ) then contains a unique closed
point whose associated Galois representation ρ is irreducible. If we delete the split
locus from this union (which, in terms of the explicit description of Remark 7.2.17,
means deleting the loci x = 0 and x = ∞), we obtain an open subset Vred of Xred
containing ρ, which admits the explicit description Vred = [Spec k[t1 , t2 ]/(t1 t2 )/µ2 ],where the µ2 -action is trivial.
If we let V denote the open substack of X corresponding to the open subset Vredof Xred , then a deformation theory argument shows that V admits the description∼ [Spf O[a, t1 , t2 ]b/µ2 ],V=
where the hat indicates (p, a, t1 t2 )-adic completion, and where the µ2 -action is again
trivial.
7.3. The Banach case for GL2 (Qp ) — II. Proof sketches. In this section
we outline the proof of Expected Theorem 7.2.1.
7.3.1. The construction of L∞ . We begin by defining our pro-coherent sheaf L∞
of GL2 (Qp )-representations. In essence this is a natural generalization of Colmez’s
GL2 (Qp )-representation D\  P1 . Some of the arguments of [Col10c] go throughessentially unchanged in our setting, but others need more serious adaptation. (Themain difficulties that arise are that Colmez’s D] and Dnr are not compatible withflat base change, so that all arguments involving them have to be reworked; andthat, when working with (ϕ, Γ)-modules with general coefficients, quotient of lattices need not be literally finite modules, so that arguments invoking such finitenessstatements must also be reworked.)
To begin with, we note that by definition there is a universal rank 2 (ϕ, Γ)module D over X . We find it convenient to introduce a twist into its definition, in
the following way. If A is a finite type O/$a -algebra, then a morphism Spec A → X
corresponds to a rank 2 (ϕ, Γ)-module with A-coefficients and determinant the112

M. EMERTON, T. GEE, AND E. HELLMANN

inverse cyclotomic character, and the formation of this (ϕ, Γ)-module is compatible
with base change in A. We write DA for the twist of this (ϕ, Γ)-module by thecyclotomic character, so that DA has determinant equal to the cyclotomic character(rather than its inverse).
We work with (ϕ, Γ)-modules for the full cyclotomic extension Qp (ζp∞ )/Qp , sothat DA is a rank 2 projective module over the ringAA = A((T )),equipped with commuting(A-linear) semilinear actions of ϕ and Γ, whereϕ(1 + T ) = (1 + T )p ,γ(1 + T ) = (1 + T )ε(γ)
for any γ ∈ Γ (where ε is as usual the p-adic cyclotomic character). There is alsoa left inverse ψ : DA → DA to ϕ; this is an A-linear surjection which commuteswith Γ, and satisfies
ψ(ϕ(a)m) = aψ(m),ψ(aϕ(m)) = ψ(a)mfor any a ∈ AA , m ∈ DA .
Following Colmez, we use the actions of ϕ, ψ andΓ on DA to explicitly
build

Z
\
{0}
Z
p
p
a GL2 (Qp )-representation. Write P + for the monoid. Then there
0
1+
is an action of P on DA given by the continuous maps
 k

p a b
z = (1 + T )b ϕk ◦ σa (z)01
ε(σa ) = a.
for a ∈ Z×
p , b ∈ Zp , k ∈ Z≥0 , where we write σa ∈ Γ for the element with


a
b
+
There is a natural action of P on Zp (respectively of P on Qp ) via
x=
0 1
ax + b, and as explained in [Col10b, §3.1], this action can be used to define a
P + -equivariant sheaf of A-modules U 7→ DA  U on Zp . This is arranged byletting DA  Zp be DA itself, and for any i∈ Zp and k ≥ 0, the sections ofkpiDA  (i + pk Zp ) on i + pk Zp are defined to be
DA ⊆ DA . The restriction
0 1
k
map Resi+pk Zp : DA  Zp → DA  (i + p Zp ) is given by




1 i
1 −i
◦ ϕk ◦ ψ k ◦
.
0 1
0 1
ψ=0
We thus have in particular DA  Z×
p = DA .
Given an affine linear map f : U → V between compact open subsets of Zp wehave an induced map f∗ : M  U → M  V . This definition extends immediately tothe case of continuous maps which are locally affine linear, and by taking limits, tofunctions which can be uniformly approximated by affine linear maps. In particular
this applies to the map z → 1/z on Z×
p , and we can then glue DA Zp := DA to itself
1
1
over DA  Z×
p via this map, giving a sheaf over P (Zp )(= P (Qp ) = Qp ∪ {∞}).
1
We denote the global sections of this sheaf by DA  P ; by functoriality, they havean action of GL2 (Qp ) with trivial central character.
Since Qp is an open subset of P1 (Zp ), we can also take the sections of DA  P1
over Qp , to obtain a B-representation that is denoted DA  Qp . Unwinding theCATEGORICAL p-ADIC LANGLANDS

113

definitions, one obtains an explicit description of DA  Qp as the inverse limit
limψ DA .
←−The action of GL2 (Qp ) on DA  P1 is compatible with flat base change in A,
but DA  P1 is not the sheaf L∞ that we are seeking. This is already apparent in
the original setting of [Col10c], where DA P1 does not realise the local Langlands
correspondence, but rather an extension of the correspondence by its (twisted) dual.
(See Section 7.3.22 below for an extension of this phenomenon to our setting.) We
will therefore construct (again following Colmez) L∞ as a subrepresentation ofDA  P1 .
In order to do so, we set A+
A = A[[T ]], which inherits actions of ϕ, ψ and Γ\from AA . It can be shown that DA contains a minimal ψ-stable A+
A -lattice DA (it+is a lattice in the sense that it is finitely generated over AA and spans DA over AA ),whose formation is compatible with flat base change. There is a natural restrictionmap
ResQp : DA  P1 → DA  Qp ,and we set\\
 Qp },
 P1 := {z ∈ DA  P1 , ResQp (z) ∈ DA
DA\\\
where DA Qp := limψ DA. It turns out that DA
 P1 is a GL2 (Qp )-stable lattice←−1
in DA  P , but this is not at all obvious; indeed it is not at all obvious (even in the\ P1 is open in DA  P1 . (Here “lattice” is inoriginal setting of [Col10c]) that DA
the sense of [Dri06]: a lattice in a Tate module over A is an open A-submodule Lwith the property that if U ⊆ L is open, then L/U is a finite A-module.)\
The proof that DA
P1 is a GL2 (Qp )-stable lattice is complicated; it involves areduction (via consideration of complete local rings) to the case that A is Artinian,and then as in [Col10c] it is ultimately proved via a further reduction to the explicit description of the p-adic local Langlands correspondence for 2-dimensional
crystalline representations of GalQp proved by Berger–Breuil [BB10]. Having established this, we have the following useful corollary.

Corollary 7.3.2 (Dotto–E.–G.). Suppose that p ≥ 3. Assume that A is a
finite type O/$a -algebra for some a ≥ 1, and that DA is a projective étale (ϕ, Γ)\
module of rank 2 with A-coefficients and determinant ε. Then DA
 P1 is a finite
A[[K]]-module.
\
Proof. This is deduced from the stability of DA
P1 under the action of GL2 (Qp ),\together with the Noetherianness of A[[K]], and the fact that DA, being a lattice,+
is a finite AA -module.
\It can then be shown that the formation of DA
 P1 is compatible with flatbase change in A, and we use this to construct a pro-coherent sheaf L∞ over X ,equipped with a continuous GL2 (Qp )-action.
7.3.3. A reinterpretation of some results of Paškūnas. We now recall some of
the main results of [Paš13], and reinterpret them in terms of our functor A. Let(sm. G)l.adm denote the full subcategory of sm. G consisting of locally admissible
representations. (By definition, a representation π ∈ sm. G is locally admissible ifevery vector v ∈ π is smooth and generates an admissible representation; in fact,
for GL2 (Qp ), locally admissible representations are automatically locally finite, sothat for every v, the representation generated by v is of finite length.)

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Unlike sm. G, the category (sm. G)l.adm admits a block decomposition, as we
now explain. By definition, a block of (sm. G)l.adm is an equivalence class of (isomorphism classes of) irreducible objects under the equivalence relation generatedby
π1 ∼ π2 if Ext1sm. G (π1 , π2 ) 6= 0 or Ext1sm. G (π2 , π1 ) 6= 0.
Strictly speaking, we should have written “irreducible admissible objects”, but
in fact the irreducible objects of sm. G are all contained in (sm. G)l.adm . Indeed, the absolutely irreducible smooth representations of GL2 (Qp ) in characteristic p admitting a central character18 have been classified by Barthel–Livné andBreuil [BL94, Bre03a], and in particular have been shown to be admissible. The
non-absolutely irreducible case is rather easily reduced to the absolutely irreducible
case (see e.g. [Ber12, Prop. 1.2] and [Paš13, §5.13]).
The blocks B are as follows:(1) B = {π} for a supersingular irreducible representation π having trivial
central character,−1−1(2) B = {IndG), IndG⊗ ω −1 χ)}, for some character χ : Q×
B (χ ⊗ χ
B (ωχ
p →
×
2
2
k such that χ 6= 1, ω ,−1χ)} for a quadratic character χ : Q×
(3) B = {χ, χ ⊗ StG , IndG
B (ωχ ⊗ ωp →×k , and(4) blocks that do not contain absolutely irreducible objects.
A key observation for us is that there is a bijection between the blocks of types(1)–(3) and X(k). (For the sake of exposition we ignore the blocks of type (4); they
are easily handled by allowing finite extensions of our coefficient field k, and indeed
correspond to points of X over extensions of k.) To describe this bijection, recall
that for any Serre weight σ, the Hecke algebraH(σ) := EndG (c-IndGK σ)
is equal to k[T ] for an explicit Hecke operator T , and the absolutely irreducible
representations are all subquotients of the representations(c-IndG
K σ)/(T − λ)for λ ∈ k. Furthermore these representations are (absolutely) irreducible unless σ =σa,0 or σ = σa,p−1 for some a and λ = ±1, in which case there are two Jordan–Hölder factors, which are of the form χ, χ ⊗ StG for some χ.
The bijection between the blocks and X(k) relies on regarding the Hecke operator T as a coordinate on an A1 in the component X(σ/σ co ) of X. We do this asfollows, restricting as above to the generic case for the sake of exposition.
Definition 7.3.4. Let σ be a generic Serre weight. Then we defined above amorphism X (σ/σ co ) → X(σ/σ co ) ⊂ X to be given by one of πσ or πσco , and we
fixed a coordinate T on X(σ/σ co ) (a copy of P1 ). In the case that this morphism
is given by πσ , we define fσ : Spec H(σ) → X by sending T → T ; and in the casethat this morphism is given by πσco , we send T 7→ T −1 .
18Subsequently, Berger showed [Ber12] that irreducible smooth representations of GL (Q )p2over Fp necessarily admit central characters. In our particular context, we are interested in
representations of PGL2 (Qp ), which is to say of GL2 (Qp ) having trivial central character, and so
this issue with central characters does not arise.

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Then if x is a k-point of X, the corresponding block is given by


[
Bx =
JH c-IndG
K (σ) ⊗H(σ) x .
σ s.t. x∈fσ (A1 )More precisely, Bx is a block of (sm. G)k , and the map x 7→ Bx (suitably extended
to the case of non-generic Serre weights) is a bijection from X(k) to the set of blocksof sm. G containing absolutely irreducible representations.
l.adm
If B is a block of (sm. G)l.adm , then we write (sm. G)B
to denote the fulll.adm
subcategory of (sm. G)
whose objects are those representations all of whose
irreducible subquotients lie in the given block B. We then have the block decomposition
Y
∼
l.adm
(sm. G)l.adm −→(sm. G)B.
B

In [Paš13] Paškūnas establishes equivalences of categories between the various
categories (sm. G)l.adm, and categories of modules over certain pseudodeformationBrings. We use Paškūnas’s results (and their proofs) as an input to the proof of
Theorem 7.2.1, but in order to do so it is necessary to reformulate these results(and in some cases slightly extend them) as statements about fully faithful functors
to categories of sheaves on stacks.
Let θ : GalQp → GL2 (k) be a semisimple representation (equivalently, a pseudorepresentation), corresponding to a point x ∈ X(k). Assume (for simplicity of
exposition) that the associated block Bx lies in one of cases (1) or (2) above; thatis, we assume that x is not a point T = ±1 of X(σ/σ co ) for some {σ, σ co } of theform {σa,0 , σa+1,p−3 }. Write (sm. G)θl.adm := (sm. G)l.adm
Bx .
We let Xθ denote Carl Wang-Erickson’s stack of Galois representations associated to θ. By definition, Xθ is characterised by the following property: if A isan O-algebra in which p is nilpotent, then X Gal (A) is the groupoid of continuous
morphisms
ρ : GalQp → GL2 (A)with det ρ = ε−1 and are such that the pseudorepresentation associated to ρ modulo $ is equal to θ (here A has the discrete topology, and GalQp its natural profinite
topology).
Wang-Erickson showed in [WE18] that Xθ is a formal algebraic stack, andby [EG22, Thm. 6.7.2] and [EG20, Thm. 10.2.2], there is a natural monomorphism Xθ ,→ X which induces a closed immersion on underlying topological spaces(with image corresponding to those representations ρ : GalQp → GL2 (Fp ) withsemisimplification isomorphic to θ). This morphism is versal at the finite typepoints of Xθ , so we naturally expect that it is some kind of completion of the formal algebraic stack X at the closed substack Zθ of Xred whose Fp -points are those ρwith semisimplification θ.
This expectation can be made precise by using the language of coherent completeness [AHR20, Defn. 2.1] in the following way. Write Iθ for the ideal sheaf[n]of Zθ in X , and Xθ for the closed substack of X cut out by Iθn+1 . Then it can be[n]shown that for each n, Xθ
(given by restriction)

is a closed substack of Xθ , and that the natural functor[n]Coh(Xθ ) → lim Coh(Xθ )
←−
n

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M. EMERTON, T. GEE, AND E. HELLMANN

is an equivalence of categories; that is, a coherent sheaf on Xθ is the same thing as
a compatible system of coherent sheaves on the infinitesimal neighbourhoods of Zθ
in X .
Given the above constructions, the following theorem is essentially a matter ofunwinding some of the proofs of the main results of [Paš13]. Let (sm. G)fg.adm
θdenote the full subcategory of (sm. G)l.adm
consistingoffinitely
generated
repreθ
sentations.
Theorem 7.3.5 (Paškūnas+ε). Suppose as above that θ : GalQp → GL2 (k) isa pseudorepresentation, corresponding to a block which lies in one of cases (1) or(2). Then the functor(7.3.6)(L∞ )|Xθ ⊗O[[G]] –is an exact fully faithful functor from (sm. G)fg.adm
to the category of coherentθOXθ -modules.
Proof. Let Rθps denote the complete Noetherian local O-algebra parameterizing deformations of θ of fixed determinant ε−1 over finite type Artinian localeO-algebras, and let θu be the universal deformation of θ over Rθps . We write Rθfor the Cayley–Hamilton algebra defined as follows: let J denote the closure of thetwo-sided ideal of Rθps [[GalQp ]] generated by the elements g 2 − θu (g)g + (ε−1 )(g),for g ∈ GalQp , and then set
e := Rps [[GalQ ]]/J.
RpθθAs in Section 3, we let V denote Colmez’s Montreal functor. Paškūnas shows
e is free of rank 4 as an Rps -module, and the resultsin [Paš13] that the ring Rθθ
of [Paš13] can be interpreted as showing that the functor π 7→ V (π) induces an
e -modules.
equivalence between (sm. G)fg.adm
and the category of finite length RθθThere is a universal two-dimensional GalQp -representation V lying over Xθ ,which can be regarded as a rank 2 locally free sheaf of OXθ -modules equipped withe . Then by using the relationships between D\ P1 and the Montreal
an action of Rθfunctor V established in [Col10c], one can show that the functor (7.3.6) is naturally
equivalent to the functor
V ⊗Re V (–),
θ

and it is now easy to deduce the result from the theorems proved in [Paš13].

co

The case of a block of type (3) (i.e. the case that x is a point T = ±1 of X(σ/σ )for {σ, σ co } of the form {σa,0 , σa+1,p−3 }) is more subtle, as in contrast to the otherblocks we see derived phenomena. This case has recently been studied by Christian
Johansson, James Newton and Carl Wang-Erickson, and we expect that their results
can be used to prove the following theorem.
Theorem 7.3.7 (Johansson–Newton–Wang-Erickson+ε). Suppose as above that
θ : GalQp → GL2 (k) is a pseudorepresentation, corresponding to a block Bθ = Bx
which lies in case (3). Then the functor(7.3.8)(L∞ )|Xθ ⊗LO[[G]] –is a fully faithful functor with amplitude [−1, 0] from D(Bfg) to the bounded derived
θcategory of coherent OXθ -modules.

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7.3.9. A useful resolution. Let N be the normalizer of the usual (upper triangular) Iwahori subgroup I, and let δ be the nontrivial quadratic character of N/ZI.
If π is any smooth G-representation, then we have a complex of O[[G]]-modules(7.3.10)G
0 → c-IndG
N (δπ) → c-IndK (π) → π → 0

where the third arrow is the tautologicalc-IndGK π → π. The second
 surjection
0 1
arrow is defined as follows. Write Π =. Then the morphism
p 0
v 7→ Π ⊗ Π−1 v − 1 ⊗ v ∈ O[G] ⊗O[K] π = c-IndG
Kπ
is an N -equivariant morphism δπ → c-IndG
K π which induces the second arrow
GG
c-IndN δπ → c-IndK π. It is easy to show that the complex (7.3.10) is acylic.
Remark 7.3.11. The resolution (7.3.10) is valid for PGL2 (F ) for any F/Qp ,and can be used to reduce certain assertions about general π to the case π =c-IndGK σ for a Serre weight σ. In particular, we can reduce the evaluation of
cocontinuous functors (such as our hypothetical functor A) on general π to theevaluation on c-IndG
K σ in the following way. Since we are assuming that p > 2,K
GG
c-IndGN δπ is a direct summand of c-IndI π = c-IndK (c-IndI π), so we can reduceG
to studying representations of the form c-IndK V .
Since compact induction is compatible with passing to filtered colimits, we thenreduce to the case when V is finitely generated. Since compact induction is exact,
we furthermore reduce to the case when π is of the form c-IndG
K σ for some Serreweight σ, as claimed.
7.3.12. The definition of the functor. We are now in a position to define ourfunctor. Since we have already defined the pro-coherent sheaf L∞ of O[[G]]representations over X , we would simply like to define this via(7.3.13)

π 7→ L∞ ⊗L
O[[G]] π.

However, it is not a priori obvious that this definition is well-behaved, and we
proceed in stages. Firstly, if π is of the form c-IndG
K V with V finitely generated,then we note that
G
L
L∞ ⊗L
O[[G]] c-IndK V = L∞ ⊗O[[K]] V,

which is easily seen to be a bounded-above complex of coherent OX -modules by
Corollary 7.3.2. We claim that it is in fact a sheaf, concentrated in degree zero. To
see this, it is enough (by descending induction on the cohomological degree, andusing the coherence of the cohomology groups) to check after restriction to each
substack Xθ , where the result follows from Theorem 7.3.5.
Passing to colimits, we see that for any V (finitely generated or otherwise),G
L ∞ ⊗LO[[G]] c-IndK V is a quasicoherent sheaf in degree zero. By Remark 7.3.11,
we deduce that for any smooth representation π, L∞ ⊗L
O[[G]] π is a complex ofquasicoherent sheaves concentrated in degrees [−1, 0], and that if π is finitely presented (equivalently, finitely generated), then this complex has coherent cohomology
sheaves. We conclude that L∞ ⊗L
O[[G]] − defines a functor with amplitude in [−1, 0]
and takes bounded complexes of finitely presented representations to bounded coherent complexes of OX -modules.

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M. EMERTON, T. GEE, AND E. HELLMANN

7.3.14. Localization and gluing. If σ is a Serre weight, we heuristically regard
1c-IndGK σ as “living over” a copy of A = Spec k[T ] = Spec H(σ) in X via Definition 7.3.4. This intuition is the basis of the localization theory for sm. G developedin [DEGb], and we now explain how to make it precise. The results described inthis section, and their proofs, are for the most part purely representation-theoretic(with some occasional uses of the classification of blocks, which is proved in [Paš13]
using the p-adic local Langlands correspondence), although they are of course motivated by the geometric picture.
If Y is a closed subset of X, we let (sm. G)Y denote the subcategory of sm. G
consisting of those representations all of whose irreducible subquotients lie in a
block corresponding to a closed point of Y . If U is an open subset of X, then we
write(sm. G)U := sm. G/(sm. G)Y ,where Y := X \ U . It is not hard to check that the subcategory (sm. G)Y is
localizing in the usual sense: namely, there is a fully faithful right adjoint(jU )∗ : (sm. G)U → sm. G
to the canonical quotient functor (jU )∗ : sm. G → (sm. G)U . Where U is understood, we write j∗ for (jU )∗ , and j ∗ for (jU )∗ .
The following result, which is [DEGb, Prop. 3.1.7], confirms some basic intuitions about how the functors j∗ , j ∗ should behave.
Proposition 7.3.15. Let Y be a closed subset of X, and write U := X \ Y .
(1) If σ is a Serre weight, and if fσ−1 (Y ) = V (g) (a closed subset of Spec H(σ))
for some g ∈ H(σ), then the natural map
Gc-IndGK σ → (c-IndK σ)[1/g]can be identified with the morphismG∗c-IndGK σ → j∗ j c-IndK σ.

(2) The functor j∗ is exact and commutes with filtered colimits.
If {U0 , . . . , Un } is any finite open cover of X, then for any object π of sm. G,
we obtain a functorial Čech resolution
Y(7.3.16)0→π→(ji )∗ (ji )∗ π → · · · → (j0,...,n )∗ (j0,...,n )∗ π → 0iwhere as usual we write Ui...k for Ui ∩ · · · ∩ Uk , we have written ji...k for jUi...k , andthe differentials are given by the usual formulas. The following is [DEGb, Prop.
3.2.3].
Proposition 7.3.17. For any object π of sm. G, the resolution (7.3.16) isacyclic.
Proof. Using the resolution (7.3.10), we reduce as in Remark 7.3.11 to thecase that π = c-IndG
K σ for a Serre weight σ. In this case Proposition 7.3.15 allows
us to reduce to the standard assertion that the Čech complex associated to a finite
open cover of an affine scheme by distinguished opens is acyclic.
From Proposition 7.3.17, we deduce that sm. G is a stack in abelian categories
over X, as claimed in Theorem 7.2.3.

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We need a gluing statement for categories of representations that corresponds
to Beauville–Laszlo gluing of sheaves. Write (sm. G)fg for the full subcategory of
sm. G consisting of finitely generated objects. Similarly, for any closed subset Y
of X, we write (sm. G)fg
Y to denote the full subcategory of (sm. G)Y consistingof finitely generated objects. We may form the category of formal pro-objects
pro((sm. G)fg
Y ).
Definition 7.3.18. If π is an object of (sm. G)fg , and if Y is a closed subset
0of X, then we write πbY for the object “ limI ”π 0 of pro((sm. G)fg
Y ), where π runs←−
over all quotients of π which lie in (sm. G)fg
Y.
The following is [DEGb, Thm. 3.8.1].
Theorem 7.3.19. If Y is finite, then the functor π 7→ πbY is exact, and thecanonical functorfg(sm. G)fg −→ pro((sm. G)fg
Y ) ×(pro((sm. G)fg )U ) (sm. G)U ,Y

induced by completion along Y and by localization over U , is an equivalence of
categories.
7.3.20. A description of A generically. Given an open subset U of |X|, we havethe localized category (sm. G)U , and an open substack U := π −1 (U ) of X . We now
take U to be the complement of the finite set Y consisting of points corresponding toblocks of types (1) or (3); in particular we have deleted the marked points. Then U
is dense in X, and as we now explain, it is in fact possible to explicitly describe therestriction of A to (sm. G)U , and to show that it is a fully faithful functor to D(U).
The rough idea of our argument is to follow the strategy of [Paš13] for blocks oftype (2), but to “algebraize in the unramified direction”.
The open set U is a disjoint union of its intersections with the various irreducible
components of |X|, and we handle each of these intersections in turn. In thepresent discussion we restrict ourselves to describing the case of U ∩ X(σ/σ co )when σ, σ co is a companion pair of very generic Serre weights. This intersection
was denoted U (σ/σ co ) in Section 7.2.21, but to simplify notation, we now redefine
U as U := U (σ/σ co ). The preimage of U in X is the open substack U := U(σ/σ co ).
Our aim, then, is to describe the composition of A with restriction to U, whichturns out to factor through (sm. G)U . Our description of this composition will be
Morita-theoretic, in terms of a projective pro-generator for the category (sm. G)U ,and the simplest way to construct this pro-generator in terms of the sheaf L∞ that
we have used to define A. Actually, we find it convenient to use a dual version L∨∞
of L∞ , which can be constructed in a similar way to L∞ itself (see also Section 7.3.22
below).
We now writeb m ] = [Spf A/µ2 ],U \ X (σ co ) = [(Spf O[s, t±1 , x, y ±1 ]∧ /G
where A = O[s, t±1 , x]∧ (the hat denoting (p, s, x)-adic completion), and the actionof µ2 is trivial. We can regard L∨∞ |U \X (σ co ) as topological A[[G]]-module Pα withan action of µ2 . With some work, and a comparison to the results
 of [Paš13],
we can show that Pα is a “pro-projective” object of pro (sm. G)U , meaning that
if π → π 0 is an epimorphism in (sm. G)U , then Hom(Pα , π) → Hom(Pα , π 0 ) issurjective. (Because projective limits of surjections of abelian groups need not besurjective, this does not imply that Pα is projective in pro (sm. G)U .)120

M. EMERTON, T. GEE, AND E. HELLMANN

Reversing the roles of σ and σ co , we obtain another pro-projective object Pβ .
Together, Pα and Pβ generate (sm. G)U . By computing their completions at closed
points using Theorem 7.3.5, we can show that A(Pα ) and A(Pβ ) are certain explicit
line bundles on U. In fact, with an appropriate interpretation of the notation,these are the line bundles corresponding to the graded modules B(1) and B(−1)
considered in Example C.4.7, and the full faithfulness of the functor A : (sm. G)U →D(U) is a straightforward consequence of Proposition C.4.9.
7.3.21. Full faithfulness of the functor. As at the beginning of Section 7.3.20,
let U denote the complement of the finite set Y consisting of points corresponding
to blocks of types (1) or (3). Ignoring as usual the issue of blocks of type (3) (which
require some additional arguments), the full faithfulness of A follows easily fromthe full faithfulness of A : (sm. G)U → D(U), together with Theorem 7.3.19 andTheorem 7.3.5.
7.3.22. Compatibility with duality. For any finite type O/$a -algebra A, there
is an obvious short exact sequence of GL2 (Qp )-representations(7.3.23)\\
0 → DA P1 → DA  P1 → (DA  P1 )/(DA P1 ) → 0.

This short exact sequence plays an important role in [Col10c], and using Colmez’s
results (via a reduction to the case that A is Artinian), we can prove that there isa perfect duality pairing (ultimately coming from the residue pairing in Tate local\\
duality) between DA
 P1 and (DA  P1 )/(DA P1 ).
bb
We write D : Df.p.
(sm. G) → Df.p.
(sm. G) for the contravariant functor definedviaD(–) = RHom(–, O[[G]])[4].
Using the resolution (7.3.10), it is easy to check that D has amplitude [0, 1]. We also
write DX for the antiequivalence of Dcoh (X ) given by Grothendieck–Serre duality,
i.e. RHomOX (–, ωX ) where ωX is the dualizing sheaf of X , placed in degree 0.
By the construction of L∞ , we can use (7.3.23) to construct an extension
0 → L∞ → ? → DX L∞ → 0.
This extension class is an element of
HomO[[G]] DX L∞ , L∞ [1] = Ext1O[[G]] DX L∞ , L∞ [1],and composition with this extension class gives us a natural morphism of functorsbon Df.p.
(sm. G)
RHomO[[G]] (–, DX L∞ ) → RHomO[[G]] –, L∞ [1]which we expect can be shown to be an isomorphism, so that we have the expectedcompatibility with duality∼

A ◦ D −→ (DX ◦ A)[3].
As discussed
in Remark 6.1.34, this implies that the various sheaves A (c-IndG
K Wλ ⊗∧σ(τ )) are Cohen–Macaulay. Since this is a property that can be checked bypulling back over versal rings, it also follows more directly from the construction of
the functor A, its relationship to Paškūnas’s results discussed in Section 7.3.3, andthe results of [Paš15].
crys,◦

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121

7.4. The Banach case for GL2 (Qp ) — III. Examples. We now describe
some explicit values of the functor A, beginning with its values on compact inductions of Serre weights and on (absolutely) irreducible representations, and thenturning to some cases that are related to tamely potentially Barsotti–Tate Galois
representations. These latter examples also illustrate the compatibility between the
Banach and the analytic case posited in Section 6.2.10.
7.4.1. Compact inductions of Serre weights. We consider a Serre weight σa,b .
Assume firstly that b 6= p − 1, i.e. that σa,b is generic. Then A(c-IndG
K σa,b ) is aline bundle on X (σa,b ). This is to be expected, for example by comparison to thecorresponding Taylor–Wiles patched modules, which are free over their support asa consequence of Fontaine–Laffaille theory, but it is less obvious precisely which line
bundle on X (σa,b ) we obtain. To describe this, we note that by Proposition 7.2.11(and the triviality of the Picard group of A2 \ {0}) we have Pic(X (σa,b )) = Z,
and it can be checked that A(c-IndG
K σa,b ) is the line bundle corresponding to −1 ∈Z. (This is in fact forced by the compatibility with duality; the discussion of
Remark 6.1.32 shows that A(c-IndG
K σa,b ) must be the unique self-dual line bundle
on X (σa,b ). See also Section 7.7 for a related computation for GL2 (Qp2 ).)If b = p − 1 then A(c-IndG
K σa,p−1 ) is a line bundle on the union(7.4.2)X (σa,0 ) ∪ X (σa,p−1 )+ ∪ X (σa,p−1 )− .

We anticipate that this line bundle can be described explicitly as follows: By Proposition 7.2.12, the stacks X (σa,p−1 )± also have Picard group isomorphic to Z, andwe expect that it can be checked that the restriction of A(c-IndG
K σa,p−1 ) to each
irreducible component in (7.4.2) corresponds to −1 ∈ Z.
7.4.3. Absolutely irreducible representations. The results of Section 7.4.1 allow
us to compute the values of A on absolutely irreducible representations. As discussed in Section 7.3.3, these are subquotients of the representations(c-IndG
K σ)/(T − λ)for λ ∈ k; and these representations are in fact absolutely irreducible unless σ = σa,0or σ = σa,p−1 for some a and λ = ±1. Supposing that we are not in this case,
we can deduce from the injectivity of T − λ on the line bundle A(c-IndG
K σ) that−1
A((c-IndG
σ)/(T
−λ))
is
a
rank
1
sheaf
supportedontheclosed
locus
π
(λ)∩X
(σ).
K
σ
(In the case λ 6= 0 we describe this rank 1 sheaf more precisely in Section 7.4.4
below.)
Unsurprisingly, the trivial and Steinberg representations exhibit more complicated behaviour, which mirrors the corresponding picture for l-adic representations
in characteristic zero (see Remark C.4.5). There are short exact sequences of Grepresentations
G
0 → c-IndG
K σ0,0 → c-IndK σ0,p−1 → St ⊕ (nr−1 ◦ det) ⊗ St → 0
G
0 → c-IndG
K σ0,p−1 → c-IndK σ0,0 → 1 ⊕ nr−1 ◦ det → 0

which in combination with the descriptions of the A(c-IndG
K σa,b ) above can beused to show that A(St) is a line bundle on X (Symp−1 )+ , while A(1) is a shift tocohomological degree −1 of another line bundle on X (Symp−1 )+ .

122

M. EMERTON, T. GEE, AND E. HELLMANN

7.4.4. Type (2) blocks. We consider the case of a block of type (2) in more detail.
−1−1
In this case B = {π1 , π2 }, with π1 = IndG
) and π2 = IndG
⊗ ω −1 χ)}
B (χ ⊗ χB (ωχ××for some suitably generic characters χ : Qp → k . In this context there are nonsplit extensions 0 → π2 → κ → π1 → 0 and 0 → π1 → κ0 → π2 → 0 (each unique
up to isomorphism), and we will describe the various sheaves A(π1 ), A(π2 ), A(κ),
and A(κ0 ).
To begin with, we note that we may write π1 = (c-IndG
K σ)/(T − λ) and π2 =G co
−1(c-IndK σ )/(T − λ ) for some very generic σ and its companion σ co , and someappropriately chosen λ. All the sheaves under consideration will then have support
contained in πσ−1 (λ).
The locus πσ−1 (λ) is a closed substack of U(σ/σ co )red . Following the notationof (7.2.22), we haveU(σ/σ co )red := [Spec k[t±1 , x, y]/(xy) /Gm ],with the Gm -action being given by u·(x, y) = (u2 x, u−2 y). The map πσ : U(σ/σ co )red →A1 is then simply projection to the t-coordinate, and so
πσ−1 (λ) = [(Spec k[x, y]/(xy)) /Gm ](the locus where t = λ).
We may give k[x, y]/(xy) its grading induced by the Gm -action (so x has degree
2 and y has degree −2), and may then describe coherent sheaves on πσ−1 (λ) as
graded k[x, y]/(xy)-modules. We then have the following descriptions of the sheaves
under consideration: A(π1 ) = k[x, y]/(y)] (−1); A(π2 ) = k[x, y]/(x)] (1); A(κ) =k[x, y]/(xy))(−1); A(κ) = k[x, y]/(xy))(1). (Here the twists indicate shifts ingrading, as usual.) Note the evident short exact sequencesy·
0 → k[x, y]/(x) (1) −→ k[x, y]/(xy) (−1) −→ k[x, y]/(y) (−1) → 0
and



x·
0 → k[x, y]/(y) (−1) −→ k[x, y]/(xy) (1) −→ k[x, y]/(x) (1) → 0;
these can be interpreted as arising from applying A to the short exact sequencesthat defined κ and κ0 .
7.4.5. Tame principal series types. In this example and the next, we treat somep-adically completed compactly induced representations that are related to tamely
potentially Barsotti–Tate representations. We content ourselves with merely sketching the various computations required to verify all our claims.
We begin with the tame principal series case, and to this end, let σ/σ co be very
generic, and let τ be a tame principal series type such that σ is a Jordan–Hölder
factor of σ crys,◦ (τ ). Note that this implies that σ co is not a Jordan–Hölder factorof σ crys,◦ (τ ); indeed, if σ = σa,b , then the Jordan–Hölder factors of σ crys,◦ (τ ) are σ
itself, together with σa+b,p−1−b . (Explicitly, the type σ crys,◦ (τ ) is inflated from a
principal series representation of GL2 (Fp ), as in Section 7.7.7.)
We may then consider the stack X crys,(1,0),τ of potentially Barsotti–Tate representations of type τ . We can describe this closed substack of X quite explicitly,
and we will proceed to do so. The first point to note is that the mod $ fibre
X crys,(1,0),τ is equal to X (σa,b ) ∪ X (σa+b,p−1−b ). (This is a precise form of thegeometric Breuil–Mézard conjecture discussed in Section 6.1.37 for the particular
type τ .) This union admits the explicit description given in Remark 7.2.17, whichembeds it as an open substack of Spec k[t1 , t2 ]/(t1 t2 ) × [P1 /Gm ]. This latter stackCATEGORICAL p-ADIC LANGLANDS

123

may be identified with the underlying reduced substack of the p-adic formal algec1 /G
b m ], and X crys,(1,0),τ is isomorphic tobraic stack (Spf(O[t1 , t2 ]/(t1 t2 − p))∧ × [Pthe open substack of this formal algebraic stack having X (σa,b ) ∪ X (σa+b,p−1−b ) as
its underlying reduced substack.

crys,◦
We now turn to describing the sheaf A (c-IndG(τ ))∧ . Of course, thisKσdepends on the particular choice of lattice σ crys,◦ (τ ). Among all such, there are twoparticular lattices Λ1 and Λ2 , determined uniquely (up to scaling) by the condition
that the cosocle of Λ1 (resp. Λ2 ) is equal to σ (resp. σa+b,p−1−b ). We may anddo normalize these lattices so that pΛ1 ⊂ Λ2 ⊂ Λ1 . Then (up to scaling) any
lattice Λ is of the form $a Λ1 + Λ2 , with 0 ≤ a ≤ e (where e denotes the absolute
ramification degree of our coefficient field L). (This description of lattices in tameprincipal series types for GL2 is due to Breuil [Bre14, §2], and is a special case of[EGS15, Prop. 4.1.4].)×××
If χ : F×denotes the tame lift of the highest weightp × Fp → Zp ⊆ Oof σ, which we inflate to a character of the (usual upper triangular) Iwahori I,sthen Λ1 = IndKI χ, while if χ denotes the conjugate of χ by the non-trivial
 Weyl0 1
K sgroup element s, then Λ2 = IndI χ . Conjugation by the elementin thep 0
normalizer (in G) of I interchanges χ and χs , and so there is an isomorphism
c-IndG Λ1 = c-IndG χ ∼= c-IndG χs = c-IndG Λ2 .
K

K

I

K
G
Following this isomorphism with the inclusion c-IndK Λ2 ,→ c-IndG
K Λ1 induced by
the inclusion of Λ2 in Λ1 gives rise to an endomorphism Up of c-IndG
K Λ1 . Interchanging the roles of Λ1 and Λ2 gives rise to another endomorphism Up0 of c-IndG
K Λ1 ,
0with the property that Up Up = p; and indeed00EndG (c-IndG
K Λ1 ) = O[Up , Up ]/(Up Up − p).

Passing to p-adic completions, we find that∧
= O[Up , Up0 ]∧ /(Up Up0 − p).
EndG (c-IndG
K Λ1 )
If Λ = $a Λ1 + Λ2 , then the inclusion of Λ in Λ1 induces an identificationGac-IndGK Λ = ($ , Up )c-IndK Λ1 , and hence a corresponding identification of p-adic∧
completions (c-IndGΛ)∧ = ($a , Up )(c-IndGofK Λ1 ) . This reduces the computation
 K

G
∧
∧
Λ
)
.
A (c-IndK Λ) for any lattice Λ to the computation of A (c-IndG
1
K
Now we choose our coefficient field L to be unramified over Qp (indeed the character χ, and hence the lattice Λ1 , are defined over Zp ), and with this hypothesis we
see that X crys,(1,0),τ
is regular. Thus, since (as noted in Section 7.3.22) the sheaf∧A (c-IndGΛ)isCohen–Macaulay,
the Auslander–Buchsbaum formula shows
K ithat it is locally free, and in fact it must be invertible (e.g. for the reason sketched
G
G
∧ ∼
∧
in Remark 6.1.34). Since Λ∨
1 = Λ2 , and since (c-IndK Λ1 ) = (c-IndK Λ2 ) , wecrys,(1,0),τ
further see that this line bundle is Grothendieck–Serre self-dual on X.
c1 /Gb m ]) = Z × Z, with the first copy of Z
Now one can check that Pic([Pc1 ) = Z, and the second copy coming from twisting by characterscoming from Pic(Pb m . We let O(m) c1 (n) denote the invertible sheaf corresponding to a pairof G
P(m, n) ∈ Z × Z.
Pulling back along the composite
f : X crys,(1,0),τ

open

c1 /G
c1 /G
b m ] proj.
b m],→ (Spf(O[t1 , t2 ]/(t1 t2 − p))∧ × [P
−→ [P

124

M. EMERTON, T. GEE, AND E. HELLMANN

induces an isomorphism∼
c1 /Gb m ]) −→Pic([P
Pic(X crys,(1,0),τ ).

c1 /G
b m ], namely O(−2) c1 , pulls back to the
Furthermore, the dualizing sheaf of [PP
crys,(1,0),τ
dualizing sheaf of X
. Thus (since it is self-dual), we find that∧ ∼ ∗A (c-IndG
c1 .
K Λ1 ) = f O(−1)PSince A is fully faithful, we also obtain an isomorphism ∼∧EndG (c-IndG−→ Γ(X crys,(1,0),τ , OX crys,(1,0),τ ) = O[t1 , t2 ]∧ /(t1 t2 − p),K Λ1 )
which is given (after choosing the ti suitably) by mapping Up to t1 and Up0 to t2 .
For any lattice Λ = $a Λ1 + Λ2 , we then find that∧a∗
A (c-IndG
c1 (−1).
K Λ) = ($ , t1 )f OP
In order to compare these values of A with those we computed above for thecompact inductions of Serre weights, it helps to note first that pullback to thespecial fibre induces an isomorphism∼
c1 /Gb m ]) −→Pic([P
Pic([P1 /Gm ]).

Then, if we let x0 and x1 denote the homogeneous coordinates on P1 (so that theinhomogeneous coordinate x of Remark 7.2.17 is equal to x1 /x0 ), and let u denotethe coordinate on Gm , each of [A1x1 /x0 /(Gm )u ] and [A1x0 /x1 /(Gm )u−1 ] embeds as anopen substack of [P1 /Gm ], and restriction to these substacks induces an embedding
Pic([P1 /Gm ]) ,→ Pic([A1x1 /x0 /(Gm )u ]) × Pic([A1x0 /x1 /(Gm )u−1 ]) = Z × Z,
given by(m, n) 7→ (m + n, m − n).
From this, we compute the restrictions
(f ∗ O(−1)Pc1 |X (σa,b )= OX (σa,b ) (−1)

and(f ∗ O(−1)Pc1 |X (σa+b,p−1−b )= OX (σa+b,p−1−b ) (−1).

Thus, if we apply A to the short exact sequence
Up

G
G
∧
∧
0 → (c-IndG
K Λ1 ) −→ (c-IndK Λ1 ) → c-IndK σa,b → 0,

respectively
Up0

G
G
∧
∧
0 → (c-IndG
K Λ1 ) −→ (c-IndK Λ1 ) → c-IndK σa+b,p−1−b → 0,

we obtain the short exact sequence of sheavest1∗
0 → f ∗ OP
c1 (−1) −→ f OP
c1 (−1) → OX (σa,b ) (−1) → 0,

respectivelyt2∗
0 → f ∗ OP
c1 (−1) −→ f OP
c1 (−1) → OX (σa+b,p−1−b ) (−1) → 0.

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125

7.4.6. Tame cuspidal types. One can make a similar analysis to that of Section 7.4.5 in the case of a tame cuspidal type. Namely, there is a tame cuspidal typeτ uniquely determined by the condition that the Jordan–Hölder factors of σ crys,◦ (τ )are σ and σ co . The potentially Barsotti–Tate substack X crys,(1,0),τ is then a p-adiccoformal lift of X (σ/σ co ) := X (σ) ∪ X (σ co ). As noted inRemark
 7.2.16, X (σ/σ )1embeds as an open substack of P × Spec k[x, y]/(xy) /Gm , and a deformation
theory argument shows that X crys,(1,0),τ is isomorphic to the open formal algebraic
c1 × Spf(O[x, y]/(xy − p) ∧ /G
bm
substack of the p-adic formal algebraic stack P
lifting X (σ/σ co ).
Just as in the principal series case, there are distinguished lattices Λ1 andΛ2 in σ crys (τ ), uniquely determined up to scaling, whose cosocles are σ and σ corespectively. Fixing a choice of Λ1 , we can then scale Λ2 so that pΛ1 ⊂ Λ2 ⊂ Λ1 ,a
and any lattice is (up to scaling) of the form Λ = $ Λ1 + Λ2 , with 0 ≤ a ≤ e. Thus,G∧
to describe the various sheaves A (c-IndK Λ) , it suffices to describe the sheaves∧∧
A (c-IndG
and A (c-IndG
K Λ1 )
K Λ2 ) , as well as the morphism


G
∧
∧
(7.4.7)A (c-IndG
K Λ2 ) → A (c-IndK Λ1 )induced by the inclusion Λ2 ⊂ Λ1 .
As in the principal series we choose our coefficient field L to be unramified over Qp (tame types may always be defined over unramified extensions),
crys,(1,0),τ
and with this hypothesis
is again regular, so that the we see that X
G∧sheaves A (c-IndK Λi ) are again invertible. Further, since Λ∨
1 = Λ2 , we find that


G
G
∧
∧
A (c-IndK Λ1 ) and A (c-IndK Λ2 ) must be interchanged under Grothendieck–
Serre duality.
Now pullback induces an isomorphism ∼c1 × Spf(O[x, y]/(xy − p) ∧ /Gb m ) −→
Pic(P
Pic(X crys,(1,0),τ ),and so any invertible sheaf on X crys,(1,0),τ is necessarily of the form L(n), where L is
c1 , and n is a an integer indicating the weight of the Gb m -action.
pulled back from P
crys,(1,0),τ
Furthermore, the dualizing sheaf on X
is obtained as the pull back of the

∧

c
1×
bm .
dualizing sheaf O(−2)PonP
Spf(O[x,y]/(xy− p) /Gc1
The short exact sequencesGG∧∧
0 → (c-IndG
K Λ2 ) → (c-IndK Λ1 ) → c-IndK σ → 0

and
p

G
G co
∧
∧
0 → (c-IndG→0
K Λ1 ) −→ (c-IndK Λ2 ) → c-IndK σ

induce corresponding short exact sequences of sheaves after applying A, and fromG co
the known values of A on c-IndG
K σ and c-IndK σ , we find that∧ ∼A (c-IndG
K Λ1 ) = L(−1),
that

∧ ∼
A (c-IndG
K Λ2 ) = L(1),and that the morphism (7.4.7) is induced by multiplication by y. This analysis
doesn’t determine the invertible sheaf L (since the restriction of any L pulled backfrom P1 to either X (σ) or X (σ co ) is necessarily trivial). However, the mutual
duality of L(±1) then shows that L must be the pullback of O(−1)P
c1 .

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Remark 7.4.8. One difference between the cuspidal and principal series cases
is that in the cuspidal case,we have Γ(X crys,(1,0),τ , OX crys,(1,0),τ ) = O (just as wehave Γ X (σ/σ co ), OX (σ/σco ) = k). This reflects (via consideration of the functor
A) the fact that the endomorphism ring of the compactly induced representation∧
crys,◦
c-IndG(τ ) (for any choice of lattice σ crys,◦ (τ )) is trivial in this case.
Kσ
7.4.9. Illustrating the compatibility with the analytic case. Allowing, for a moment, τ to denote either a tame principal series or tame cuspidal type, then, as
indicated in (5.2.12) in the semi-stable case, there is a morphisman
π : (X crys,(1,0),τ )rig
η → (Fil(1,0) WDτ ) ,whose target is the (rigid analytic) moduli stack of two-dimensional Weil–Deligne
representations having inertial type τ , of fixed inverse cyclotomic determinant,equipped with a filtration by a one-dimensional subspace.
Suppose now that τ is a principal series type. Then∼
(X crys,(1,0),τ )rig
η =




b m )rig ,{(t, x) ∈ (Gm )t ×P1x | 1 ≥ |t| ≥ |p|}\({(x, t) | 1 > |t|, |x|}∪{(x, t) | |t| > |p|, |x| > 1}) /(Gηb m )rig = {u | |u| = 1} ⊂ Gm , acting via u · (t, x) = (t, u2 x), while
where (Gη(Fil(1,0) WDτ )an ∼= [(Gm )t × P1x /(Gm )u ],where19 again the action is given by u · (t, x) = (t, u2 x). The map π is then theobvious one.
In terms of this description, the “forget the filtration” map prWD is the morphism[(Gm )t × P1x /(Gm )u ] → [(Gm )t /(Gm )u ](the right hand stack being formed with respect to the trivial action of (Gm )uon (Gm )t ). The twist ((−ρ0 )σ ) which appears in (6.2.5) is, in this case, the twist
by the pullback of the sheaf O(−1)P1 along the the “forget the Weil–Deligne representation” map
prFil : [(Gm )t × P1x /(Gm )u ] → P1x / SL2 ,that is induced by projection to the second factor together with the inclusion of(Gm )u as the diagonal torus in SL2 .
Now the smooth categorical Langlands correspondence associates the trivial
crys
invertible sheaf, with trivial (Gm )u -equivariant structure, to c-IndG(τ ). TheKσ
expected compatibility (6.2.12) in Section 6.2.10 (taking into account also Conjecture 6.2.4 (1)) then predicts thatrig
crys,◦A (c-IndG(τ ))∧ η ∼= pr∗Fil O(−1)P1 .
Kσ
The line bundle O(−1)P1 on P1x pulls back to the trivial line bundle on the open
subset{(t, x) | 1 ≥ |t| ≥ |p|} \ ({(x, t) | 1 > |t|, |x|} ∪ {(x, t) | |t| > |p|, |x| > 1})
19Here we write just G and P1 for the multiplicative group and the projective line viewedm
as rigid analytic spaces.

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of (Gm )t × P1x under the projection to P1x , but it has non-trivial (Gm )u -action: infact, (Gm )u acts on it with weight −1. Thus (6.2.12) predicts an isomorphismrigcrys,◦A (c-IndG(τ ))∧ η ∼(−1),= O(X crys,(1,0),τ )rigKση
in accordance with the computation of Section 7.4.5.
Now suppose that τ is cuspidal. In this case∼
(X crys,(1,0),τ )rig
η =




b m )rig ,{(t, x) ∈ P1 ×Gm | 1 ≥ |x| ≥ |p|}\({(t, x) | 1 > |t|, |x|}∪{(t, x) | |t| > 1, |x| > |p|}) /(Gηb m )rig acts via u · (t, x) = (t, u2 x), while
where (Gη(Fil(1,0) WDτ )an ∼= [P1t /µ2 ],where µ2 acts trivially. The map π is then the obvious one, given by forgetting x.
The map prWD is the projection[P1t /µ2 ] → [∗/µ2 ],while we again have the map
prFil : [P1t /µ2 ] → [P1t / SL2 ]that forgets the Weil–Deligne representations and which is induced by the central
embedding of µ2 into SL2 . The expected compatibility (6.2.12) then predicts an
isomorphismrigcrys,◦A (c-IndG(τ ))∧ η ∼= (π ∗ pr∗Fil O(−1)P1 )(−1)
Kσb m )rig ), which is in(the outer twist by −1 indicating the equivariant action of (Gη
accordance with the computations of Section 7.4.6. (Note that the twists by 1 andby −1 give non-isomorphic invertible sheaves on X crys,(1,0),τ , but that these line
bundles become isomorphic upon being pulled back to (X crys,(1,0),τ )rig
η , since theinvertible sheaf O(X crys,(1,0),τ )rig
(2)
istrivial,admitting
as
it
does
the
coordinate x
ηas a nowhere zero global section.)
7.5. Semiorthogonal decompositions. We now explain an explicit semiorthogonal decomposition of IndCoh(X ) (where X is as in Section 7.2), which allows us todescribe the essential image of the functor A. We also discuss the essential image ofthe analogous expected functor for representations of D× , where D is the nonsplit
quaternion algebra with centre Qp , as well as the case of GL2 (Qp2 ).
We begin by noting that we have a morphism f : µ2 × X → IX (the inertia
stack of X ) given by the action of ζ ∈ µ2 on a (ϕ, Γ)-module by multiplication by ζ(note that since ζ 2 = 1, this acts trivially on ∧2 D). This morphism is furthermore
central (as one sees from its construction) with respect to the groupoid structure
on IX (which is a groupoid over X ).
The morphism f induces a canonical action of µ2 on Coh(X ), and thus on Ind Coh(X ),as follows: We may find a formal scheme U with a smooth surjective morphism
U → X (in other words, a smooth chart for X ), and if R := U ×X U, then we may
write X = [U/R]. The morphism f above then induces a morphism U × µ2 → R
of groupoids over U , where µ2 acts trivially on U . A coherent sheaf on X may
be regarded as a coherent sheaf on U with an R-equivariant structure (encoding
the descent data back down to X ); in particular any such sheaf is equipped with

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M. EMERTON, T. GEE, AND E. HELLMANN

a µ2 -equivariant structure, which (by the centrality of f ) commutes with the Requivariant structure. Thus coherent sheaves on X are indeed equipped with acanonical µ2 -equivariant structure.
We let IndCoh(X )± be the subcategory of IndCoh(X ), on which −1 ∈ µ2
acts as ±. We sometimes refer to IndCoh(X )+ as the even subcategory, andIndCoh(X )− as the odd subcategory.
As a first step towards a semiorthogonal decomposition of IndCoh(X ), it is easy
to see that we have an orthogonal decomposition into the two pieces IndCoh(X )+and IndCoh(X )− . It follows from the construction of L∞ that L∞ is odd, so theessential image of A is contained in IndCoh(X )− .
7.5.1. Motivation. We refer to Appendix A.8 for some generalities on semiorthogonal decompositions. Our semiorthogonal decomposition is motivated by the possibility of a “p-adic Fargues–Scholze conjecture” as in Remark 1.4.6, and by the
structure of the stack BunG of G-bundles on the Fargues–Fontaine curve in thecase G = GL2 (F ). This stack admits a stratification with strata corresponding
to particular isocrystals, and as in the `-adic case, it is reasonable to expect thatwhatever categories of p-adic sheaves on BunG are considered, there will be a corresponding semiorthogonal decomposition into pieces which admit a description interms of the representation theory of the automorphism groups of isocrystals. We
warn the reader that we do not, however, expect that the semiorthogonal decomposition on Ind Coh(X ) that we consider in this section to match precisely with the(anticipated) semiorthogonal decomposition coming from BunG ; rather we expectthat the two are related “by an upper triangular unipotent change of basis matrix”
which may be quite complicated.20
Remark 7.5.2. It is expected that the formulation of p-adic local Langlands as
an equivalence categories should involve a nilpotent singular support condition on
Ind Coh(X ). We ignore this throughout this section; note that for the most part we
confine ourselves to the generic part of the stack X , where we expect this condition
to be automatic.
7.5.3. Considerations from BunG . As already indicated, the structure of thesemiorthogonal decomposition of Ind coh(X ) that we aim to exhibit is motivatedby a parallel structure on the category of constructible sheaves on BunG (the case
G = GL2 (F ) being the one of interest to us), arising from a natural stratification
on BunG that we now recall.
Namely, following [FS21, §I.4] and [FS22, §2.6], the stack BunGL2 admits
a stratification into locally closed strata as follows. The connected componentsof BunGL2 are indexed by elements α ∈ 21 Z, and each connected component contains
a unique closed stratum. If α ∈ Z, this closed stratum is isomorphic to [∗/ GL2 (F )],while if α ∈/ Z, it is isomorphic to [∗/D× ], where D is the non-split quaternion
algebra with centre F . (Here and below we informally confuse the p-adic points ofgroup schemes with the corresponding sheaves.) The other strata are indexed by
the pairs of integers (m, n) with m > n and m + n = 2α, and are identified with
the quotient stacks [∗/Gm,n ], where ×F
BC(O(m − n))Gm,n =
0F×
20Xinwen Zhu has remarked to the authors that the situation is somewhat analogous to thatof a sheaf-theoretic Radon transform as considered in [Yun09, §4].

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is the semidirect product of a unipotent group diamond (in fact a Banach–Colmez
space) BC(O(m − n)) by the torus F × × F × . (The stack BunPGL2 can be describedby similar data, by identifying those isocrystals which differ by a twist by O(1); forexample, the connected components are now indexed by α ∈ 21 Z/Z.)
In the `-adic case there is a corresponding (infinite) semiorthogonal decomposition of the category of sheaves on BunGL2 , by [FS21, Thm. I.5.1], and we expectthat the same will be true in the p-adic case. (Indeed, the semiorthogonal decomposition of IndCoh(X ) that we exhibit below was motivated by this expectation,and retrospectively justifies it.) In the `-adic case one can furthermore ignore the
Banach–Colmez spaces BC(O(m − n)), but it is a priori unclear that this should be
the case for p-adic coefficients. As we explain below, we expect that they affect our
semiorthogonal decomposition if and only if [F : Qp ] ≥ m − n. (In particular, forGL2 (Qp ), we find that they give rise to a single constituent of the semiorthogonal
decomposition of IndCoh(X )+ , and do not intervene at all in the decompositionof IndCoh(X )− — and so in particular do not intervene in the description of theessential image of A.)
As we have already indicated, in the case F = Qp , we construct below asemiorthogonal decomposition of IndCoh(X ), and in particular we describe theimage of A as the left orthogonal of a category of sheaves pushed forward fromthe reducible locus in X . We furthermore give a semiorthogonal decomposition of
the sheaves on the residually reducible locus. As indicated above, we anticipate
that this latter decomposition should ultimately be related to the representation
theory of Q×
p . Furthermore, it should have a description in terms of a “spectral
Eisenstein functor” involving the analogues in the Banach setting of the stacks ofBorel (ϕ, Γ)-modules as in Remark 5.3.15. The geometry of these Borel stacks for
GLd (F ), and the construction of a corresponding spectral Eisenstein functor, isthe subject of work in progress of T.G. and Jack Sempliner (although in the caseat hand, they can easily be constructed and studied directly, particularly on thegeneric components of X ).
7.5.4. Restriction to the reducible locus. We return to the setting F = Qp . The
residually reducible locus U ⊂ X is a union of connected components U(σ/σ co ),where as always for the sake of exposition we assume that σ is very generic. As
b m ], where the wedge
we saw in (7.2.22), we have U(σ/σ co ) = [Spf O[s, t±1 , x, y]∧ /Gdenotes (p, s, xy)-adic completion, and x and y have weights 2 and −2 respectively.
Write A := O[s, t±1 ], and write B = A[x, y], regarded as a graded A-algebraby giving x weight 2 and y weight −2. Let I := (p, s) ⊂ A and J := (I, xy) ⊂ B
denote the indicated ideals, and let BJ denote the full subcategory of the category ofgraded B-modules consisting of those modules each element of which is annihilated
by a power of J. There is then a canonical equivalence ∼Ind Coh U(σ/σ co ) −→ Ind Coh(BJ ),which respects the ±-decomposition on its source and target (with respect to theµ2 -action on the source, and with respect to parity of grading on the target).
It follows straightforwardly from Proposition C.4.17 that each ±-component
Ind Coh(BJ± ) admits a semiorthogonal decomposition into semiorthogonal pieces
D0± and E0± = (D0± )⊥ , whereD0+ = hB/Ji,
E0+ = h(A/I)[x](−2), (A/I)[x](−4), · · · , (A/I)[y](2), (A/I)[y](4), · · · i,130

M. EMERTON, T. GEE, AND E. HELLMANN

D0− = hB/J(1), B/J(−1)i,and
E0− = h(A/I)[x](−3), (A/I)[x](−5), · · · , (A/I)[y](3), (A/I)[y](5), · · · i.
By our Morita-theoretic construction of the restriction to U(σ/σ co ) of the functor A
given in Section 7.3.20, we see that the image of this restriction is precisely equal
to D0− .
7.5.5. The essential image of A in the case of PGL2 (Qp ). We now consider the
pushforward along j : U → X of our semiorthogonal decomposition of IndCoh(U)(ignoring as ever the non-generic components, and therefore the issue of nilpotent
singular support), as in Appendix A.8.13. By (A.8.14), we have a semiorthogonal
decomposition A1 , A2 of IndCoh(X )− , where A2 = j∗ E0− , and A1 = (j ∗ )−1 (D0− ).
We claim that A1 is the essential image of A. To see this, note that for any x
in A1 , we have the morphism
x → j∗ j ∗ x(from the unit of the adjunction between j∗ and j ∗ ), whose cofiber is supported on
the complement of U. This cofiber is therefore in the essential image of A, because
this essential image contains the (odd parity) skyscraper sheaves at the supersingular points (these are the images of the supersingular irreducible representations,
via Theorem 7.3.5). It therefore suffices to show that the restriction to U of theessential image of A is D0− , which, as we already observed, follows from the explicit
description of the restriction to U in Section 7.3.20.
7.5.6. Representations of D× . We now consider the sheaves of even parity. We
continue to assume that p ≥ 5 for simplicity.
We begin by describing some features of a conjectural functor AD× taking
smooth representations of D× with trivial central character to the even category
IndCoh(X )+ . We anticipate that such a functor exists, and enjoys similar properties to those of our functor A for PGL2 (Qp ). In particular, we expect it to becocontinuous and fully faithful, and we expect it to satisfy a compatibility with
Taylor–Wiles patching.
Using this latter expectation, we can predict the values of AD× on a set ofgenerators. In order to do so, we briefly recall some of the basic representation
theory of D× . To avoid having to discuss rationality issues, we work with Fp coefficients. Writing Π for a uniformizer of D, and OD for the usual maximal
order, we see that OD /Π is a quadratic extension of Fp , which we identify with Fp2 .
×are given by the characters of F×Thus the irreducible Fp -representations of OD
p2 .
×

×
→ Fp , and every
Fixing an embedding Fp2 ,→ Fp defines a character η : OD
i2other character is of the form η for some i ∈ Z/(p − 1)Z; these representations
have trivial central character if and only if (p − 1) divides i, which we assume from×
now on; then we can extend them to representations of Π2Z OD
with trivial centralicharacter, which we also denote by η .
×If (p + 1) - i then σi := IndD× η i is an irreducible 2-dimensional represenΠ2Z OD
tation of D× (with trivial central character); we have σi ∼= σ−i , and σi 6∼= σj if×

i2 6= j 2 . If i = 0 or i = (p2 − 1)/2, then IndD
η is the direct sum of two repreΠ2Z O × iD±sentations η ±i with η i (Π) = ±1. These representations give a set of generators of
the category of smooth D× representations with trivial central character.

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We anticipate that the values of AD× on these representations take a particularly simple form. The mod p Jacquet–Langlands correspondence of T.G. andGeraghty [GG15] associates to η i a companion pair of Serre weights σ, σ co (in thesense of Definition 7.2.13). Indeed, by Remark 7.2.14, the pair σ, σ co is the set of
Jordan–Hölder constituents of a cuspidal tame type, and this cuspidal tame type isthe one corresponding to (the tame lift to characteristic zero of the representationof IQp corresponding via local class field theory to) η i ⊕ η −i .
Write X (σi ) := X (σ) ∪ X (σ co ). (Note that this does not always agree withX (σ/σ co ) as defined in Definition 7.2.15, because we are omitting components ofthe form X (σa,p−1 )± .)Then we expect that (where j = 0 or 1)(7.5.7)±
AD× (η ±(p2 −1)j/2 ) = OX (σ(p−1)j/2,p−1 ) ,while if (p + 1) - i we have
(7.5.8)

AD× (σi ) = OX (σi ) .

Our justification for this expectation is as follows. By multiplicity one results
for mod p quaternion modular forms, i.e. freeness results for the corresponding
patched modules (see Section 7.5.13), and the expected compatibility of AD× with
Taylor–Wiles patching, we expect that all of these sheaves are invertible sheaves
over their supports (which are the indicated closed substacks). (As far as we know,
these multiplicity one results are not in the literature, so we sketch a proof of thembelow.) Since they are also in IndCoh(X )+ , it is natural to guess that they are
given by the structure sheaves of their supports.
Remark 7.5.9. It seems plausible that the compatibility of the functor AD×with Conjecture 9.3.2 in Eisenstein situations (for a quaternion algebra over Q
which is ramified at p) would imply that these invertible sheaves are indeed justthe structure sheaves of their support, adding further evidence for our expectation.
Remark 7.5.10. If we drop the assumption that our representations have trivial
central character, then the above statements extend in the obvious fashion, once wedecree that a Serre weight σa,p−2 is its own companion (so the “companion pair”
is a singleton in this case).
We can now repeat the analysis of Section 7.5.5. We again have a semiorthogonal decomposition A1 , A2 of IndCoh(X )+ , where A2 = j∗ E0+ , and A1 = (j ∗ )−1 (D0+ ).
Using expectations (7.5.7) and (7.5.8), we see that the essential image of AD× iscontained in A1 , and that A1 is generated by this essential image, together withthe (even parity) skyscraper sheaves at the irreducible points.
However, in contrast to the case of GL2 (Qp ), the essential image of AD× does
not contain these skyscraper sheaves! Instead, we expect that these skyscraperaccounts are accounted for by the first non-open stratum of BunPGL2 , i.e. by the
representation theory of BC(O(1)). We are therefore suggesting that BC(O(1))
behaves differently from BC(O(n)) for n > 1; as some justification for this, we notethat BC(O(n)) is representable by a perfectoid space if n = 1, but not if n > 1.
7.5.11. The case of GL2 (Qp2 ). It is possible to make a similar analysis in thecase of GL2 (Qpf ). We can give generators for the essential image of the hypothetical functor A in a similar fashion to our predictions for D× above, and we candescribe the irreducible components of the corresponding stack XQpf ,2 (for generic

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M. EMERTON, T. GEE, AND E. HELLMANN

Serre weights, we do this in Section 7.6). One can construct a semiorthogonal
decomposition on the reducible locus by an analysis similar to the one given above.
The case GL2 (Qp2 ) is very similar to the case of D× considered above. We again
find that the essential image of (the hypothetical) A is contained in the even partof the corresponding category of sheaves, and is contained in the left orthogonalof an explicit category of sheaves pushed forward from the reducible locus. We
again find that the skyscraper sheaves at irreducible points are contained in thisleft orthogonal, but seem not to be in the essential image of A.
We again anticipate that these skyscraper sheaves are accounted for by the
representation theory of a Banach–Colmez space, in this case BC(O(2)); again,
this Banach–Colmez space is represented by a perfectoid space. (For a general F ,BC(O(n)) is represented by a perfectoid space if and only if 0 < n ≤ [F : Qp ],[FS21, Prop. II.2.5(iv)].)Remark 7.5.12. The failure of the skyscraper sheaves at irreducible points tobe contained in the essential image of A is related to the failure of supersingular
representations of GL2 (Qp2 ) to be finitely presented; see Section 7.7.59 below. We
do not know if there is a purely representation-theoretic interpretation of this failureof finite presentation in terms of BC(O(2)).
7.5.13. Multiplicity one. We end this section by sketching a proof of the multiplicity one result for D× mentioned above. For the sake of completeness we drop
the assumption that our representations have trivial central character. Let DQ bea quaternion algebra over Q which is ramified precisely at p, ∞, and let r : GalQ →GL2 (Fp ) be such that r|GalQ(ζp ) is irreducible. Let m be the corresponding maximal
ideal of an appropriate prime-to-p Hecke algebra. (For simplicity the reader might×
we have a
imagine that r is unramified outside p.) For each character η i of OD
×corresponding space of quaternionic modular forms M (η i ) for the group DQ, andia patched module M∞ (η ), which is the reduction modulo p of a patched module
M∞ (η i ), where η is the lift of η.
These patched modules have been studied in the papers [GS11, GG15, Roz15],
and by modularity lifting theorems, one can show that M∞ (η i ) is maximal Cohen–
Macaulay over a ring R∞ (η i ). The ring R∞ (η i ) is formally smooth over a framed
deformation ring Rρ (η i ) for ρ := r|GalQp , which can be described as follows.
Firstly, if (p + 1)|i then after twisting we may suppose that i = 0, in whichcase Rρ (η i ) is a framed deformation ring for non-crystalline semistable representations of Hodge–Tate weights 0,1. Secondly, if (p + 1) - i, then Rρ (η i ) is a framed
deformation ring for potentially crystalline representations of Hodge–Tate weights
0,1 and inertial type ηi ⊕ η−i .
With this description at hand, one can easily check that the supports of thepatched modules agree with the prescriptions (7.5.8), (7.5.7). (This is the weightpart of Serre’s conjecture for quaternionic modular forms, originally proved by T.G.
and Savitt in [GS11]; the refinement to distinguish between the η ±i was provedby Rozensztajn in [Roz15, Lem. 6.1.1], by considering “extended types”, i.e. bykeeping track of an action of a Frobenius.)
Now, if Rρ (η i ) is regular, M∞ (η i ) is automatically free (this is the usual argument for deducing multiplicity one results from Taylor–Wiles patching, due independently to Diamond and Fujiwara). In particular, this holds if Rρ (η i ) is either
formally smooth or is formally smooth over O[[x, y]]/(xy − p) where O is the ring

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133

of integers in an appropriate unramified extension of Qp . The deformation ringsRρ (η i ) have been computed in many cases. In the case (p + 1)|i, they are formallysmooth (and can be described very explicitly, as all the lifts are ordinary) while if(p + 1) - i then from [Sav05, Thm. 1.4] and [EGS15, Thm. 7.2.1], we know thatthey are regular except possibly for the cases that ρ is unramified, or ρ|Ip a twistof 1 ⊕ ε; we refer to these below as the exceptional cases.
We do not know of a reference which computes the rings Rρ (η i ) in these exceptional cases, but it can be proved by “pure thought” (via a comparison to patched
modules for crystalline representations of Hodge–Tate weights 0, p − 1) that in thecase that ρ is a direct sum of distinct unramified characters, Rρ (η i ) is not regular.
In fact unpublished calculations of Bao Le Hung which will appear in [HMM22]
show that it is formally smooth over O[[x, y]]/(xy − p2 ) (at least if p > 7). In thiscase it seems unlikely that one can prove the freeness of M∞ (η i ) without using
some geometric input. In the remaining unramified cases the deformation rings areagain computed by Le Hung, and they are also not regular; note that the underlying cycles of their special fibres can also be read off from [San14] via the geometric
Breuil–Mézard conjecture. In the case that ρ is a twist of λ ⊕ ε where λ is a nontrivial unramified character, Le Hung’s calculations show that Rρ (η i ) is formallysmooth over O[[x, y]]/(xy − p), but we do not know what happens for λ = 1.
The following argument for freeness in these exceptional cases is taken from
unpublished work of George Boxer, Frank Calegari and T.G. (It was independently
discovered by Chengyang Bao, Andrea Dotto and Yulia Kotelnikova, and will appear in a forthcoming paper by them.) By Nakayama’s lemma, it is enough toprove that the corresponding eigenspace of quaternionic modular forms M (η i )[m]is one-dimensional over Fp . Now, we interpret M (η i ) as a space of functions on the
supersingular points SS of a modular curve X. More precisely, after twisting, we
can identify M (η i )m with the term H 0 (SS, ω n+p−1 )m in the short exact sequence
0 → H 0 (X, ω n )m → H 0 (X, ω n+p−1 )m → H 0 (SS, ω n+p−1 )m → 0
where either n = 3 and ρ|IQp ∼= ε ⊕ ε, or n = 4 and ρ|IQp ∼= ε ⊕ ε2 . In eithercase it follows from the weight part of Serre’s conjecture (and the assumption that∼
ρ is reducible) that H 0 (X, ω n )m = 0, so H 0 (X, ω n+p−1 )m −→ H 0 (SS, ω n+p−1 )m ,and we need only show that H 0 (X, ω n+p−1 )[m] is 1-dimensional. Suppose that thisis not the case; then there is an eigenform with a1 = 0, and by the q-expansion
principle, such an eigenform is in the kernel of the θ operator. But this kernel istrivial by a theorem of Katz [Kat77], so we are done.
7.6. Preliminaries in the Banach case for GL2 (Qpf ). Following the caseof GL2 (Qp ), a natural next case to consider is that of G := GL2 (Qpf ), for f > 1;
here, as usual, Qpf denotes the unramified extension of Qp with residue field Fq :=Fpf , and we write K := GL2 (Zpf ) and Z = Z(G). For example, the mod p local Langlands correspondence has been studied quite intensively in this case, as
has the problem of mod p local-global compatibility. (We recall some of the history of these investigations as we go along; the reader who is unfamiliar with it
may wish to consult [Bre10a] and the introduction to [BHH+ 21].) On the other
hand, the structure of the category of mod p (or, more generally, mod pn ) smooth
G-representations remains rather mysterious, the classification of supersingular irreducible admissible representations remains unknown, and as of yet there is no134

M. EMERTON, T. GEE, AND E. HELLMANN

precise formulation of a mod p or p-adic Langlands correspondence in the “traditional” mode as exemplified by [Col10c] and [Paš13] in the GL2 (Qp )-case.
We believe that the categorical perspective sheds quite a bit of light on thiscase, helping both to illuminate known results and to indicate why a traditionalmode of formulation for the p-adic Langlands correspondence has proved so elusive.
In the following discussion we elaborate on these points, by describing some of our21
preliminary results and expectations for the categorical Langlands program for thegroups GL2 (Qpf ). To ease notation and fix ideas, we primarily focus on the casewhen f = 2, but we begin in this section with some considerations that are valid
for any f .
Remark 7.6.1. The reader who is familiar with the various difficulties andmysteries in the p-adic local Langlands program for GL2 (Qp2 ), and who is anxious
to see what light we can shed on them, may wish to jump to Section 7.7.59 andRemark 7.8.5.
Remark 7.6.2. The similarities and differences between the case of GL2 (Qp )and GL2 (Qp2 ) can be illustrated by the following example. In either case, for ageneric Serre weight σ, the functor A takes c-IndG
KZ σ to an explicit line bundle
supported on the corresponding component X (σ) of Xred . In the case of GL2 (Qp ),the correspondence is “not very derived”; in particular, a simple calculation showsthat the supersingular irreducible admissible representation (c-IndG
KZ σ)/T is taken
to a (twisted) skyscraper sheaf supported at the corresponding irreducible Galois
representation.
For GL2 (Qp2 ), the same calculation shows that (c-IndG
KZ σ)/T is taken to asheaf supported on the closed locus on X (σ) corresponding to T = 0. However,this support is now one-dimensional, and the representation (c-IndG
KZ σ)/T has
infinite length. In fact, we know (see (7.7.53)) that it admits as subrepresentations
0
0
the full compact inductions c-IndG
KZ σ for certain weights σ 6= σ, and it follows
easily that the supersingular irreducible admissible representations of GL2 (Qp2 )cannot possibly be taken to actual sheaves (as opposed to complexes of sheaves),
in complete contrast to the case of GL2 (Qp ) (or the case ` 6= p).
7.6.3. The generic structure of Xred . As always we fix our coefficients O to be
the ring of integers in some finite extension L of Qp , which we furthermore assume
admits an embedding of Qpf . (In other words, we may regard L as an extensionof Qpf .) We let k denote the residue field of O. We also fix a character ζ : Q×→pf
O× , and let X denote the Noetherian formal algebraic stack classifying rank 2
projective étale (ϕ, Γ)-modules for GalQpf over p-adically complete O-algebras,
having their determinant fixed to equal ζε−1 .
7.6.4. The components X (σ). Just as in the case of GL2 (Qp ), the underlying
reduced algebraic substack Xred is of finite presentation over Spec k, and is a unionof finitely many irreducible components X (σ) labelled by Serre weights σ having
central character ζ|O× , with the exception that there is a pair of components X(σ)±
when σ is a twist of the Steinberg weight. And again, as in that case, for a generic
21Many of these ideas have been worked out in joint work of M.E. and T.G. with Ana
Caraiani, Michael Harris, Bao Le Hung, Brandon Levin, and David Savitt, under the auspices of
the NSF grant 1952556 “Collaborative Research: Geometric Structures in the p-adic Langlands
Program”, and we thank them for their permission to describe these results here.

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135

σ (one of the formif −1

σ = (Syma0 ⊗ det b0 ) ⊗ · · · ⊗ (Symai ⊗ det bi )Fr ⊗ · · · ⊗ (Symaf −1 ⊗ det bf −1 )Fr

with all the ai satisfying 0 ≤ ai ≤ p − 3) we have that X (σ) can be described as amoduli stack of Fontaine–Laffaille modules with coefficients in k-algebras.
In general, for a reductive group G over Fp , the corresponding Fontaine–
Laffaille moduli space can be described as the quotient (U \G)/T , where U is theunipotent radical of some fixed Borel in G, acting by left translation, and T is the
maximal torus of the the same Borel, acting by Frobenius twisted conjugation. In
our case, the group G is the restriction of scalars of SL2 from Fpf to Fp . Since k
is assumed to contain Fpf , we have
G = G/k = SLf2 ,1
U = U/k = (
0


Ga f
) ,
1

and
T = T/k = Gfm
with Frobenius
cyclicallypermuting the factors.




1 Ga
∗ ∗
2
Now
\ SL2 = A \ {0} (via
7→ (x, y)). Thus we have
0 1
x y
U \G :=fY−1(A2 \ {0})(xi ,yi )

i=0(where the subscripts indicate our notation for the coordinates). The torus T thenacts via(t0 , . . . , tf −1 ) · (xi , yi ) = (ti−1 ti xi , ti−1 t−1
i yi ).
So (for generic σ) we have find that X (σ) is equal to the quotient stackX (σ) = [fY
−1(A2 \ {0})(xi ,yi ) /Gfm ]i=0with the torus action being given by the preceding formula.
The coordinates (xi , yi ) on X (σ) are evidently not completely canonical, butthey are not completely without meaning either; in particular, we will shortly seethat their zero loci describe the loci of intersection of X (σ) with other components.
A special role is played by the product
T := y0 · · · yf −1 .
Evidently, T is Gfm invariant, and so defines a regular function on X (σ). In fact,
it generates the ring of functions on X (σ), as we record in the following lemma.
Lemma 7.6.5.
(1) Γ X (σ), OX (σ) = k[T ].
(2) The distinguished open subset D(T ) (i.e. the non-vanishing locus of T ) in
X (σ) is precisely the niveau 1 ordinary locus in X (σ).

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Concretely, we see that
D(T ) = [Af(x0 ,...,xf −1 ) /(Gm )t ] × (Gm )T ,where t acts viat · (x0 , . . . , xf −1 ) = (t2 x0 , . . . , t2 xf −1 );so (Gm )t can be identified with the diagonal copy of Gm in Gfm . There is acharacter χ : GalQpf → k × , depending on σ, with the property that the point(T, x0 , . . . , xf −1 ) of D(T ) corresponds to a two-dimensional representation ρ of
GalQpf of the form


urT χ
∗
,
0
urT −1 ζε−1 χ−1where ∗ is an element of Ext1GalQ (urT −1 ζε−1 χ−1 , urT χ), an f -dimensional vectorpf

space with coordinates x0 , . . . , xf −1 .
The zero locus Z(T ) := X (σ) \ D(T ) is more complicated in its geometry; we
describe it in detail in the case f = 2 in Section 7.7.
Suppose now that σ is very generic in the sense that each ai satisfies 0 < ai <p − 3. Then there are 2f Serre weights σ 0 such Ext1GL2 (Fq ) (σ, σ 0 ) 6= 0, and eachsuch σ 0 is generic. One then finds that X (σ) and X (σ 0 ) intersect in one of thecodimension 1 closed substacks of X (σ) cut out by either xi = 0 or yi = 0 (for somechoice of i).
7.6.6. Picard groups. Continue to assume that σ is generic.
the descrip Fromtion of X (σ) as a quotient stack, it is easy to see that Pic X (σ) ∼= Zf (the characterlattice of Gfm ). This isomorphism is not completely canonical, but it is canonicalup to permuting the coordinates of Zf , as we now explain. For this, we assume
that σ is in fact very generic.
To begin with, we consider the open substack D(T ) of X (σ). As already noted,fthis admits
 the description D(T ) = (A /Gm ) × Gm , from which one deduces that
Pic D(T ) = Z (the character lattice of Gm ). Furthermore, this identification
is canonical, since the Gm that appears is naturally oriented (corresponding to
the fact that D(T ) parameterizes extensions of urT −1 ζε−1 χ−1 by urT χ in a fixed
direction).
For each i = 0, . . . , f − 1, write σi for the weight such that X (σ) ∩ X (σi ) =X (σ)yi =0 ; since σ is very generic each σi is again generic. We write Ti for the
analogue of the function T on X (σi ); thenxi−1 Y(Ti )X (σi )yi =0 =yjxij6=i−1,i

Thus
X (σ)yi =0 ∩ D(Ti ) = X (σ)xi−1

Q

j6=i−1,i yj 6=0,yi =0.

Now
D(xi−1 xiYyj ) = [Af(x0 ,...,xi−2 ,yi−1 ,yi ,xi+1 ,...,xf −1 ) /(Gm )t ],j6=i−1,i

where t acts viat · (x0 , . . . , xi−2 , yi−1 , yi , xi+1 , . . . , xf −1 )= (t2 x0 , . . . , t2 xi−2 , t2 yi−1 , t−2 yi , t2 xi+1 , . . . , t2 xf −1 );CATEGORICAL p-ADIC LANGLANDS

137

the torus (Gm )t can be regarded as the copy of Gm embedded into Gfm via
t 7→ (t, . . . , t, t−1 in the i − 1 slot, t, . . . , t).
The closed substack X (σ)yi =0 ∩ D(Ti ) is then the copy of [Af −1 /Gm ] cut out by
yi = 0. Of course the embedding
X (σ)yi =0 ∩ D(Ti ) ,→ D(Ti )fembeds [Af −1 /Gm ] into a different copy of [A /Gm ]. f
We’ve already observedi ) = Pic([A /Gm ]) = Z, and the re that Pic fD(T−1striction map Pic D(T)→
Pic([A/Gm ]) = Z is an isomorphism. Thus
i
yi =0
Pic X (σ)∩ D(Ti ) = Z (canonically). We now consider the product of therestriction morphisms(7.6.7)

−1

 fY

Zf ∼
Pic
X(σ)
→
Pic
D(T
)
×
Pic X (σ)yi =0 ∩ D(Ti )=i=0= Z × Zf = Zf +1 .
Recalling the descriptions of the D(T ) and the various X (σ)yi =0 ∩D(Ti ) as quotients[Af /(Gm )t ] and [Af −1 /(Gm )t ], and of the embeddings of these various copies of(Gm )t into (Gm )f , we find that this map is given by(a0 , . . . , af −1 ) 7→Xjaj , (

Xf−1
X
X
f −1 
f −1 
aj − ai )i=0=
aj , (
aj − 2ai )i=0.
j

j6=i−1

j=0

So we see that (7.6.7) is an embedding, and that we can recover the various ai fromthe coordinates in Zf +1 ; explicitly, the (left) inverse is given by1(b − b0 , . . . , b − bf −1 ).
2
Now the coordinates (b, b0 , . . . , bf −1 ) are canonical, up to permuting the bi (corre
sponding to a relabelling of the σi ), and so we see that the isomorphism Pic X(σ) ∼=f
Z is canonical, up to permuting the coordinates ai .
7.6.8. Tame types. If τ is a tame inertial type for Qpf , then the associatedK-representation σ crys (τ ) factors through GL2 (kF ) = GL2 (Fpf ). If σ crys (τ )◦ isa K-invariant lattice in σ crys (τ ), then the multi-set of Jordan–Hölder factors of
its reduction is in fact a set of distinct Serre weights which is independent of theparticular choice of lattice, and which we denote by JH σ crys (τ ) . If τ is furthermorefcrys
sufficiently generic, thenS JH σ (τ ) consists of exactly 2 weights.
We write X (τ ) =X (σ); this is a closed substack of Xred .
crys(b, b0 , . . . , bf −1 ) 7→σ∈JH σ(τ )Theorem 7.6.9. If τ is a sufficiently generic tame type, then Pic X (τ ) ∼= Z2f .
In fact, the product of the restriction mapsYPic X (τ ) ,→σ∈JH σ crys (τ )
Pic X (σ)
is injective, and one can identify the image of Pic X (τ ) as a certain specific subgroup of the target.

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7.6.10. Values of the functor. The results of [EGS15] show that if σ is a generic
Serre weight, or an appropriately chosen lattice σ crys (τ )◦ for a sufficiently generictame type τ (having determinant equal to ζ|O× ), then the patched modules M∞ (σ)
Fand M∞ σ crys (τ )◦ (constructed from an appropriate Shimura curve context) are
free of rank one over their support. The weight part of Serre’s conjecture and the(geometric) Breuil–Mézard conjecture for potentially Barsotti–Tate representations
imply that the support of M∞ (σ) is precisely the pullback tothe corresponding
versal ring of X (σ), and similarly the support of M∞ σ crys (τ )◦ is the pullback of
X (τ ). (See [EG22, Thm. 8.6.2].) Thus we expect that the value of the functor A on
Gcrys (τ )◦ will be a line bundle on X (σ) or X (τ ) respectively.
c-IndGKZ σ or c-IndKZ σThis explains our interest in computing the Picard groups of these stacks.
More generally, if V is a representation of GL2 (kF ) over F which is multiplicity
free (i.e. each of whose Jordan–Hölder factors occurs with multiplicity exactly one),and which has central character equal toSζ, then the value of A on c-IndG
KZ V should
be scheme-theoretically supported on σ∈JH(V ) X (σ), and in particular on Xred .
(This follows from the expected compatibility with the Breuil–Mézard conjecture,and the expected maximal Cohen–Macaulay property of A(c-IndG
KZ V ).)
Expected Theorem 7.6.11 (Caraiani–E.–G.–Harris–Le Hung–Levin–Savitt).
There is a fully faithful functor A from the additive category of finitely presented
smooth representations generated by the objects c-IndG
KZ V (with V multiplicity free,and having all Jordan–Hölder factors being sufficiently generic) to the abelian category Coh(Xred ).
The functor A of this theorem, which we certainly believe to be the restrictionof the functor A of Conjecture 6.1.14 for GL2 (Qpf ), is produced by an explicit
construction, building on the description of the various Picard groups given above.
In particular, we have the following expectation.
Expectation 7.6.12. For G = GL2 (Qfp ), and for a very generic Serre weight σ,
the sheaf A(c-IndG
KZ σ) is the line bundle O(−1, . . . , −1) on X (σ), i.e. the line bundle corresponding to the element (−1, . . . , −1) ∈ Pic(X (σ)) ∼= Zf .
(Note that (−1, . . . , −1) is invariant under permutation, and so does indeed
give a canonically defined element of Pic X (σ) .)The functor A of Expected Theorem 7.6.11 is defined by imposing Expectation 7.6.12 as a definition; it turns out that the extension of A to the additive
subcategory in its statement is then essentially forced.
Remark 7.6.13. In the case f = 1 Expectation 7.6.12 agrees with the resultsof [DEGa], as explained in Section 7.4.1. In general the expected compatibility withpatching, and the weight part of Serre’s conjecture, imply that A(c-IndG
KZ σ) will bea line bundle on X (σ). One can then check that the full faithfulness of A, and thepossible extensions between the various c-IndG
KZ σ, imply that if the corresponding
element of the Picard group — identified with Zf as above — is independent of σ,then it is necessarily (−1, . . . , −1). The interested reader is invited to check this in
the case f = 2 using the calculations of Section 7.7 below. (Alternatively, this canbe seen from the expected compatibility with duality, as mentioned in Section 7.4.1
in the case F = Qp .)We also note that the discussion of Remark 7.8.4 below shows that Expectation 7.6.12 is compatible with the philosophy introduced by Breuil in [Bre11].

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139

7.7. Partial results and examples in the Banach case for GL2 (Qp2 ). In
order to continue, we restrict to the case f = 2, in which case we can make muchof the preceding discussion quite explicit. To begin with, we simplify the notation
by writing (u, v), (x, y) (rather than (x0 , y0 ), (x1 , y1 )) for the coordinates on G/U ,and (s, t) (rather than t0 , t1 ) for the coordinates on T .
Then, if σ is a generic Serre weight, we have
(7.7.1)

X (σ) = [(A2 \ {0})(u,v) × (A2 \ {0})(x,y) /(Gm × Gm )(s,t) ],

where


(s, t) · (u, v), (x, y) = (stu, s−1 tv), (stx, st−1 y) .
Because the diagonal copy of µ2 in T acts trivially in this formula, and because (asit furthermore turns out) these copies of µ2 (for the various generic Serre weights σ)
act trivially on all the sheaves that we will consider, it is notationally convenient
to work instead on stacks that we will denote Cσ , defined as follows.
We set a = st, b = s−1 t, so that
(7.7.2)(s, t) 7→ (a, b)

is a self-isogeny of Gm × Gm having the diagonal copy of µ2 as its kernel. As
already noted, the action of Gm × Gm on (A2 \ {0})(u,v) × (A2 \ {0})(x,y) occurringin the definition of X (σ) then factors through this isogeny, and we write Cσ for thequotient of (A2 \ {0})(u,v) × (A2 \ {0})(x,y) by the corresponding isogenous copy of
Gm × Gm . Concretely, we write(7.7.3)Cσ = [(A2 \ {0})(u,v) × (A2 \ {0})(x,y) /(Gm × Gm )(a,b) ],

where


(a, b) · (u, v), (x, y) = (au, bv), (ax, b−1 y) .
There is then a natural map X (σ) → Cσ , and the various stacks Cσ glue together,in a manner that emulates the gluing together of the X (σ) in Xred , to produce a
reduced algebraic stack that we will denote Cred .
There are two open covers of Cσ that we wish to introduce. To describe them,
we first remark that, as usual, if f is a function on a stack, or more generally, asection of a line bundle, then we let D(f ) denote the non-vanishing locus of f ;this is an open subset of the stack in question. We then set Uσ := D(vy), andVσ := D(u) ∪ D(x), and note that Cσ = Uσ ∪ Vσ . We also write Zσ := Cσ \ Uσ .
There is an evident isomorphism
Uσ = [(A2(u,x) /(Gm )a ] × (Gm )vy(here the subscripts denote the coordinates), where of course a · (u, x) = (au, ax).
The Galois-theoretic interpretation of this locus is straightforward: the points correspond to extensions of a pair of characters of some fixed inertial weights; thequantity vy corresponds to a certain Frobenius eigenvalue appearing in these characters, and the A2 is the 2-dimensional Ext1 between these characters.

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There is an isomorphismZσ = P1[u:x] × [{(v, y) | vy = 0} /(Gm )b ] \ ({u = v = 0} ∪ {x = y = 0}),where b acts via b · (v, y) = (bv, b−1 y). If we consider the open substack of Zσ whereux 6= 0, we obtain the stack(Gm )x/u × [{(v, y) | vy = 0} /(Gm )b ],which is a family of extensions of characters of certain fixed inertial weights, the twolines corresponding to extensions in either possible directions, but which are “split
along one of the two possible embeddings”. The variable x/u again parameterizes
a Frobenius eigenvalue appearing in these characters. The points {u = y = 0} and{v = x = 0} each correspond to an irreducible Galois representation.

CATEGORICAL p-ADIC LANGLANDS141

The open cover {Uσ , Vσ } of Cσ is useful when we want to think about pointsof X (σ) in terms of the Galois representations that they correspond to. However,there is another open cover we will work with, which at first seems less conceptually
meaningful, but which leads to simple computations (especially when we consider
how the various components X (σ) intersect one another); this is the open cover{D(u), D(v)}.
Concretely, we see that
(7.7.4)D(u) = [ A1v × (A2 \ {0})x/u,y /(Gm )b ],where (Gm )b acts via b · (v, x/u, y) = (bv, x/u, b−1 y), while(7.7.5)D(v) = [ A1u × (A2 \ {0})x,vy /(Gm )a ],where (Gm )a acts via a · (u, x, vy) = (au, ax, vy). We can also describe their intersection:D(u) ∩ D(v) = D(uv) = (A2 \ {0})x/u,vy .
Since (A2 \ {0}) × (A2 \ {0}) has trivial Picard group, we see from (7.7.1) that
Pic(X (σ)) = X• (Gm × Gm )s,t = Z × Z,
and from (7.7.3) that
Pic(Cσ ) = X• (Gm × Gm )a,b = Z × Z.
The pull back map Pic(Cσ ) → Pic(X (σ)) is just the map of character lattices−1 1
induced by the isogeny (7.7.2), and so is given by the matrix.
−1 −1Similarly, we see that
Pic D(u) = X • (Gm )b = Z,
while


Pic D(v) = X • (Gm )a = Z.

142

M. EMERTON, T. GEE, AND E. HELLMANN

Thus the product of the restriction maps
Pic(Cσ ) → Pic D(v) × Pic D(u)is an isomorphism, indeed is just the identity map from Z × Z to itself. This is one
reason that the open cover {D(u), D(v)} is useful: we can describe a line bundle on
Cσ simply by describing its restrictions to each of D(u) and D(v), and line bundles
on each of these open substacks are given by prescribing a single twist (i.e. a single
integer).
Following Expectation 7.6.12 (and taking into account our change of coordinates (7.7.2)), for a generic Serre weight σ we defineA(c-IndG
KZ σ) := OCσ (−1, 0).
The endomorphism T of c-IndG
KZ σ will then correspond to the endomorphism ofOCσ (−1, 0) given by multiplication by vy.
7.7.6. Intersections and unions of generic components. Two generic components X (σ) and X (σ 0 ) intersect in a codimension one locus precisely if σ and σ 0
admit a non-trivial extension as representations of GL2 (Fp2 ). For a sufficiently
generic σ, there are four such σ 0 . The corresponding loci of intersection on X (σ)
are respectively {u = 0} , {v = 0}, {x = 0} , and {y = 0}. On each of the corresponding components Xσ0 , these same loci are given by {v 0 = 0}, {u0 = 0}, {y 0 = 0},
and {x0 = 0}. (Here we have used primes to denote the variables on Xσ0 .) Theseloci are compatible with passing from the X (σ) to the Cσ , and indeed the whole
picture of Xred that we will describe descends to the Cσ level, and this is where wework from now on.
7.7.7. A brief interlude on representation theory. If ξ is a character of the torus××
T (Fq ) := F×
q × Fq in GL2 (Fq ) (with values in k , say, although the following
discussion is independent of the particular field or ring of coefficients), then we can
GL (F )
define the principal series PS(ξ) := IndB(F2 q ) q ξ (where B denotes the subgroupscheme of upper triangular matrices in GL2 ). We may inflate the GL2 (Fq )-action
on PS(ξ) to a K-action, and, assuming that ξ is suitably compatible with ourcentral character ζ, we can then extend this K-action to a KZ-action by having
Z act via ζ. We may then form the compact induction c-IndG
KZ PS(ξ). If we lets denote the non-trivial Weyl group involution of T (Fq ), then we may similarlysconsider PS(ξ s ) and c-IndG
KZ PS(ξ ).
s
Now if ξ 6= ξ ,then PS(ξ) and PS(ξ s ) are not isomorphic as GL2 (Fq )-representations(and hence are not isomorphic as KZ-representations). However, there is an isomorphism(7.7.8)∼Gsc-IndGKZ PS(ξ) −→ c-IndKZ PS(ξ ).

To see this, writeGc-IndGKZ PS(ξ) = c-IndIZ ξ

and
G
s
s
c-IndGKZ PS(ξ ) = c-IndIZ ξ ,
0 1
and note that the matrix∈ G normalizes I and induces the action of s
p 0
on T (Fq ).

CATEGORICAL p-ADIC LANGLANDS

143

The isomorphism (7.7.8) is perhaps the most fundamental example of a morphism in the category of smooth G-representations which is not obtained by inducing a morphism of KZ-representations, and plays an important (if sometimes only
implicit) role in our analysis below.
7.7.9. A union of two components, and extensions of Serre weights. We begin
by describing Cσ ∪Cσ0 in the case when these two components meet in codimension 1(i.e. in the case that σ, σ 0 admit a non-trivial extension). Up to switching σ and σ 0
and swapping the variables u and v or x and y, we may assume that the intersectionis along the locus u = 0 (in Cσ ), which coincides with the locus v 0 = 0 (in Cσ0 ). Thislocus is evidently disjoint from (indeed, equal to the complement in Cσ of) D(u),and so is contained in D(v). Similarly, this locus is disjoint from D(v 0 ) in Cσ0 , andso is contained in D(u0 ). Thus
(7.7.10){D(u), D(v) ∪ D(u0 ), D(v 0 )}is an open cover of Cσ ∪ Cσ0 .
Recall from (7.7.5) that
D(v) = [ A1u × (A2 \ {0})x,vy /(Gm )a ],while applying (7.7.4) to Cσ0 shows thatD(u0 ) = [ A1v0 × (A2 \ {0})x0 /u0 ,y0 /(Gm )b0 ].
The union D(v) ∪ D(u0 ) is then equal to[{(u, x, vy, v 0 ) | (x, vy) 6= (0, 0), uv 0 = 0} /(Gm )a ],where the locus {v 0 = 0} is identified with D(v) in the evident way, while the locus{u = 0} is identified with D(u0 ) via b0 = a−1 , x0 /u0 = vy, y 0 = x.
Since A1 × (A2 \ {0}) has trivial Picard, we see that restriction to either D(v)or D(u0 ) induces isomorphisms
 ∼

 ∼
Pic D(v) ∪ D(u0 ) −→ Pic D(v) , Pic D(u0 ) −→ Z,
the two different identifications with Z differing by a sign (since b0 = a−1 ). A consideration of the open cover (7.7.10) then shows that the product of the restriction
maps induces an isomorphism∼
Pic(Cσ ∪ Cσ0 ) −→ Pic D(v 0 ) × Pic D(v) ∪ D(u0 ) × Pic D(u)∼−→ Pic D(v 0 ) × Pic D(v) × Pic D(u) = Z × Z × Z.
In terms of these coordinates, we find that the restriction
Pic(Cσ ∪ Cσ0 ) → Pic(Cσ ) = Pic D(v) × D(u) = Z × Z
is given by(l, m, n) 7→ (m, n),whilePic(Cσ ∪ Cσ0 ) → Pic(Cσ0 ) = Pic D(v 0 ) × D(u0 ) = Z × Z
is given by(l, m, n) 7→ (l, −m).
We now present some examples of morphisms of coherent sheaves on Cσ ∪ Cσ0
which incorporate some aspects of the preceding discussion. These examples willplay an important role in our discussion of the functor A.

144

M. EMERTON, T. GEE, AND E. HELLMANN

Example 7.7.11. Consider the short exact sequence
0 → ICσ → OCσ ∪Cσ0 → OCσ → 0;
here, of course, ICσ denotes the ideal sheaf cutting out Cσ as a closed substack of
the union Cσ ∪ Cσ0 . The middle term is the trivial element of Pic(Cσ ∪ Cσ0 ), theright-hand term is the trivial element of Pic(Cσ ), and the ideal sheaf is the element(0, −1) in Pic(Cσ0 ) (as one sees by considering its restriction to the members of
the open cover (7.7.10): its restriction to D(u) is the zero ideal, its restriction toD(v) ∪ D(u0 ) is the ideal sheaf spanned by v 0 , which is an invertible sheaf on D(u0 )on which the (Gm )b0 -action has been twisted by weight −1, and its restriction toD(v 0 ) is the structure sheaf). Thus we may rewrite this short exact sequence as(7.7.12)v0 ·0 → OCσ0 (0, −1) −→ OCσ ∪Cσ0 → OCσ → 0.

(Note that v 0 is a well-defined global section of OCσ ∪Cσ0 (0, −1, n) for any n — andthat OCσ ∪Cσ0 (0, −1, n) restricts to OCσ0 (0, 1) on Cσ0 — so that multiplication by v 0
is a well-defined morphism OCσ0 (0, −1) → OCσ ∪Cσ0 .)Example 7.7.13. We also have the short exact sequence
0 → ICσ0 → OCσ ∪Cσ0 → OCσ0 → 0,
which can be rewritten as the short exact sequence(7.7.14)u·

0 → OCσ (−1, 0) −→ OCσ ∪Cσ0 → OCσ0 → 0.

Example 7.7.15. The element x = y 0 is a well-defined global section of OCσ ∪Cσ0 (0, 1, 0).
Thus multiplication by y 0 , followed by restriction to Cσ0 , induces a morphism(7.7.16)y0 ·OCσ ∪Cσ0 −→ OCσ0 (0, −1).

The composite of this morphism with the first non-trivial arrow of (7.7.12) (i.e.
with the “multiplication by v 0 ” map) gives a morphism(7.7.17)(v 0 y 0 )·OCσ0 (0, −1) −→ OCσ0 (0, −1).

By assumption, σ and σ 0 admit a nonzero extension as GL2 (Fq )-representations,and in fact the corresponding Ext1 is one-dimensional. We let U denote thecorresponding non-split extension of σ by σ 0 , and we define A(c-IndG
KZ U ) :=OCσ ∪Cσ0 (−1, −1, 0). The short exact sequenceGG0
0 → c-IndG
KZ σ → c-IndKZ U → c-IndKZ σ → 0

corresponds to the short exact sequence(7.7.18)

v0 ·0 → OCσ0 (−1, 0) −→ OCσ ∪Cσ0 (−1, −1, 0) → OCσ (−1, 0) → 0,

which is a twist of (7.7.12).
G
0There is a well-known morphism22 c-IndG
KZ U → c-IndKZ σ (which is not the0
compact induction of a morphism U → σ ; indeed, there is no non-zero such morphism) with the property that the composite of this morphism with the obvious
22We may choose a character ξ so that σ 0 is the socle of PS(ξ) and such that the extension U
embeds as a GL2 (Fq )-subrepresentation of PS(ξ). Then σ 0 is the cosocle of PS(ξ s ), and so we may(7.7.8)
∼

G
G
G
s
0
form the composite morphism c-IndG
KZ U ,→ c-IndKZ PS(ξ) −→ c-IndKZ PS(ξ ) → c-IndKZ σ .

CATEGORICAL p-ADIC LANGLANDS

145

G
G
0
0
inclusion c-IndG
KZ σ ,→ c-IndKZ U is equal to the endomorphism T of c-IndKZ σ .
This well-known morphism corresponds under A to the twisty0 ·OCσ ∪Cσ0 (−1, −1, 0) −→ OCσ0 (−1, 0)of (7.7.16). Composing this with the first non-trivial arrow in (7.7.18) then gives
the twist(v 0 y 0 )·OCσ0 (−1, 0) −→ OCσ0 (−1, 0)0of (7.7.17). This does indeed correspond to the endomorphism T of c-IndG
KZ σ ,and so we have verified an instance of the functoriality of A.
Similarly, if we let V be the non-split extension of σ 0 by σ, then we defineA(c-IndG
KZ V ) := OCσ ∪Cσ0 (−1, 0, 0). Then the short exact sequenceGG0
0 → c-IndG
KZ σ → c-IndKZ V → c-IndKZ σ → 0

corresponds to the short exact sequenceu·

0 → OCσ (−1, 0) −→ OCσ ∪Cσ0 (−1, 0, 0) → OCσ0 (−1, 0) → 0
obtained by twisting (7.7.14).
7.7.19. Union of four components — case one (principal series). Typically (i.e.
for the induction of a suitably generic k × -valued character), the parabolic induction
PS(ξ) of a character ξ of the torus in GL2 (Fq ) can be denoted schematically as
follows:
σ3

σ2

σ0

σ1
Here the various σi are the Jordan–Hölder factors of PS(ξ), the rows of the diagram
are the layers of its socle filtration (starting from the bottom of the diagram) or
equivalently of the cosocle filtration (starting from the top of the diagram), and thelines indicate non-trivial extensions.
We write Ci := Cσi ; our aim now is to describe C0 ∪ C1 ∪ C2 ∪ C3 , and then todescribe its Picard group. We begin by noting that this union admits the open
cover
(7.7.20){D(u0 x0 ), D(v1 y1 ), D(u2 x2 ), D(v3 y3 ), D(v0 x0 ) ∪ D(u1 y1 ),D(v1 x1 ) ∪ D(u2 y2 ), D(v2 x2 ) ∪ D(u3 y3 ), D(v3 x3 ) ∪ D(u0 y0 ),D(v0 y0 ) ∪ D(u1 x1 ) ∪ D(v2 y2 ) ∪ D(u3 x3 )}.

(Here and below we use the notation u, v, x, y, a, b as before, but add a subscripti ∈ {0, 1, 2, 3} to indicate the component to which the variable pertains.) Here are
explicit descriptions of these various open subsets:D(ui xi ) = (Gm )ui /xi × [A2vi ,yi /(Gm )bi ];D(vi yi ) = [A2ui ,xi /(Gm )ai ] × (Gm )vi yi ;D(vi xi ) = A1ui /xi × A1vi yi ,D(ui+1 yi+1 ) = A1xi+1 /ui+1 × A1vi+1 yi+1 ,146

M. EMERTON, T. GEE, AND E. HELLMANNand for i = 0, 2, we haveD(vi xi ) ∪ D(ui+1 yi+1 ) = {ui /xi , vi yi , vi+1 yi+1 ) | (ui /xi )(vi+1 yi+1 ) = 0} ,with the identification vi yi = xi+1 /ui+1 , while for i = 1, 3 we haveD(vi xi ) ∪ D(ui+1 yi+1 ) = {ui /xi , vi yi , xi+1 /ui+1 ) | (vi yi )(xi+1 /ui+1 ) = 0} ,with the identification ui /xi = vi+1 yi+1 ; and finallyD(v0 y0 ) ∪ D(u1 x1 ) ∪ D(v2 y2 ) ∪ D(u3 x3 )= [{(u0 , x0 , v1 , y3 , v0 y0 ) | u0 v1 = x0 y3 = 0, v0 y0 6= 0} /(Gm )a0 ],where we have the identifications
x0 = y1 , x2 = y3 , u0 = v3 , u2 = v1 ,v0 y0 = x1 /u1 = (v2 y2 )−1 = u3 /x3 ,−1
a0 = b−1
1 = a2 = b3 .

Using the open cover (7.7.20), we compute(7.7.21)Γ(OC0 ∪C1 ∪C2 ∪C3 ) = k[v1 y1 , v3 y3 ]/(v1 y1 v3 y3 ).

(To give slightly more detail: the function v1 y1 is well-defined on C1 , and extends
by zero over the union of all four components. Similarly, the function v3 y3 is welldefined on C3 , and again extends by zero.)
One can show that there is an embedding
Pic(C0 ∪ C1 ∪ C2 ∪ C3 ) ,→ Z × Z × Z × Z × Z,where the factors are given by Pic D(v 1 y1 ) , Pic D(v3 y3 ) , Pic D(v0 y0 )∪D(u1 x1 )∪D(v2 y2 ) ∪ D(u3 x3 ) , Pic D(u0 x0 ) , and Pic D(u2 x2 ) , respectively, with the morphisms being defined by restriction. The image of this embedding is equal to(7.7.22){(j, k, l, m, n) | j − k + m − n = 0} ;in particular, Pic(C0 ∪ C1 ∪ C2 ∪ C3 ) ∼= Z4 .
The simplest way to fix our choice of coordinates precisely is to describe thecorresponding embedding(7.7.23)(7.7.24)

Pic(C0 ∪ C1 ∪ C2 ∪ C3 ) ,→ Pic(C0 ) × Pic(C1 ) × Pic(C2 ) × Pic(C3 );it is given by(j, k, l, m, n) 7→ (l, m), (j, −l), (−l, n), (k, l) .

(7.7.25)

Example 7.7.26. We have the following diagram of ideal subsheaves of OC0 ∪C1 ∪C2 ∪C3 :OC0 ∪C1 ∪C2 ∪C3O
OC3

OC2 (−1,0)*IC2 ∪C3

?
8 IC3 hIC2 ∪C3 + IC0 ∪C3
OC0 (−1,0)
5U
6 IC0 ∪C3f)
4T
IC0 ∪C2 ∪C3

IC2 ∪C3 ∩ IC0 ∪C3

v1 y1 OC0 ∪C1 ∪C2 ∪C3 o

v1 y1 ·∼

OC1

CATEGORICAL p-ADIC LANGLANDS

147

where the sheaf labels on the arrows are indicate the corresponding cokernel. Thus
we may describe OC0 ∪C1 ∪C2 ∪C3 via the following extension diagram:
OC3(7.7.27)

OC2 (−1, 0)

OC0 (−1, 0)

OC1Similarly, we have the diagram
OC0 ∪C1 ∪C2 ∪C3
O
OC1

?
I
8 C1 h*IC0 ∪C1

IC0 ∪C1 + IC1 ∪C2OC2 (−1,0)OC0 (−1,0)f)
4TIC0 ∪C1 ∪C2

5U
6 IC1 ∪C2
v3 y3 OC0 ∪C1 ∪C2 ∪C3 o

IC0 ∪C1 ∩ IC1 ∪C2

v3 y3 ·∼

OC3

which allows us to describe OC0 ∪C1 ∪C2 ∪C3 via the extension diagram
OC1(7.7.28)

OC0 (−1, 0)

OC2 (−1, 0)

OC3
Example 7.7.29. We also have the following diagrams of ideal subsheaves ofOC0 ∪C1 ∪C2 ∪C3 , namely
OC0 ∪C1 ∪C2 ∪C3O
OC0

OC3 (0,1)*
IC0 ∪C3

?
I
8 C0 hIC0 ∪C3 + IC0 ∪C1
OC1 (0,−1)
5U
6 IC0 ∪C3f)
4T
IC0 ∪C1 ∪C3

IC0 ∪C3 ∩ IC0 ∪C1 o∼

OC2 (−2, 0)

148

M. EMERTON, T. GEE, AND E. HELLMANNandOC0 ∪C1 ∪C2 ∪C3O
OC2OC1 (0,1)*IC1 ∪C2

?
8 IC2 hIC1 ∪C2 + IC2 ∪C3
OC3 (0,−1)
5U
6 IC2 ∪C3f)
4T
IC1 ∪C2 ∪C3

IC1 ∪C2 ∩ IC2 ∪C3 o∼

OC0 (−2, 0)

which allows us to describe OC0 ∪C1 ∪C2 ∪C3 via the extension diagrams
OC0(7.7.30)

OC3 (0, 1)

OC1 (0, −1)

OC2 (−2, 0)
andOC2(7.7.31)

OC1 (0, 1)

OC3 (0, −1)

OC0 (−2, 0)
We now explain the representation-theoretic interpretation of the above calculations. We write U := PS(ξ), and recall the description(7.7.32)

σ3

σ2

σ0

σ1
of U in terms of its Jordan–Hölder factors. We then defineA(c-IndGKZ U ) := OC0 ∪C1 ∪C2 ∪C3 (−1, −1, 0, 0, 0),
where the twisting is understood to be in terms of the embedding (7.7.22); notethat −1 − (−1) + 0 = 0 = 0, so that the constraint of (7.7.23) is satisfied. If we form

CATEGORICAL p-ADIC LANGLANDS

149

the corresponding twist of the diagram (7.7.27), and take into account the formulaof (7.7.25), we obtain the diagram
OC3 (−1, 0)(7.7.33)

OC2 (−1, 0)

OC0 (−1, 0)

OC1 (−1, 0)
which shows that our definition is compatible with the functoriality of A and diagram (7.7.32).
If we induce the character ξ s (the non-trivial Weyl group conjugate of ξ) instead,
we obtain a representation U s := PS(ξ s ), which can be described by the extension
diagram(7.7.34)

σ1

σ0

σ2

σ3
Although U and U s are not isomorphic representations of GL2 (Fp2 ), their compact
inductions to GL2 (Qp2 ) are isomorphic (via the isomorphism (7.7.8)), and so wesalso have that A(c-IndG
KZ U ) = OC0 ∪C1 ∪C2 ∪C3 (−1, −1, 0, 0, 0). Note that if we twist
diagram (7.7.28) by (−1, −1, 0, 0, 0), we obtain the diagram
OC1 (−1, 0)(7.7.35)

OC0 (−1, 0)

OC2 (−1, 0)

OC3 (−1, 0)This verifies the functoriality of A when applied to the diagram (7.7.34).
∼
GGs
The composite c-IndG→ c-IndG
KZ σ1 ,→ c-IndKZ U −→ c-IndKZ U
KZ σ1 isknown to coincide with the Hecke operator T . A consideration of the diagramsin Example 7.7.26 shows that this map is given by multiplication by v1 y1 , whichis indeed A(T ). Related to this computation, we note that (7.7.21) is compatible
with the full faithfulness of A and the known structure of the endomorphism ringof c-IndG
KZ U .

150

M. EMERTON, T. GEE, AND E. HELLMANN

As is well-known, by reducing lattices in integral principal series, we can also
obtain representations V and W of GL2 (Fq ), which can be described by the extension diagrams(7.7.36)

σ0

σ3

σ1

σ2
and
(7.7.37)

σ2

σ1

σ3

σ0
respectively.
We defineA(c-IndG
KZ V ) := OC0 ∪C1 ∪C2 ∪C3 (−1, −1, −1, 0, 0)
andA(c-IndG
KZ W ) := OC0 ∪C1 ∪C2 ∪C3 (−1, −1, 1, 0, 0).
Then twisting the diagrams (7.7.30) and (7.7.31) by (−1, −1, −1, 0, 0) and (−1, −1, 1, 0, 0)
respectively, we obtain diagrams
OC0 (−1, 0)(7.7.38)

OC3 (−1, 0)

OC1 (−1, 0)

OC2 (−1, 0)
and
OC2 (−1, 0)(7.7.39)

OC1 (−1, 0)

OC3 (−1, 0)

OC0 (−1, 0)
confirming the functoriality of A when applied to (7.7.36) and (7.7.37).

CATEGORICAL p-ADIC LANGLANDS

151

7.7.40. Union of four components — case two (cuspidal). The computations inthis case follow similar lines to those in the principal series case. For a tame cuspidal
type, there is no preferred lattice, though. Rather, the reduction (generically)
contains four weights σ0 , σ1 , σ2 , and σ3 , each with multiplicity one, and one canarrange for them to appear in the extension diagram
σ0

σ3

σ1

σ2
or more generally in any cyclic permutation of this diagram.
For an appropriate choice of labelling of the weights, the union Cσ0 ∪ Cσ1 ∪ Cσ2 ∪Cσ3 admits the open cover(7.7.41){D(v0 x0 ), D(u1 y1 ), D(v2 x2 ), D(u3 y3 ), D(v0 y0 ) ∪ D(u1 x1 ),D(v1 y1 ) ∪ D(u2 x2 ), D(v2 y2 ) ∪ D(u3 x3 ), D(v3 y3 ) ∪ D(u0 x0 ),D(u0 y0 ) ∪ D(v1 x1 ) ∪ D(u2 y2 ) ∪ D(v3 x3 )}.

We haveD(ui yi ) = A1xi /ui × A1vi yi ;D(vi xi ) = A1ui /xi × A1vi yi ;D(ui xi ) = (Gm )xi /ui × [A2vi ,yi /(Gm )bi ],D(vi yi ) = [A2ui ,xi /(Gm )ai ] × (Gm )vi yi ,and for i = 0, 2, we haveD(vi yi ) ∪ D(ui+1 xi+1 ) = [{(ui , xi , vi yi , yi+1 ) | vi yi 6= 0, xi yi+1 = 0} /(Gm )ai ],with the identifications ui = vi+1 , vi yi = ui+1 /xi+1 , and ai = bi+1 , while for
i = 1, 3 we haveD(vi yi ) ∪ D(ui+1 xi+1 ) = [{(ui , xi , vi yi , vi+1 ) | vi yi 6= 0, ui vi+1 = 0} /(Gm )ai ],with the identifications xi = yi+1 , vi yi = xi+1 /ui+1 , and ai = b−1
i+1 ; and finallyD(u0 y0 )∪D(v1 x1 )∪D(u2 y2 )∪D(v3 x3 ) = {(v0 y0 , v1 y1 , v2 y2 , v3 y3 ) | (v0 y0 )(v2 y2 ) = (v1 y1 )(v3 y2 ) = 0} ,with the identifications x0 /u0 = v3 y3 , u1 /x1 = v0 y0 , x2 /u2 = v1 y1 , and u3 /x3 =v2 y2 .
Using the open cover (7.7.41), we compute(7.7.42)Γ(OC0 ∪C1 ∪C2 ∪C3 ) = k.

We also compute that
Pic(C0 ∪ C1 ∪ C2 ∪ C3 ) = Z × Z × Z × Z,where the factors are given
by Pic D(v2 y2 ) ∪ D(u3 x3 ) , Pic D(v1 y1 ) ∪ D(u2 x2 ) ,
Pic D(v0 y0 ) ∪ D(u1 x1 ) , and Pic D(v3 y3 ) ∪ D(u0 x0 ) , respectively. We fix our
choice of coordinates by describing the embedding(7.7.43)

Pic(C0 ∪ C1 ∪ C2 ∪ C3 ) ,→ Pic(C0 ) × Pic(C1 ) × Pic(C2 ) × Pic(C3 );152

M. EMERTON, T. GEE, AND E. HELLMANN

it is given by(k, l, m, n) 7→ (m, n), (−l, m), (k, l), (−n, k) .

(7.7.44)

Example 7.7.45. We have the following diagrams of ideal subsheaves of OC0 ∪C1 ∪C2 ∪C3 :OC0 ∪C1 ∪C2 ∪C3O
OC0

OC3 (−1,0)?
I
8 C0 hIC0 ∪C3 + IC0 ∪C1
OC1 (0,1)
5U
6 IC0 ∪C3*
IC0 ∪C3f)
4T
IC0 ∪C1 ∪C3

IC0 ∪C3 ∩ IC0 ∪C1 o∼

OC2 (−1, −1)∼

OC3 (−1, 1)∼

OC0 (−1, −1)

OC0 ∪C1 ∪C2 ∪C3
O
OC1

?
8 IC1 h*OC0 (−1,0)IC0 ∪C1

IC0 ∪C1 + IC1 ∪C2OC2 (0,−1)f)
4T
IC0 ∪C1 ∪C2

5U
6 IC1 ∪C2
IC0 ∪C1 ∩ IC1 ∪C2 o

OC0 ∪C1 ∪C2 ∪C3O
OC2OC1 (−1,0)*IC1 ∪C2

?
8 IC2 hIC1 ∪C2 + IC2 ∪C3
OC3 (0,1)
5U
6 IC2 ∪C3f)
4T
IC1 ∪C2 ∪C3

IC1 ∪C2 ∩ IC2 ∪C3 o

CATEGORICAL p-ADIC LANGLANDS

153

OC0 ∪C1 ∪C2 ∪C3O
OC3

OC2 (−1,0)*IC2 ∪C3

?
I
8 C3 hIC2 ∪C3 + IC0 ∪C3
OC0 (0,−1)
5U
6 IC0 ∪C3f)
4T
IC0 ∪C2 ∪C3

IC2 ∪C3 ∩ IC0 ∪C3 o∼

OC1 (−1, 1)

The following representations (described via their extension diagrams) arise as
the reduction of lattices in cuspidal types:(7.7.46)

σ0

σ3

σ1

σ2
(7.7.47)

σ1

σ0

σ2

σ3
(7.7.48)

σ2

σ1

σ3

σ0
and
(7.7.49)

σ3

σ2

σ0

σ1

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The corresponding values of A are OC0 ∪C1 ∪C2 ∪C3 (0, 1, −1, 0), OC0 ∪C1 ∪C2 ∪C3 (−1, 1, 0, 0),OC0 ∪C1 ∪C2 ∪C3 (−1, 0, 0, 1), and OC0 ∪C1 ∪C2 ∪C3 (0, 0, −1, 1). If we twist each of the diagrams in Example 7.7.45 by the corresponding twist (and compute using the formula (7.7.44)), we obtain an extension diagram constituted from the A(σi ) in theappropriate order; this confirms the functoriality of A on the various diagrams(7.7.46), (7.7.47), (7.7.48), and (7.7.49) in turn.
∼7.7.50. A non-compactly induced example. The isomorphism c-IndG
KZ PS(ξ) −→G
c-IndKZ PS(ξ (s) ) allows us to construct an interesting representation. (Here PS(ξ)
is the representation denoted U in Section 7.7.19, and PS(ξ) is the representation
denoted U s .) If we let σξ denote the socle of PS(ξ), and σξ(s) denote the socle ofPS(ξ (s)) , then there are embeddingsGc-IndGKZ σξ ,→ c-IndKZ PS(ξ)

and
∼

G
(s) ) −→ c-IndG PS(ξ),
c-IndGKZ
KZ σξ (s) ,→ c-IndKZ PS(ξinducing
GG
c-IndGKZ σξ ⊕ c-IndKZ σξ (s) ,→ c-IndKZ PS(ξ).

Define Π to be the cokernel of this embedding.
We can compute A(Π) using the calculations of Section 7.7.19. Indeed, notingthat σξ and σξ(s) can be taken to be σ1 and σ3 in the notation of that discussion,
we find that A(Π) is the cokernel of the embedding
OC1 (−1, 0) ⊕ OC3 (−1, 0) ,→ OC0 ∪C1 ∪C2 ∪C3 (−1, −1, 0, 0, 0).
The image of this embedding is identified with the twist by (−1, −1, 0, 0, 0) of theideal sheaf IC0 ∪C2 , and one finds that A(Π) is a certain invertible sheaf on C0 ∪ C2 .
(In particular, it is not Cohen–Macaulay.)
7.7.51. An example — weight cycling. In the 2006 AIM meeting, the participants found an argument which has come to be known as “weight cycling”. Hereit is in its original form. (For more details, the interested reader could consult thecontemporary reports https://aimath.org/WWN/padicmodularity/emertonrepo
rt.pdf and https://www.ma.ic.ac.uk/~buzzard/maths/research/notes/notes on mod p local langlands.pdf.)
Proposition 7.7.52. Suppose that π 0 is a representation of GL2 (Qp2 ) with thefollowing properties:(1) The constituents of the K-socle of π 0 are among the constituents of the
reduction of some sufficiently generic cuspidal type τ .
(2) Each constituent of the K-socle of π 0 appears with multiplicity one.
(3) Each constituent of the K-socle of π 0 is supersingular.
Then there is a (necessarily irreducible) subrepresentation π of π 0 with the following
properties:(a) π admits a central character.
(b) Every G-subrepresentation of π 0 contains π.
(c) The K-socle of π coincides with the K-socle of π 0 .
(d) All four constituents of the reduction of τ actually appear in the common
K-socle of π 0 and π.

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Proof. Any G-subrepresentation of π 0 contains at least one of the constituentsof the K-socle of π 0 , and thus contains the G-subrepresentation that this constituent
generates. Thus if we show that the G-subrepresentation π generated by any oneof these constituents contains all of these constituents — that is, if we show that
π satisfies (c) — then we will also have shown that π is independent of the initialchoice of generating constituent, and thus that π also satisfies condition (b). (Andthe fact that π satisfies (b) implies that it is irreducible.)
So we turn to proving that π does indeed satisfy (c). Along the way, we will
also show that it satisfies (a) and (d).
Let σ0 denote the chosen constituent of the reduction of τ that appears in
the K-socle of π 0 and generates π. Because σ0 appears in the K-socle of π withmultiplicity one, the centre Z of G acts on σ0 , and hence also on the representation
π that it generates, through a character. This proves (a).
Recalling that σ0 is one of the weights appearing in the reduction of τ , wethen label the remaining weights in the reduction of τ as σ1 , σ2 , σ3 . Since τ hasa central character as a representation of K, each σi has a central character as arepresentation of K, and these characters all coincide (they are just the reductionof the central character of τ ). In particular the central character of π is compatible
with the central character of the σi (i.e. the restriction of the former to K ∩ Z
coincides with the latter), and so we can and do extend each σi to a representationof KZ by having Z act through the central character of π.
We can then choose our labels so that for each i = 0, . . . , 3, we have an embedding (originally constructed by Paškūnas)(7.7.53)GG
0(c-IndGKZ σi+1 ⊕ c-IndKZ σi+1 ) ,→ c-IndKZ σi /T,0where i + 1 is to be computed mod 4, and σi+1
is another Serre weight which is notisomorphic to any of the σj (j = 0, . . . , 3). (See Remark 7.7.54 below for a slightelaboration on this.)Let π 0 (σi ) denote the cokernel of (7.7.53). Then a basic fact is that π 0 (σi ) =
(s)
0 (s)
π (σi ), where σi denotes the Serre weight whose highest weight is obtained fromthat of σi by applying the Weyl group involution s. In particular, π 0 (σi ) is not only(s)Ga quotient of c-IndG
KZ σi , but is also a quotient of c-IndKZ σi . Another basic fact(s)is that none of the σi (for i = 0, . . . , 3) is isomorphic to any of the σj ; i.e. noneof the former representations are constituents of the reduction of τ . (Again, see
Remark 7.7.54 below for an elaboration on this.) By our assumption on the K-socle(s)of π 0 , this means that any morphism c-IndG→ π 0 (for any value of i) must
KZ σi0vanish, and consequently any morphism π (σi ) → π 0 must vanish. Since π ⊆ π 0 ,the same applies with π in place of π 0 .
Now consider the morphism c-IndG
KZ σ0 → π induced by the inclusion σ0 ⊆ π.
The supersingularity assumption implies (indeed, more-or-less means) that thisfactors through c-IndG
KZ σ0 /T. Since it is non-zero (it contains σ0 in its image)it must not factor through π 0 (σi ), and so it must not vanish identically on thedomain of (7.7.53). However, it must vanish on the second such summand (by ourassumption on the K-socle of π 0 ), and hence it is non-zero on the first summand.
Thus we obtain a non-zero morphism c-IndG
KZ σi+1 → π. Equivalently, we obtainan embedding σi+1 ,→ π.

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Continuing in this manner, we cycle through all the weights σi , and find thatthe π 0 contains all of them in its K-socle, and that in fact they are all containedin π. This completes the proof of the proposition.
Remark 7.7.54. Just to say slightly more about the facts regarding Serreweights that we used above: if χi denotes the highest weight of σi , then the weights(s)0, and σi . And the reduction of ain the reduction of PS(χi ) are σi , σi+1 , σi+1
(sufficiently generic) cuspidal type and of a (sufficiently generic) principal series
type have either no or two weights in common. So, since τ contains σi and σi+1 in(s)
0
its reduction, it does not contain σi+1
or σi .
Remark 7.7.55. The original aim of the participants at AIM in 2006 hadbeen to apply Proposition 7.7.52 with π 0 equal to the m-torsion in the completed
cohomology of a Shimura curves (where m denotes a maximal ideal of an appropriate
Hecke algebra). In this case the hypotheses of the proposition were known to holdunder appropriate genericity hypotheses by [Gee11]. Much more recently it has
been proved that in this case we even have π 0 = π (see [BHH+ 21, Cor. 1.3.10]).
Shortly after the 2006 AIM meeting, Breuil and Paškūnas [BP12] gave a direct
construction of an irreducible admissible representation π as in Proposition 7.7.52.
In fact, they exhibited infinitely many pairwise non-isomorphic such representations.
The proof of Proposition 7.7.52 shows that there is actually a scalar invariant
attached to π 0 — or, really, to its irreducible subrepresentation π. Namely, if we
begin with the inclusion σ0 ⊆ π and cycle four times, we return to an embedding
σ0 ,→ π, which must then (by our multiplicity one assumption) be equal to a nonzero scalar multiple of the original embedding. In order to make this well-defined,
one has to pin down the embeddings (7.7.53), but with a bit of care one can dothat (we explain below how to do this after applying A), and one obtains a scalar
invariant of π.
More precisely, we choose a non-zero scalar λ = λ0 ∈ k × , write λi = 1 if
i = 1, 2, 3,, and consider the two step complex(7.7.56)

3
M

3
M

G
0
c-IndG
σ
→
σ
⊕
c-Ind
c-IndGKZ σi /T,
KZ i
KZ i
i=0

i=0

G
0
where the arrow is defined on the summand c-IndG
KZ σi ⊕ c-IndKZ σi via
0c-IndGKZ σi ⊕ c-IndKZ σi 3 (v, w)
G
7→ (v + w, −λi v mod T ) ∈ c-IndG
KZ σi−1 /T ⊕ c-IndKZ σi /T ;here we regard v and w as elements of c-IndKZ σi−1 /T via (7.7.53).
Lemma 7.7.57. The arrow of (7.7.56) is injective.
Proof. If the kernel is non-zero, then it has a non-zero K-socle, and so there0
is a non-zero morphism from c-IndG
σ 0 . But if
KZ σ to this kernel, for some weight
L3
G
G
G
0
0
0
c-IndKZ σ → i=0 c-IndKZ σi ⊕ c-IndKZ σi is non-zero, then σ is equal to oneof the σi or σi0 (we are assuming that these weights are sufficiently generic), andthe morphism is just an embedding into the corresponding summand. The lemmathen follows from the fact that the arrow of (7.7.56) is injective on each summandin its domain.
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Let Πλ denote the cokernel of (7.7.56). Then giving a morphism Πλ → π, forsome other representation π, is equivalent to giving a copy of σ0 (or indeed, any
one of the σi ) in π, such that the weight cycling of Proposition 7.7.52 “works” inπ, and such that the corresponding scalar is equal to λ.
We can compute the coherent sheaf (or rather, complex of coherent sheaves)associated to Πλ . The answer depends on λ. Indeed, the geometry allows us tonormalize the embeddings (7.7.53). Namely, each of the arrows c-IndG
KZ σi+1 →GG0c-IndG
σ
/T
and
c-Ind
σ
→
c-Ind
σ
/T
corresponds,
onthecoherent
KZ i
KZ i+1
KZ isheaf side, to an arrow OCσi+1 (−1, 0) → OCσ (−1)/(vy), respectively OCσ0 (−1, 0) →i+1

OCσ (−1, 0)/(vy), and we can take these to simply be the restriction maps induced0by the inclusions Cσvy=0 ,→ Cσi+1 , Cσi+1. Applying A to the morphism (7.7.56), withthe morphisms being interpreted in the manner just described, one can (with alittle effort) explicitly compute the complexes A(Πλ ).
We summarize the results of this computation in the following statement, whichshows in particular that the value λ = 1 is distinguished.
Conclusion 7.7.58.
(1) H 0 A(Π1 ) = skyscraper supported at x, where x is the closed point obtained as theC (i = 0, . . . , 3), while intersection of the components
S3 σi
H −1 A(Π1 ) is a line bundle supported on i=0 Cσi ∪ Cσi0 .
(2) If λ 6= 1, then H 0 A(Πλ ) = 0, while H −1 A(Πλ ) is a torsion-free sheaf
S3
supported on C := i=0 Cσi ∪ Cσi0 . More precisely, it is the pushforward to
C of a line bundle on C \ {x} which depends on the parameter λ.
7.7.59. Irreducible quotients. Consider the short exact sequence
0 → K → Πλ → π → 0(where π is an irreducible admissible representation as above, and K is defined tobe the kernel of the surjection). Passing to the (now merely hypothetical) sheaves,and bearing in mind that A has amplitude [−1, 0] (see Remark 6.1.27), we obtaina long exact sequence
0 → H −1 A(K) → H −1 A(Πλ ) → H −1 A(π)

→ H 0 A(K) → H 0 A(Πλ ) → H 0 A(π) → 0.
If λ 6= 1, then we find that H 0 A(π) = 0 (since H 0 A(Πλ ) = 0), and soA(π) is supported in degree −1. If λ = 1, thensince H 0 A(Π1 ) is the skyscraper
supported at x, we see either that H 0 A(π) is again the skyscraper supported at x,
or else that H 0 A(π) = 0.
On the other hand,since π is not finitely presented [Sch15c], Lemma A.3.2 (3)implies that H −1 A(π) is not coherent, no matter the value of λ.
What can we say about the sheaf H −1 A(π) ? Here is one property it has:00since Hom(c-IndG
KZ σ , π) = 0 if σ 6= σi for some i = 0, . . . , 3, we find that
Ext1 OCσ0 , H −1 A(π) = 0
for any such σ 0 .
If λ = 1, a representation π as above (that is, an irreducible quotient of Πλwhose K-socle consists precisely of the σi , each with multiplicity one) can also beconstructed globally, via the completed cohomology of definite quaternion algebras

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and/or Shimura curves. (That λ necessarily equals 1 in this case was proved by
Dotto–Le [DL21a]. The representations in completed cohomology have been studied by many authors, using Taylor–Wiles patching as a key ingredient, and we referto the introduction to [BHH+ 21] for a brief overview of what is known.)
In the case when π is constructed via completed cohomology, the expected
relationship between
the functor A and completed cohomology suggests that themap H 0 A(Π1 ) → H 0 A(π)
 should be non-zero, and so at least in this case
we anticipate that H 0 A(π) is equal to the skyscraper at x. We explore thesheaf A(π) further in Section 7.8, where we see that our conjectures imply thatthe irreducible representation π should be uniquely determined by the requirementthat H 0 A(π) 6= 0 (see Remark 7.8.5).
7.8. The adjoint functor. Return to the situation of Conjecture 6.1.14, andbthe cocontinuous functor A : Ind Df.p.
(sm. G) → Ind Coh(Xd ). By Lemma A.3.2 (1)(i.e. the adjoint functor theorem), A admits a right adjoint B : Ind Coh(Xd ) →b
Ind Df.p.
(sm. G). By part (2) of this same lemma (which applies, because A preserves compact objects by assumption; see Remark 6.1.28), we see that B is again
cocontinuous.
Remark 7.8.1. In the setting of Remark 1.4.6 (i.e. of a p-adic version of the
Fargues–Scholze conjecture, probably formulated using the results of Mann [Man22]),the functors A and B will presumably be induced by the functors i! and i! , where
i is the inclusion of the open stratum [∗/G(F )] ,→ BunG .
We can use the calculations of the previous section to investigate the functor B
in the cases of PGL2 (Qp ) and PGL2 (Qp2 ). (Of course, this is not precisely thesetting of Conjecture 6.1.14, but the above remarks go over unchanged to thiscase.) This will allow us to connect the conjectures of this paper to the more
traditional approach to the p-adic and mod p local Langlands correspondences via
the cohomology of Shimura curves, and in particular to the hope that it is possible
to associate to each representation ρ : GalQp2 → GL2 (Fp ) a specific admissible
representation of GL2 (Qp2 ), which is determined by its occurrence in any (that is
to say, every!) global context related to a global lift of ρ (see for example [Bre10a,Q. 3.11]). We return to this theme in Remark 7.8.5 below.
We can study the expected properties of B by means of the expected compatibility of A with Taylor–Wiles patching. Let F/Qp be arbitrary, and let X be
the stack of rank 2 (ϕ, Γ)-modules with determinant ε−1 ; and let L∞ be the (hypothetical, for F 6= Qp ) pro-coherent sheaf which is a kernel for A, in the sensethat
A(π) := L∞ ⊗L
O[[G]] π.
Then the adjoint functor is given by(7.8.2)

B(F) := RHompro −OX (L∞ , F).

Suppose now that x : Spec k → X is a finite type point, corresponding to acontinuous representation ρ : GF → GL2 (k), and let f : Spf R∞ → X be the versal
morphism arising from patching at some globalization of ρ. By Remark 6.1.30 we
expect that f ∗ L∞ is equal to the patched module M∞ . For generic choices of ρ,
the ring R∞ is formally smooth, and it follows from [GN22, Prop. 4.3.1] and theresults of [BHH+ 20, HW20, Wan22] that M∞ is pro-flat over R∞ . We expect

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that it is even pro-free. Accordingly, we expect that we can compute B(F) via theformulaB(F) = RHompro −OX (L∞ , F)
= RΓ ◦ RHompro −OX (L∞ , F)
= RΓ(X , Hompro −OX (L∞ , F)).
Write Gx = AutGalF (ρ). Assume further that ρss is not scalar (this is automatically the case if as above ρ is assumed to be generic); so Gx = µ2 if ρ is irreducible
or non-split reducible, and equals Gm if ρ is a sum of two distinct characters. Thenthe residual gerbe at x is an affine immersion23
i : [Spec k/Gx ] ,→ X ,and we can form the quasicoherent sheaf δx := i∗ O[Spec k/Gx ] . If, as above, f :Spf R∞ → X is a versal morphism at x arising from Taylor–Wiles patching, and m∞
denotes the maximal ideal of R∞ , then we see that∨
Hompro −OX (L∞ , δx ) = M∞
[m∞ ]equipped with some Gx -action. (To see this, note that L∞ ⊗OX δx = M∞ /m∞ M∞
equipped with some Gx -action, and that the continuous k-dual of M∞ /m∞ M∞∨[m∞ ].) Note that by the construction of M∞ by Taylor–Wiles patching,equals M∞∨
M∞ [m∞ ] is equal to the m-torsion in completed cohomology, where m is a certain
maximal ideal of an appropriate Hecke algebra. Since Gx is linearly reductive (being
equal to either µ2 or Gm ), we then see that∨
B(δx ) = (M∞[m∞ ])Gx .
∨We can recover all of M∞[m∞ ] by considering the valuesG x∨
B δx (n) = M∞
[m∞ ](n)as n varies. Or, rather than considering the various twists δx (n) individually, if we
let g : Spec k → X denote the compositeiSpec k → [Spec k/Gx ] ,→ X ,then we can apply B to the quasicoherent sheaf ∆x := g∗ OSpec k . (This is thequasicoherent sheaf over the residual gerbe of x which corresponds to the regular
representation of Gx .) Indeed, we compute thatB(∆x ) = RHompro −OX (L∞ , g∗ OSpec F )∨= RHompro −OSpec k (g ∗ L∞ , OSpec F ) = Homcont (M∞ /m∞ , k) = M∞
[m∞ ].

Thus x 7→ B(∆x ) should recover the hoped-for “traditional” mod p local Langlands∨
correspondence alluded to above. The Gx -action on M∞[m∞ ] that arises by considering the sheaf-Hom Hompro −OX (L∞ , δx ) as above is some additional structurein the mod p Langlands correspondence that (as far as we are aware) has not beenpreviously considered (although one could say that it is implicit in the ideas of
Breuil and his collaborators expressed in papers such as [Bre11] and [BH15]).
23If ρ is semisimple then i is a closed immersion; otherwise i factors as [Spec k/µ ] open,→
2

closed[A1k /Gm ] ,→ X , which exhibits it as a composite of affine immersions.

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M. EMERTON, T. GEE, AND E. HELLMANN

7.8.3. Some examples. In the cases that F = Qp or F = Qp2 , we know some∨thing about M∞[m] (in the former case by M.E.’s local-global compatibility theorem [Eme11a], and in the latter case by [BHH+ 21]), and we either know or canguess something about how Gx acts. We have the following examples.
±1∨(1) If F = Qp and ρ = χ1 ⊕ χ2 with χ1 χ−1, then M∞[m] is a2 6= triv or εdirect sum π1 ⊕ π2 of two irreducible principal series representations, andin fact for an appropriate ordering of π1 , π2 we have

Hompro −OX (L∞ , δx ) = π1 (1) ⊕ π2 (−1)(where the twists denote the weight of the Gm -action; note that the apparent asymmetry is due to the need to choose an ordering of χ1 , χ2 inorder to identify Gx with Gm ).
In this case ∆x := ⊕∞n=−∞ δx (n), and soB(∆x ) = B δx (−1) ⊕ B δx (1) = π1 ⊕ π2 .
If σ1 , σ2 are (a chosenordering of) the two Serre weights of ρ, we find that(A ◦ B) δx (−1) = A(π1 ) is supported on X (σ1 ) while (A ◦ B)(δx (1)) =A(π2 ) is supported on X (σ2 ). As a side remark, we note that although δx
is supported on the intersection X (σ1 ) ∩ X (σ2 ), these representations arenot; the representations A(πi ) are supported on the closed locus in therespective component X (σi ) corresponding to Galois representations withsemisimplification ρ. Thus (A ◦ B)(∆x ) = A(π1 ) ⊕ A(π2 ) is supported on
the union of these two loci, which is the locus of all Galois representations
having ρ as their semisimplification.
(2) If F = Qp and ρ is a non-split extension of characters χ1 and χ2 , again±1∨with χ1 χ−1, then M∞[m] is a non-split extension of the two
2 6= triv or εirreducible principal series π1 and π2 appearing in case (1), and∨Hompro −OX (L∞ , δx ) = M∞[m](1),

the twist indicating that µ2 acts non-trivially.
In this case ∆x := δx ⊕ δx (1), and so∨B(∆x ) = B δx (1) = M∞[m](the non-split extension of π1 and π2 ). Furthermore, (A ◦ B)(∆x ) =(A ◦ B) δx (1) is a non-split extension of A(π1 ) and A(π2 ), and in particular has support equal to the closed locus corresponding to Galois representations with semisimplification equal to χ1 ⊕ χ2 .
∨(3) If F = Qp and ρ is irreducible, then M∞[m] = π is a supersingularirreducible representation, andHompro −OX (L∞ , δx ) = π1 (1)(the twist indicating that the action of µ2 is nontrivial).
As in case (2), we have ∆x := δx ⊕ δx (1), and soB(∆x ) = B δx (1) = π,and (A ◦ B)(∆x ) = (A ◦ B) δx (1) = δx (1).
(4) If F = Qp2 and ρ = χ1 ⊕χ2 with χ1 , χ2 suitably generic, then by [BHH+ 21,∨
Cor. 1.3.10], we have M∞[m] = π1 ⊕ π ⊕ π2 , where π1 , π2 are explicit

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irreducible principal series representations, and π is a (non-explicit!) supersingular irreducible representation. We expect that
Hompro −OX (L∞ , δx ) = π1 (2) ⊕ π ⊕ π2 (−2).
In this case, as in case (1), we have ∆x := ⊕∞n=−∞ δx (n), and soB(∆x ) = B δx (−2) ⊕ B(δx ) ⊕ B δx (2) = π1 ⊕ π ⊕ π2 .
In particular, B(δx ) = π is irreducible and supersingular. (A similar analysis could be performed in the case of a nonsplit extension of characters,
using the results of [HW22].)(5) If F = Qp2 and ρ is irreducible and suitably generic, then by [BHH+ 21,∨
Thm. 1.3.11], we have M∞[m] = π, where π is a supersingular irreducible
representation. We expect that
Hompro −OX (L∞ , δx ) = π(with trivial µ2 -action).
As in case (3), we have ∆x := δx ⊕ δx (1), and soB(∆x ) = B(δx ) = π
is irreducible and supersingular.
Remark 7.8.4. The twists in examples (4) and (5) were, at first, arrived atheuristically, based on examples (1), (2), and (3) and the philosophy suggested
by Breuil’s paper [Bre11]. However, one can confirm them (within our conjectural framework, i.e. assuming that A exists with its conjectured properties, andthat the values of A(c-IndG
KZ σ) for sufficiently generic Serre weights σ are as
conjectured in Expectation 7.6.12) in the following manner: The Serre weights
σ of ρ are known, and are precisely the irreducible constituents of the K-socle∨of M∞[m∞ ], and are known to appear with multiplicity one. It is also known∨[m∞ ].
how they distribute themselves among the various direct summands of M∞(See [GLS12, GK14, EGS15, Bre14] for these results.) Thus, as σ ranges over∨
the Serre weights of ρ, the multiplicity spaces HomG (c-IndG
KZ σ, M∞ [m∞ ]) are one∨dimensional, and it is known which direct summand of M∞ [m∞ ] any non-zeromorphism factors through.
On the other hand, we can write this Hom space asG∨
HomG (c-IndG
KZ σ, M∞ [m∞ ]) = HomG c-IndKZ σ, B(∆x )




G
= HomG c-IndG
σ,
B
⊕
δ
(n)
=
Hom
c-Ind
σ,
⊕
B
δ
(n)
n
x
G
n
x
KZ
KZ


= ⊕n HomG c-IndG,
KZ σ, B δx (n)the sum ranging over either n ∈ Z or n ∈ Z/2Z, depending on whether ρ is a direct
sum of characters or is irreducible. Now the adjunction between A and B shows
that



HomG c-IndG= HomG A(c-IndG
KZ σ, B δx (n)KZ σ), δx (n)= HomG OX (σ) (−1, −1), δx (n) = i∗ OX (σ) (−1, −1) (n)Gx ,where, as above, i : [Spec k/Gx ] ,→ X is the closed immersion of the residual gerbe
at x into X .

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Since OX (σ) (−1, −1) is an invertible sheaf, we see that i∗ OX (σ) (−1, −1) isa character of Gx , and so has the form k(m) for some integer m (which we see isdetermined by σ and x). Thus i∗ OX (σ) (−1, −1) (n)Gx = k(m + n)Gx is non-zerofor exactly one value of n (namely n = −m), and this determines the twist of∨
the corresponding constituent of M∞[m∞ ]; namely, it appears with a twist by m.
Computing the values of m for the various possible choices of x (which is to say,of ρ) and of σ then yields the twists appearing in examples (4) and (5).
Remark 7.8.5. Since the results of Breuil and Paškūnas [BP12] it has beenclear that the formulation of the p-adic local Langlands correspondence for GL2 (Qp2 )would have to be very different from the (classical) correspondence for GL2 (Qp ),for the reason that there appear to be many more representations of GL2 (Qp2 )than there are 2-dimensional representations of GalQp2 . In particular, as alreadydiscussed in Section 7.7.59, there are infinite families of supersingular irreducible
admissible mod p representations of GL2 (Qp2 ), all of which appear to correspondto the same irreducible representation ρ of GalQp2 .
Our conjectural functor A gives one explanation of this phenomenon: namely,
the apparent lack of Galois representations is made up for by an abundance ofcoherent sheaves on X , and in particular, these supersingular representations allcorrespond to Ind-coherent sheaves on X , which are distinguished by their (quasicoherent, but not coherent) H −1 sheaves.
There is, however, an earlier approach to reconciling the results of [BP12]
with the expectation that there is indeed a p-adic local Langlands correspondence for GL2 (Qp2 ), which is to hope that there is a uniquely determined canonical element πρ of the infinite family of supersingular representations constructedin [BP12]. Furthermore, the representation πρ is determined by local-global compatibility, in the sense that it occurs in the cohomology of Shimura curves, for any
globalisation of ρ; so the existence of πρ can be rephrased as showing that the representations occurring in Shimura curves depends only on ρ (and that it is indeed ansupersingular irreducible representation of the type constructed by [BP12]). This
question has been investigated by many authors, and there is a considerable amountof evidence in favour of this hope; we refer again to the introduction to [BHH+ 21]
for a brief account of these investigations. (This hope motivated the proposedconstruction of a p-adic local Langlands correspondence via Taylor–Wiles patchingin [CEG+ 16], which in turn motivated the conjectures of these notes.)
The computations in this section are also consistent with this expectation,and indeed we see from examples (4) and (5) above that if x is the closed point
corresponding to a suitably generic ρ, then we should have πρ = B(δx ). We claimthat πρ can be characterised among all irreducible representations π of GL2 (Qp2 )by the following property: it is the unique irreducible π for which the support ofH 0 (A(π)) contains x.
Indeed, suppose that π is irreducible and that the support of H 0 (A(π)) contains x. Since π and πρ = B(δx ) are irreducible, it is enough to show that
HomGL2 (Qp2 ) (π, B(δx )) 6= 0.
By adjunction, this is equivalent to showing that
Hompro −OX (A(π), δx ) 6= 0.

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To see that this holds, we consider the canonical morphism
A(π) → τ ≥0 A(π),
and note that τ ≥0 A(π) is coherent (by Remark 6.1.28), and thus admits a morphism
to δx by the assumptions that the support of H 0 (A(π)) contains x (here we usethat A(π) is contained in the even part of Ind Coh(X ), see Section 7.5.11).
Note that since the supersingular representation πρ admits a surjection froma representation c-IndG
KZ σ (where σ is some Serre weight), the coherent sheafτ ≥0 A(π) admits a surjection from a line bundle. In particular in case (5) we seethat H 0 (A(π)) should equal δx .
Remark 7.8.6. In cases (4) and (5) above we do not have an explicit descriptionof A(π), except that in case (5) its H 0 is equal to δx (as discussed in Remark 7.8.5).
(Similarly, we expect that in case (4) the H 0 is the structure sheaf of the closed
locus of all points whose semisimplification is x.) However, A(π) itself could in
principle be computed via the machinery of Section A.8.12, because it is equal to(A◦B)(δx ), and we expect to be able to write down a collection of generators for theimage of the hypothetical functor A (namely, the various line bundles corresponding
to the A(c-IndG
KZ σ), which in the generic case we made explicit in Section 7.7).
Remark 7.8.7. One anticipated feature of the the traditional mod p local
Langlands correspondence for GL2 (F ) is that, for reducible ρ, and writing (in termsof the notation introduced above) π(ρ) := B(∆x ), one should have
∼ π(ρ)ss .
(7.8.8)π(ρss ) =It is natural to ask if this can be seen from our present perspective.
To this end, we note that if ρss is a direct sum of two suitably generic characters
χ1 and χ2 of GQpf , then the locus of ρ which are extensions of χ1 by χ2 is equal
to a copy of [Af /Gm ]. If we let F denote the structure sheaf of [Af /Gm ], pushedforward to a coherent sheaf on X , then Hom(L∞ , F) can be interpreted as a Gm equivariant flat family of smooth GL2 (F )-representations over Af . If we forget
the Gm -equivariant structure, this family interpolates the various π(ρ) as ρ runs
over the Galois representations (i.e. extensions of characters) parameterized by Af .
It seems plausible that the existence of this flat family (with its Gm -equivariant
structure!) could imply the anticipated isomorphism (7.8.8).
7.9. Global functions and Hecke algebras. The full faithfulness of the putative functor A has some concrete consequences for the geometry of the stacks Xd .
For example, we saw above that if G = GL2 (F ) with F/Qp unramified, then for ageneric Serre weight σ we expect A(c-IndG
K σ) to be a line bundle on the component
X (σ); so that in particular the (underived) Hecke algebra EndG (c-IndG
K σ) should
be identified with the global functions on X (σ).
This expectation follows immediately from the explicit description of X (σ) interms of Fontaine–Laffaille theory, and similar results can be proved for tame types
by explicit integral p-adic Hodge theory (again in the generic case for GL2 (F ) with
F/Qp unramified).
More generally, the following result is work in progress of Eivind Hjelle, Louis
Jaburi, Rachel Knak, Yulia Kotelnikova, Billy Lee, and Shenrong Wang, under theremit of the NSF grant 1952556 “Collaborative Research: Geometric Structures in
the p-adic Langlands Program”: if σ is a suitably generic Serre weight for GLd (F ),164

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with F/Qp unramified, and X (σ) is the irreducible component of the stack XF,dlabelled by σ, then the global functions on X (σ) are given by a ring of the formk[T1 , . . . , Td−1 , Td , Td−1 ]. Note that this ring is isomorphic to the Hecke algebra
EndG (c-IndG
K σ), but there is no obviously defined direct map between the Hecke
algebra and the ring of global functions; rather, it is possible to characterize Ti asa renormalized Hecke operator on the ordinary locus in X (σ).
In the non-generic case, the paper [GKK+ 22] of Anthony Guzman, Kalyani
Kansal, Jason Kountouridis, Ben Savoie, and Xiyuan Wang (also under NSF grant
1952556) proves under mild hypotheses that for F/Qp unramified and σ a notnecessarily-generic Serre weight of GL2 (F ), the global functions on the normalization of X (σ) can be identified with the corresponding Hecke algebra.
8. Categorical local Langlands in the case ` 6= p
Although our focus so far has been on the categorical formulation of the padic local Langlands conjecture for the group GLd over a p-adic local field, we also
wish to address the global context; indeed, there is a case to be made that themain point of developing the local aspects of number theory is to provide tools forinvestigating global problems. We will explain the global theory in Section 9; thisnaturally involves the local ` 6= p case, which we study in this section.
The main global problem we will focus on is the conjectural description of thecohomology of Shimura varieties (and, even more speculatively, of general congruence quotients) in terms of the sheaves arising from the categorical local conjecture.
On the one hand, this is related to the completed cohomological approach to the
Fontaine–Mazur conjecture (as in [Eme11a, Pan22a]). On the other, it is related
to Taylor–Wiles patching (and in particular, provides a conjectural explanation for
the expected relationship between patched modules and the coherent sheaves of thecategorical conjecture).
In order to study global problems, especially those involving Shimura varieties,
we have to be able to consider reductive groups that are not necessarily split.
Although currently our understanding of p-adic local Langlands in the non-GLdcase is even more fragmentary than our understanding for GLd , the local situation
at primes away from p is better understood; indeed, there are precise conjectures inthis case, on which significant progress has been made, in particular in the papers[FS21, Zhu20, Hel20, BZCHN20, HZ]. In this section we briefly recall some ofthese local conjectures and results, referring to these papers and to the notes [FS22]
for further details.
We begin by briefly recalling the formalism of C-groups; these will be the targetsof the Galois representations arising in the local context when non-split groups are
involved. We then describe the moduli stacks of local Langlands parameters in the` 6= p case, and very briefly recall the categorical local Langlands conjectures that
we need for our global discussion.
8.1. C-groups. As always, we have fixed a finite extension L of Qp , whosering of integers O serves as our coefficients.
If F is a finite extension of either Q` (for some prime `) or of Q, and G is aconnected reductive linear algebraic group over F , then we can form the C-groupc
G of G as in [BG14]; see also [Zhu20, §3.0]. This is a variant of the L-group L G
which is better adapted to issues relating to fields of definition. By construction, itCATEGORICAL p-ADIC LANGLANDS

165

is a semi-direct productcb o Gm × Gal(Fe/F ) ,G := G

where Fe is a certain finite Galois extension of F , chosen “large enough” for theb is the usual Langlands dual group. (In particular, Fe
purposes at hand, and G
should contain a splitting field for the quasi-split inner form of G; the definitions
involving c G that appear below are easily seen to be independent of the particularb ad defined by one-halfchoice of Fe.) Here Gm acts via the cocharacter ρad : Gm → Gcb
the sum of the positive coroots of G. We regard G as a group scheme over the
ring of integers O.
Remark 8.1.1. The C-group can also be defined as the L-group of a certain
e of G, see [BG14, Defn. 5.3.2].
Gm -extension G
8.2. Langlands parameters in the local case — ` 6= p. If F is an `-adic
field with ` 6= p, then one can define a stack XG,F which parameterizes representations of a Weil–Deligne-type group of F (suitably interpreted) into c G. There arein fact various ways of defining XG,F , depending on the base over which one wishes
to have it live. We will consider XG,F as a stack over Spf O, and then we can makethe following definition: we first choose elements σ, τ ∈ GalF := Gal(F /F ) that
respectively lift Frobenius and a topological generator of tame inertia, and let ΓFdenote the subgroup of GalF generated by PF (the wild inertia subgroup of GalF ),σ, and τ . We have a short exact sequence of groups
1 → PF → ΓF → Γtame:= hσ, τ στ σ −1 = τ q i → 1
F(where q denotes the order of the residue field of F ). Note that PF has a neighbourhood basis of the identity consisting of open subgroups Q which are normal in ΓF ,so that we may write ΓF = limQ ΓF /Q as a projective limit of discrete quotients,←−
with each Q being a subgroup of PF (and hence pro-`).
For any O-algebra A in which p is nilpotent, and for any of the precedinglyQ
mentioned subgroups Q, we define XG,F(A) to be the groupoid of homomorphisms
ΓF /Q → c G(A) which are C-homomorphisms, i.e. which satisfy the C-group versionof the usual “L-homomorphism” condition: the composite(8.2.1)

ΓF → ΓF /Q → c G(A) → A× × Gal(Fe/F )(the second arrow being the tautological projection) should equal the product of
the inverse cyclotomic character ΓF → GalF → Z×p and the canonical surjection
QΓF → GalF → Gal(Fe/F ). We then set XG,F := limQ XG,F
; each transition map−→
is an open and closed embedding (because each Q is pro-` and ` 6= p), and in thisway XG,F is given the structure of an Ind-algebraic stack.
Actually, while this discussion defines XG,F as a classical stack, we reallywant to define it as a derived stack, which we can do by following the ideas explained in Appendix F. A key point is that, even when XG,F is defined as a derived
stack, it turns out to be classical, and even a local complete intersection ([Zhu20],
Prop. 3.1.6). Since local Galois cohomology is supported in cohomological degrees[0, 2], this amounts to the classical stack underlying XG,F being of the expecteddimension, which can be proved by an explicit computation, which we now briefly
recall.

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The first point is that, for an A-valued point ρ : ΓF → c G(A) of XG,F (hereA is a (classical!) O-algebra in which p is nilpotent), the tangent complex to XG,Fat ρ is computed by the shifted continuous cochain complex C • (ΓF , g ⊗O A)[1],
where ΓF acts on g ⊗O A via the composite of ρ with the adjoint action of c G on
its Lie algebra. This complex has cohomology supported in degrees [−1, 1], and so
we find that the cotangent complex of XG,F at any point has cohomology vanishingin degrees ≤ −2. This implies that XG,F is quasi-smooth, i.e. a (possibly derived)
local complete intersection. Furthermore, XG,F is of its expected dimension (theEuler characteristic of C • (ΓF , g ⊗O A)[1] equals zero, essentially by the local Euler
characteristic formula, and XG,F is flat over Spf O of relative dimension zero —see [Zhu20, Prop. 3.2.7]). This implies that XG,F is actually a classical local
complete intersection, as claimed (see the proof of [Zhu20, Prop. 2.2.13], on which
the proof of [Zhu20, Prop. 3.1.6] ultimately depends).
8.3. Categorical local Langlands. Let F be an `-adic local field, for some
prime ` 6= p, and let G be a connected reductive group over F . We let D G(F )denote the stable ∞-category of complexes of smooth representations of G(F ) on
locally p-torsion O-modules (locally p-torsion meaning that each element of the0module is annihilated
 by some power of p). We let D G(F ) denote a “renormalization” of D G(F ) analogous to the one describedin [Zhu20, §4.1] (in whichren
G(F ), O , the key difference between that
Rep G(F ), O is replaced by Rep
context and ours being that no locally p-torsion hypothesis is applied there); in0D G(F ) , all compactly induced representations are compact objects (by definition).
We then have the following categorical form of the local Langlands conjecturein the ` 6= p context.
Conjecture 8.3.1. There is a fully faithful cocontinuous functor of stable ∞categories
0
AG : D G(F ) → Ind Coh(XG,F ).
Remark 8.3.2. As in Remark 6.1.23, we expect that the functor AG of Conjecture 8.3.1 should be given by the (derived) tensor product with a kernel. In thecase G = GLn this expectation was first made explicit by E.H. in [Hel20]; here
for the Iwahori block of L-representations of GLn (F ), the functor is expected to
be given by the derived tensor product over the Iwahori Hecke algebra with the
Iwahori invariants in the Emerton–Helm–Moss local Langlands correspondence infamilies.
Conjecture 8.3.1, as stated, is rather imprecise. A much more precise version
is stated by Zhu as [Zhu20, Conj. 4.5.1]. (In that statement there is also norestriction to locally p-torsion representations, and the target category is the Indcoherent sheaves on an algebraic O-stack, of which our stack XG,F is the p-adic
completion.) The analogous conjecture in the case of L-coefficients has been madeindependently by E.H. [Hel20]. When G is split, and continuing to work with
L-coefficients, an analogous conjecture was made, yet again independently, by BenZvi–Chen–Helm–Nadler [BZCHN20], who furthermore construct the restriction of
the conjectural functor to the Iwahori block. This restriction (and an extension of it
to the general case of unipotent parameters) was also constructed independently byHemo–Zhu [HZ]. We remark that when G is not of the form GLd (for some d), theCATEGORICAL p-ADIC LANGLANDS

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functor AG depends on the choice of auxiliary data (e.g. a pinning and associatedWhittaker data, in the case that G is quasi-split). There should be line bundles on
the stack XG,F that allow one to twist between the functors arising from different
choices of auxiliary data (see [Zhu20, Rem. 4.5.7]).
In the case G = GLd , the construction of [BZCHN20] extends to the entire
category of smooth GLd (F )-representations (essentially because all the Hecke algebras appearing in the theory of the Bernstein centre for GLd are Iwahori Hecke
algebras). As already explained in Remark 8.3.2, E.H. gives another conjecturaldescription of this functor (still in the case of L-coefficients) in terms of the “local
Langlands in families” whose existence was conjectured in [EH14], and proved byHelm and Moss [HM18]. It seems likely that E.H.’s conjectural description should
apply also in the case of O-coefficients; see e.g. [Zhu20, Rem. 4.4.4], where Zhu
also gives yet another conjectural description of the functor in the case of GLd ,inspired by a conjecture made by Braverman–Finkelberg [BF19]. in the context
of geometric Langlands. (As far as we know, even for L-coefficients, the construction of [BZCHN20] has not been shown to match with either E.H.’s conjecturaldescription via local Langlands in families, or Zhu’s conjectural description à la
Braverman–Finkelberg.)
In fact, the fully faithful functor of Conjecture 8.3.1 should actually arise as the
restriction of an equivalence of categories coming from an arithmetic analogue ofthe geometric Langlands correspondence in its categorical form. Such a conjecture
has been made by Fargues and Scholze [FS21]. Another such conjecture has also
been made by Zhu [Zhu20, §4.6, esp. Conj. 4.6.4 and Rem. 4.6.7].
Remark 8.3.3. Let G = GLd (F ). In characteristic zero (over a fixed p-adicbase field L) there should be a version of Conjecture 8.3.1 involving smooth Grepresentations on L-vector spaces and (ind-)coherent sheaves on the stack[
YG,F =
Modd,ϕ,N,F 0 /F
F0of Section 5.2 instead of the stack of L-parameters XG,F . Note that the stack of
L-parameters for GLd agrees with the moduli stack of d-dimensional Weil–Deligne
representations WDd,F . The point is that we expect local Langlands in families
to make sense over YG,F . Indeed, following [Hel16] local Langlands in families isa family of representation V over XG,F constructed using a choice of a Whittakerdatum (N, ψ) and an isomorphism from the ring of invariant functions on XG,F to
the Bernstein center Z of the category of smooth G-representations on L-vector
spaces. Then V is the unique quotient of(c-IndG
N ψ) ⊗Z OXG,F
with a certain prescribed quotient at the generic points of XG,F . Using Proposition
5.2.8 and Proposition 5.2.10 the same construction should carry over to YG,F . There
are two difficulties in this strategy: we need to fix a Whittaker datum (N, ψ) which
is only possible if L contains all the p-power roots of unity (and hence in particular L
can’t be finite over Qp ). And the stacks in question are not geometrically irreducible(and their connected components are not geometrically connected). Hence oneneeds to prove some rationality results in order to define the family V over a given
base field L.

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9. Moduli stacks of global Langlands parameters and the cohomologyof Shimura varieties
In this section we briefly describe the global aspects of the theory. We sketch theconstruction of moduli stacks of global Galois representations, and of the associatedglobal-to-local restriction maps, and then explain how these maps can be used,along with the conjectural categorical local Langlands correspondences, to give a
conjectural formula for the cohomology of Shimura varieties. We next explain someresults in the case of GL2 /Q obtained in [EGZ], which explain the connection
between these conjectures, the Taylor–Wiles method, and M.E.’s results on localglobal compatibility in the p-adic Langlands program [Eme11a].
9.1. Global moduli stacks. We begin with some basic facts regarding moduli stacks of global Langlands parameters. Although one can usually reduce to thecase of considering reductive groups defined over Q (via an application of restriction of scalars), it is no more difficult to consider the case of general number fields,
and we do this. Thus we fix a number field F and a connected reductive group G
defined over F . We also fix a finite set S of places of F containing all places above
p and ∞.
We can define a derived stack XG,F,S which lies over Spf O, and whose groupoid
XG,F,S (A) of A-valued points, for a classical O-algebra A in which p is nilpotent,
coincides with the groupoid of continuous representations ρ : GalF,S → c G(A) (thetarget being given its discrete topology) which are C-homomorphisms (i.e. thosewhich satisfy (8.2.1)). (This defines the points of XG,F,S on classical rings, i.e.
cl; the definition of the points
it characterizes the underlying classical stack XG,F,S
of XG,F,S on derived rings is more involved — see [Zhu20, §2.4] for the generalformalism, the discussion at the end of [Zhu20, §3.4] for a summary of the specifics
in this context, and forthcoming paper [EZ] for the precise details.)
In the case when G = GLd for some d ≥ 1, the underlying classical stackclhas already been introduced by Wang-Erickson [WE18]. He shows that it
XGL
d ,F,Sis a disjoint union of formal algebraic stacks, each of finite type over the formal Specof complete Noetherian local ring (in fact, an appropriate pseudodeformation ring).
In general that XG,F,S has an analogous structure, and in particular is a formal
derived algebraic stack. More precisely, if θ is a residual pseudorepresentation of
GalF,S , then we let Xθ denote the associated formal deformation space of θ. Passing
from a Galois representation
to its associated pseudorepresentation gives rise to a`morphism XG,F,S → θ Xθ , and we let Xθ denote the preimage of Xθ . We thus
obtain a decompositiona(9.1.1)
XG,F,S :=Xθ ,θwhich is precisely the decomposition of XG,F,S into its connected components.
Just as in the local case, if ρ : GalF,S → c G(A) is an A-valued point of XG,F,S ,for some classical O-algebra A in which p is nilpotent, then the tangent complex
to XG,F,S at ρ is computed via group cohomology, in this case the complex ofcontinuous cochains C • (GalF,S , g)[1]. From now on we assume that p > 2. Then,just like in the local case, the cohomology of GalF,S vanishes in degrees > 2, andso we find that XG,F,S is formally quasi-smooth over Spf O. However, unlike in thelocal case, XG,F,S need not be of its expected dimension (the expected dimension at

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a particular point being computable as the Euler characteristic of C • (GalF,S , g)[1],
which is given by the global Euler characteristic formula, and so depends on thelocal behaviour at the real places of F of the Galois representation corresponding to
the point under consideration); more precisely, it can have components that are ofgreater than the expected dimension. Because it is formally quasi-smooth, though,it is a derived local complete intersection, i.e. can be written as the quotient by anappropriately defined conjugation action of c Gb(the p-adic completion of c G) on aderived formal affine scheme of the form
Spf O[[X1 , . . . , Xd ]]/(f1 , . . . , fr ),cwhere 1+d−r −dim G is equal to the expected dimension of XG,F,S . (Here the firstsummand “1” corresponds to the dimension of O.) The relations f1 , . . . , fr need
not form a regular sequence, though, and so the quotient ring must be understoodin a derived sense. The derived structure of XG,F,S is thus supported on thoseclcomponents of XG,F,S
which are of greater-than-expected dimension.
9.2. The global-to-local map. The restriction of Galois representations is
expected to give rise to a morphismY(9.2.1)
f : XG,F,S →XG,Fv .
v∈S

We can only define this morphism when G = GLd locally at places v|p, since itis only in this case that we have a definition of the local stacks XG,Fv . Assumingthat this is the case, the definition of this morphism is non-trivial, since the local
stack XG,Fv is defined in terms of (ϕ, Γ)-modules rather than directly in terms of
local Galois representations, and even in these cases, is defined only at the classicallevel. However, there is a tautological restriction morphism to a derived version of
Wang-Erickson’s p-adic local stacks [WE18], and the results of [BIP21] show thatthese stacks are actually classical (cf. Remark 4.1.7); and these classical stacks aresubstacks of our local stacks XG,Fv by Remark 4.1.2.
In principle, we should also consider local stacks XG,Fv for the real archimedean
primes v of F . However, since we are assuming that p > 2, these will simply beclassical stacks [Zhu20, Prop. 2.3.2], and classify conjugacy classes of involutions
lying above the complex conjugation cv under the projection of c G onto its Galois
factor. In fact, we only want to consider a particular conjugacy class, namely theodd“odd” class in c G — see [BV13, Prop. 6.1]. We let XG,F,S
denote the union ofconnected components of XG,F,S classifying Galois representations which are odd
at every real prime of F .
Remark 9.2.2. As will be explained in [EZ], standard Fontaine–Mazur-type
conjectures imply that the morphism f is representable by algebraic (derived)
stacks.
Remark 9.2.3.
The tangent complex of f (i.e. the relative tangent complexQ
oddof XG,F,S
over v∈S XG,Fv ) at a point is computed (up to a shift) by the cone of
the restriction morphism from global-to-local Galois cohomology; namely we havea distinguished triangleY· · · → tf → C • (GalF,S , g)[1] →C • (ΓFv , g)[1] → · · · ,
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so that the relative tangent complex tf is a Selmer-type complex of the sort considered in [GV18, App. B], which can possibly admit a non-zero H 3 term, which inturn, by global duality, is dual to another Galois H 0 , namely H 0 GalF,S , bg(1) .
(A similar analysis, in the more traditional language of lifting rings, is madein [Kis07b, §4]. Indeed, this local analysis of f is a more geometric version of
the traditional method of presenting a global deformation ring or lifting ring over atensor product of local lifting rings. Such presentations are important in modularity lifting theorems (and in their applications to potential modularity); an example
in our earlier presentation is the morphism 3.1.3.
The condition of f being relatively quasi-smooth at ρ is aninteresting one. For 
example, if G = GLd , so that b
g = End(ρ), then H 0 GalF,S , bg(1) = EndGalF,S ρ, ρ(1) ,and so we are considering whether or not ρ admits a non-trivial GalF,S -equivariant
map to its Tate twist ρ(1). If F = Q and G = GL2 , then (recalling p > 2) thisis impossible unless either ρss is a twist of the direct sum 1 ⊕ χ (where χ is themod p cyclotomic character) or p = 3 and ρ is induced from GalQ(√−3) . Theseare precisely the cases excluded by the local-to-global principle for liftings of ρ ofDiamond–Taylor [DT94, Thm. 1], and is known that such a local-to-global principle fails in this case. In the case that ρ is induced from GalQ(√−3) , this goes backto work of Carayol and Serre, and an explicit example is given in [All19, §6.4]. In
the reducible case the question of the possible levels has been investigated by Ribet
and Yoo, see in particular [Yoo19]. (See also [LHL16] for analogous results in thecase p = 2.)More generally, the condition that H 0 GalF,S , bg(1) = 0 is closely related to the“big image” conditions in the Taylor–Wiles method, and to the genericity conditionof [CS17].
9.3. Computing the cohomology of Shimura varieties via categorical
Langlands. We state a conjectural formula for the cohomology of Shimura varieties, which is related to the various conjectures of [Zhu20], §4.7. (This conjecture
will be discussed in more detail in [EZ].) While the general conjecture seems far
out of reach (since, for example, it implies all kinds of automorphy theorems!), we
discuss the case of the modular curve in Section 9.4.
9.3.1. Cohomology of Shimura varieties. Suppose now that F = Q, and thatthe reductive group G admits a Shimura datum. We let F (which is now freed
up as a piece of notation!) denote the reflex field of this Shimura datum, and letµ : (Gm )/F → G/F denote the associated Hodge cocharacter (well defined up to
conjugacy). In the definition of c G, we choose the extension Fe of Q large enoughthat it contains F , and then writecb o Gm × Gal(Fe/F ) ⊆ G
b o Gm × Gal(Fe/Q) =: c G.
GF := Gb gives rise to an irreducible
The Hodge cocharacter µ, thought of as a weight of G,b(highest weight) representation Vµ of G, which naturally extends to a representationof c GF . For any set S of primes (containing p and ∞ as usual), the moduli stack
XG,Q,S parameterizes a universal family of (derived) C-homomorphisms GalQ →c
G, which we can restrict to GalF so as to obtain a family of representations
GalF → c GF . Composing this family with the representation Vµ of c GF , we obtaina vector bundle Vµ over XG,Q,S , which is endowed with an action of GalF .
Choose a level subgroup Kf , i.e.Q
a compact open subgroup of G(A), which we
suppose may be factored as Kf = v Kv , with Kv compact open in G(Qv ) forCATEGORICAL p-ADIC LANGLANDS

171

each finite place v. We choose (as we may) S large enough that G is unramified
outside S, and so that, furthermore, there is an extension G of G to a reductive
group scheme over Z[1/S] such that Kv = G(Zv ) for v 6∈ S. The locally symmetricquotient Y (Kf ) := G(Q)\ X × G(Af ) /Kf is a Shimura variety.
For each finite prime v ∈ S, we let Wv N
be a smooth representation of Kv
on a finite torsion O-module. We let W :=v∈S Wv denote
Q the corresponding
representation of Kf (which acts through its projection onto v∈S Kv ; the tensor
product is taken over O, of course),
and we let W denote the associated local systemW := G(Q)\ X ×G(Af )×W /Kf over Y (Kf ). For each v, the compact induction
G(Q )c-IndKv v Wv is a smooth G(Qv )-representation. Applying the functors of Conjectures 6.1.14 and 8.3.1, we obtain (conjecturally!) a coherent sheaf on XG,Qv , which
we Q
denote simply by Av . We may form the exterior tensor product A := v∈Sf Av
on v∈Sf XG,Qv .
Q
odd
As above, let f : XG,Q,S→ v∈S XG,Qv denote the global-to-local restriction
morphism. We then have the following conjectural formula for the cohomology of
the Shimura variety.
Conjecture 9.3.2. Let d be the complex dimension of Y (Kf ). Then there is
an isomorphism(9.3.3)∼odd
RΓc Y (Kf ), W)[d] −→ RΓ(XG,Q,S, f ! A ⊗ Vµ )compatible with the GalF and (derived) Hecke actions on source and target.
By taking the inverse limit of the mod pn cases, the conjecture naturally extends
to the case of cohomology with p-adic coefficients. We can also take a limit over
the tame levels at places in S, to study completed cohomology, as in Expected
Theorem 9.4.2 below.
Remark 9.3.4. As briefly mentioned in our discussion of Conjecture 8.3.1 in
Subsection 8.3, the local functors appearing in the statement of Conjecture 9.3.2
depend on auxiliary choices of Whittaker-type data. In the statement of the conjecture, we should make a global choice of this data, and then compute the local
functors Av in terms of the induced local data. The resulting sheaf f ! A should thenbe well-determined independent of the global choice. (A different choice of global
data should cause each value of Av to be twisted by a certain line bundle, but
odd
the tensor product of the pullbacks to XG,Q,Sof these various line bundles should
then be trivial; see the discussion at the top of p. 63 of [Zhu20] for the analogous
discussion in the function field context.)
Remark 9.3.5. Conjecture 9.3.2 implies in particular that various representations of GalF occur in the cohomology of Shimura varieties, so in particular it
implies cases of the Fontaine–Mazur conjecture. To make this implication precise,
one needs to know something about the supports of the sheaves Av . For v|p, Conjecture 6.1.14 (4) is the relevant statement: it shows that the Av will detect all
oddpoints of XG,Q,S
whose restrictions to Xv satisfy an appropriate p-adic Hodge theoretic condition. Analogous statements in the case v - p, describing the support of
Av in terms of ramification conditions, should also hold. We don’t spell these out
here, but refer to the discussions of the v - p case in [BZCHN20], [FS21], [Hel20],
and [Zhu20]; the essential point in this case is that the categorical theory should becompatible, in a suitable sense, with the classical local Langlands correspondence.

172

M. EMERTON, T. GEE, AND E. HELLMANN

Remark 9.3.6. The Hecke algebras that act naturally on the source and target
G(Q )
G(Q )
of (9.3.3) are the derived Hecke algebras RHomG (c-IndKv v Wv , c-IndKv v Wv ) for
finite places v (where we interpret Wv simply as the trivial representation of Kv
in the case v 6∈ S; to see that the derived Hecke algebras do indeed act at placesv 6∈ S, one compares the situations at S and S ∪ {v} just after [Zhu20, Conj.
4.7.5]). We refer the reader to [Zhu20] for a discussion of these derived Hecke
algebras, and several conjectures related to them, in the case when v 6= p.
Remark 9.3.7. While the factor Vµ has a global aspect, since it carries anaction of the global absolute Galois group GalF , if we ignore this then it can be
regarded as the contribution from ∞ to the formula of Conjecture 9.3.2. We believethat an analogous isomorphism should hold for any connected reductive group G(i.e. for the cohomology of all the locally symmetric spaces Y (Kf ), whether or notthey admit the structure of Shimura variety), with an appropriate choice of factor
odd
coming from ∞ (which is now presumably just a vector bundle on XG,Q,S, pulledback from an appropriately chosen vector bundle on the odd moduli stack at ∞, andnot admitting the additional structure of a Galois representation). In particular,
we expect to recover the derived phenomena observed and conjectured by Galatius
and Venkatesh in [GV18, Ven19].
9.4. An example in the global case: (PGL2 )/Q . In [EGZ], M.E., T.G. andXinwen Zhu use the Taylor–Wiles–Kisin patching method to study Conjecture 9.3.2
in the case of the modular curve. We briefly explain some of this work in progressin a special case, which is adapted to our discussion of the patching method in
Section 3.
9.4.1. Relationship to patching. To fix ideas, set G = PGL2 , so we are in themodular curve case. Choose r : GalQ,{p,∞} → GL2 (k) as in Section 3.1, so thatin particular r|GalQ(ζp ) is absolutely irreducible, and r has determinant ε−1 . Writeρ := r|GalQp . We consider (9.2.1) in the special case that S = {p, ∞}, so that wehave the morphism−1−1

ε ,oddεf : XGL
→ XGL
,
2 ,Qp
2 ,Q,{p,∞}where we are only considering representations with determinant ε−1 , so that thetarget is the stack denoted X in Section 7.2. Write V for the rank 2 vector bundle
ε−1 ,odd
on XGL
given by the universal GalQ -representation.
2 ,Q,{p,∞}As in Section 3.1, we write R(ρ) for the universal deformation ring of ρ with
fixed determinant ε−1 . By definition there is a versal morphism g : Spf R(rp ) → Xp
that maps the closed point in its domain to the local restriction f (r). The patching
construction gives an R(r|GalQp )-module M∞ with an action of PGL2 (Qp ).
We expect to prove the following result in [EGZ], describing the localization
at the maximal ideal m of the prime-to-p Hecke algebra corresponding to r ofe 1 . As in (9.1.1) we write Xr for the irreducible
the completed homology group H−1
ε ,oddcomponent of XGL
corresponding to r.
2 ,Q,{p,∞}Expected Theorem 9.4.2 (M.E.–T.G.–Zhu). Let L∞ be the sheaf of PGL2 (Qp )representations of Expected Theorem 7.2.1. Then there is a PGL2 (Qp )-equivariant
isomorphism(9.4.3)∼e 1,m −→H
RΓ(Xr , f ! L∞ ⊗ V)[−1]CATEGORICAL p-ADIC LANGLANDS

173

compatible with the actions of GalQ and the (derived ) Hecke algebras at primes` 6= p on each side.
∼
In addition, there is a PGL2 (Qp )-equivariant isomorphism M∞ −→ g ∗ L∞ .
Remark 9.4.4. We anticipate that the hypotheses in Expected Theorem 9.4.2
can be considerably relaxed. In particular, we can allow r to be at ramified atplaces not dividing p, and we hope to relax the other assumptions on r as much as
possible in the final version of [EGZ].
We expect to prove Expected Theorem 9.4.2 as follows. The basic idea is thatM∞ is “patched” out of the cohomology of modular curves, and we want to showthat the patching process “undoes” the f ! appearing in the conjectural formulafor cohomology. We show (using existing modularity lifting theorems) that Xr isequal to [Spf R{p,∞} (r)/µ2 ]. We then show that R{p,∞} (r) is the quotient of R(ρ)
by a certain regular sequence which is also a regular sequence on M∞ , in order tocompute the pullback f ! via a Koszul complex, and ultimately reduce (9.4.3) to the
local-global compatibility results originally proved by M.E. in [Eme11a], or more
precisely to their interpretation and reproof in [CEG+ 18].
odd
9.5. The structure of XGL. Another aim of [EGZ] is to use Taylor–
2 ,Q,S
odd
Wiles patching and Iwasawa theory to describe the structure of the stacks XGL.
2 ,Q,S
We briefly describe some of our expected results. For simplicity we again restrict
ε−1 ,odd
to the stacks XGLwith determinant ε−1 .
2 ,Q,S
Similar to (9.1.1) we have the decomposition
a
odd
Xθ
XGL
:=
,Q,S
2
θ

into connected components, where θ runs over the odd residual pseudorepresentation of GalQ,S with determinant ε−1 . The simplest case is when θ arises froman irreducible odd representation r : GalQ,S → GL2 (Fp ). We then write Xθ =Xr = Spf Rr , where Rr is the deformation ring of r, and we find that Xθ = Xr =b m ], where Gb m denotes the mR -adic completion of Gm , acting trivially[Spf Rr /Gron Spf Rr . In this case, then, there is no essential difference between the theory of
the moduli stack Xr and usual Galois deformation theory. Serre’s conjecture tells
us that r is modular, and then known “big R = big T” theorems imply (at least
under the usual Taylor–Wiles hypotheses on r) that Rr may be identified with anappropriately defined Hecke algebra acting on a space of p-adic modular forms.
The case when θ is reducible is more subtle, and our goal in this case is again
to describe Xθ as much as possible in terms of modular forms. To give a sense ofwhat this means, set θ = 1 + ε−1 , and let Yθ be the (derived) substack obtained by
imposing the local conditions of being unipotently tame at the primes
Q ` ∈ S\{p, ∞},as well as the local condition of being finite flat at p. Let N = `∈S\{p,∞} `, andchoose O large enough that each cuspidal eigenform f of weight two and level
N has coefficients in O. Then each such cuspform f whose associated mod p
pseudorepresentation θf is equal to 1 + ε−1 contributes an irreducible componentb m ], where c is a certain modulus of
to Yθ of the form [Spf OhX, Y i/(XY − c)/Gb m denotes
reducibility, OhX, Y i denotes the p-adic completion of O[X, Y ], and G2−2
the p-adic completion of Gm , acting via t · (X, Y ) = (t X, t Y ). This component
parameterizes lattices in the p-adic Galois representation attached to f .

174

M. EMERTON, T. GEE, AND E. HELLMANNIn addition to such components, there is an “Eisenstein” component, whoseb m ], where n denotes
underlying classical stack is of the form [Spf OhX1 , . . . , Xn i/Gb m acts via t · Xi =the order of S \ {p, ∞} (i.e. the number of factors of N ), and G
t2 Xi . This component parameterizes extensions of ε−1 by 1; the space of suchextensions is spanned by the Kummer classes associated to the n factors of N . Ifn > 1, then this component is of larger-than-expected dimension, and so (following
the discussion above) will carry non-trivial derived structure.
There can be yet another irreducible component of X , parameterizing extensions of 1 by ε−1 (i.e. extensions “the wrong way”). Assuming p ≥ 5 for simplicity,
these are governed by those ` ∈ S for which ` ≡ −1 mod p. If there are m such ` ,then the family of such extensions is (m−1)-dimensional (there is an m-dimensionalb m -action)
space of extensions in characteristic p, which is then quotiented by a G
and is supported (set-theoretically) purely in characteristic p. If m > 1 it is acomponent of Yθ . If m > 2 it is also of greater-than-expected dimension, and so
admits derived structure.
We expect that the preceding discussion enumerates all possible componentsof Yθ . We hope that this (and more general statements) can be proved by suitableapplications of (known cases of the) the Fontaine–Mazur conjecture (especially inthe residually reducible case, where it has been proved by Skinner–Wiles [SW97]
and Pan [Pan22a]), together with the main conjecture of Iwasawa theory [MW84].
9.6. Eigenvarieties and sheaves of overconvergent p-adic automorphic
forms. We now state an analogue of Conjecture 9.3.2, giving a conjectural description of the cohomology of some (overconvergent) p-adic coefficient systems. This
gives a conjectural description of the sheaves of overconvergent p-adic automorphic
forms that appear on eigenvarieties. We maintain the notation from the preceding
section.
There are basically two approaches to p-adic automorphic forms and eigenvarieties: one can use the cohomology of locally symmetric spaces with coefficientsin p-adic coefficient systems (see e.g. [Che04] or [Han17]), or one can use thecompleted cohomology groups introduced by M.E. and further developed by M.E.
and Frank Calegari (see [Eme06b, Eme14, CE12]) and apply the locally analytic
Jacquet functor introduced by M.E. in [Eme06a] (see [Eme06b] for a constructionof eigenvarieties using this approach). We briefly discuss the relation between the
two approaches and their relation with the functor Arig
G of Conjecture 6.2.4.
Remark 9.6.1. In general an eigenvariety can be associated with a tamelevel K p , Bernstein blocks Ωv = [Mv , σv ] of Repsm
L G(Fv ) at places v|p, and algebraic representations Wv (defined up to twist with an algebraic character of Mv )of the supercuspidal support Mv of Ωv . We will focus on the classical case where
all the Bernstein blocks are principal (and hence Wv is not an extra datum) andrefer to [BD21] for the treatment of arbitrary Bernstein components. We pointout that in this more general context it is also possible to state similar conjectures
to Conjectures 9.6.8 and 9.6.27 below.
9.6.2. The case of Shimura sets. Let us first discuss the special case where thereductive group G is compact at infinite places, and hence the locally symmetricspaces Y (Kf ) are just point sets. We refer to these sets as Shimura sets eventhough strictly speaking they are not Shimura varieties. In order to fix the setup
we assume from now on that F is totally real and the group G is

CATEGORICAL p-ADIC LANGLANDS

175- either a definite unitary group associated to a hermitian space over a CM
extension E of F ,- or the group associated to a definite quaternion algebra over F ,and that G is split at places dividing p.
Let us fix for each place v|p a split maximal torus and a Borel Tv ⊂ Bv ⊂Gv = G(Fv ) = GLd (Fv ). We also fix the choice of an Iwahori subgroup Iv adaptedto Bv and write Tv0 = Tv ∩ Iv for the maximal compact subgroup of the torus Tv ,and Bv,0 = Bv ∩ Iv . Moreover, we fix a tame level K p which is a compact open
0×
subgroup of G(A∞,pF ). Let κv : Tv → L be a locally analytic character. Then we
can define the following space of module valued overconvergent p-adic automorphic
forms(9.6.3))(N∞pc (IndIv κv )an
f
:
G(F
)\G(A
)/K
→
F
Bv,0
v|p
Q
M † (G, K p , (κv )v|p ) =s. th. f (gu) = u−1 f (g) for all g ∈ G(A∞F ), u ∈v|p Iv
compare e.g. [Loe11, 3.3]. This space comes equipped with an action of the socalled Atkin–Lehner ring A+ (Gv ) whose definition (and action) we now briefly
sketch.
For each place v|p the choice of the Borel subgroup Bv and its subgroup
Nv,0 = Nv ∩ Bv,0 cuts out a submonoid Tv+ ⊂ Tv consisting of all t ∈ Tv suchthat tNv,0 t−1 ⊂ Nv,0 . Clearly Tv+ contains the maximal compact subgroup Tv0 .
We fix the choice of a free abelian subgroup Σv ⊂ Tv such that Σv ∼= Tv /Tv0 (inthe case of the diagonal torus and after choosing a uniformizer $v we might for
example choose the diagonal matrices all of whose entries are powers of $v ) and+set Σ+
v = Σv ∩ Tv , which is a free abelian monoid. Then we define
A+ (Gv ) = L[Σ+v]
O
A+ (G) =A+ (Gv ).
v|p+

It turns out [Loe11, 3.4] that A (Gv ) is canonically isomorphic to the subalgebraof the Hecke algebra H(Gv ) consisting of functions that are Iv -biinvariant and that+are supported on the submonoid Σ+v Iv of Gv generated by Σv and Iv . The action
Iv+
of Iv on IndBv,0 κv canonically extends to Σv Iv [Loe11, Theorem 2.4.7] and hence
this monoid acts on M † (G, K p , (κv )v|p ) via (γ.f ) = γf (gγ). Viewing elements of
Q
A+ (G) as locally constant functions on G( v|p Fv ) we hence obtain an action of
A+ (G) on M † (G, K p , (κv )v|p ).
We now explain a different characterization of the above space of overconvergent
p-adic automorphic forms together with its A+ (G)-action that involves completed
cohomology and is in fact less involved to write down (in particular it is easier towrite down the action of the Atkin–Lehner ring in this setup). We however preferred
to give the above definition first as it is directly related to the cohomology withcoefficients in a p-adic coefficient system defined by the representations IndIBvv,0 κv(which makes it easier to formulate Conjecture 9.6.8 below).
Consider the spacep
Ŝ(K p , L) = {f : G(F )\G(A∞F )/K → L continuous}

which is an admissible Banach space representation of G(Fp ) and coincides withthe completed cohomology (base changed to L) of Y (K p ) as our group is compact

176

M. EMERTON, T. GEE, AND E. HELLMANN

at infinity. The locally analytic vectors in this Banach space representations are
given by
(9.6.4)p
Ŝ(K p , L)an = {f : G(F )\G(A∞F )/K → L locally analytic}.

As above we write
Q Nv ⊂ Bv for the unipotent radical of Bv and Nv,0 = Nv ∩ Bv,0 .
If we set N0 = v|p Nv,0 we can then define an action of T + on the N0 -invariants
Ŝ(K p , L)N0 by(9.6.4)

X

t.f =

ntf.

n∈N0 /tN0 t−1

The following proposition compares the modules (9.6.3) and (9.6.4) equipped with
their A+ (G)-action (where A+ (Gv ) acts on the N0 -invariants of (9.6.4) via the+inclusion L[Σ+v ] ⊂ L[Tv ]).
Q
Proposition 9.6.5. Let N0 = v Nv,0 . Then there is an isomorphism∼

M † (G, K p , (κv )v|p ) −→ (Ŝ(K p , L)an )N0 [(κv )v|p ]Nequivariant for the action of the Atkin–Lehner ring A+ (G) = v|p A+ (Gv ) and for
the Hecke-action away from p. Here the (−)[(κv )v|p ] on the right hand side means
QQ
the subspace of the N0 -invariants on which T 0 = v Tv0 acts via v κv .
Proof. This is for example proven in [Loe11, Prop. 3.10.1, Prop. 3.10.2]. Instead of a fixed locally analytic character one can apply the same construction
to a family of characters of T 0 or even to the universal characterκuniv : T 0 → Γ(T̂ 0 , OT̂ 0 )× ,where T̂ 0 denotes the space of continuous characters of T 0 . This means that in
the definition of M † (G, K p , κuniv ) we replace the representation IndIBvv,0 κv by theIv -representation IndINvv,0 1 (which contains all the representations IndIBvv,0 κv for thevarious characters κv ).
Proposition 9.6.6. There is an isomorphism∼

M † (G, K p , κuniv ) −→ (Ŝ(K p , L)an )N0 ,equivariant for the action of L[T + ] and for the action of the Hecke-operators awayfrom p.
Proof. This is a special case of [Fu21, Theorem 1.2]; see also the discussionof [Eme06b, §3.2].

We now conjecturally relate these spaces of overconvergent p-adic automorphic
rigforms to the functor Arig
G . On the Galois side, let us write XG,F,S = XG,F,S for the
rigid analytic generic fiber of the stack XG,F,S . Note that XG,F,S in general is a
derived stack (not a classical stack), and we hence have to adapt the definition ofits generic fiber (which will be a derived rigid analytic Artin stack), generalizing(5.1.11) to the derived setting. As we do not want to discuss derived rigid analytic
geometry here, we do not spell out the precise definition of XG,F,S .
We note that similarly to (9.2.1), given v|p there is a morphism
frig,v : XG,F,S = (XG,F,S )rig
η → XG,Fv ,CATEGORICAL p-ADIC LANGLANDS

177

that is given by composing the corresponding map to (XG,Fv )rig
η with the morphism(5.1.27). We writeYfrig : XG,F,S →XG,Fvv

for the product of the maps frig,v . By abuse of notation we also write frig for theodd
rig
restriction of frig to Xodd
G,F,S = (XG,F,S )η ⊂ XG,F,S .
Iv
Let us write σκv = IndBv,0 κv and σv = IndINvv,0 1. We can use these representation and the functors Arig
G(Fv ) to define the coherent sheaves(9.6.7)O
rig O rig
G(F )
G(F )
⊗
AG(Fv ) (c-IndIv v σκv ),Arig ((κv )v|p ) =AG(Fv ) (c-IndKv v 1)
v-p

O

Auniv
rig =

G(F )

AG(Fv ) (c-IndKv v 1)

rig

⊗

O

G(Fv )Arig
G(Fv ) (c-IndIv

σv ).

v-p

Conjecture 9.6.8. There are isomorphisms
M † (G, K p , (κv )v|p ) ∼= RΓc (Xodd , f ∗ Arig ((κv )v|p ),
G,F,S

†

p

univ

M (G, K , κ

rig

∗
univ
)∼
= RΓc (Xodd
G,F,S , frig Arig )compatible with the Hecke action and the action of the Atkin–Lehner ring.
Remark 9.6.9. The Atkin–Lehner ring acts on the right hand side via thev
canonical action of L[T + ] on c-IndG
Iv σκv , see for example [KS12, Lemma 2.2,
Lemma 2.3].
We point out that this conjecture is somehow dual in nature to Conjecture
9.3.2: we use cohomology on the side of the symmetric space (which is just a point
set in this case) and cohomology with compact support on the side of coherent
sheaves (note that the stack Xodd
G,F,S is usually the quotient of a Stein space by theaction of a reductive group and hence we can define cohomology with compact
support as in [Chi90], similar to its use in Section 7.1). This duality also explains
why in this conjecture we use the ∗-pullback along frig instead of the !-pullback ofConjecture 9.3.2.
Remark 9.6.10. In the definition of M † (G, K p , (κv )v|p ) we could also haveinserted non-trivial Kv -representations σv for some places v - p as coefficients. We
then expect the same isomorphism as in Conjecture 9.6.8, but then of course thetrivial Kv -representation 1 in (9.6.7) has to be replaced by σv .
9.6.11. Finite slope spaces and Emerton’s locally analytic Jacquet functor. We
recall the definition of finite slope spaces and of the locally analytic Jacquet module[Eme06a].
The spaces M † (G, K p , (κv )v|p ) and (Ŝ(K p , L)an )N0 are locally convex topological L-vector spaces (of compact type) that come equipped with an action of themonoid L[T + ]. The construction of a finite slope space Vfs attached to a locallyconvex L-vector space V equipped with a topological T + -action (or a Tv+ -action
for a place v|p) is a canonical procedure to extend the L[T + ]-action to a locallyanalytic representation of T ⊃ T + . Its formal definition is given by
Vfs = HomL[T + ],b (Γ(T̂ , OT̂ ), V )178

M. EMERTON, T. GEE, AND E. HELLMANN

compare [Eme06a, Definition 3.2.1]. Here T̂ is the rigid analytic space of characters of T (note that Γ(T̂ , OT̂ ) contains the distribution algebra D(T ) but is strictly
larger) and the subscript b indicates that we take the space of continuous homomorphisms and equip it with the strong topology. Then the Γ(T̂ , OT )-module structure
on Vfs restricts to an action of D(T ) and hence to an action of T which in fact islocally analytic [Eme06a, Proposition 3.2.4]. If this representation is (essentially)
admissible, the strong dual of the finite slope space is a coadmissible module overthe Frechet–Stein algebra Γ(T̂ , OT̂ ). As T̂ is a Stein space this implies that there
exists a (necessarily unique) coherent sheaf M∗V on T̂ such that
(9.6.11)(Vfs )0 ∼= Γ(T̂ , M∗V )as Γ(T̂ , OT̂ )-modules.
Remark 9.6.12.
(i) If V = ΠNv,0 for a locally analytic Gv -representation Π, then V is equippedwith a T + -action defined by (9.6.4). Then essential admissibility of Vfs (as alocally analytic Tv -representation) can for example be assured by assuming that
Π|H = C an (H, L)m for some open compact subgroup H ⊂ Gv and some m > 0,see [Eme06a, Proposition 3.2.24] and [BHS17b, Proposition 5.3] (which relies on[Eme06a]). In this situation [BHS17b, Proposition 5.3] moreover shows that locally on T̂v the sheaf M∗V is finite projective over T̂v0 .
(ii) Assume that the T -representation on Vfs is essentially admissible. Then using
Serre duality on the Stein space T̂ (as done for the open unit disc U in (7.1.4)) we
can rewrite (9.6.11) as
Vfs = RΓc (T̂ , MV ),where MV = D(M∗V ) is the Serre dual (on T̂ ) to M∗V . A priori this is only a complex of coherent sheaves. However, in good situations (e.g. in the situation of (i))
the sheaf M∗V turns out to be a Cohen–Macaulay module and hence MV is concentrated in one degree (though this degree might be strictly negative, dependingon the dimension of the support of M∗V ).
Definition 9.6.13. The locally analytic Jaquet module JBv (Πv ) of a locallyN
analytic Gv -representation Πv is the Tv -representation on (Πv v,0 )fs .
With this definition at hand Proposition 9.6.6 directly implies the following:
Proposition 9.6.14. There is an isomorphism∼=

M † (G, K p , κuniv )fs −→ JB (Ŝ(K p , L)an )of locally analytic T -representations.
If we restrict attention to the finite slope subspace of M † (G, K p , κuniv ), thenwe can formulate a conjecture that gives a precise description of the coherent sheaf
whose compactly supported cohomology conjecturally computes M † (G, K p , κuniv )fs
and that does not involve the conjectural functors Arig
G (though of course the conjecture is inspired by these conjectural functors and their expected properties withrespect to parabolic induction).
rigv
As above, for v - p, let us write Arig,v = (AG(Fv ) (c-IndG
Kv 1)) . For v|p consider
the morphism
frig,v × ιv : Xodd
G,F,S × T̂ −→ XG,Fv × T̂v ,CATEGORICAL p-ADIC LANGLANDS

179

where ιv is given by projection to the v-th factor followed by the automorphism(δ1 , . . . , δn ) 7→ δBv · (δ1 , δ2 (ε ◦ recFv ), . . . , δn (ε ◦ recFv )n−1 )of T̂v , where δB,v denotes the modulus character. Moreover consider the diagram
XBv
πv

αv

z
XG,Fv × T̂v

!
XTv .

and define
Arig,v = πv,∗ αv∗ (OXTv ([Fv : Qp ](0, −1, . . . , −n + 1))[rkQp Tv ]).
Here, as in Conjecture 6.2.4 (2), the twist (0, −1, . . . , −n + 1) means that we takea sheaf on T̂v and consider it as a sheaf on XTv = T̂v /Gnm by equipping it with theGnm -equivariant structure given by the algebraic character (0, −1, . . . , −n + 1).
Remark 9.6.15. We note that it would be more canonical to talk about πv,!
instead of πv,∗ . But as πv is proper by Theorem 5.3.11 these two functors agree
and we do not need to worry about the definition of πv,! (that most likely would
involve condensed structures).
Writing
grig =

Yv6|p

frig,v ×Ygrig,v : Xodd
G,F,S × T̂ →Y

XG,Fv × T̂v

v|pwe can formulate the following conjecture:
Conjecture 9.6.16. There is an isomorphism∗
M † (G, K p , κuniv )fs ∼= RΓc (XG,F,S × T̂ , grigOArig,v ).

v

equivariant for the action of the Hecke-operators away from p and for the action of
the Atkin–Lehner ring at the places dividing p.
Remark 9.6.17. Note that the degree shift in the definition of Arig,v is necessary: it isPexpected that the coherent sheaf in Conjecture 9.6.16 has support ofdimension v|p rkQp Tv . Hence with this degree shift we should expect that thecompactly supported cohomology sits in degree 0. This should also explain why weactually need some degree shift in the analytic case of Section 7.1.
9.6.18. Description of Hecke eigenspaces of p-adic automorphic forms. We assume that Πv is a Gv -representation satisfying the assumptions in Remark 9.6.12,
that is we assume that Πv |Hv is isomorphic to C an (Hv , L)m for some m > 0 andsome compact open subgroup Hv ⊂ Gv . Given a locally analytic character δ of T
we can compute the eigenspacev,0
HomT (δ, JBv (Πv )) = HomT + (δ, ΠN)vusing the coherent sheaf M∗Πv := M∗ Nv,0 defined as in (9.6.11). Indeed we canΠvconsider δ as a point in T̂ and compute that
(9.6.18)

HomT (δ, JBv (Πv )) = (M∗Πv ⊗OT̂ k(δ))0180

M. EMERTON, T. GEE, AND E. HELLMANN

is the dual space to the fiber of M∗ at the point δ. We remark that there is also aderived version of this formula given by
0
RHomT (δ, JBv (Πv )) = (RΓ(T̂ , M∗Πv ⊗L
OT̂ k(δ))) .

Remark 9.6.19.
(i) At least if δ is a locally algebraic character the eigenspace HomT (δ, JBv (Πv ))can be computed using Breuil’s adjunction formula [Bre15, Remarque 4.4]
Gv
HomT (δ, JBv (Πv )) = HomGv (FB(δ), Πv ),v
Gv
where FB(−) is the functor defined by Orlik and Strauch, see Section 6.2.13 above.
v
Roughly this adjunction says that the finite slope partM † (G, K p , κuniv )fs = JB (Ŝ(K p , L)an )consists of those forms f ∈ Ŝ(K p , L)an that generate a locally analytic subrepresentation all of whose subquotients occur in parabolically induced representations.
(ii) If δv is a locally analytic character of Tv it induces by restriction a character κv
of Tv+ and a character χ : A+ (Gv ) = L[Σ+v ] → L. Conversely any such pair (κv , χ)canonically defines a character δv of Tv . With these notations and using (9.6.18)
we can rewrite
HomT (δ, JB (Ŝ(K p , L)an )) = HomTv+ (δ, (Ŝ(K p , L)an )N0 )+= M † (G, K p , (κv )v )A (G)=χ .
9.6.20. Eigenvarieties. We indicate (slightly informally) the relation with theconstruction of eigenvarieties as e.g. in [Eme06b]. The Hecke action in Conjecture
9.6.8 is in particular the action of a big Hecke algebra T that is a tensor product ofspherical Hecke algebras at good places away from p. Let us write T̂(K p ) for the
inverse limit of the images of T in the endomorphism rings ofpmŜ(K p , O/$m ) = {f : G(F )\G(A∞locally constant}.
F )/K → O/$

Then T̂(K p ) is a noetherian semi-local ring and we may form its attached formal
scheme Y(K p ) = Spf T̂(K p ) as well as its rigid analytic generic fiber Y(K p ) =Y(K p )rig
η . Following [Eme06b] (or our Remark 9.6.12 above and the discussion
preceding it) the dual of the Jacquet module JB (Ŝ(K p , L)an )0 becomes in a natural
way a coherent sheaf M∗ (G, K p ) on Y(K p ) × T̂ . The support of this coherent
sheaf is called the eigenvariety E(G, K p ) associated with K p (and the choice of theprincipal Bernstein blocks that we omit from the notation, compare Remark 9.6.1).
Remark 9.6.21. Note that of course one could as well use M † (G, K p , κuniv )fs
to construct the eigenvariety. This is the point of view of [Che04] or [Loe11].
Conjecturally, as indicated for example in Section 1.3, the completed big Hecke
algebra T̂(K p ) should coincide with a product of pseudodeformation rings, andthese pseudodeformation rings should either be defined as or proven to be isomorcl
phic to the invariant global sections of the classical stack XG,F,S
underlying XG,F,S ;that is, conjecturally one has an isomorphism
clT̂(K p ) ∼, OXG,F,S).
cl= Γ(XG,F,S

CATEGORICAL p-ADIC LANGLANDS

181

ps
Let us write XG,F,S
for the disjoint union of the formal spectra of pseudodeps
formation rings corresponding to (G, F, S). Then, using a map Y(K p ) → XG,F,S(constructed using the construction of Galois representations associated to automorphic forms) we may regard M∗ (G, K p ) as a coherent sheaf on the Stein space
ps
ps
rig
Xps
G,F,S × T̂ , with XG,F,S = (XG,F,S )η . Conjecture 9.6.16 now roughly says that∗pthe coherent sheaf M (G, K ) in some sense localizes over XG,F,S × T̂ , namely that
there is a coherent sheaf on this stack (the Serre dual of the explicitly described
coherent sheaf in the conjecture) whose (derived) global sections are concentrated
in degree zero and agree with the global sections of the sheaf M∗ (G, K p ) on the
Stein space Xps
G,F,S × T̂ .

Remark 9.6.22.
(i) We remark that we can in fact strengthen the computation of eigenspaces (9.6.18)
in order to not just compute the eigenspace for the Hecke action at p (i.e. the actionof the Atkin-Lehner ring A+ (G)) but also for the Hecke action away from p, i.e. forthe action of the Hecke operators in T̂(K p ). Let mρ ⊂ T(K p ) be the ideal attached
to a Hecke eigensystem with associated Galois representation ρ : GalF,S → GLn (L).
Then mρ defines a point xρ ∈ Y(K p )rig
η and we findYGb v FB
HomG (⊗(δv ), Ŝ(K p , L)an [mρ ]) = HomT ( δv , JB (Ŝ(K p , L)an )[mρ ])v(9.6.22)= (M∗ (G, K p ) ⊗ k(x))0 ,where x = (xρ , (δv )v ) ∈ Y(K p )rig
η × T̂ .
(ii) In good situations the spaces Y(K p )rig
and XG,F,S agree (up to the trivialηaction of a copy of Gm ) after passing to connected components where for example
the residual Galois representation is a fixed irreducible and odd representation.
In these cases (up to using Serre duality) Conjecture 9.6.16 gives a very precise
description of the coherent sheaf M∗ (G, K p ) that only involves stacks of Galois
representations (or (ϕ, Γ)-modules). Hence we also obtain a (conjectural) purely
Galois-theoretic description of the eigenspaces (9.6.22).
9.6.23. More general Shimura varieties. We return to the more general situation, i.e. we drop the assumption that G is compact at infinity. In particular the
locally symmetric spaces (Shimura varieties) Y (Kf ) can have cohomology in several
degrees, and it is better to work with complexes rather than individual cohomology
groups.
Again we fix
Qsplit maximal tori Tv and Borel subgroups Bv at the places dividing p. Let κ = v|p κv : T → L denote a locally analytic character. We fix K p andQ
Iv as above and let Kf = K p × v|p Iv . Then the representationb v|p IndIBv κv
W (κ) = ⊗v,0defines a “locally analytic” coefficient system again denoted by W (κ) over Y (Kf ),see [Han17, 2.2, 3] (the coefficient system W (κ) is denoted Aκ in loc. cit.). The
cohomology complex RΓ(Y (Kf ), W (κ)) is then equipped with an action of the
Atkin–Lehner ring A+ (G). Of course there is also a variant W (κuniv ) of this construction involving the universal character κuniv .
Remark 9.6.24. The complex RΓ(Y (Kf ), W (κ)) should be seen as an object
in the derived category of solid L-vector spaces, in particular it is only well defined

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M. EMERTON, T. GEE, AND E. HELLMANN

up to quasi-isomorphism. Its cohomology groups are L-vector spaces of compact
type. The action of A+ (G) should also be regarded as an action in the derived
category, i.e. the endomorphism defined by a given element is only well defined up
to homotopy, etc.
Similarly we can consider the completed cohomology, compare [Fu21, 5] forthe discussion of this in the derived sense. Again we consider complexes instead ofindividual cohomology groups, that is we consider the complex
RΓ̃(Y (K p ), L) = lim lim RΓ(Y (K p Kp ), O/$m ) ⊗O L.
←,m →,Kp

Again we regard this as an object in the derived category of solid L-vector spaces.
In this derived category it is equipped with an action of G such that the cohomology
groups are admissible G-representations. Even better, the dual of RΓ̃(Y (K p ), L),i.e. the complex computing completed homology, is a perfect complex, when we
only consider it as a complex of O[[K]]-modules. Similar to (9.6.4) we may pass to
the locally analytic vectors RΓ̃(Y (K p ), L)an . We then have a comparison similar
to Proposition 9.6.5.
Proposition 9.6.25. There is a quasi-isomorphism
RΓ(Y (Kf ), W (κuniv )) ∼= (RΓ̃(Y (K p ), L)an )N0
equivariant for the action of the Atkin–Lehner ring.
Proof. This is basically [Fu21, Theorem 1.2].



Remark 9.6.26. There is of course a comparison of the individual complexes
RΓ(Y (Kf ), W (κ)) with a fixed κ instead of the universal one. For that comparison
one has to take a derived eigenspace on the right hand side. We omit the technicaldetails here.
Fu [Fu21, 6] continuous the parallels with Section 9.6.2 and gives a derived
construction of eigenvarieties building upon the complex (RΓ̃(Y (K p ), L)an )N0 , respectively its dual. We will not discuss this here, but we point out that (at least0
on a heuristic level) the dual of the finite slope part (RΓ̃(Y (K p ), L)an )N
fs conjecturally should localize to an object of the derived category of coherent sheaves on
XG,F,S × T̂ . Instead of making the construction of eigenvarieties more precise westate the following analogue of Conjecture 9.6.8 (and hence of Conjecture 9.3.2) inthis context. Similarly to Section 9.6.2 we have a stack XG,F,S and a global-to-local
map frig .
Conjecture 9.6.27. Let d denote the dimension of Y (Kf ). Then there is an
isomorphism∗
RΓ(Y (Kf ), W (κ))[d] ∼= RΓc (Xodd
G,F,S , frig Arig ((κv )v|p ) ⊗ Vµ )compatible with the (derived) Hecke action and the action of the Atkin–Lehner ring.
Remark 9.6.28. In fact, as in Section 9.3.1, one should actually replace G by
its Weil restriction to Q and obtain an additional action of GalF (F now being thereflex field of the Shimura datum) on the cohomology complexes. As in Conjecture
9.3.2 the isomorphism in Conjecture 9.6.27 should then also be equivariant for the
GalF -action.

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183

9.6.29. Companion points and Breuil’s socle conjecture. We return to the setup
of Section 9.6.2 and assume in addition that G is a definite unitary group (of course
similar results are expected in a more general context). In [Bre15, Conj. 5.3,
Conj. 6.1] Breuil has made a precise conjecture about the socles (i.e. the largest
semisimple subrepresentations) of the (admissible) locally analytic representations
Ŝ(G, K p )an [mρ ] for Galois representations ρ (with coefficients in L) that are crystalline at places dividing p and are such that the eigenvalues ϕ1,v , . . . , ϕn,v ∈ L of
the crystalline Frobenius on the Weil–Deligne representation WD(ρ|GalFv ) satisfy
ϕi /ϕj ∈/ {1, qv } for i 6= j. Breuil’s conjecture describes which (tensor products indexed by the places dividing p of) irreducible subquotients of locally analytic prinG(F )cipal series representations (IndBv v δv )an embed into the space Ŝ(G, K p )an [mρ ].
A weaker
Q version of this conjecture is the precise description of the characters(δv )v ∈ v T̂v such thatYT̂v .
(ρ, (δv )v ) ∈ E(G, K p ) ⊂ Xps
G,F,S ×v

We give a precise version of the weaker conjecture and refer to [Bre15] for the
precise formulation of Breuil’s socle conjecture. Using Remark 9.6.22 one can easily
show that Breuil’s conjecture implies the weaker version presented here.
We continue to assume that ρv = ρ|GalFv is crystalline with pairwise distinct
Frobenius eigenvalues on WD(ρv ) that satisfy the above regularity assumption. Assume that ρv has regular Hodge–Tate weights λv and write ξv for the corresponding
highest weight. To each Frobenius stable flag Fv ⊂ WD(ρv ) (defined over Qp ) we
may associate- an unramified characterδFv = unrϕ1,v ⊗ · · · ⊗ unrϕn,v : Tv → L×
where ϕ1,v , . . . , ϕn,v is the ordering of the Frobenius eigenvalues given by
Fv .
- for each embedding τ : Fv ,→ Q̄p an element wτ ∈ Sn giving the relative
position of the (τ -part of the) Hodge-Filtration on DdR
Q(ρv ) ⊗F,τ Q̄p withrespect to Fv . We write w(Fv ) for the tuple (wτ )τ ∈ τ :Fv ,→Qp Sn .
Conjecture 9.6.30. Let ρ be a representation as above and let (δv )v|p ∈ T̂ (L).
Then (ρ, (δv )v ) ∈ E(G, K p ) if and only if each δv is of the form
δv = z wv w0 ·ξv δsm
for some choice of a collection of Frobenius stable flags (Fv )v|p and Weyl groupelements w(Fv )  wv = (wτ )τ :Fv ,→L . Here w0 denotes the longest Weyl groupelement and  denotes the usual Bruhat order.
We point out that this conjecture would be a direct consequence of Conjecture
9.6.16: we only need to compute the support of the coherent sheaf occurring on theright hand side of Conjecture 9.6.16. However, this support is basically computed
using Theorem 5.3.18.
Remark 9.6.31. The points
xwv = (ρ, (δFv z wv w0 ·ξv )v )for wv as in the conjecture are often referred to as companion points of the pointxw0 = (ρ, (δFv z ·ξv )v ). Note that the latter point conjecturally corresponds to (the184

M. EMERTON, T. GEE, AND E. HELLMANN

Hecke eigensystem of) a classical automorphic form f . The p-adic automorphic
forms in the Hecke eigensystem defined by xwv are then often referred to as companion forms of f .
Under additional assumptions, so-called Taylor–Wiles assumptions related tothe patching construction in 3, this conjecture is proven in [BHS19, Theorem 5.3.3,
Remark 5.3.2].
Theorem 9.6.32. Assume that
- p>2
- the CM extension E of F is unramified and does not contain a p-th rootof unity.
- the group G is quasi-split at finite places and Kv is hyperspecial if v is aplace of F inert in E.
- the residual Galois representation ρ̄ of ρ is adequate.
Then Conjecture 9.6.30 holds true.
Remark 9.6.33.
(i) In fact Conjecture 6.2.4 (iii) and Conjecture 9.6.8 imply the full conjecture in thelocally analytic socle and [BHS19, Theorem 5.3.3] proves the full socle conjecture
under the assumptions in the above theorem. The main input into this is the (proofof) the locally analytic Breuil–Mézard conjecture, Theorem 6.2.35.
(ii) There are generalizations of the above Theorem in the case of non-regular
Hodge–Tate weights by Zhixiang Wu [Wu21a] and to the case of eigenvarieties
associated to more general Bernstein components by Breuil–Ding [BD21].
9.6.34. Classical and non-classical p-adic automorphic forms. As in Section 9.6.29
we continue to work in the fixed
Q global setup of a definite unitary group G. Let
x = (ρ, (δv )v ) ∈ Xps
G,F,S ×v T̂v be a point. We assume that the charactersδv = z ξv δv,sm are locally algebraic with unramified smooth part δv,sm . Recall
from Remark 9.6.19 and Remark 9.6.22 that
Q we may view δ as a tuple ((κv )v , χ)consisting of a character κ = (κv )v : T 0 = v|p Tv0 → L× and a characterOχ=χδv,sm : A+ (G) → Lvof the Atkin–Lehner ring. Moreover, we have the identification of eigenspaces(9.6.35)+G
b v FBM † (G, K p , (ξv )v|p )[mρ ]A (G)=χ ∼(δv ), Ŝ(K p , L)an [mρ ]).
= HomG (⊗Assume that ξv is the algebraic character associated with a regular Hodge–Tateweight λv . Then the locally analytic representation IndIBvv,0 z ξv contains the algebraic representation Vλv of highest weight ξv . In particular we find that the
space M † (G, K p , (z ξv )v|p ) of overconvergent p-adic automorphic forms of weightξ = (z ξv )v|p contains the spaceNpf : G(F )\G(A∞F )/K →
v|p Vλv
Q
S(K p I, (κv )v ) =s. th. f (gu) = u−1 f (g) for all g ∈ G(A∞F ), u ∈v|p Ivof classical automorphic forms of weight ξ.

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185

Lemma 9.6.36. Under the identification (9.6.35) an overconvergent p-adic automorphic form+f ∈ M † (G, K p , (z ξv )v|p )[mρ ]A (G)=χ
is classical (i.e. lies in S(K p I, (z ξv )v )) if and only if the corresponding morphismGb v FB⊗(δv ) −→ Ŝ(K p , L)an [mρ ]factors through the locally algebraic quotient24
O
G(F )
G
b v FB⊗(δv ) −→
Vλv ⊗ (IndBv v δsm )sm .
v|pProof. The proof is similar to the proof of [BHS17a, Proposition 3.4].

The following conjecture is basically implied by standard Fontaine–Mazur typeconjectures.
Conjecture 9.6.37. Let ρ ∈ XG,F,S be a crystalline representation with regular
Hodge–Tate weights λ and let χ be a character of the Atkin–Lehner ring such thatthe eigenspace+M † (G, K p , (z ξv )v|p )[mρ ]A (G)=χ
is non-zero, i.e. ρ is associated to a p-adic automorphic form f . Then the eigenspace+S(K p I, (z ξv )v )[mρ ]A (G)=χ
is non-zero, i.e. ρ is associated to a classical automorphic form.
In view of our conjectures on functors from locally analytic representations
to coherent sheaves on stacks of Galois representations, one can approach thisconjecture by studying the morphism of coherent sheaves obtained by applying theconjectural functor Arig
G to the map
O
G(F )
G
b v FB⊗(δv ) −→
(Vλv ⊗ IndBv v δsm )sm .
v|pof locally analytic representations. In fact this strategy led to a proof of Conjecture 9.6.37 under the additional Taylor–Wiles assumptions as in Theorem 9.6.32,[BHS19, Theorem 5.1.3].
It is an interesting question whether the p-adic automorphic form f in Conjecture 9.6.37 is automatically classical. Note that this is not covered by the phenomenon of companion points or companion forms, as companion forms are p-adic
forms of non-classical weight (that hence can’t be classical!), but here we have fixed
the weight to be classical. Still our conjectures imply that one cannot expect f
to be classical. Conjecture 9.6.16 and Remark 9.6.22 give a precise conjecturaldescription of the eigenspaces++

S(K p I, (z ξv )v )[mρ ]A (G)=χ and M † (G, K p , (z ξv )v|p )[mρ ]A (G)=χ
in terms of a coherent sheaf that can be computed using the Drinfeld compactification XB . It turns out that this compactification is not Gorenstein in generaland these singularities have the consequence that the fiber dimension of the sheafM∗ (G, K p ) can jump. This basically implies that+M † (G, K p , (z ξv )v|p )[mρ ]A (G)=χ .
24See Remark 6.2.20 for the description of that quotient map.

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M. EMERTON, T. GEE, AND E. HELLMANN

should sometimes have larger dimension than its classical subspace. This expectation can be made into a concrete example in the case of a definite unitary group in
three variables (and will appear in [HHS]).
Appendix A. Background on category theory
We recall various results in (∞-)category theory that will be applied in themain body of the notes.
A.1. Stable ∞-categories. The functors we consider in the categorical local
Langlands program will necessarily be derived, in so far as they won’t necessarily
take objects from the abelian category of smooth representations of the p-adic
group G under consideration to an abelian category of coherent or quasicoherent
sheaves on the corresponding moduli stack of Langlands parameters; rather, certain
representations will be taken to complexes of sheaves. (This is already forced by thedesired compatibility with Kisin–Taylor–Wiles patching, and so is an inevitable partof the story.) For this reason, it is necessary to work with appropriate categories
of complexes of sheaves.
If we were writing these notes in the twentieth century, we would then proceed
to explain that this obliges us to work with various derived or triangulated categories, and we would go on to explain which ones. But we will take advantage ofcontemporary advances in homological algebra by working with stable ∞-categories,
which avoid the well-known technical drawbacks of triangulated categories, and areespecially useful to work with when one wants to apply various gluing or limiting
processes (both of which we will be applying, e.g. since we are working with sheaves
on formal algebraic stacks, which have both a topological structure — necessitating
gluing arguments — and an Ind-stack structure, necessitating limiting arguments).
For the basics of stable ∞-categories, and of t-structures on them, we refer
to [Lur17, Ch. 1]. We note, though, that we will use cohomological indexing,
while this reference uses homological indexing. We very briefly recall some of thesebasics, mainly for the purpose of orienting those readers (and those authors!) who
are more familiar with the traditional theory of derived and triangulated categories
than with the theory of stable ∞-categories.
If C is a stable ∞-category, then its underlying homotopy category has a canonical triangulated category structure. In practice one can often just imagine that
one is working in this underlying triangulated category, but with the improvementthat cones (and thus homotopy limits and colimits) are canonical.
One important example of a stable ∞-category is the derived ∞-category D(A)associated to a Grothendieck abelian category A, whose corresponding homotopy
category is the usual (unbounded) derived category of A. (Recall that A is a
Grothendieck abelian category if it is cocomplete, the formation of filtered colimits
in A is exact, and A admits a set of generators.) It is defined in [Lur17, Def. 1.3.5.8]
as the differential graded nerve of the full subcategory of the category of chain
complexes in A consisting of objects which are fibrant for a certain model structureon the category of chain complexes. (See [Lur17, Prop. 1.3.5.3] for the constructionof this model structure, and [Lur17, Prop. 1.3.5.6] for a partial description of the
fibrant objects.) We can also describe D(A) as the ∞-category localization of thecategory of chain complexes with respect to quasi-isomorphisms. (See [Lur17,
Prop. 1.3.5.15] and its proof.)CATEGORICAL p-ADIC LANGLANDS

187

If X and Y are objects of an ∞-category C, then we have a mapping anima25
MapsC (X, Y ). When C is furthermore stable, this mapping anima will in fact be aconnective spectrum. In our contexts our categories will be furthermore O-linear(for some ring of coefficients O), and so MapsC (X, Y ) will be a connective Omodule spectrum, or equivalently an animated O-module, which (by the Dold–Kancorrespondence) can be thought of as an object of D≤0 (O − Mod), the stable ∞category overlying the derived category of non-positively graded cochain complexesof O-modules.
Again using stability, one can furthermore define a (typically non-connective)
spectrum RHomC (X, Y ), from which the mapping space can be recovered via theformula
MapsC (X, Y ) = τ ≤0 RHomC (X, Y ).
Again, in the contexts we consider, our categories will be O-linear, and so thevarious RHomC (X, Y ) will be O-module spectra, which we can regard as objects ofD(O − Mod) (unbounded in both directions).
Placing a t-structure on C amounts (by definition) to placing a t-structure on
its underlying triangulated category. The full subcategories26 C ≤n , C ≥n (for n ∈ Z)
are then defined as pullbacks of the corresponding subcategories of the homotopy
category. The key facts are that:(1) MapsC (X, Y ) = 0 (i.e. is contractible) if X ∈ C ≤n and Y ∈ C ≥n+1 ;(2) The inclusion C ≤n ,→ C admits a right adjoint τ ≤n .
(3) The inclusion C ≥n ,→ C admits a left adjoint τ ≥n .
Definition A.1.1. If C is a stable ∞-category endowed with a t-structure,then we define the heart of C to be the intersection C ♥ := C ≥0 ∩ C ≤0 .
The heart C ♥ is an abelian category [Lur17, Rem. 1.2.1.12]. (More precisely, it
is equivalent to the nerve of the heart of the associated t-structure on the homotopy
category of C, and this latter heart is literally an abelian category. Recall that forming nerves is the technical device for converting usual categories into ∞-categories,
in the framework of [Lur09a].)
If A is a Grothendieck abelian category, then the stable ∞-category D(A) is
endowed with a canonical t-structure, arising from the usual t-structure on thederived category of A [Lur17, Prop. 1.3.5.21]. There is a fully faithful functor
A ,→ D(A) which induces an equivalence between A and the heart of the t-structureon D(A). (By definition the heart of D(A) coincides with the heart of its fullsubcategory D+ (A) discussed below, and then this claim follows from the dual
version of [Lur17, Prop. 1.3.2.19].) We sometimes refer to A as the subcategoryof D(A) consisting of static objects (sometimes known as “discrete” or “classical”
objects).
S
We may form the full subcategory D+ (A) = n∈Z D≥n (A). This category
may also be constructed as the differential graded nerve of the full subcategoryof bounded below chain complexes of injective objects of A. (This is because thebounded below fibrant complexes are precisely the bounded below complexes of
injectives; see [Lur17, Prop. 1.3.5.6]. This construction of D+ (A) is then dual
25In [Lur09b] and [Lur17], the terminology spaces is used for the ∞-category arising from
simplicial sets under the usual Quillen model structure, and for its objects, but we follow [CS21]
in using the terminology anima instead.
26Meaning sub-∞-categories, of course!

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to the construction of D− (A) for abelian categories with enough projectives given
in [Lur17, §1.3.2]. Note also that this construction of D+ (A) only requires that A
has enough injectives, and doesn’t require A to be Grothendieck.)
A.2. Associative algebra objects. The notion of E1 -rings (sometimes known
as E1 -ring spectra or A∞ -ring spectra) is defined in [Lur17, Defn. 7.1.0.1]. By[Lur17, Rem. 7.1.0.3], static E1 -rings are associative rings in the usual sense.
There is a theory of left and right modules over an E1 -ring A, and we write LModA ,RModA respectively for these stable ∞-categories. In the case that A is static, theseidentify with the derived ∞-categories of the usual abelian categories of modules,[Lur17, Rem. 7.1.1.16].
If x is an object of a stable ∞-category C, then by [Lur17, Rem. 7.1.2.2]
the spectrum RHomC (x, x) acquires the structure of an E1 -ring which we denote
EndC (x), so that in particular πn EndC (x) = Ext−n
C (x, x) for all n ∈ Z.
A.3. Cocontinuous functors.
Definition A.3.1. We say that a functor between ∞-categories is cocontinuous
if it preserves small colimits. (Note that in some parts of the literature, this is thedefinition of a continuous functor.)
The notes [Cam] provide a useful overview of categorical notions related tocompact objects. The following lemma is partly based on some of the results
discussed in these notes.
Lemma A.3.2. Let F : C → D be a functor between cocomplete and compactly
generated stable ∞-categories.
(1) F is cocontinuous if and only if it admits a right adjoint G.
(2) If F is cocontinuous, then F preserves compact objects if and only if its
adjoint G is cocontinuous.
(3) If F is cocontinuous, fully faithful, and preserves compact objects, thenF also reflects compact objects, i.e. F (X) compact implies X itself iscompact.
Proof. (1) This is the adjoint functor theorem [Lur09a, Cor. 5.5.2.9].
(2) Suppose firstly that F preserves compact objects. In order to show that G
is cocontinuous, it is enough to show that it preserves finite colimits and filteredcolimits. For the former, recall that a functor between stable ∞-categories preserves
finite colimits if and only if it preserves finite limits; and since G is a right adjoint,
it in fact preserves all limits.
∼
Thus for any filtered colimit colimj Yj −→ Y in D, we have to show that theinduced morphism colimj G(Yj ) → G(Y ) is an isomorphism. By Yoneda’s lemma,
its enough to do this after applying Mor(X, –) for any object X of C, and since C iscompactly generated, we can further assume that X is compact. We then find that ∼
Mor X, colimj G(Yj ) = colimj Mor X, G(Yj ) −→ colimj Mor F (X), Yj
 ∼
 ∼

∼
−→ Mor F (X), colimj Yj −→ Mor F (X), Y −→ Mor X, G(Y ) ,as required.
Conversely, if G is cocontinuous, it in particular preserves filtered colimits.
Accordingly, if X is compact, and colimj Yj is a filtered colimit in D, then we may

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write
 ∼
 ∼

Mor F (X), colimj Yj −→ Mor X, G(colimj Yj ) −→ Mor X, colimj G(Yj ) ∼= colimj Mor X, G(Yj ) −→ colimj Mor F (X), Yj ,so that F (X) is compact, as required.
(3) We must show that F (X) compact implies that X itself is compact. Hereare two arguments:∼(i) Write X −→ limi Xi , with Xi compact. Applying the functor F , we obtain−→∼
F (X) −→ limi F (Xi ). If F (X) is compact, then this latter isomorphism factors−→
through one of the F (Xi ). Since F is fully faithful, we find that our original
isomorphism similarly so factors, and so the identity automorphism of X factors
as X → Xi → X. Thus X is a retract of the compact object Xi , and so is itself
compact.
(ii) Since F is fully faithful, its right adjoint is a retract, i.e. the unit of adjunc∼tion is a natural isomorphism idC −→ G ◦ F ; in particular, any object of C is in theessential image of G. Thus it suffices to show that the natural map
MM
Mor X, G(Yj ) → Mor(X,G(Yj )jj

is an isomorphism, for any collection {Yj } of objects of D. (Here we use the factthat in a stable ∞-category, to test compactness of an object it suffices to test
Homs against coproducts.) By (2), the adjoint G preserves filtered colimits, and so
we find that
M
M
 ∼ M
 ∼

Mor X, G(Yj )) −→Mor F (X), Yj ) −→ Mor F (X),
Yj )
j

j

j
∼

M
M
 ∼

−→ Mor X, G(
Yj ) −→ Mor X,
G(Yj ) ,j

as required.

j

A.4. Pro-categories (and Ind-categories). We very briefly recall the definitions of Pro- and Ind-∞-categories; see [Lur09a, §5.3.5] for more details. (Moreprecisely, this reference treats the case of Ind-∞-categories, but for any ∞-category C,
there is an equivalence Ind(C) ∼= Pro(C op )op .)If C is a (small) ∞-category, one can define its associated pro-category Pro(C)to be the category whose objects are the diagrams F : I → C indexed by a cofilteredsmall ∞-category I, with the morphisms between two diagrams F : I → C, G : J →C being defined by the formula
MorPro(C) (F, G) = lim lim MorC (F (i), G(j)).
←− −→
J

I

(In fact, as explained in the introduction to [Lur09a, §5.3], this is not the mostconvenient way to set up the theory of Ind- and Pro- ∞-categories, but we ignore
this point.)
In the case that C is an ordinary category, we will sometimes denote the diagram
F : I → C via “ limI ”F (i), and refer to it as a pro-object of C. The point of this←−notation is to distinguish the pro-object from the limit limI F (i) in C itself, if this
←−

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limit happens to exist. Often, when employing this notation, we write Xi rather
than F (i), and so denote the object F of Pro(C) as “ limI ”Xi .
←−A.5. Complete t-structures. References for this material are [BZNP17,§6] and [Lur17, §§1.2, 1.3]. (Important caution: these references use homologicalindexing, rather than the cohomological indexing that we use!)
Definition A.5.1. If C is a stable ∞-category equipped with a t-structure,then it admits a left completion
Cb := lim C ≥n .
The limit is indexed by the totally ordered set Z, and the transition functors are
given by the truncations τ ≥n : C ≥n−1 → C ≥n .
b defined by mapping an object X to the
There is a canonical functor C → C,≥nb We saysequence (τ X), which has a natural interpretation as an object of C.
that C is left complete if this functor is an equivalence.
There are dual notions of right completion and right completeness. We don’t
introduce notation for the right completion, but to be explicit: C is right complete
if the canonical functor C → lim C ≤n defined by X 7→ (τ ≤n X) is an equivalence.
Related to these notions of completeness are corresponding notions of separatedness.
Definition A.5.2. If C is a stable ∞-category equipped with a t-structure,then we say it is left separated if\
C −∞ :=
C ≤nn∈Z

consists only of zero objects. Dually, we say it is right separated if\
C ∞ :=
C ≥nn∈Z

consists only of zero objects.
If A is a Grothendieck abelian category, the stable ∞-category D(A) is left andright separated, and also right complete [Lur17, Prop. 1.3.5.21]. It need not beleft complete in general [Nee11].
If A admits countable products, and if these are furthermore exact, then D(A)admits countable products (take products level wise on complexes), and it follows
from [Lur17, Prop. 1.2.1.19] that D(A) is left complete (since Condition (2) thereis satisfied). For example, if A is the abelian category of modules over a ring, thenD(A) is left complete as well as right complete. We will be interested in contexts
in which products in A, although they exist, may not be exact, and so we note thefollowing mild generalization of [Lur17, Prop. 1.2.1.19]. (The possibility of sucha generalization is noted in [BZNP17, Rmk. 6.1.5], where the hypothesis of rightt-exactness of products is required to hold only “up to a finite shift”.)
Proposition A.5.3. Let C be a stable ∞-category equipped with a t-structure.
Suppose that C admits countable products, and that the formation of countable products is of bounded amplitude (i.e.
Q there is some a ≥ 0 such that if {Xn } is any
sequence of objects of C ≤0 , then n Xn lies in C ≤a .) Then C is left complete if andonly if C is left separated.

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Proof. One follows the proof of [Lur17, Prop. 1.2.1.19], noting now that if· · · → Xn → Xn−1 → · · · is a tower of objects in C ≤0 , then lim Xn lies in C ≤a+1 .
Now the proof works just as written, except that rather than considering thefactorization lim f → f (n − 1) → f (n), we consider the factorization lim f →f (n − a − 1) → f (n).
Remark A.5.4. We elaborate slightly on the construction of [Nee11] alluded
to above: it gives examples of Grothendieck abelian categories A for which both the
hypothesis of Proposition A.5.3 (that countable products in D(A) have bounded
cohomological amplitude), and the conclusion (that D(A) is left complete) fail.
Example E.2.3 below gives an explicit example of this construction, which arises
naturally in the context of smooth representation theory of (non-analytic) pro-pgroups.
One of the advantages of the framework of stable ∞-categories, compared tothat of triangulated categories, is the following simple and very general result (whichis the dual version to [Lur17, Thm. 1.3.3.2]) on the existence of right derived
functors. Recall that we say that a functor F : C → D between stable ∞-categories
endowed with t-structures is left t-exact if F (C ≥0 ) ⊆ D≥0 , that it is right t-exact if
F (C ≤0 ) ⊆ D≤0 , and t-exact if it is both left and right t-exact.
Theorem A.5.5. If A is an abelian category with enough injectives, and if C is astable ∞-category equipped with a right complete t-structure, then F 7→ τ ≤0 F|A (therestriction being taken by identifying A with the heart of D+ (A)) is an equivalence
between the ∞-category of left t-exact functors D+ (A) → C which carry injective
objects of A into C ♥ , and the ordinary category of left exact functors from A to C ♥ .
As a particular consequence we have the following result [Lur17, Prop. 1.3.3.7](which notoriously does not hold in the same level of generality in the triangulated
category context, because of the non-functoriality of the formation of cones in that
context).
Proposition A.5.6. Suppose that C is a stable ∞-category equipped with a rightcomplete t-structure, and that its heart C ♥ has enough injectives. Then the inclusionC ♥ ,→ C extends essentially uniquely 27 to a t-exact functor F : D+ (C ♥ ) → C.
Furthermore, the following properties are equivalent:(1) This functor is fully faithful.
(2) For any objects X and Y of C ♥ the natural map ExtiC ♥ (X, Y ) → ExtiC (X, Y )is an isomorphism for i > 0.
(3) For any objects X and Y of C ♥ with Y injective, ExtiC (X, Y ) = 0 for
i > 0.
(4) For any objects X and Y of C ♥ with Y injective, there is an epimorphism
Z → X such that ExtiC (Z, Y ) = 0 for i > 0.
Finally, if these equivalent properties hold, then the essential image of F is equal
to C + := ∪n C ≥n .
27Meaning that the sub-∞-category of the ∞-category of functors Fun D + (C ♥ ), C  whose

objects consist of such extensions, whose 1-morphisms consist of natural transformations that
restrict to the identity on the restriction of F to C ♥ , and whose higher morphisms consist ofarbitrary higher morphisms between these 1-morphisms, is contractible. This is the ∞-categoricalanalogue of being unique up to a unique natural isomorphism that restricts to the identity on the
restriction of F to C ♥ .

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Proof. Except for the insertion of item (2) (which we add just for expositional clarity) this is simply the dual version of [Lur17, Prop. 1.3.3.7]. To see
the equivalence of (2) with the other conditions, note that (1) certainly implies (2)(even for i = 0; however this case is in any event automatic, since C ♥ is full in C byconstruction), while (2) clearly implies (3).
Below we will use this proposition both as stated, and in the dual form for
projective objects and categories bounded above (which is the case literally treatedin [Lur17, Prop. 1.3.3.7]).
A.6. Compact and coherent objects and regular t-structures. A reference for this material is [BZNP17, §6]. (Reminder: this reference uses homologicalindexing, rather than the cohomological indexing that we use.)
Definition A.6.1. Let C be an ∞-category admitting filtered colimits. We
say that an object x ∈ C is compact if MapsC (x, –) preserves filtered colimits(see [Lur09a, Defn. 5.3.4.5]).
Remark A.6.2. If C is a stable ∞-category admitting filtered colimits, then it
is cocomplete. Then x ∈ C is compact if and only if RHomC (x, –) preserves filteredcolimits, if and only if RHomC (x, –) preserves all (small) colimits. (This is because
RHomC (x, –) automatically preserves all limits and is therefore exact.)
For stable ∞-categories equipped with a t-structure, we have the related notionof coherent objects. (See [BZNP17, §6.2].) The following is [BZNP17, Def. 6.2.2].
Definition A.6.3. Let C be a stable ∞-category that admits filtered colimits,
and is equipped with a t-structure that is compatible with filtered colimits (which is
to say that C ≥0 , or equivalently every C ≥n , is closed under the formation of filteredcolimits in C). We say that an object x ∈ C is coherent if x is bounded below, i.e.
x lies in C ≥n for some n, and if x is furthermore compact in C ≥m for all m ≤ n.
Definition A.6.4. In the context of Definition A.6.3, we write Coh(C) for
the full sub-∞-category of coherent objects of C. This is a stable sub-∞-category
by [BZNP17, Rem. 6.2.3.].
Remark A.6.5. In the context of Definition A.6.3, if m ≥ n, then maps fromx to objects of C ≥m factor through τ ≥m x, and so if x is compact in C ≥n then we
see that τ ≥m x is compact as an object of C ≥m for m ≥ n.
Remark A.6.6. If C is as in Definition A.6.3 and is furthermore right complete,and if x ∈ C is coherent, then x is bounded above as well as below. (Apply thecompactness assumption to the identity map x → x = colim τ ≤n x.)
The following result combines parts of [BZNP17, Lems. 6.2.4, 6.2.5].
Lemma A.6.7. Let C be a stable ∞-category endowed with a right complete
t-structure that is also compatible with filtered colimits. Then the following are
equivalent:(1) The inclusion C ≥0 ,→ C ≥−1 preserves compact objects.
(2) The full subcategory of compact objects of C ♥ (the heart of the t-structureon C) is an abelian subcategory of C ♥ .
If these conditions hold, then the coherent objects in C are precisely those objects
x which are bounded both above and below and such that each H n (x) is a compact
object of C ♥ .

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193

Definition A.6.8. A Grothendieck abelian category is locally coherent if it
is compactly generated, and if its full subcategory of compact objects is abelian.
(See for example [Her97, Thm. 1.6] for some equivalent characterisations of thisnotion.)The following definition is [BZNP17, Def. 6.2.7].
Definition A.6.9. We say that a t-structure on a stable ∞-category admitting
filtered colimits is coherent if it is right complete and is compatible with filteredcolimits, and if furthermore C ♥ is locally coherent.
Since the compact objects of a locally coherent abelian category form an abeliansubcategory (by definition), a coherent t-structure in particular satisfies the equivalent conditions of Lemma A.6.7.
We now define what it means for a coherent t-structure to be regular (following [BZNP17, §6.3]). We first note that if C is equipped with a coherent t-structure,then since by assumption C admits filtered colimits, there is a canonical functor(A.6.10)

Ind Coh(C) → C.

This functor is cocontinuous (essentially by construction).
Definition A.6.11. We say that a coherent t-structure on C is regular if C iscompactly generated by its coherent objects (i.e. its coherent objects are compact,
and generate), or if (equivalently) the functor (A.6.10) is an equivalence.
Remark A.6.12. By [BZNP17, Prop. 6.3.2], if C has a regular coherent tstructure, then the compact objects of C are exactly the coherent objects.
If C is equipped with a coherent t-structure, then the pullback of this t-structure
along (A.6.10) endows Ind Coh(C) with a t-structure, which is again coherent andis in fact regular [BZNP17, Prop. 6.3.2]. The functor (A.6.10) is t-exact by construction. For any n, it restricts to an equivalence∼

Ind Coh(C)≥n −→ C ≥n .
If C is equipped with a coherent t-structure, then the t-structure on its left
completion Cb is again coherent [BZNP17, Prop. 6.3.4]. Again, the canonical functor
C → Cbinduces equivalences∼
C ≥n −→ Cb≥n .
Thus the formation of Ind Coh(C) and of Cb are two different ways of “filling out”
C + = ∪n C ≥n in the leftward direction.

Remark A.6.13. If the t-structure on C is left separated (e.g. because it
is left complete), then the kernel of the functor (A.6.10) consists precisely of
Ind Coh(C)−∞ . In examples when (A.6.10) is not an equivalence, i.e. when thet-structure on C is not regular, it is often the case that Ind Coh(C)−∞ is non-zero,i.e. that the t-structure on Ind Coh(C) is not left separated.
For later reference, we note the following general fact about t-structures andInd-categories [AGH19, Prop. 2.13].

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Proposition A.6.14. If C is a stable ∞-category endowed with a t-structure,then Ind C inherits a t-structure, characterized by the requirements that C ,→ Ind C∼
be t-exact, and that the inclusion C ≥0 → C induce an equivalence Ind(C ≥0 ) −→≥0
Ind(C) . If the t-structure on C is furthermore right bounded, then the t-structure
on Ind C is right complete.
A.7. Derived functors and adjoints. We recall the following result.
Proposition A.7.1. Suppose that F : A → B is an exact functor between
Grothendieck abelian categories which is compatible with colimits, and let G : B → A
denote the right adjoint to F (which exists by the adjoint functor theorem). Then
(1) F induces a t-exact cocontinuous functor D(A) → D(B), which we again
denote by F .
(2) The right derived functor RG : D(B) → D(A) exists, and is right adjoint
to F .
Example A.7.2. A typical context in which one might apply Proposition A.7.1
is when B is a Grothendieck category, and A is a localizing subcategory, i.e. a Serresubcategory which is furthermore closed under the formation of arbitrary directsums (and hence arbitrary colimits) in B. Then A is also a Grothendieck abeliancategory, and we may take F : A → B to be the inclusion.
Although F is fully faithful by construction, its extension to a functor D(A) →D(B) need not be fully faithful. (Example E.2.3 below gives an illustration of this.)
However, we have the following positive result in this direction,
Proposition A.7.3. As in the preceding discussion, let F : A ,→ B be theinclusion of a localizing subcategory into a Grothendieck abelian category, and consider its t-exact extensions D+ (A) → D+ (B) and D(A) → D(B).
(1) If F preserves injectives, then the extension D+ (A) → D+ (B) is again+
fully faithful, and its essential image consists of the full subcategory DA(B)consisting of objects whose cohomologies lie in A.
(2) Suppose that the extension D+ (A) → D+ (B) is fully faithful. Suppose,
furthermore, that the formation of products in B is exact, and that thederived right adjoint RG of Proposition A.7.1 has finite cohomologicaldimension. Then the extension D(A) → D(B) is again fully faithful, withessential image equal to DA (B) (the subscript A again denoting the fullsubcategory consisting of objects whose cohomologies lie in A).
+
Proof. The t-structure on D(B) induces a t-structure on DA(B), whose heart
is precisely A. If we assume that F preserves injectives, then Proposition A.5.6
gives rise to a t-exact equivalence∼+
D+ A) −→ DA(B)which extends the identity functor on hearts. The essential uniqueness statementof that proposition shows that this equivalence must essentially coincide with thecanonical extension of F . This proves (1).
Suppose now that the extension of F to D+ (A) is fully faithful; as above, we
continue to denote this extension, as well as the extension to D(A), by F . For
any object X of D(A), the unit of adjunction gives rise to a natural morphism
X → RG F (X) . Thus if n is any integer, applying this to the exact triangle

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195

τ <n X → X → τ ≥n X (and recalling that F is t-exact), we obtain a morphism ofexact triangles
τ <n X/X/ τ ≥n X
RG τ <n F (X)/ RG F (X)/ RG τ ≥n F (X)Since the functor F is assumed to be fully faithful on D+ (A), we see that the righthand vertical arrow is an isomorphism. Of course τ <n X has vanishing cohomology
in degrees ≥ n,
 and if a ≥ 0 denote the cohomological amplitude of RG, then
RG τ <n F (X) has vanishing cohomology in degrees ≥ n + a. Thus if i ≥ n + a,
we obtain a commutative square/ H i (τ ≥n X)∼

H i (X)∼
H i RG F (X)



∼



/ H i RG τ ≥n F (X)in which (as indicated) all but the left hand vertical arrow are known to be isomorphisms. Thus this arrow is an isomorphism as well.

Since n was arbitrary, we find that the unit of adjunction X → RG F (X)induces an isomorphism on cohomology in all degrees, and hence is an isomorphismin D(A). Thus RG is a left quasi-inverse to F , and so the latter is fully faithful, as
claimed.
Thus D(A) is equivalent to a full subcategory of D(B), which is evidently
contained in DA (B). At this point there may well be a direct argument to showthat DA (B) is actually equal to the essential image of F , but we finish the argument
in a slightly roundabout way.
First, we note that since the formation of products in B is exact, the category
D(B) admits products, and their formation is t-exact. Proposition A.5.3 shows that
D(B) is furthermore left complete. One then sees that DAQ(B) is also left complete.
∞≤0Now, if {Xn }∞isasequenceofobjects
of
D(A)
,
then
n=0
n=0 F (Xn ) is an object

Q∞
≤0
≤a
of D(B) , and so RG n=0 F (Xn ) is an object of D(A) (where, as above, adenotes the cohomological amplitude of RG). Furthermore, one checks
Q∞ using theadjointness of F and RG and the full faithfulness of F that RG n=0 F (Xn )is the product of the Xn in D(A). Thus countable products exist in D(A), andtheir formation has bounded cohomological amplitude. Applying Proposition A.5.3
again, we find that that D(A) is left complete as well.
Thus we may consider the square
\
D(A)
O

b
F

∼

D(A)

\
/D
A (B)
O
∼

F

/ Dsm (B)in which the hats denote left completions, the vertical arrows, which are the canonical functors from the indicated categories to their left completions, are (as we havejust noted) equivalences, the bottom horizontal arrow is the functor F , and the top

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M. EMERTON, T. GEE, AND E. HELLMANN

horizontal arrow is the functor Fb induced by F on left completions. We claim that
this diagram commutes (up to natural isomorphism). Since Fb is an equivalence (asF restricts to an equivalence D(A)≥n → Dsm (B) for each n), we conclude that thebottom horizontal arrow is an equivalence, as required.
To see the claimed commutativity, note that if X is an object of D(A), mapping\ then we have a natural isomorphism
to the element lim τ ≥n X in D(A),∼Fb(lim τ ≥n ) := lim F (τ ≥n X) −→ lim τ ≥n F (X),using the t-exactness of F , and lim τ ≥n F (X) is precisely the image of the object\
F (X) of DA (B) in D
A (B).
Remark A.7.4. In the context of Proposition A.7.3 (2), the proof shows thatboth D(A) and D(B) are left complete, and that one may compute the extensionof F from D+ (A) to D(A) via the formula F (X) := lim F (τ≥n X).
A.8. Semiorthogonal decompositions. In this section we outline the basic concepts related to semiorthogonal decompositions of stable ∞-categories. The
stable ∞-categories under consideration will always be cocomplete. Furthermore,when we speak of a cocomplete subcategory, we will always mean this in thestrongest sense, i.e. that the subcategory in question is closed under the formation of colimits in the ambient category.
We begin by proving some generalities about generators in cocomplete stable∞-categories.
Definition A.8.1. Let C be a stable ∞-category, and X a set of objects of C.
Then we define the left and right orthogonals⊥

X := {y ∈ C| RHom(y, x) = 0 for all x ∈ X},

X ⊥ := {y ∈ C| RHom(x, y) = 0 for all x ∈ X}.
Lemma A.8.2. If C is a cocomplete stable ∞-category, and X is a set of objectsof C, then ⊥ X is a cocomplete stable subcategory of C.
Proof. The cocompleteness of ⊥ X is immediate from
RHom(colimi yi , x) = lim RHom(yi , x),iso we only need to check stability. By [Lur17, Lem. 1.1.3.3], it is enough tocheck that C is stable under the formation of cofibers and translations. The claim
for cofibers is immediate from the definitions, and that for translations from theequality
RHom(y[1], x) = Ω RHom(y, x)
which shows that y ∈ ⊥ X if and only if y[1] ∈ ⊥ X.

Lemma A.8.3. If C is a cocomplete stable ∞-category, and X is a set of compact
objects of C, then X ⊥ is a cocomplete stable subcategory of C.
Proof. Each x ∈ X is compact, so by Remark A.6.2 we have
RHom(x, colimi yi ) = colimi RHom(x, yi ),⊥showing that X is cocomplete. It is immediate from the definitions that X ⊥is stable under the formation of fibers, and the result follows as in the proof
of Lemma A.8.2 (using a variant of [Lur17, Lem. 1.1.3.3] for fibers rather thancofibers, which follows from loc. cit. by passage to the opposite category).
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We will use Lemma A.8.2 to define the subcategory hXi of C generated by X.
Before doing so, we find it useful to consider the following more general situation.
Let F : C → D be a fully faithful and cocontinuous functor between cocomplete
stable ∞-categories. By Lemma A.3.2 (1), the functor F admits a right adjoint
G : D → C. Let A ⊆ D denote the kernel of G, and let B ⊆ D denote the essential
image of F . These are both thick full subcategories of D (obviously in the caseof A, and as a consequence of Lemma A.8.4 (6) in the case of B).
Lemma A.8.4. In the preceding situation we have the following statements.
∼(1) F induces an equivalence C −→ B.
∼(2) G induces an equivalence D/A −→ C.
(3) The composite B ⊆ D → D/A is an equivalence.
(4) B is closed under the formation of colimits in D.
(5) A = B ⊥ .
(6) B = ⊥ A.
Proof. (1) follows from the very definition of B as the essential image of F ,together with the assumption that F is fully faithful. To prove (2), note first thatby the definition of A as the kernel of G, the functor G induces a functor D/A → C.
∼
Since the unit of adjunction idC −→ GF is an equivalence (by full faithfulness of F ),∼and the counit is an equivalence modulo A (i.e. gives an equivalence GF G −→ G)
we deduce that this induced functor is indeed an equivalence. (3) is a reformulation,∼using (1) and (2), of the fact that GF −→ idC . (4) follows from the fact
 that F is
To prove (5), note that
Y
∈
A
iff
G(Y
)
=
0
iff
RHom
X,
G(Y
)
= 0 for allX ∈ C iff RHom F (X), Y = 0 for all X ∈ C (by the adjunction between F and G)
iff Y ∈ B ⊥ . To prove (6), note first that (5) gives the inclusion B ⊆ ⊥ A. Now let
Y ∈ D, and let Z denote the cofiber of the counit of adjunction F G(Y ) → Y . Since∼
GF −→ idC , the morphism GF G(Y ) → G(Y ) is an equivalence, so that Z ∈ A. In
particular, if Y ∈ ⊥ A, then as F G(Y ) ∈ B ⊆ ⊥ A, we must also have Z ∈ ⊥ A; andsince Z ∈ A, we see that idZ = 0, so Z = 0, as required.
Definition A.8.5. If C is a cocomplete stable ∞-category, and X is a set ofobjects of C, then we define the subcategory generated by X to be hXi := ⊥ (X ⊥ ).
If hXi = C then we say that X is a set of generators of C.
This may look like a rather strange definition, but we note that by Lemma A.8.2,
hXi is a cocomplete stable subcategory of C. The following lemma shows that it is
the smallest cocomplete stable subcategory containing X, justifying the definition.
Lemma A.8.6. Suppose that C is a cocomplete stable ∞-category, that X is aset of objects of C, and that E is a cocomplete stable subcategory of C which contains
every object in X. Then E contains hXi.
Proof. Applying Lemma A.8.4 with F given by the inclusion E ⊆ C, we seethat E = ⊥ (E ⊥ ). Since hXi = ⊥ (X ⊥ ) (by definition), we need to show that ⊥ (X ⊥ )is contained in ⊥ (E ⊥ ). By definition, it suffices to prove that X ⊥ contains E ⊥ ,which is immediate from our assumption that E contains X.
The following corollary is immediate.
Corollary A.8.7. Let D be a cocomplete stable ∞-category, and let X be a
set of objects of D. Then the following are equivalent.

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M. EMERTON, T. GEE, AND E. HELLMANN(1) X is a set of generators of D.
(2) The only cocomplete stable subcategory of D containing every element of X
is D itself.
(3) If y is an object of D satisfying RHom(x, y) = 0 for every x ∈ X, theny = 0.
We now recall what we mean by a semiorthogonal decomposition (which may
differ slightly from other definitions in the literature, in that we allow the decomposition to be infinite). Let C be a cocomplete stable ∞- category, and let A1 , A2 , . . .
be cocomplete stable subcategories (either finite or infinite in number).
Definition A.8.8. We say that A1 , A2 , . . . is a semiorthogonal decompositionof C if(1) The subcategories A1 , A2 , . . . generate C, and(2) if x ∈ Ai , y ∈ Aj with i < j, then RHom(x, y) = 0.
Remark A.8.9. By definition, condition (2) in Definition A.8.8 is equivalent toasking that if i < j then Ai ⊆ ⊥ Aj , and is also equivalent to asking that Aj ⊆ A⊥i .
We can define semiorthogonal decompositions in terms of generators, as in thefollowing definition and lemma.
Definition A.8.10. Let C be a cocomplete stable ∞-category. We say that a(possibly infinite) sequence x1 , x2 , . . . of objects of C is weakly exceptional if:(1) The xi are all compact objects of C.
(2) The xi generate C.
(3) RHom(xi , xj ) = 0 for i < j.
Lemma A.8.11. Let C be a cocomplete stable ∞-category, let x1 , x2 , . . . be aweakly exceptional sequence of objects of C, and let Ai := hxi i be the subcategory
generated by xi . Then A1 , A2 , . . . is a semiorthogonal decomposition of C.
Proof. By assumption the sequence x1 , x2 , . . . generates C. Since xi is compact, we see from Lemma A.8.3 that x⊥
i is cocomplete and stable, and since it
contains xj by assumption, we see that it must contain hxj i = Aj .
Now let y ∈ Aj be arbitrary. We have just shown that RHom(xi , y) = 0, sothat xi ∈ ⊥ y, which is cocomplete and stable by Lemma A.8.2. Thus ⊥ y contains
Ai = hxi i, and since y is arbitrary, we have shown that RHom(x, y) = 0 for any
x ∈ Ai , y ∈ Aj , as required.

A.8.12. Generators and adjunctions. We now return to the setting of Lemma A.8.4.
∼∼
Let F 0 : C −→ B denote the equivalence of part (1), and let G0 : B −→ C denote
its quasi-inverse. Let ι : B ⊆ D denote the inclusion, and π : D → D/A denote∼∼the projection. Then F −→ ιF 0 , while G −→ G0 (πι)−1 π (noting that πι is anequivalence, by (3)). Now, by (5) and (6), either of the subcategories A and B
determines the other, and the functors ι and π are defined canonically in termsof A and B. Thus the original data of the functor F is determined by giving eitherof the subcategories A or B, together with the equivalence F 0 .
Suppose that {Xi }i∈I is a collection of generators of C. Then we see thatB ⊥ = ∩i∈I F (Xi )⊥ = {Y ∈ D | RHom F (Xi ), Y = 0 for all i ∈ I}.
By Lemma A.8.4 (5), we then see that A = ∩i∈I F (Xi )⊥ .

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Thus, suppose that we know the values of the functor F on the generatingcollection {Xi }i∈I , say Yi := F (Xi ). Then we can characterize B as the stable
full sub-∞-category generated by the Yi . Given B, we can then determine A,
and although from this data we can’t determine F itself, we can determine the∼composite F G. Namely, we have F G −→ ι(πι)−1 π.
Another way to describe this is as follows: any Y in D fits into a cofiber sequence(equivalently, fiber sequence, or exact triangle) — the semiorthogonal decomposition
of Y — which is unique up to equivalence, of the form
X→Y →Z
with X ∈ B and Z ∈ A. Namely, X = F G(Y ), and Z is the cofiber of the counitof the adjunction F G(Y ) → Y . We then recover F G(Y ) as the term X in this
distinguished triangle.
We can apply this formalism in applications where the functor F is conjectural.
More precisely, we can suppose given a collection of objects {Yi }i∈I of D (which we
imagine arise as the image of a generating set of objects Xi ∈ C under a functor F ;but we won’t actually be given F ). Then we can define B to be the subcategory
generated by the Yi , just as in the preceding paragraph. By construction the inclusion B ⊂ D is fully faithful and preserves colimits, and so by Lemma A.3.2 (1)admits a right adjoint. Lemma A.8.4 then applies: we may form A := B ⊥ , every
object Y of D admits a functorial semiorthogonal decomposition given by a cofiber
sequence
X→Y →Z
with X ∈ B and Z ∈ A, and the functor Y 7→ X is the right adjoint to the inclusionof B.
A.8.13. Restriction to open substacks. Suppose further that we have a cocomplete stable ∞-category D0 with a functor j ∗ : D → D0 , together with a fully faithfulright adjoint j∗ : D0 → D. (In applications, D will be a category of sheaves on astack X , D0 will be the category of sheaves on an open substack j : U → X , andj ∗ , j∗ will have their usual meanings.) Let B 0 , A0 be sub-stable ∞-categories of D0
which give a semiorthogonal decomposition as above: so A0 , B 0 together generate
D0 , and we have B 0 = ⊥ A0 .
Set A := j∗ A0 . Then (since the full faithfulness of j∗ implies that j ∗ j∗ isnaturally equivalent to the identity) we have A0 = j ∗ A, and by adjointness we have
(A.8.14)⊥A = (j ∗ )−1 (⊥ A0 ) = (j ∗ )−1 (B 0 ).

Appendix B. Coherent, Ind-Coherent, and Pro-Coherent sheaves on
Ind-algebraic stacks
B.1. Modules over I-adically complete rings. We will establish someresults relating various stable ∞-categories of modules. These provide the commutative algebra background for the theory of (Ind-)coherent sheaves on formal
algebraic stacks. In fact our results apply in certain non-commutative contexts aswell, and in this case we will use them below to deduce certain facts about stable∞-categories of smooth representations of p-adic analytic groups.
The setup is as follows: we let A be a (not necessarily commutative) Noetherianring, and I a two-sided ideal in A with respect to which A is I-adically complete.
We furthermore assume that I satisfies the Artin–Rees property: namely, that any
A-submodule M of A⊕n (for any n ≥ 0) is closed with respect to the I-adic topology

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M. EMERTON, T. GEE, AND E. HELLMANN

on A⊕n , and the topology on M induced by the I-adic topology on A⊕n coincides
with the I-adic topology on M . It follows from this hypothesis that any finitely
generated A-module is complete and separated with respect to its I-adic topology,
and that morphisms between finitely generated A-modules have I-adically closed
closed image and are strict with respect to the I-adic topologies on their source
and target.
If A is commutative, then the Artin–Rees property holds automatically. If A
is non-commutative, then it need not hold (as far as we know), but it does hold in
one particular case that we are interested in.
Example B.1.1. If H is a compact p-adic analytic group, if P is a normal
open pro-p subgroup of H (the identity has a neighbourhood basis consisting ofsuch P ), and if I is the kernel of the projection O[[H]] → k[H/P ], then the I-adic
topology on O[[H]], and more generally on any finitely generated O[[H]]-module,
coincides with the canonical topology (as discussed e.g. in Section D.2 below). Thus
I satisfies the Artin–Rees property.
Definition B.1.2. We write D(A) := D(A − Mod), and similarly for any n wewrite D(A/I n ) := D(A/I n − Mod).
Since A is Noetherian, the t-structure on D(A) is coherent. Indeed the compact
objects in its heart, which is the abelian category A − Mod, are precisely thefinitely presented A-modules, and they form a generating abelian subcategory of
A − Mod. The full subcategory of D(A) consisting of coherent objects is thusb
precisely the category Df.p.
(A) consisting of (cohomologically) bounded complexes
whose cohomologies are finitely presented (or, equivalently, finitely generated, since
A is Noetherian). Similar remarks apply with A/I n in place of A.
There are obvious exact inclusions(B.1.3)

A/I m − Mod ,→ A/I n − Mod ,→ A − Mod,for m ≤ n, which induce t-exact functors
b
m
b
n
b
Df.p. (A/I ) → Df.p. (A/I ) → Df.p. (A).

Since these functors are compatible in an evident way as m and n vary, we obtaina functor(B.1.4)bbcolimn Df.p.
(A/I n ) → Df.p.
(A)(the colimit being taken in the ∞-category of stable ∞-categories). We will show
below that (B.1.4) is fully faithful, and describe its essential image. To begin with,
we consider an analogous construction on the level of abelian categories.
Definition B.1.5. We denote by (A − Mod)I the full abelian subcategory(indeed Serre subcategory) of A − Mod consisting of modules each element of which
is annihilated by some powerSof I. Equivalently, (A − Mod)I is the closure underfiltered colimits of the union n A/I n − Mod, the union taking place in A − Mod.
Remark B.1.6. Since (A − Mod)I is a Serre subcategory of the Grothendieckabelian category A − Mod which is furthermore closed under the formation ofarbitrary direct sums, it is itself a Grothendieck abelian category. In particular
D (A − Mod)I is defined, and is equipped with its canonical t-structure, having(A − Mod)I as its heart.

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The compact objects in the category (A − Mod)I are precisely the finitely presented modules (i.e. those that are compact simply as A-modules), and these forma generating
abelian subcategory of (A − Mod)I . Thus the t-structure on D (A −bMod)I is coherent. Its full subcategory of coherent objects is equal to Df.p.
(A −Mod)I ), the subcategory consisting of complexes whose cohomologies are finitely
presented.
There is one more collection of categories we need to introduce.
Definition B.1.7. We let let DI (A) denote the full subcategory of D(A) conb
sisting of objects whose cohomologies lie in (A − Mod)I , and set Df.p.,I(A) =b
Df.p. (A) ∩ DI (A).
The heart of the t-structure on DI (A) is precisely (A − Mod)I , which is locallybcoherent, and hence Df.p.,I(A) is precisely the full subcategory of coherent objectsof DI (A).
The inclusions A/I n −Mod ,→ A−Mod of (B.1.3) evidently factor through (A−Mod)I , and so (B.1.4) factors as(B.1.8)bbbbcolimn Df.p.
(A/I n ) → Df.p.
(A − Mod)I ) → Df.p.,I(A) ,→ Df.p.
(A).

We will show that the first two functors in this factorization are equivalences.
Proposition B.1.9. The functor(B.1.10)bb
colimn Df.p.
(A/I n ) → Df.p.
(A − Mod)I )(the first arrow of (B.1.8)) is an equivalence.
Proof. We will prove this by constructing a quasi-inverse to (B.1.10). But
we begin by noting that (B.1.10) is essentially surjective, since we will need to use
this in our construction of the quasi-inverse. This essential surjectivity amounts tothe fact that if X • is a bounded complex of objects in (A − Mod)I , each of whose
cohomologies is finitely generated, then we can find a quasi-isomorphism Y • → X •of bounded complexes with each Y • being a finitely presented object of (A − Mod)I(so that Y • is in fact a complex of A/I n -modules for some n). (This is standard,using that since A is Noetherian, all submodules of finitely generated A-modules are
finitely presented, but for the convenience of the reader we sketch a proof. Assume
that X m is finitely presented for m > n, and let (X n )0 be a finitely generated
A-submodule of X n which surjects onto both H n (X • ) and im(dn : X n → X n+1 ).
Let Y • be the subcomplex of X • with Y r = X r for r 6= n − 1, n, Y n = (X n )0 ,Y n−1 = (dn−1 )−1 (Y n ). Then the inclusion Y • → X • is a quasi-isomorphism, andY m is finitely presented for all m ≥ n, and we may proceed by downward induction
on n.)
We now turn to constructing the required quasi-inverse. Proposition A.6.14
bendows Ind colimn Df.p.
(A/I n ) with a right complete t-structure, whose heart is
precisely (A − Mod)I , so that Proposition A.5.6 gives rise to a t-exact functorb(B.1.11)
D+ (A − Mod)I → Ind colimn Df.p.
(A/I n ).
We claim that (B.1.11) restricts to a functor
bbn(B.1.12)
Df.p. (A − Mod)I → colimn Df.p. (A/I ),which is furthermore quasi-inverse to (B.1.10).

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M. EMERTON, T. GEE, AND E. HELLMANN

The composite of (B.1.11) with the functor D+ (A/I m ) → D+ (A − Mod)I )(for some fixed m) is a t-exact functor(B.1.13)b
D+ (A/I m ) → Ind colimn Df.p.
(A/I n )whose restriction to A/I m − Mod coincides with the canonical functorbbb
A/I m − Mod = Ind Df.p.
(A/I m )♥ ,→ Ind Df.p.
(A/I m ) → Ind colimn Df.p.
(A/I n ).

The same is true of the canonical functor(B.1.14)+∼bb
D+ (A/I m ) −→ Ind Df.p.
(A/I m ) → Ind colimn Df.p.
(A/I n ).

Thus Proposition A.5.6 shows that the two functors (B.1.13) and (B.1.14) essentially coincide.
bConsequently, the composite of (B.1.11) with the canonical functor Df.p.
(A/I m ) →+
D (A − Mod)I , which we can factor as(B.1.13)

b
m
+
m
b
n
Df.p. (A/I ) → D (A/I ) −→ Ind colimn Df.p. (A/I ),essentially coincides with the composite(B.1.14)

b
m
+
m
b
n
Df.p. (A/I ) → D (A/I ) −→ Ind colimn Df.p. (A/I ),which in turn evidently essentially coincides with the composite
b
m
b
m
b
n
Df.p. (A/I ) → colimn Df.p. (A/I ) ,→ Ind colimn Df.p. (A/I ).

Varying m, and relabelling it as (the dummy variable) n, we find that the composite
of (B.1.11) with the functor
(B.1.10)bb
colimn Df.p.
(A/I n ) −→ Df.p.
(A − Mod)I ) ,→ D+ (A − Mod)I
coincides (up to natural equivalence) with the inclusionbbcolimn Df.p.
(A/I n ) ,→ Ind colimn Df.p.
(A/I n ).

Since (B.1.10) is essentiallysurjective, we also find that the restriction of (B.1.11)
bb
to Df.p.
(A − Mod)I does have image lying in colimn Df.p.
(A/I n ), and thus does
induce a functor (B.1.12). Furthermore, what we have shown so far is that the comb
posite of (B.1.12) with (B.1.10) essentially coincides with the identity on colimn Df.p.
(A/I n ).
It remains to consider the composite of (B.1.10) with (B.1.12). This composb
ite is a t-exact functor from Df.p.
(A − Mod)I ) to itself, which restricts to theb♥identity on Df.p. (A − Mod)I ) . Theorem A.5.5 shows that it essentially coincides
with the identity functor. (Technically, we should be applying Theorem
 A.5.5b
to D+ (A − Mod)I ) rather than its full subcategory Df.p.
(A − Mod)I . But
we can do this completely formally, by forming the canonicalextension of ourbfunctor to a t-exact endofunctor of Ind Df.p.
(A − Mod)I , restricting this to +b
Ind Df.p.
(A − Mod)I, and then using the canonical equivalence D+ (A − ∼bMod)I −→ Ind Df.p.
(A − Mod)I )+ .)Before showing that the second functor in (B.1.8) is an equivalence, we recall
what is presumably a standard result comparing injective objects in A − Mod and(A − Mod)I .
The inclusion (A − Mod)I ,→ A has a right adjoint, namely M 7→ M [I ∞ ]. Since
the right adjoint to an exact functor preserves injectives, we see that if M is an

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203

injective A-module, then M [I ∞ ] is an injective object of (A − Mod)I . But in fact
more is true, as the following lemma shows.
Lemma B.1.15. If M is an injective A-module, then M [I ∞ ] is again an injective
A-module.
Proof. By Baer’s criterion, we must show that if J is an ideal of A, then any
morphism f : J → M [I ∞ ] extends to a morphism A → M [I ∞ ]. Since J is finitely
generated (A being Noetherian), the morphism f has image lying in M [I m ] forsome m, and so factors through the quotient J/I m J of J. Now our Artin–Rees
hypothesis implies that I n ∩ J ⊆ I m J for some n ≥ m. Thus f factors througha morphism g : J/I n ∩ J → M [I ∞ ]. If we rewrite the domain of g as I n + J/I n ,and regard this as a submodule of R/I n , then the hypothesis that M is injective
ensures that g extends to a morphism R/I n → M, which obviously has image
lying in M [I n ] ⊆ M [I ∞ ]. Composing this extension of g with the quotient map
R → R/I n gives the required extension of f .
Corollary B.1.16. The inclusion (A−Mod)I ,→ A−Mod preserves injectives.
Proof. Let M be an injective object of (A − Mod)I , and let M ,→ N be an
inclusion of M into an injective A-module. This inclusion evidently factors throughan inclusion M ,→ N [I ∞ ]. This is a morphism of injective objects of (A − Mod)I ,and hence is a split inclusion. On the other hand, Lemma B.1.15 shows that N [I ∞ ]is again an injective A-module, and hence so is its direct summand M .
Proposition B.1.17. The canonical functor D+ (A − Mod)I ) → DI+ (A) is ∼bban equivalence, which restricts to an equivalence Df.p.
(A − Mod)I −→ Df.p.,I(A)(and this is the second arrow of (B.1.8)).
Proof. The first claimed equivalence follows by combining Corollary B.1.16
with Proposition A.7.3 (1). Restricting this equivalence to the compact objects inthe truncations τ ≥n for each n, we see that the equivalence then restricts to an∼bbequivalence Df.p.
(A − Mod)I −→ Df.p.,I(A).

B.2. Ind-coherent sheaves on Ind-algebraic stacks. In this section we
introduce (or, really, recall from the literature) some basic notions related to derived
categories of sheaves of O-modules on algebraic and formal algebraic stacks. Our
discussion is slightly unsatisfactory, for the reason that we wish to work with stable∞-categories, rather than merely the underlying triangulated categories, but ourbasic reference for the theory of stacks is [Sta], which does not use ∞-categoricalnotions.
For example, in the ∞-categorical world, equivariance is a perfectly good notion
for objects in the (stable ∞-categorical version of the) derived category of quasicoherent sheaves on a scheme, and so by writing an algebraic stack X as a quotient[U/R] for some scheme U and groupoid R, one can define the stable ∞-categoryof quasicoherent complexes on X as the R-equivariant objects of the corresponding
category for U . (Cf. Remark B.2.4 below.) In this way one can even define the derived category of quasicoherent complexes on X prior to having defined the abeliancategory of quasicoherent sheaves on X ; the latter can then be defined as the heartof a t-structure on the former. In this approach, the more general concept of OX module, or the derived category of complexes of such, does not even arise. (And204

M. EMERTON, T. GEE, AND E. HELLMANNthe question of, e.g., the relationship between D QCoh(X ) and Dqc (OX ) doesn’tcome up.) However, in the framework of [Sta], such a definition is not feasible.
We don’t really try to reconcile these conflicting approaches here. We simply
state various possible definitions, with pointers to the literature, and do our best to
indicate the relationships between them. Ultimately, the ∞-categorical perspective
will be the one best suited to our goals.
B.2.1. The case of algebraic stacks. If X is an algebraic stack, then we will let
D(X ) denote the stable ∞-category of quasicoherent complexes on X . Let us briefly
indicate what this means. (Note that the citations we are making typically workin the context of derived/triangulated categories, rather than stable ∞-categories,
but their results are readily adapted to this latter setting.)
There are actually (at least!) two possible approaches to defining this category(for any algebraic stack X ): we can first consider the abelian category QCoh(X ) of
quasicoherent sheaves on X (and even this category admits multiple definitions: via
small lisse-étale sites, small fppf sites (called the flat-fppf site in [Sta, Tag 0786]) or
large fppf sites, which however lead to equivalent notions [Sta, Tag 07B1]), which
is a Grothendieck abelian category, and then form D QCoh(X ) .
Alternatively, we can consider the larger Grothendieck abelian category OX −Mod of all sheaves of OX -modules (again on some appropriate site of X ), form
its associated stable ∞-category D OX ), and then (attempt to) consider the fullsubcategory Dqc OX ) consisting of objects whose cohomologies are quasicoherent,
i.e. lie in QCoh(X ). If we work with small sites, this definition is valid and reasonable, because the inclusion QCoh(X ) → OX − Mod is exact in that context. If
we work with big sites, then the inclusion QCoh(X ) → OX − Mod is generally notexact (see point (6) of [Sta, Tag 06VE]), and so this construction is not valid; but
a more elaborate version of it can be carried out instead [Sta, Tag 07B5]). Again,
the resulting category is independent, up to canonical equivalence, of the methodof construction.

There is a canonical t-exact functor D QCoh(X ) → Dqc (OX ). We don’tknow if this is an equivalence, even under reasonable finiteness hypotheses on X .
However, if X is quasi-compact and geometric (meaning that is has affine diagonal),
then this functor does restrict to an equivalence(B.2.2) ∼+D+ QCoh(X ) −→ Dqc(OX );

this is [ATJLPRVG18, Prop. 1.6].
The heart of the t-structure on either D QCoh(X ) or Dqc (OX ) is the category QCoh(OX ). If X is a Noetherian algebraic stack, then this category is locally coherent, and its full subcategory of compact objects is precisely the subcategory Coh(X) of coherent sheaves on X . In particular, the t-structures on both
D QCoh(X ) and Dqc (OX ) are then coherent. Suppose that X is furthermore
geometric; then the equivalence (B.2.2) induces an equivalence of subcategories
 ofcompact objects (which are precisely the coherent objects in D QCoh(X ) andDqc (OX ) respectively). We identify these stable ∞-categories of coherent objectsb
via this equivalence, and denote either one by Dcoh
(OX ). This is reasonable, sincethese objects (being coherent objects for a coherent t-structure) are bounded, andbthe heart of Dcoh
(OX ) is precisely Coh(X ) (again by the general properties of coherent t-structures).

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Definition B.2.3. Suppose that X is a Noetherian and geometric algebraicbstack (so that, by the preceding discussion, Dcoh
(OX ) is unambiguously defined).
b
We then define Ind Coh(X ) := Ind Dcoh (OX ), and sometimes refer to it as thestable ∞-category of Ind-coherent complexes on X .
Remark B.2.4. If one works in the ∞ (rather than triangulated) categoricalcontext from the beginning,then it is possible to give yet another definition of (aversion of) D QCoh(X ) , via descent from the affine case; see e.g. [Gai, §1.1.3].
(Working ∞-categorically is crucial for this definition to work, since the coherent
higher homotopies that are implicit in that context are required for descent to bea valid process on the derived level.) This stable ∞-category is usually denoted
QCoh(X ), and in fact is defined in much greater generality — namely for derived
prestacks. Unwinding the definition of [Gai, §1.1.3] for an algebraic stack that can∼be written in the form X −→ [U/R] where U = Spec A is an affine scheme and R
is an fppf groupoid on M , we find that Dqc (OX ) is equal to the stable ∞-categoryof R-equivariant objects in Dqc (U ) = D(A).
Note that the definition given in Section 5.1.7 of the stable ∞-category Dcoh (X)on a rigid analytic Artin stack X is analogous to this definition.
One can similarly define Ind Coh(X ) in the same style, and so also in muchgreater generality, see e.g. [Gai12] and [GR11, §2.3.1].
B.2.5. The case of Ind-algebraic stacks. Suppose that X is an Ind-algebraic
stack which admits a description(B.2.6)∼X −→ colimn Xn ,with the Xn being geometric Noetherian algebraic stacks, and the transition morphisms Xn → Xn+1 being closed immersions. These closed immersions then induce
t-exact functorsbb
Dcoh(Xn ) → Dcoh
(Xn+1 ),and thus (by passing to Ind categories)
Ind Coh(Xn ) ,→ Ind Coh(Xn+1 ).
Definition B.2.7. We define Ind Coh(X ) := colimn Ind Coh(Xn ), the colimit
being taken in the ∞-category whose objects are cocomplete stable ∞-categories
and whose morphisms are cocontinuous functors.
Remark B.2.8. The pushforward functors Ind Coh(Xn ) → Ind Coh(Xn+1 ) used
to define the colimit in Definition B.2.7 are t-exact, and so Ind Coh(X ) is naturally
equipped with a t-structure, which is furthermore coherent, and with respect towhich Ind Coh(X ) is left regular.
We define Coh(X ) to be the abelian category of compact objects in Ind Coh(X )♥ .
Equivalently, Coh(X ) := colimn Coh(Xn ).
Remark B.2.9. Any two descriptions of X of the form (B.2.6) are mutuallycofinal in an evident sense, and thus Ind Coh(X ) and Coh(X ) are well-defined up
to canonical equivalence independent of the choice of such a description.
Remark B.2.10. The definition of Ind Coh(X ) given in [Gai12] and [GR11,§2.3.1], which we recalled in Remark B.2.4 above, can also be applied directly to
the Ind-algebraic stack X . We’ve given the more concrete definition above simply
because we find it easier to think about.

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B.3. Pro-coherent sheaves on formal algebraic stacks. Suppose that X
is a geometric Noetherian formal algebraic stack, described as an Ind-algebraic stack∼
via colimn Xn −→ X , where the Xn are geometric Noetherian algebraic stacks, andthe transition maps Xn → Xn+1 are thickenings, i.e. closed immersions inducing
isomorphisms on underlying reduced substacks. We may then define Ind Coh(X )and Coh(X ) following Definition B.2.7 and Remark B.2.8.
There are natural “sheaves” on X that one might want to consider, which do
not arise as objects of Coh(X ) according to our definition. For example, if A is an
I-adically complete Noetherian ring, then the usual definition of coherent sheaves
on Spf A yields a category equivalent (via passage to global sections) to the categoryof finitely generated A-modules, whereas according to our definition, Coh(Spf A) isequivalent (via passage to global sections) to the subcategory of finitely generated
A-modules annihilated by I n for some n. In particular, the structure sheaf on Spf A,
corresponding to the ring A itself, is not a coherent sheaf on Spf A. Instead, we
identify it with the formal pro-object “ limn ”A/I n , and regard it as a pro-coherent←−
sheaf on Spf A.
Definition B.3.1. The abelian category of pro-coherent sheaves on X is defined to be the formal pro-category of Pro Coh(X ). This category has enoughprojectives, and so we may form D− Pro Coh(X ) , following [Lur17, §1.3.2]. Thisis the (bounded above) stable ∞-category of pro-coherent complexes on X .
Example B.3.2. If A is a Cohen–Macaulay I-adically complete Noetherianring, with dualizing module D, then D also gives rise to a pro-coherent sheaf on
Spf A. More generally, if a Noetherian formal algebraic stack admits a dualizing
complex, this will naturally be an object of D− (Pro Coh(X ) . We adopt this pointof view on dualizing sheaves/complexes in the body of our notes, since we will want
to apply Grothendieck–Serre duality in the context of pro-coherent sheaves.
We note, though, that if we restrict our duality theory to objects of Ind Coh(X ),then we can construct an alternative version of the dualizing complex, which is∼itself an object of Ind Coh(X ). Namely, if we write X −→ colimn Xn , and let
in,n+1 : Xn ,→ Xn+1 denote the transition maps (assumed to be closed immersions)and in : Xn ,→ X denote the canonical closed immersions, then if ωn denotes the∼dualizing complex on Xn , the natural isomorphism ωn −→ i!n,n+1 ωn+1 induces amorphism (in,n+1 )∗ ωn → ωn+1 , and hence a morphism (in )∗ ωn → (in+1 )∗ ωn+1 . If
we set ω := colimn (in )∗ ωn , then ω is an object of Ind Coh(X ) which serves as adualizing sheaf (for objects of Ind Coh(X )).
As a concrete example, consider A = k[[T ]] (with its T -adic topology). Thenk[[T ]] is a dualizing module for A. On the other hand, if we write A = limn k[T ]/(T n ),←−then the construction of the preceding paragraph gives rise to the A-module colimn k[T ]/(T n ),the transition maps being given by multiplication by T , which is isomorphic tok((T ))/k[[T ]]. This module corresponds to an object of Ind Coh(Spf A), and aconsideration of the short exact sequence
0 → k[[T ]] → k((T )) → k((T ))/k[[T ]] → 0shows that RHomA –, k[[T ]] and RHomA –, k((T ))/k[[T ]] coincide (up to a shift)on objects of Ind Coh(Spf A) (since RHomA –, k((T )) vanishes on T -power torsion
A-modules).

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Appendix C. Localization and Morita theory“Localization” refers to the idea of interpreting modules M over some ringA as sheaves (of some sort; for us they will always be quasicoherent sheaves, orsome variant thereof, such as complexes of quasicoherent sheaves or ind-coherent
sheaves) on an appropriate sort of space (for us, a scheme or variant thereof, suchas an algebraic stack or a formal algebraic stack). The basic intuition comes fromspectral theory: if A is a k-algebra, then an A-module M is a k-vector spaceendowed with an action of A as a ring of operators. We can then try to decompose
M in terms of its various simultaneous eigenspaces with respect to the action of A
(if A is commutative) or more generally into a direct sum or direct integral of simple
A-modules (if A is non-commutative). We can hope that the collection of simple
A-modules has the structure of a space of some sort, and that this decomposition iseffected by spreading M out over the space of simple A-modules, with the fibre of
the associated sheaf at a given point corresponding to the multiplicity space thatthe measures the contribution of that particular simple module to M .
C.1. Localizing A-modules over Spec A (the paradigmatic example).
The most basic example of localization in algebraic geometry is the theory of quasicoherent sheaves over affine varieties: if X = Spec A is an affine variety, then thefunctor of global sections(C.1.1)

Γ(X, −) : QCoh(X) → Γ(X, OX ) − Modis an equivalence. The quasi-inverse is, of course, given by the localization functor
M 7→ OX ⊗A M.
Another way to describe the global sections functor, when applied to sheavesof OX -modules, is as HomOX (OX , –). Thus this fundamental example of localization
illustrates a general paradigm: the sheaf OX is naturally an (OX , A)-bimodule(in a somewhat trivial manner, just via the identification of A with the ring of
global sections of OX ), and so we have the pair of adjoint functors OX ⊗A – and
HomOX (OX , –), which together induce an equivalence between the category of Amodules and the category of quasicoherent A-modules. This is an instance of amore general structure provided by Morita theory, which we discuss more formallybelow.
To elaborate even further on this example, we remark that we can regard any
commutative ring A as an “abstract” ring of commuting operators, and an Amodule M as a concrete realization of A as a ring of operators (namely, via its
operation on M ). Note then that X = Spec A can be thought of as parameterizing
all the possible systems of simultaneous eigenvalues that the elements of A canassume (since the points of Spec A correspond, via taking the kernel, to equivalenceclasses of morphisms A → F with F a field, two morphisms A → E and A → F
being equivalent if we can find a common field extension K of E and F suchthat the induced morphisms A → K coincide); hence the designation of X as the“spectrum” of A. Roughly, the support of the quasicoherent sheaf M associated to
M consists of those systems of simultaneous eigenvalues that appear in M . (Moreprecisely, the fibre of M at a point of X is precisely the co-eigenspace of M forthe corresponding system of eigenvalues.) Thus localization provides an algebraic
interpretation of spectral theory.

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Example C.1.2. If we set A = k[x] (for some field k) and apply the preceding
discussion, we recover the usual derivation of the spectral theory of a single operator
via an analysis of the structure of k[x]-modules.
We are interested in other, less obvious, examples of localization. But in eachcase, just as in the paradigmatic example of localizing A-modules, we hope thatthe structure of the sheaf associated to a module (its fibres, its support, and so on)will illuminate, and separate out, various pieces of internal structure that are more
obscure when considered purely in terms of modules.
C.2. Morita theory. Morita theory, at least as we intend it here, is a slightly
catch-all term for the theory that describes functors, especially equivalences, between categories of modules over various rings. We recall the classical (i.e. abelian,i.e. module-theoretic) case of the theory first, before turning to the derived context(where the ideas of the theory easily extend to provide a flexible and convenient
context for studying functors that are not necessarily equivalences).
C.2.1. The abelian case. One way to phrase the classical (i.e. module theoretic)
version of Morita theory is that if A and B are (not necessarily commutative)
associative rings, with respective categories of left modules A − Mod and B − Mod,
then equivalences of categories A − Mod → B − Mod are all of the form
M 7→ P ⊗A M,
where P is a finitely generated projective B-module which generates B − Mod∼equipped with an isomorphism A −→ EndB (P ). A quasi-inverse is given by
N 7→ HomB (P, N ).
(Of course, Morita theory applies to this adjoint functor as well; we find that
Q := P ∨ := HomB (P, B) — which is an (A, B)-bimodule — is necessarily projective
as an A-module, and the quasi-inverse can be rewritten as N 7→ N ⊗B Q.)
More generally, any functor F : A − Mod → B − Mod which commutes withcolimits is of the form
(C.2.2)

F : M 7→ P ⊗A M

for some (B, A)-bimodule P . (The bimodule P is given by evaluating the functor
on A. The formula (C.2.2) is then obtained by evaluating F on some presentationof M .) The right adjoint of F (which exists since F commutes with colimits) isevidently given by HomB (P, –).
If F is furthermore fully faithful, then we see that A → EndB (P ) must be an
isomorphism. If it is in fact an equivalence, then P must inherit all the purely
categorical properties that A has in A − Mod (compactness, projectivity, being agenerator), which gives the statement of classical Morita theory.
Example C.2.3. A standard example of classical Morita theory is given by the⊕ncase B = Mn (A) and(thought of as length n column vectors) with its
 P =A∼
obvious Mn (A), A -bimodule structure. Since B −→ P ⊕n as a module over itself,
we see that P is finitely generated and projective, and so M 7→ P ⊗A M = M ⊕ngives an equivalence between the categories A − Mod and Mn (A) − Mod.
Going the other way, if we set Q = A⊕n (now thought of as length n row
vectors), with its obvious A, Mn (A) -bimodule structure, then N 7→ Q ⊗Mn (A) N
gives the quasi-inverse equivalence between Mn (A) − Mod and A − Mod.

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Example C.2.4. When A is commutative, the preceding example can be turned
into an example of localization, by writing X = Spec A, and considering the functor⊕nN 7→ OX⊗Mn (A) N,

which induces an equivalence of categories between Mn (A) − Mod and the categoryof quasicoherent sheaves on X.
Of course we are interested in studying modules over non-commutative rings
that are more complicated than matrix rings over commutative rings. In such contexts, it is generally not reasonable to expect to obtain an equivalence of categories
with the category of modules over a commutative ring, or with the category of
quasicoherent sheaves on a scheme. As we will see, it is more plausible to hope fora fully faithful embedding than an equivalence, and it also makes sense to expand
the possibilities for the “space” X which will carry the sheaves, e.g. by allowing it
to be an algebraic stack.
It is also more natural to pass to a derived setting. Indeed, if one considers
the problem of determining when a functor of the form (C.2.2) is fully faithful(but not necessarily an equivalence), one immediately runs into questions about
the homological nature of F which are more naturally considered in the derived
world. We will thus briefly discuss derived Morita theory next, with an emphasison the extra flexibility it provides (in comparison to the classical abelian setting)for investigating functors which are not necessarily equivalences.
C.2.5. The case of stable ∞-categories. If A, B are E1 -rings (see Appendix A.2
for the terminology of E1 -rings), then by [Lur17, Prop. 7.1.2.4], the cocontinuousfunctors LModA → LModB are given by tensoring with (A, B)-bimodules. In
particular if A and B are static (i.e. are associative rings in the usual sense), thenequivalences of stable ∞-categories D(A − Mod) → D(B − Mod) are all of the form
(C.2.6)

M 7→ T ⊗L
A M,

where T is a perfect complex of B-modules which generates D(B − Mod), and∼is equipped with an isomorphism of E1 -rings A −→ REndB (T ), where we write
REndB (T ) for the E1 -ring EndD(B−Mod) (T, T ). (See [Lur17, Prop. 7.1.2.1, Rem.
7.1.2.3].)
If we drop the assumption that T generates D(B − Mod), but continue to∼assume that A −→ REndB (T ), then (C.2.6) still defines a cocontinuous fully faithful
functor F : D(A − Mod) → D(B − Mod), with a right adjoint G : D(B − Mod) →D(A − Mod) given by(C.2.7)

N 7→ RHomB (T, N ).

More generally, if T is a perfect complex of B-modules and S is a perfect complexof A-modules which generates D(A − Mod) and is equipped with an isomorphism∼
REndA (S) −→ REndB (T ) := H, then combining the equivalence of D(A − Mod)
with H−Mod given by S and the cocontinuous fully faithful functor from H−Mod toD(B−Mod) given by T , we have a cocontinuous fully faithful functorD(A−Mod) →D(B − Mod) given by(C.2.8)

M 7→ T ⊗L
H RHomA (S, M ),with a right adjoint D(B − Mod) → D(A − Mod) given by
N 7→ S ⊗L
H RHomB (T, N ).

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There are, of course, many generalisations of this framework. In particular, inthe body of the text we consider examples where D(B − Mod) is replaced by anappropriate derived category of sheaves on a stack, as in the examples below (seealso Remark 6.1.23).
Remark C.2.9. The notation H for the derived endomorphism ring is intended
to be suggestive of a (derived) Hecke algebra. See for example [Hel20, (1.1)] for afunctor from a category of representations to a category of sheaves defined in thisfashion.
Remark C.2.10. In the literature, derived Hecke algebras are often thoughtof as dgas, rather than E1 -rings. The correspondence between these notions is asfollows: by [Lur17, Prop. 7.1.4.6], if A is a (static) commutative ring, then the ∞category of E1 -algebras over A can be identified with the ∞-category of differential
graded A-algebras. For example, if we take A = Fp , Schneider’s derived Hecke
algebra [Sch15a] can equally well be regarded as an Fp -dga or an Fp -E1 -algebra.
C.3. Non-commutative phenomena. There are some phenomena that canoccur in the theory of modules over non-commutative rings that don’t occur in thecommutative case, which have to be taken into account when trying to localize in the
non-commutative setting. We briefly recall some of these, and explain why allowing
localization over algebraic stacks (rather than just schemes) helps to ameliorate thesituation.
There is one basic phenomenon that modules over a non-commutative ring A
can exhibit which does not occur in the commutative context, namely, if S1 andS2 are non-isomorphic simple A-modules, then although HomA (S1 , S2 ) evidently
vanishes, the higher Exts, i.e. ExtiA (S1 , S2 ) for i > 0, need not vanish. This meansthat if A − Mod is mapped functorially into a category of sheaves in such a mannerthat induces an isomorphism on higher Exts as well as just Homs, then the imagesof S1 and S2 cannot have disjoint support.
Remembering that the simple objects in the category of quasicoherent sheaves
on a Noetherian28 scheme are precisely the skyscraper sheaves at closed points,
so that non-isomorphic simple objects have disjoint support, we see for example
that if A is a non-commutative ring admitting simple modules S1 and S2 withthis property (i.e. such that HomA (S1 , S2 ) = 0 but RHomA (S1 , S2 ) 6= 0), then the
category A − Mod certainly cannot be equivalent to the category of quasicoherent
sheaves on such a scheme.
Remark C.3.1. The preceding discussion helps to explain why in Beilinson–
Bernstein localization theory, which does give an equivalence (between certain categories of Lie algebra representations and the category of algebraic D-modules on aflag variety), it is quasicoherent D-modules that appear, rather than merely quasicoherent O-modules. The point is that the Riemann–Hilbert correspondence showsthat D-modules behave like perverse sheaves, in which (to first approximation) the
simple objects are described by their support (a stratum in some stratification of
the ambient scheme X), and there are interesting extension classes between sheaves
supported on different strata. (The most basic example is the sequence
0 → j! C → C → i∗ C → 0,
28We make this assumption just to ensure that the discussion is not overburdened withtechnical complications.

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211

with j : A1 \ {0} → A1 , resp. i : {0} → A1 the natural open, resp. closed,
immersion.)
If we only want fully faithful functors, the various (im)possibilities are less clear,since then a simple A-module needn’t map to a simple quasicoherent sheaf (it mightjust be that the proper quasicoherent subsheaves of its image are not themselves in
the image of the functor). If we pass to the derived context, the situation becomesyet murkier, because then the notions of simple module and submodule aren’t defined. Nevertheless, the difference in behaviour between the non-commutative andcommutative situations creates a tension which makes it difficult to construct interesting localization functors for modules over non-commutative rings.
One way to lower this tension is by allowing the target category to be a categoryof quasicoherent sheaves on an algebraic stack X , as we now explain.
If x ∈ |X | is a closed point of (the underlying topological space of) an algebraic
stack X , then it typically does not correspond to a closed immersion Spec k ,→ X
for some field k, but rather (under mild finiteness hypotheses, such as having quasicompact diagonal; see [Sta, Tag 06RD], as well as the discussion of [EG22, App. E])
to a closed immersion Z ,→ X with Z being a one-point gerbe. If, for example, ourstack X is of finite type over an algebraically closed field k, then Z = [Spec k/G]
for some algebraic group over G.
Suppose then, to fix ideas, that we have a closed embedding [Spec k/G] ,→X for some field k and some algebraic group G over k, giving rise to the closed
point x ∈ |X |. Representations of G then correspond to quasicoherent sheaves
on X that are supported at x; irreducible representations in particular correspondto simple objects in the category of quasicoherent sheaves on X . In examples, we
can often produce non-split extensions of these sheaves by considering the higher
infinitesimal neighbourhoods of x in X .
Example C.3.2. Consider X = [A1 /Gm ], with A1 = Spec k[x], Gm = Spec k[t±1 ],and Gm acting on A1 via t · x = tx. Let I ⊂ OX denote the ideal sheaf cuttingout the closed point {x = 0}. Then we have a non-split short exact sequence ofcoherent sheaves
0 → k(−1) ∼= I/I 2 → OX /I 2 → OX /I ∼= k(0) → 0(where as usual we identify Gm representations with graded k-modules).
In short, in the category of coherent sheaves on algebraic stacks, it is possible
to find non-isomorphic simple objects that admit non-split extensions. This makes
categories of coherent sheaves on algebraic stacks amenable targets for localization
functors defined on categories of modules over non-commutative rings.
C.4. Examples. We now give some examples of Morita theory and localization, building up from some very simple ones to some which occur in the representation theory of GL2 (Qp ). We work throughout over some commutative ring k.
Example C.4.1. As we saw in Example C.1.2 above, the category k[x] − Modis equivalent to QCoh(A1 ); this Morita equivalence is mediated by the structure
sheaf OA1 (which is of course a module over its ring of global sections, namelyk[x]). Slightly less trivially, as a special case of Example C.2.3, these categories areequivalent to the categories of left modules for the matrix algebrak[x] k[x],k[x] k[x]212

M. EMERTON, T. GEE, AND E. HELLMANN

via the bimodule OA1 ⊕ OA1 .
Example C.4.2. Consider instead the algebrak[x] k[x]A :=xk[x] k[x]Its category of left modules is not Morita equivalent to the category of modules overany commutative ring. However, there is a fully faithful functor from D(A − Mod)to D(QCoh(X )), where
X := [(Spec k[x, y]/(xy))/Gm ],with the Gm -action given by29t · (x, y) = (x, t2 y),and the functor being given by (derived!) tensoring with
P := OX ⊕ (OX /y).
In order for this to make sense, we need P to be a right A-module. We claim thatin fact
REndOX (P ) = EndOX (P ) = Aop ,so that the functor P ⊗A − is fully faithful.
In general, for an object X ⊕ Y in an abelian category, we have
REnd(X)
RHom(Y, X)
REnd(X ⊕ Y ) =,RHom(Y, X)
REnd(Y )acting on the left on X ⊕ Y , thought of as column vectors. Then
REnd(X)
RHom(X, Y )opREnd(X ⊕ Y ) =,RHom(Y, X)
REnd(Y )acting on the right on X ⊕ Y (now thought of as row vectors).
In our example, X is an affine scheme modulo a linearly reductive group, infact the torus Gm , and so the global sections functor on X is exact; equivalently,
OX is projective, and RHomOX (OX , –) = HomOX (OX , –) is simply the functor of
global sections. In particular
RHomOX (OX , OX ) = RHomOX (OX , OX /y) = k[x].
To compute RHomOX (OX /y, –), we use the projective resolutionyyx· · · → OX (−2) → OX (−2) → OX → OX /y → 0.
(Here as usual our notation for invertible sheaves on the quotient stack X corresponds to our notation for Z-graded modules, with the convention that Gri M (n) =Gri+n M .)We have HomOX (OX (−n), OX ) = k[x] if n = 0 and = k · y n if n > 0, andsimilarly HomOX (OX (−n), OX /y) = k[x] if n = 0 and = 0 if n > 0. It follows that
RHomOX (OX /y, OX ) is computed by the complex
0∼0

0 → k[x] → k → k −→ k → k → . . .
29The appearance of t2 rather than t in the formula for the action is of no intrinsic importancefor this example; its only significance is that it yields an example that connects directly with an
instance of categorical Langlands, as we note below.

CATEGORICAL p-ADIC LANGLANDS

213(where the morphism k[x] → k is given by evaluation at 0) and that RHomOX (OX /y, OX /y)is computed by the complex
0 → k[x] → 0 → 0 → . . . ,so that
RHomOX (OX /y, OX ) = HomOX (OX /y, OX ) = xk[x]and that
RHomOX (OX /y, OX /y) = HomOX (OX /y, OX /y) = k[x].
Thus indeed REndOX (P )op = A.
Continuing with this example a little more, writing M = k[x] ⊕ xk[x] (thoughtof as column vectors, with the evident left action of A) and N = k[x] ⊕ k[x] (again
thought of as column vectors, with the evident left action of A) then A = M ⊕ N asleft A-modules, so that M and N are both projective as A-modules, and P ⊗A M =OX , while P ⊗A N = OX /y.
The evident inclusion(C.4.3)

M ,→ N

has cokernel S1 := 0 ⊕ k (whose A-module structure comes by thinking of elements
as column vectors and having x act via 0), while the the morphism(C.4.4)x

N −→ M

has cokernel S2 := k⊕0 (which again has the A-module structure given by regarding
its elements as column vectors and having x act via 0). The two A-modules S1 , S2
are simple and not isomorphic (and, up to isomorphism, are precisely the two
simple A-modules on which the the element x in the centre k[x] of A acts via 0).
The quotient M/xM is a non-split extension of S2 by S1 , while N/xN is a non-split
extension of S1 by S2 .
If we pass to sheaves on X , the inclusion (C.4.3) corresponds to the surjection
OX → OX /y, and thus S1 corresponds to the cone of this morphism, which isx(OX /x)(−2). The morphism (C.4.4) corresponds to the morphism OX /y −→ OX ,and thus S2 corresponds to the cone of this morphism, which is OX /x. Thus we seean example of the possibility discussed in Section C.3, i.e. of two non-isomorphic
simple modules, admitting a non-split extension between them, corresponding tocoherent sheaves which are non-isomorphic but have the same support.
Remark C.4.5. Example C.4.2 is closely connected to the `-adic (` 6= p) representation theory of PGL2 (Qp ), see for example [Hel20, Rem. 4.43]. Indeed the
Iwahori Hecke algebra HI for PGL2 (Qp ) is a matrix algebraQ` [T ]
Q` [T ]HI =(T 2 − (p + 1)2 )Q` [T ] Q` [T ]and there is a fully faithful functor (determined by the coherent Springer sheaf)
from the derived category of left H-modules to the category of ind-coherent sheaves
on the moduli stack X2unip of 2-dimensional unipotently ramified Weil–Deligne representations with inverse cyclotomic determinant.
This stack has three irreducible components; the unramified component, andtwo “Steinberg” components which are related by a quadratic twist. The two Steinberg components do not intersect, but each intersects the unramified component, at
respectively the representation 1 ⊕ ε−1 and its quadratic twist. In a neighbourhood

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M. EMERTON, T. GEE, AND E. HELLMANN

of either one of these intersection points we can identify X2unip with (a corresponding neighbourhood of the singular point on) the stack X above. The locus y = 0
then corresponds to the Steinberg component, and P corresponds to the coherent
Springer sheaf (see e.g. [Zhu20, Ex. 4.4.4]).
The module M corresponds to the Steinberg representation (or its quadratic
twist), whose associated sheaf is OX /y, the structure sheaf of the Steinberg component; while N corresponds to the trivial representation (or its quadratic twist),
whose associated sheaf is (OX /y)(−2)[1], a shift of a twist of the structure sheaf of
the Steinberg component.
Remark C.4.6. Note that the object P of Example C.4.2 is not a perfectcomplex, but it is coherent; thus already in this example we see that it is natural
to consider ind-coherent sheaves on stacks of Langlands parameters, rather thanquasicoherent sheaves.
The following variant on Example C.4.2 plays a similar role in the p-adic Langlands correspondence for GL2 (Qp ), as described in Section 7.3.
Example C.4.7. Let A be a (commutative) ring, and consider the graded ring
B := A[x, y],where the grading is defined by giving x and y weights 2 and −2 respectively. Let
B denote the category of graded B-modules (equivalently, quasicoherent sheaves on
the quotient stack [Spec B/Gm ]). If M is a graded module, and n ∈ Z, then as
usual we let M (n) denote the graded module whose underlying module coincides
with that of M , and for which Gri M (n) = Gri+n M.
We consider the module P := B(1) ⊕ B(−1), and define R := EndB (P )op ; thenR acts on the right on P . Concretely, we find that

 


End B(1)
Hom B(1), B(−1)
Gr0 B
Gr0 B(−2)
R==
Hom B(−1), B(1)End B(−1)
Gr0 B(2)
Gr0 B
 0



Gr B Gr−2 B
A[xy] yA[xy]==xA[xy] A[xy]Gr2 B Gr0 B
acting on P via the usual right multiplication of matrices on row vectors.
Certainly P is a projective object of B. Slightly less obviously, we have thefollowing lemma.
Lemma C.4.8. P is projective as a right R-module.
Proof. We first note that as a right module over itself, we have R = R1 ⊕ R2 ,where R1 = A[xy] ⊕ yA[xy] (the first row of R regarded as above as a matrix order)
and R2 = xA[xy] ⊕ A[xy] (the second row of R). Each of R1 and R2 is projective(being a direct summand of R).
L∞
We next note that P = m=−∞ Gr1+2m P ; since R acts as graded endomorphisms of P , this is an isomorphism of R-modules. If 1 + 2m ≥ 1 (i.e. m ≥ 0), then∼Gr1+2m P = xm A[xy] ⊕ xm+1 A[xy] −→ A[xy] ⊕ xA[xy] = R2 , while if 1 + 2m ≤ −1
∼
1+2m
−m−1
(i.e. m ≤ −1), then Gr
P =y
A[xy]⊕y −m A[xy] −→ A[xy]⊕yA[xy] = R1 .
Thus P is a direct sum of projective R-modules, and so is itself a projective Rmodule.

We have the following immediate consequence.

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Proposition C.4.9. The functor N 7→ P ⊗R N induces a fully faithful andexact embedding from the abelian category of R-modules into B, whose essential
image is equal to the full subcategory C of B generated by P .
In fact, since P is projective over B, so that RHomB (P, P ) = R, the preceding
proposition has a derived analogue (following the discussion of C.2.5).
Proposition C.4.10. The functor N 7→ P ⊗R N induces a fully faithful andt-exact embedding of stable ∞-categories with t-structures D(R) → D(B), whose
essential image is equal to the full subcategory of D(B) generated by P .
C.4.11. Characterizing an image. As already remarked, Example C.4.7 has anapplication to the categorical p-adic Langlands correspondence in the GL2 (Qp )case. In fact, in this application we will study a variant of this functor, in which
we work with stable ∞-categories and also impose a support condition.
We now describe our setup precisely. To begin with, we maintain the notationof Example C.4.7, but furthermore assume that the ring A is Noetherian. We also
introduce additional notation. Namely, we let R denote the abelian category of Rmodules, and let R0 denote the full subcategory of R consisting of those modules,
each element of which is annihilated by some power of xy. Similarly, we let B0
denote the subcategory of the category B of graded B-modules consisting of thosemodules satisfying the same condition, i.e. each element is annihilated by somepower of xy.
Proposition B.1.17 (and its evident variant for graded rings) shows that theembedings R0 ,→ R − Mod and B0 ,→ B induce equivalences(C.4.12)∼+
D+ (R0 ) −→ D(xy)(R)

and(C.4.13)∼+
D+ (B0 ) −→ D(xy)(B).

(The subscript (xy) here denotes the full subcategory consisting of those objects
each of whose cohomologies is annihilated elementwise by some power of xy.)
Lemma C.4.14. The equivalences (C.4.12) and (C.4.13) extend to equivalences∼∼
D(R0 ) −→ D(xy) (R) and D(B0 ) −→ D(xy) (B).
Proof. This will follow from Proposition A.7.3 (2), once we show that theadjoint functors to the inclusions R0 ,→ R − Mod and B0 ,→ B have bounded cohomological dimension. However, in either case, the derived functor of this adjoint(xy)nis computed via X 7→ colimn (X −→ X), and so has a cohomological amplitudeof 1.
The t-exact fully faithful functor N 7→ P ⊗R N of Proposition C.4.10 evidently
restricts to a t-exact functor fully faithful functor D(xy) (R) → D(xy) (B), which by
Lemma C.4.14 we may also regard as a t-exact fully faithful functor D(R0 ) →D(B0 ), We will characterize the image of this functor in terms of a certain semiorthogonal decomposition, which we now explain.
The adjoint functor HomB (P, –) : D(xy) (B) → D(xy) (R) admits the more explicit description (remembering that P := B(1) ⊕ B(−1))
M 7→ Gr−1 M ⊕ Gr1 M.

216

M. EMERTON, T. GEE, AND E. HELLMANN(The right hand side being equipped with its evident R-structure: each summandis naturally an A[xy]-module; multiplication by x takes the first summand to thesecond; and multiplication by y takes the second summand to the first.)
Proposition C.4.15. Let K denote the full subcategory of D(xy) (B) consistingof those objects M such that Gr−1 M = Gr1 M = 0. Then the image of P ⊗R – isequal to ⊥ K.
Proof. This follows from Lemma A.8.4 (6).

For applications to categorical p-adic Langlands, we wish to have an alternative description of the kernel K, in terms of a set of generators. In order to do
this, we change the context slightly, replacing the unbounded stable infinity categories D(R0 ) and D(B0 ) by their regularizations Ind Coh(R0 ) and Ind Coh(B0 ),following the procedure described in Section A.6. The functor P ⊗R – and its adjoint HomB (P, –) are both exact and cocontinuous (the latter because P is finitely
generated and projective as a B-module), and preserve compact objects. Thus
they induce an adjoint pair of functors between Ind Coh(R0 ) and Ind Coh(B0 ), andthe results proved above in the context of D(R0 ) and D(B0 ) carry over to theInd-coherent context.
The category B0 splits into a product B0+ ×B0− , where B0+ (resp. B0− ) consists ofobjects whose graded pieces are supported purely in even (resp. odd) degrees. Correspondingly, Ind Coh(B0 ) decomposes as a product Ind Coh(B0+ ) × Ind Coh(B0− ).
The functor P ⊗R – evidently factors through Ind Coh(B0− ), and we let D0−
denote its essential image in this category. We also consider the analogous functor
B ⊗A[xy] – : Ind Coh(A0 ) → Ind Coh(B0+ ), where A0 denotes the subcategory ofA[xy] − Mod consisting of modules annihilated elementwise by some power of xy.
We let D0+ denote the essential image in Ind Coh(B0+ ) of this latter functor.
∼
We regard A[x] as a graded B-module via the isomorphism B/(y) −→ A[x], andsimilarly for A[y]. Similarly we regard A as a graded B-module via the isomorphism∼
Bi /(x, y) −→ A. All three of A[x], A[y], and A are then objects of B0 .
Using the graded resolution(C.4.16)y

0 → B(2) −→ B → A[x] → 0,
we see (for any choice of twist n) that RHomB A[x], A[x](n) is computed by thecomplex (in cohomological degrees 0 and 1)0

Gr0 (A[x](n)) −→ Gr0 (A[x](n − 2)),
while RHomB A[x], A[y](n) is computed by the complex (in cohomological degrees
0 and 1)
0 → Gr0 A(n − 2).

Similarly, we find that RHomB A[y], A[y](n) is computed by the complex (incohomological degrees 0 and 1)0

Gr0 (A[y](n)) −→ Gr0 (A[y](n + 2)),
while RHomB A[y], A[x](n) is computed by the complex (in cohomological degrees
0 and 1)
0 → Gr0 A(n + 2).
Consequently, we find that the collections of objects {A[x](−2), A[x](−4), . . . }and {A[y](2), A[y](4), . . . } each form a weakly exceptional collection in Ind Coh(B0+ )CATEGORICAL p-ADIC LANGLANDS

217

in the sense of Definition A.8.10, and are furthermore mutually orthogonal. By
Lemma A.8.11, they induce a semiorthogonal decomposition of the full subcategory
E0+ of Ind Coh(B0+ ) that they generate.
Similarly, the collections {A[x](−3), A[x](−5), . . . } and {A[y](3), A[y](5), . . . }each form a weakly exceptional collection in Ind Coh(B0− ), and are mutually orthogonal. They similarly induce a semiorthogonal decomposition of the full subcategoryE0− of Ind Coh(B0− ) that they generate.
Proposition C.4.17.
(1) D0+ is the cocomplete full subcategory of Ind Coh(B0+ ) generated by theobject B/(xy).
(2) E0+ = (D0+ )⊥ .
(3) D0− is the cocomplete full subcategory of Ind Coh(B0+ ) generated by theobjects B/(xy)(1) and B/(xy)(−1).
(4) E0− = (D0− )⊥ .
Proof. The cocomplete stable ∞-category Ind Coh(A0 ) is generated by A :=A[xy]/(xy). Thus D0+ , which is defined to be the image of this category under thefunctor B ⊗A[xy] –, is generated by the image of A, namely B/(xy). This proves (1).
Lemma A.8.4, together with the Ind-coherent analogue of Proposition C.4.15,
shows that (D0+ )⊥ consists of those objects M for which Gr0 M = 0. On the other
hand, Lemma A.8.6 shows that E0+ can be described as the smallest cocomplete
stable subcategory of D(B0+ ) which contains the set of objects X. Since Gr M0 = 0
for each object in X, we thus find that E0+ ⊆ (D0+ )⊥ .
Now, a consideration of (the twist by −2 of) the presentation (C.4.16), alongwith the fact that Gr M0 = 0 for objects of (D0+ ), shows that RHomB A[x](−2), M =
Gr2 M for objects M of (D0+ )⊥ . In particular, RHomB A[x](−2), A[x](−2) = A.
Thus N 7→ A[x](−2)⊗A N is a fully faithful functor Ind Coh(A) → (D0+ )⊥ , with adjoint given by M 7→ Gr2 M . In particular, the cone of A[x](−2)⊗A Gr2 M → M has
vanishing Gr2 . Continuing inductively with a consideration of the various A[x](−n),
and then also of the A[y](n) (for positive even values of n), we find that if M is anobject of (E0+ )⊥ ∩ (D0+ )⊥ , then Gri M = 0 for all values of i, and thus that M = 0.
Consequently, E0+ = (D0+ )⊥ , proving (2).
The proofs of (3) and (4) are entirely analogous to those of (1) and (2), and so
we omit the details.
Appendix D. Topological abelian groups, rings and modules
D.1. Functional analysis. Let {Gn } be a sequence of linearly topologized30
Hausdorff topological abelian groups, equipped with closed embeddings Gn ,→Gn+1 . We can then define the inductive limit G := limn Gn , endowed with the−→inductive limit linear topology (i.e. the topology for which a subset is open if andonly if its inverse image in each Gn is open); this is again a linearly topologized
topological abelian group. We recall some basic properties that G satisfies; these
are analogues of standard properties of inductive limits of sequences of locally convex topological spaces in various contexts (LF -spaces, spaces of compact type, . . . ),and our arguments are an adaptation of the arguments of [RR73, Ch. VII].
Remark D.1.1.
30Meaning that 0 admits a neighbourhood basis of subgroups.

218

M. EMERTON, T. GEE, AND E. HELLMANN(1) To simplify the notation, we regard each Gn as a subset of G via thecanonical embedding. We will see below that this embedding of groups isfurthermore a topological embedding, so this should cause no confusion.
(2) By definition, a subgroup U of G is open if each pullback Gn ∩ U is an
open subgroup of Gn for each n. A typical way to construct such open
subgroups is togiven an open subgroup Vn of Gn for each n, and toPbendefine Un := i=1 Vi . Then Un is Sagain anP
open subgroup of Gn , and∞evidently Un ⊆ Un+1 , so that U := n Un = i=1 Vi is a subgroup of G.
Since Un ⊆ Gn ∩ U, we see that U is an open subgroup of G. If theVn satisfy the additional condition Gn ∩ Vn+1 ⊆ Vn , then one sees that
Un = Gn ∩ Un+1 , and thus Un = Gn ∩ U.
If we are given open subgroups Vn , then we may inductively construct0
open subgroups Vn0 ⊆ Vn such that furthermore Gn ∩ Vn+1
⊆ Vn0 . Namely,0
assuming Vi is constructed, for i = 1, . . . , n, and and since Gn ,→ Gn+1 is
a topological embedding, we may find an open subgroup Wn+1 such that0
Gn ∩ Wn+1 ⊆ Vn0 . We then set Vn+1:= Vn+1 ∩ Wn+1 .
Lemma D.1.2. If X ⊆ G has the property that X ∩ Gn is finite for each n, then
X is closed and discrete as a subset of G.
Proof. We first show that X is discrete. To this end, choose x ∈ X; we mustfind a neighbourhood of x in G which is disjoint from X \ {x}. To ease notation,
replace X by X − x, and x by 0. Since Gn ∩ (X \ {0}) is finite for each n, we may
inductively choose open subgroups Vn of Gn so that Un := V1 + · · · + Vn is disjoint
S
from Gn ∩ (X \ {0}). Then (following Remark D.1.1 (2)), we see that U := Un is
an open subgroup of G which is disjoint from X \ {0}.
Now suppose that x is an element of G \ X. Then X 0 := X ∪ {x} again satisfies
the hypothesis of the lemma, and thus, by what we have already proved, is discrete.
Thus we may find a neighbourhood of x which is disjoint from X, and so x does
not lie in the closure of X. This proves that X is closed, as claimed.
Proposition D.1.3.
(1) Each of the canonical maps Gn ,→ G is a closed embedding.
(2) G is Hausdorff, and if each Gn is furthermore complete, then G is complete.
(3) Any compact subset of G is contained in some Gn .
(4) If H is a subgroup of G, and if we set Hn := Gn ∩ H, then each of thenatural maps limn Hn → H and limn Gn /Hn → G/H is a topological−→−→isomorphism. Furthermore, H is closed in G if and only if Hn is closed
in Gn for each n.
Proof. Fix n0 , and let Un0 be any open subgroup of Gn0 . If we set Vn :=Un0 ∩ Gn for n ≤ n0 , and choose Vn such that Vn ∩ Gn−1 ⊆ Vn−1 for n > n0 ,then we may follow Remark D.1.1 (2) to find an open subgroup U ⊂ G such thatU ∩ Gn0 = Un0 . Thus the topology of Gn0 is induced by that of G, proving (1).
To see that G is Hausdorff, we have to show that if g ∈ G\{0}, then we can findan open subgroup of G not containing g. Choose n0 such that g ∈ Gn0 \{0}, and let
Un0 be any open subgroup of Gn0 such that g 6∈ Un0 . By the result of the preceding
paragraph, we may find an open subgroup U of G such that Gn0 ∩ U = Un0 . Thus
we see that g 6∈ U, as required. This proves the first claim of (2).

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b denote the completion of G.
Suppose now that each Gn is complete, and let G
b is a topological embedding (since G is Hausdorff,
The tautological morphism G → G
as we’ve just seen). Since each Gn is complete, the composite embedding Gn ,→b realizes Gn as a closed subgroup of G.
b If G is not complete, then we may
G ,→ Gbb (as we’ve just noted) andfind x ∈ G \ G, and then, since each Gn is closed in Gb such that x + Wn isdoes not contain x, we may find an open subgroup Wn of G
disjoint from Gn ; P
we may furthermore choose that Wn so that Wn+1 ⊆ Wn .
∞Now let U := n=1 Wn ∩ Gn ; by Remark D.1.1, this is an open subgroup of G.
b (i.e. its closure in G)
b is an
By the construction of completions, its completion Ubbb
open subgroup of G. Since G is dense in G, we see that (x + U ) ∩ G is non-empty,b ) ∩ Gn is non-empty for some value of n. On the other hand,
and thus that (x + U
we see thatb ⊆ U + Wn =U∞X

(Wi ∩ Gi ) + Wn

i=1

=

X

(Wi ∩ Gi ) +X(Wi ∩ Gi ) + Wn ⊆

i>n

i≤n

X
i≤n

Gn +

X

W i + Wni>n

= Gn + Wn+1 + Wn = Gn + Wn ,Thus (x + Gn + Wn ) ∩ Gn is non-empty, and so (x + Wn ) ∩ Gn is non-empty,
b completing thecontradicting our choice of Wn . This shows that necessarily G = G,proof of (2).
Let X ⊆ G be a compact subset; since G is Hausdorff (by what we’ve justproved), X is closed. Suppose that X ( Gn for every Sn; then we may find a∞sequence xn of elements of X such that xn 6∈ Gn . Then n=1 {xn } is an infinite
subset of X which is closed and discrete in G (and thus in X), by Lemma D.1.2.
This contradicts the compactness of X, and so (3) is proved.
Now let H be a subgroup of G. By definition of the induced topology, the open
subgroups of H are of the form H ∩ U , with U an open subgroup
S∞of G. Write
Un := Gn ∩ U, so that Un is an open subgroup of G and U = n=1 Un . ThenH ∩ Un = Hn ∩ Un = Hn ∩ U is an open subgroup of Hn , and Remark D.1.1 shows
that
∞
[
[
Hn ∩ Un = H ∩Un = H ∩ U
n=1

n=1

is also an open subgroup of limn Hn , proving the first claim of (4). The topological−→∼isomorphism limn Gn /Hn −→ G/H is immediately verified via a comparison of the−→universal mapping properties of the source and target.
Certainly, if H is closed in G then Hn is closed in Gn for each n. Conversely,
suppose that Hn is closed in Gn for each n. Then each quotient Gn /Hn is again
Hausdorff, and so part (2) shows that limn Gn /Hn is Hausdorff. Thus H, which is−→
the kernel of the canonical morphism G = limn Gn → limn Gn /Hn , is closed in G,−→−→
as claimed, completing the proof of (4).

D.2. Countably generated topological modules. We first note the following simple lemma.

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M. EMERTON, T. GEE, AND E. HELLMANN

Lemma D.2.1. If A is a left Noetherian (not necessarily commutative) ring,then the category of countably generated A-modules is an abelian subcategory of thecategory of all A-modules.
Proof. The only point that is not completely clear is closure under passage
to subobjects. To see this, note that if N is countably generated, then we may
write N = limn Nn , with each Nn finitely generated. Then if M is an A-submodule−→of N , we have M = limn Mn , where Mn := M ∩ Nn . Since A is left Noetherian by−→assumption, we see that each Mn is again finitely generated, and so M is indeed
countably generated.
Suppose now that A is a compact Hausdorff left Noetherian topological (possibly non-commutative) ring. Any finitely generated A-module then admits a uniquecompact Hausdorff topology — its canonical topology — with respect to which
it becomes a topological A-module, and all morphisms between finitely generated
A-modules are continuous and strict (i.e. induce topological quotient maps fromtheir domains onto their images) with respect to the canonical topologies on theirsource and target. (See [Eme10, Prop. 2.1.3] for a proof in a particular case,which immediately generalizes. The key point is that, given a finite presentationA⊕m → A⊕n → M → 0, the image of the first arrow is necessarily closed, being acompact subset of a Hausdorff space, and so M obtains a natural compact Hausdorff quotient topology, which is easily checked to be independent of the choice offinite presentation. Similar arguments with finite presentations establish the claim
about morphisms.)
In the cases we are interested in, the topology on A will be defined by a neighbourhood basis of 0 consisting of open ideals, and then one sees that the canonicaltopology on any finitely generated A-module is similarly defined by a neighbourhood basis of 0 consisting of open A-submodules. In this case we say that A and
M admit linear topologies, and we assume from now on that the topology on A isindeed linear.
We let A⊕N denote a countable direct sum of copies of A. Writing A⊕N :=limn A⊕n (with respect to the evident transition maps), we endow A⊕N with its−→
inductive limit linear topology.
Proposition D.2.2.
(1) If M is a countably generated A-module, then M admits a unique topology— which we call its canonical topology — with respect to which some(equivalently any) surjection A⊕N → M is a topological quotient map.
Endowed with its canonical topology, M becomes a linear topological Amodule.
(2) Morphisms between countably generated A-modules are necessarily continuous and strict with respect to the canonical topology on source andtarget.
Proof. If M is a countably generated A-module, then we may write M =limn Mn , where {Mn } is an increasing sequence of finitely generated A-submodules−→of M . If {Mn0 } is any other increasing sequence of finitely generated A-submodulesof M , then evidently {Mn } and {Mn0 } are mutually cofinal, in the sense that each
Mn is contained in some Mn0 0 , and conversely. Thus, if we endow each Mn with

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221

its canonical topology, the resulting inductive limit linear topology on M , coming from writing it as the colimit of the Mn , is independent of the choice of thesequence {Mn }. We declare this topology to be the canonical topology of M .
Since each Mn admits a neighbourhood basis of 0 consisting of open A-submodules,
it follows from Remark D.1.1 (2) that the same is true of M . One then easily deduces that M is a topological A-module.
If f : M → N is a morphism of finitely generated A-modules, then one may
choose the {Mn } and {Nn } such that f restricts to morphisms Mn → Nn . Thesemorphisms are necessarily continuous with respect to the canonical topology on
source and target, and hence so is the morphism
f : M = lim Mn → lim Nn = N.
−→
−→
n

n

If f is injective, then we may in fact first choose the Nn , and then set Mn :=M ∩Nn . Then Mn is closed in Nn , and so Proposition D.1.3 (4) shows that M → N
is a closed embedding.
If f is surjective, then we may in fact first choose the Mn , and then set Nn :=f (Mn ). Again by Proposition D.1.3 (4), we see that M → N is a topologicalquotient map.
Altogether, we have now proved (2), and (1) is a special case of the strictness
claim of (2).
Appendix E. Representations of p-adic analytic groups
E.1. Smooth G-representations and modules over the ring O[[G]]. We
suppose that G is a p-adic analytic group, and fix a compact open subgroup H of G.
Then the usual completed group ring O[[H]] := limJ/H open O[H/J] is a compact←−(two-sided) Noetherian linear topological ring by a result of Lazard [Laz65].
Definition E.1.1. We write
O[[G]] := O[G] ⊗O[H] O[[H]] = c-IndG
H O[[H]];
this is a priori a (left) representation of G with a compatible right module structure
over O[[H]], but following [Koh17, §1] (see also [Sho20, §3]), it actually has anatural ring structure, uniquely determined by the requirement that each of O[G]and O[[H]] is a subring. The O-algebra O[[G]] is furthermore independent of the
choice of H up to canonical isomorphism.
Any O[[G]]-module is in particular an O[G]-module, and so can be thoughtof as an O-linear G-representation. Not every O-linear G-representation admits astructure of O[[G]]-module compatible with its O[G]-action (which is just to saythat not every O[G]-module extends to a O[[G]]-module), but an important fact isthat smooth G-representations (whose definition we now recall) do.
Definition E.1.2.
(1) If M is an O-linear G-representation, we say that an element m ∈ M issmooth if m is O-torsion, and if M is fixed by some open subgroup of G.
We let Msm denote the subset of smooth elements of M ; it is a G-invariant
O-submodule of M .
(2) A representation of G on a O-module M is called smooth if Msm = M .

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M. EMERTON, T. GEE, AND E. HELLMANN

If M is a smooth G-representation, then the O[G]-action on M extends canonically to an O[[G]]-action. Thus the abelian category of smooth O-linear G-representations
may be regarded as a full subcategory of the category of O[[G]]-modules. (See [Sho20,
Lem. 3.5].)
Remark E.1.3. By definition, an element of an O-linear G-representation M
is smooth if and only if it is smooth when M is regarded as an H-representation
for some (or, equivalently, any) compact open subgroup H of G. By definition of
the topology on O[[H]], we see that an element m ∈ M is smooth if and only if
the cyclic O[[H]]-module generated by m is discrete when regarded as a quotientof O[[H]].
Remark E.1.4. Just from the definitions, we see that if M is an O-linear
G-representation, then Msm is the maximal smooth sub-G-representation of M .
Furthermore, if M itself is an O[[G]]-module, then Msm (defined in terms of theunderlying O[G]-module structure) becomes a sub-O[[G]]-module of M , when endowed with its canonical O[[G]]-module structure. (This follows from the fact that
if H is any compact p-adic analytic group, e.g. a compact open subgroup of G, thenthe augmentation ideal of O[H] generates the augmentation ideal of O[[H]].)
Remark E.1.5. Most of the preceding discussion (the definitions of O[[G]]and of smooth G-representations, and the realization of smooth representations asparticular kinds of O[[G]]-modules) applies more generally to any topological group
G which contains a profinite open subgroup. However, many of the results that weprove below (for p-adic analytic G) do not hold in this more general context. (SeeExample E.2.3 below for an illustration of this.)
We let (O[[G]]−Mod)c.g. denote the full subcategory of O[[G]]−Mod consistingof countably generated modules.
Lemma E.1.6. (O[[G]]−Mod)c.g. is a Serre subcategory of O[[G]]−Mod, which
has enough projectives. Its projective objects are precisely the countably generated
projective O[[G]]-modules.
Proof. The only non-trivial point to check regarding the Serre subcategory
claim is that submodules of countably generated O[[G]]-modules are again countably generated. To see this, note first that, for any compact open subgroup H
of G, the ring O[[G]] is countably generated as a O[[H]]-module. Thus any countably generated O[[G]]-module is also countably generated as an O[[H]]-module.
Lemma D.2.1 shows that any O[[G]]-submodule is then countably generated overO[[H]], and thus also over O[[G]].
Evidently any countably generated O[[G]]-module is a quotient of a countably
generated free O[[G]]-module. Since the latter are projective even in O[[G]] −Mod, they are projective in (O[[G]] − Mod)c.g. . This proves the claim about enoughprojectives. Any projective object of (O[[G]] − Mod)c.g. is then a retract of acountably generated free module, and so is a countably generated projective module.
As already noted in the proof of Lemma E.1.6, any countably generated O[[G]]module is also countably generated as an O[[H]]-module, and so admits a canonicaltopology (independent of the choice of H), as in Proposition D.2.2.
We note the following elementary lemma.

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Lemma E.1.7. A countably generated topological O[[H]]-module M is discretein its canonical topology if and only if M is a smooth H-representation.
Proof. If M is discrete, and if m ∈ M , then the sub-O[[H]]-module of M
generated by m is discrete. Remark E.1.3 then shows that m is smooth. Thus
Msm = M .
Conversely, suppose that M is smooth, and choose a surjection O[[H]]⊕N → M.
For any n ≥ 0, the image of O[[H]]⊕n factors through a discrete, and so finite,quotient, and so has finite image Mn ⊆ M . We have M = limn Mn , and by−→construction, the canonical topology on M is the inductive limit linear topology
on M . Lemma D.1.2 implies that M is discrete, as claimed.

We now define a functor from the category (O[[G]] − Mod)c.g. of countably generated O[[G]]-modules to the category Pro sm. G of pro-smooth G-representations
via
M 7→ Pro(M ) := “ lim ”M/U,←−
where U runs over the codirected set of O[[G]]-submodules of M that are open withrespect to the canonical topology on M . It follows from Lemma E.1.7 that M/U
is in fact a smooth G-representation. The functoriality of the formation of Pro(M )follows from the fact that if f : M → N is a morphism of countably generated
O[[G]]-modules, then f is continuous with respect to the canonical topologies on
its source and target. Indeed, we then see that f −1 (V ) is an open O[[G]]-submoduleof M whenever V is an open O[[G]]-submodule of N , so that we have an induced
morphism
Pro(M ) := “ lim ”M/U → “ lim ”M/f −1 (V ) → “ lim ”N/V =: Pro(N ).
←−←−←−Proposition E.1.8.
(1) The functor M 7→ Pro(M ) is right exact.
(2) If H is a compact open subgroup of G, and if N is a countably generated
O[[H]]-module, then O[[G]]⊗O[[H]] N is acyclic for the left-derived functors
Li Pro .
(3) All countably generated smooth representations are acyclic for the leftderived functors Li Pro .
Proof. Let 0 → M 0 → M → M 00 → 0 be a short exact sequence of countably
generated O[[G]]-modules. If U is an open O[[G]]-submodule of M , we let U 0 =M 0 ∩ U, and let U 00 denote the image of U in M 00 . Since M → M 00 is an open Gequivariant mapping, we see that, as U runs over all G-invariant open submodulesof M, the images U 00 run over all G-invariant open submodules of M 00 . Thus we
obtain a short exact sequence
0 → “ lim ”M 0 /U 0 → Pro(M ) → Pro(M 00 ) → 0.
←−
There is an evident epimorphism Pro(M 0 ) → “ lim ”M 0 /U 0 , so that←−
(E.1.9)Pro(M 0 ) → Pro(M ) → Pro(M 00 ) → 0
is exact, as claimed.
We now prove (2). Namely, if N is a countably generated O[[H]]-module, thenwe may find a resolution P • → N consisting of countably generated free O[[H]]modules, and O[[G]] ⊗O[[H]] P • is then a resolution of M := O[[G]] ⊗O[[H]] N by

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countable generated free O[[G]]-modules. (Recall that O[[G]] is flat over O[[H]].)
Using this resolution to compute the left derived functors Li Pro, the claim of (2)follows by applying Lemma E.1.10 (2) below.
Finally, we turn to proving (3). Suppose to begin with that M 00 is a countably generated smooth G-representation, and let M be a countably generated
free O[[G]]-module admitting a surjection M → M 00 . Lemma E.1.7 shows thatthe kernel M 0 of this surjection is open in M , and so Lemma E.1.10 (1) below
shows that L1 Pro(M 00 ) = 0; thus L1 Pro vanishes on countably generated smooth
representations.
Now let H be a compact open subgroup of G. We may write find a surjection
M := O[[G]] ⊗O[[H]] N → M 00 for some smooth representation N of H. The kernel
N 0 of this surjection is again countably generated and smooth, and since (2) shows∼that M is L• Pro-acyclic, we find that Li+1 Pro(M 00 ) −→ Li Pro(M 0 ) for i ≥ 1. An
obvious dimension shifting argument, taking into account the result of the preceding
paragraph, gives (3).

It is not clear to us that (E.1.9) is left-exact in general; such left-exactness
would be equivalent to the intersections U 0 being cofinal in the collection of open
G-invariant submodules of M 0 . (Without the G-invariance condition, this would
always hold, since M 0 → M is necessarily a topological embedding.) The following
lemma gives some situations in which this does hold.
Lemma E.1.10. (E.1.9) is left-exact if either(1) M 0 is open in M , or(2) the embedding M 0 ,→ M is of the form
M 0 := O[[G]] ⊗O[[H]] N 0 ,→ O[[G]] ⊗O[[H]] N =: M,
where N 0 ,→ N is an embedding of O[[H]]-modules (for some compact
open subgroup H of G).
Proof. (1) If M 0 is open in M , then any open G-invariant submodule U 0 of
M is itself an open G-invariant submodule of M , and so we can set U = U 0 . Thusin this case (E.1.9) is exact.
(2) If V 0 is an open G-invariant submodule of M 0 , then W 0 := N 0 ∩ V 0 is an
open O[[H]]-submodule of N 0 , and so U 0 := O[[G]]⊗O[[H]] W 0 is an open G-invariant
submodule of M 0 contained in V 0 ; thus submodules of this latter form are cofinal
among all open G-invariant submodules of M 0 . Now since N 0 ,→ N is necessarily atopological embedding, we may find an open O[[H]]-submodule W of N such thatW 0 = N 0 ∩ W. Then U := O[[G]] ⊗O[[H]] W is an open G-invariant submodule of
M for which M 0 ∩ U = U 0 . Thus in this case (E.1.9) is again exact.
0E.2. Derived categories of smooth representations. Let G be a p-adic
analytic group, and let O[[G]] be the ring of Definition E.1.1. We let D(O[[G]])denote the stable ∞-category associated to the abelian category of O[[G]]-modules.
It admits its standard t-structure, with respect to which it is both left and rightcomplete. (This is true for the stable ∞-category associated to the abelian categoryof modules over any ring, as we noted in Section A.5.)
Definition E.2.1. We let Dsm (O[[G]]) denote the full subcategory of D(O[[G]])consisting of objects whose cohomologies in every degree are smooth G-representations.

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The t-structure on D(O[[G]]) induces a t-structure on Dsm (O[[G]]), whose heartis the abelian category sm. G of smooth G-representations. The full subcategory
sm. G of O[[G]]−Mod is closed under passage to subobjects, quotients, and colimits
in O[[G]]. Thus sm. G is again a Grothendieck abelian category, and in particular
admits enough injectives.
∼

Proposition E.2.2. There is a canonical equivalence D(sm. G) −→ Dsm (O[[G]]).
Proof. The inclusion of smooth G-representations as a full subcategory ofO[[G]]-modules (which is an exact functor!) induces a t-exact functor F : D(sm. G) →D(O[[G]]), which evidently factors through Dsm (O[[G]]). We begin by showing thatthe restriction of F to D+ (sm. G) is fully faithful.
Firstly, we note that sm. G is precisely the heart of Dsm (O[[G]]), and so therestriction of F to D+ (sm. G) is exactly the functor of Proposition A.5.6. A consideration of that proposition shows that to show that this restriction of F is fully
faithful, it suffices to show that if X is a smooth G-representation, and Y is aninjective smooth G-representation, then we may find a smooth G-representation Z
admitting a surjection Z → X such ExtiO[[G]] (Z, Y ) = 0 for i > 0. For this, choose
a compact open subgroup H of G, and set Z = c-IndG
H X = O[[G]] ⊗O[[H]] X, which
admits a natural surjection to X.
Now the forgetful functor O[[G]] − Mod → O[[H]] − Mod admits an exact left
adjoint, namely O[[G]] ⊗O[[H]] –. Thus it preserves injectives. This same functor,
restricted to sm. H, where it can be interpreted as c-IndG
H , provides a left adjoint
to the forgetful functor sm. G → sm. H; thus this latter functor also preserves ini
jectives. Consequently ExtiO[[G]] (c-IndG
H X, Y ) = ExtO[[H]] (X, Y ), with Y injective
in sm. H, and so we reduce to verifying that if X and Y are objects of sm. H withY injective, then ExtiO[[H]] (X, Y ) = 0 for i > 0. In short, we have replaced G in
the original problem by its compact open subgroup H. In this case, the required
statement follows from Corollary B.1.16 (applied in the context of Example B.1.1).
We next note that the inclusion sm. G ,→ O[[G]] − Mod admits a right adjoint(as it must, by the adjoint functor theorem, since colimits of smooth representations
are smooth), which admits a quite explicit description: it is
M 7→ Msm := colimH,n M [$n ]H ,where in the colimit, n ranges over positive integers and H ranges over open subgroups of G. By Proposition A.7.1, we may then form the corresponding rightderived functor RΓsm : D(O[[G]]) → D(sm. G), which is right adjoint to F . Furthermore, RΓsm has finite cohomological amplitude.31 The proposition now follows
from Proposition A.7.3 (2).

Example E.2.3. As the following (counter)example will show, Proposition E.2.2
genuinely depends upon the hypothesis that G is p-adic analytic, or at least on various properties of the completed group ring O[[G]] as a pro-Artinian ring which
need not hold for more general locally
Q∞pro-p-groups.
For our example, we take G = i=0 Z/pZ, with its natural profinite topology.
To ease the comparison with the results we cite, we work with coefficients in krather than O, but the reader may easily convince themselves that this alteration
31If G has dimension d, and if H is sufficiently small (so that it is a uniform pro-p-group),
then the derived functor of M 7→ M [π n ]H as cohomological amplitude d + 1, and so the same istrue of RΓsm (which is the filtered colimit of these various derived functors).

226

M. EMERTON, T. GEE, AND E. HELLMANNof the coefficients plays no essential role, and that the discussionQ
readily extends tonthe case of O-coefficients. The group ring of the product Gn := i=0 Z/pZ admitspp
the description k[Gn ] = k[x0 , . . . , xn ]/(x0 , . . . , xn ), and so the completed group ringk[[G]] admits the descriptionk[[G]] = k[[x0 , . . . , xn , . . .]]/(xp0 , . . . , xpn , . . .) := lim k[[x0 , . . . , xn ]]/(xp0 , . . . , xpn ).
←−n
If we let m denote the maximal ideal (i.e. the augmentation ideal) of k[[G]], thenalthough m is open in k[[G]], the natural profinite topology on k[[G]] is weaker thanits m-adic topology. E.g. m/m2 = limn khx0 , . . . , xn i is infinite dimensional, and←−the profinite topology on k[[G]] endows m/m2 with its natural profinite topology,rather than the discrete topology.
Let R denote the polynomial subring k[x0 , . . . , xn , . . .] of k[[G]]. The categoryof smooth G-representations may be identified with the category of R-modules inwhich each element is killed by one of the ideals (xi )i≥n for some n ≥ 0. Thisis an example of one of the categories denoted A that are introduced in [Nee11,
Construction 1.1], and [Nee11, Thm. 1.1] then shows that D(sm. G) is not left
complete, while [Nee11, Rem. 1.2] shows that the formation of countable productsin D(sm. G) is of unbounded cohomological amplitude (illustrating the necessity ofsome bounded amplitude criterion in Proposition A.5.3). Thus the argument in the
proof of Proposition E.2.2 which extends the result from D+ (sm. G) to D(sm. G)
breaks down.
In fact even the canonical functor F : D+ (sm. G) → D+ (k[[G]]) is not fully
faithful. Indeed, the one-dimensional trivial representation of G corresponds to thek[[G]]-module k := k[[G]]/m, and so Ext1k[[G]] (k, k) = Homk (m/m2 , k) (the k-linear
dual to k). On the other hand Ext1sm. G (k, k) corresponds to the proper subspace of
Homk (m/m2 , k) consisting of functionals that are continuous when m/m2 is endowedwith its natural profinite topology.
E.3. Compact and coherent objects. We begin with some preliminaries
based on the results of [Sho20, §§2,3].
Lemma E.3.1. If M is a smooth O-linear G-representation, then the following
are equivalent:(1) M is finitely generated as an O[G]-module.
(2) M is finitely generated as an O[[G]]-module.
(3) There is a compact open subgroup H of G, a smooth representation of H ona finitely generated torsion O-module N , and a surjection c-IndG
H N → M.
(4) For every compact open subgroup H of G, there is a smooth representation of H on a finitely generated torsion O-module N , and a surjection
c-IndG
H N → M.
Proof. Clearly (4) implies (3) implies (1) implies (2). Suppose then that (2)holds, and let S ⊂ M be a finite generating set for M as an O[[G]]-module. If
H is any compact open subgroup of G, and if N is the O[[H]]-submodule of M
generated by S (which coincides with the O[H]-submodule generated by S, since
the smoothness of M implies that an open subgroup of H acts trivially on S), thenc-IndG
H = O[G] ⊗O[H] N surjects onto M , proving (4).
Definition E.3.2. We say that a smooth G-representation is finitely generated
if it satisfies the equivalent conditions of Lemma E.3.1.

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Definition E.3.3. A smooth representation of G on an O-module M is offinite presentation if there is an exact sequence
M1 → M2 → M → 0
of smooth G-representations with M1 and M2 finitely generated.
The following result is [Sho20, Prop. 3.8].
Lemma E.3.4. A smooth G-representation is of finite presentation (in the senseof Definition E.3.2) if and only if it is finitely presented as an O[[G]]-module.
We now have the following result.
Proposition E.3.5. The category sm. G is compactly generated. The compact
objects of sm. G are precisely the smooth representations of finite presentation.
Proof. We sketch the proof, which is standard. To begin with, note that
if M is a smooth O-linear G-representation, and {Mi } is the S(directed!) set offinitely generated subrepresentations of M , then evidently M = i Mi = colim Mi .
For each value of i, choose a surjection c-IndG
Hi Ni → Mi , with Hi compact openin G and Ni a smooth representation of Hi on a finitely generated torsion Omodule. If Mi0 denotes the kernel of this surjection, then, as we have just seen,
Mi0 = colim(Mi0 )j , where (Mi0 )j runs over the finitely generated subrepresentations
0of Mi0 . Then M = colimi,j (c-IndG
Hi Ni )/(Mi )j , which exhibits M as a colimit ofrepresentations of finite presentation. Thus, if we prove that representations of finite
presentation are compact, we will have proved that sm. G is compactly generated.
Now if M is compact, then applying its compactness property to its representation as a colimit of representations of finite presentation, we find that M is a direct
summand of a representation of finite presentation, and so is easily seen itself tobe a representation of finite presentation (either directly, or via an application ofLemma E.3.4). Conversely, if M is of finite presentation, then Lemma E.3.4 showsthat M is compact (even in the category of O[[G]]-modules).

We don’t know in general whether the smooth G-representations of finite presentation form an abelian subcategory of sm. G, although it seems plausible thatthey do; see Conjecture 6.1.4, as well as Remark 6.1.6 for a discussion of known
results in this direction. If they do, then we find that sm. G is a locally coherent abelian category, and thus that the t-structure on D(sm. G) is coherent. In
this case we find (applying Lemma A.6.7) that the subcategory of coherent objectsbin D(sm. G) is precisely the full subcategory Df.p.
(sm. G) of (cohomologically)bounded complexes whose cohomologies are finitely presented.
E.4. Complexes of O[[G]]-modules as pro-smooth complexes. The formal Pro-category Pro sm. G admits enough projectives. Thus we may form its
associated stable ∞-category D− (Pro sm. G), with its natural t-structure.
We may also from the formal Pro-∞-category Pro Db (sm. G). The t-structure
on Db (sm. G) induces a left complete t-structure on Pro Db (sm. G) (this is thedual statement to Proposition A.6.14), whose heart is equivalent to Pro sm. G.
Thus (by the dual version of Proposition A.5.6, i.e. [Lur17, Prop. 1.3.3.7]) there isa canonical functor(E.4.1)D− (Pro sm. G) → Pro Db (sm. G).

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Recall that (O[[G]] − Mod)c.g. denotes the abelian category of countably generated O[[G]]-modules. By Lemma E.1.6, this is an abelian category with enoughprojectives (namely, the countably generated free O[[G]]-modules), and so we mayconsider its stable ∞-category of bounded above complexes D− (O[[G]]−Mod)c.g. .
Since in fact (by Lemma E.1.6) (O[[G]]−Mod)c.g. is a Serre subcategory of O[[G]]−−Mod, it also makes sense to consider Dc.g.
(O[[G]]), the full sub-stable ∞-category−of D (O[[G]]) consisting of complexes with countably generated cohomologies. The−
canonical t-structure on D− (O[[G]]) induces a t-structure on Dc.g.
(O[[G]]), whosec.g.
heart is precisely (O[[G]] − Mod) .
Lemma E.4.2. There is a canonical equivalence ∼−(E.4.3)D− (O[[G]] − Mod)c.g. −→ Dc.g.
(O[[G]]).
Proof. This follows from the dual version of Proposition A.5.6, i.e. [Lur17,
Prop. 1.3.3.7], since the projective objects in (O[[G]] − Mod)c.g. are precisely thecountably generated projective O[[G]]-modules, which can be used to compute Exts
in O[[G]] − Mod.
Now left deriving the functor of Proposition E.1.8 gives a functor

L Pro : D− (O[[G]] − Mod)c.g. → D− (Pro sm. G).
Precomposing with an inverse to the equivalence of (E.4.3), and postcomposingwith (E.4.1), we obtain a functor(E.4.4)−Dc.g.
(O[[G]]) → Pro Db (sm. G).

b(sm. G) denote the full subcategory of Db (sm. G) consisting of obLet Dc.g.
b(O[[G]]) denote
jects whose cohomologies are countably generated, and let Dc.g.smb
the full subcategory of D (O[[G]]) consisting of complexes whose cohomologies arecountably generated and smooth. The equivalence of Proposition E.2.2 evidently
induces an equivalence(E.4.5)

∼

b
b
Dc.g.
(sm. G) −→ Dc.g.sm(O[[G]]).

It follows from Proposition E.1.8 (3) that the composite of the equivalence (E.4.5)
b(O[[G]]) of) the functor (E.4.4) is equivalent to thewith (the restriction to Dc.g.sm
composite of the canonical functorsbDc.g.
(sm. G) ,→ Db (sm. G) ,→ Pro Db (sm. G).
−
In summary, we have the functor Dc.g.
(O[[G]]) → Pro Db (sm. G) of (E.4.4),b−which when restricted to Dc.g. (sm. G) (thought of as a subcategory of Dc.g.
(O[[G]])
via the equivalence (E.4.5)) is equivalent to the evident inclusion.

Appendix F. Derived moduli stacks of group representations
In this appendix we give a brief introduction to and motivation of derived
moduli stacks of Galois representations. We refer to [Zhu20, §2] for the details.
LetΓ := hg1 , . . . , gm | r1 = · · · = rn = 1i
be a finitely presented group, with generators g1 , . . . , gm and relations r1 , . . . , rn .
(The motivational example to bear in mind is the Galois group of a finite extensionof local or global fields; in the body of the notes we will consider instead various
topologically finitely presented profinite groups, given for example by the absolute

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Galois groups of p-adic local fields, but for the sake of motivation the case of aliterally finite group suffices.)
Then giving a representation Γ → GLd (A) (for some ring A) amounts to giving
elements x1 , . . . , xm ∈ GLd (A) satisfying the relations rj . Let VΓ be the closed
subscheme of (GLd )m classifying m-tuples satisfying the relations rj . There is anaction of GLd on VΓ via simultaneous conjugation, and the quotient stack XΓ :=[VΓ / GLd ] is then the moduli stack of d-dimensional representations of Γ (overarbitrary commutative rings, as we have written it here, although of course we
could work over any given base ring O, and then consider representations overO-algebras). It is an algebraic stack (over Z, or whichever other base ring O wechoose).
If we fix a particular representation ρ : Γ → GLd (l) (defined over some field l),then this representation gives rise to a l-valued point of XΓ , and the lifting ringRρ of ρ is a versal ring to XΓ at the point ρ. The tangent space to XΓ at ρ canbe described in terms of group cohomology as H 1 (Γ, Adρ), while H 0 (Γ, Adρ) computes the infinitesimal automorphisms of ρ. Passing to higher degree cohomology,
one knows that elements of H 2 (Γ, Adρ) give rise to obstructions to higher orderdeformations of ρ. However, this obstruction theory is an extra piece of data — it
is not determined intrinsically by the stack XΓ itself. For example, in general one
can’t “compute” H 2 (Γ, Adρ) in terms of XΓ (unlike H 0 and H 1 ).
This comes up concretely when one uses group cohomology to describe the lifting ring Rρ : if we consider liftings over l-algebras, and write hi := diml H i (Γ, Adρ),
then we can write Rρ as a quotient l[[x1 , . . . , xh1 −h0 +d2 ]]/(f1 , . . . , fh2 ). When XΓ
has the “expected dimension” −h0 + h1 − h2 at ρ, the fi must form a regular sequence in l[[x1 , . . . , xh1 −h0 +d2 ]], and Rρ is a local complete intersection. But if XΓ
has larger-than-expected dimension, then the ring-theoretic structure of Rρ is lessclear, and less clearly related to the quantity h2 .
One reason, then, for considering a derived analog of XΓ is that it tightens therelationship between group cohomology and the infinitesimal structure of XΓ at its
points ρ. The precise definition of the derived representation stack is somewhat
involved (see [Zhu20, §2]), but it proceeds roughly as follows: as is usual in a derived setting, rather than just taking into account the relations ri appearing in Γ,
we consider the “higher relations”, i.e. the relations between the relations and soon. In the non-abelian setting of not-necessarily-commutative groups, this can be
expressed formally by finding a simplicial resolution Γ• of Γ by free groups (or, infact, for technical reasons, by free monoids). The schemes VΓn classifying homomorphisms Γn → GLd (A) (for coefficient rings A) are then of the form Spec RΓnfor certain rings RΓn that are localizations of polynomial rings (since the Γn are
free monoids), and which are themselves organized into a simplicial commutative
ring RΓ• . We may consider the corresponding derived affine scheme VΓ• , and form
the derived algebraic stack XΓ• := [VΓ• / GLd ], whose underlying classical substackcoincides with XΓ . In a suitable ∞-categorical framework (see e.g. the discussionof [Zhu20, §2]) the derived stack XΓ• is well-defined independently of the choice ofresolution Γ• .
There is a standard useful analogy, in which one compares the passage fromclassical to derived rings (or, in the terminology of [CS21], which we employ, “animated rings”) to the passage from reduced rings to arbitrary rings that occurs inpassing from varieties to schemes, and the passage from classical to derived schemes

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or stacks to the passage from varieties to not-necessarily-reduced schemes. Even
for a classical variety over a field l, it is interesting to consider its points overnon-reduced rings: e.g. its tangent spaces are computed via l[ε]/(ε2 )-valued points.
Similarly, for a derived stack X (even one which is actually classical), there are
non-classical analogues of l[ε]/(ε2 ) whose points in a classical or derived stack X
determine the obstruction theory of X , and using such points, we can extend theclassical computation of tangent spaces to compute the cotangent complex LX ,xof X at points x of X . In particular, when X = XΓ• , one finds that
LXΓ• ,ρ = C • (Γ, Adρ)∗ [−1](a shift of the dual to the cochain complex that computes the group cohomology
H • (Γ, Adρ)) — see e.g. [Zhu20, §2.2]. This extends the geometric interpretationsof H 0 and H 1 in the classical context.
This is especially useful in cases when H i (Γ, Adρ) = 0 for i > 2, since thenthe cohomology of LXΓ• ,ρ appears only in degrees [−1, 1]. This implies that XΓ• isquasi-smooth (i.e. derived l.c.i.) in a neighbourhood of ρ. Concretely, this meansthat there is a formally smooth morphism
Spec l[[x1 , . . . , xh1 −h0 +d2 ]]/(f1 , . . . , fh2 ) → XΓ• ,taking the closed point of the domain to ρ, provided that the domain is understood in a derived sense: if the fi do not form a regular sequence, then the ringl[[x1 , . . . , xh1 −h0 +d2 ]]/(f1 , . . . , fh2 ) will be a non-classical animated ring (its higher
homotopy groups will be computed by the Koszul complex of the sequence (fi )).
We then see that XΓ• is in fact classical in a neighbourhood of ρ if and only if theunderlying classical stack XΓ is of the expected dimension −h0 +h1 −h2 = χ(LXΓ• ,ρ )in such a neighbourhood.
In Sections 8 and 9 we apply these ideas when Γ is (at least closely relatedto) a local or global absolute Galois group. (Their extension to the context of
profinite groups is treated in [Zhu20, §2.4].) To this end, we recall that local
Galois cohomology always vanishes in degrees > 2, and that the same is true for
global Galois cohomology, provided we work with odd residue characteristics (aswe always will).

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Email address: emerton@math.uchicago.edu
Department of Mathematics, University of Chicago, 5734 S. University Ave.,
Chicago, IL 60637, USA
Email address: toby.gee@imperial.ac.uk
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Email address: e.hellmann@uni-muenster.deMathematisches Institut, Westfälische Wilhelms Universität Münster, Einsteinstraße 62, 48149 Münster, Germany