LOCALIZATION OF SMOOTH p-POWER TORSIONREPRESENTATIONS OF GL2 (Qp )ANDREA DOTTO, MATTHEW EMERTON, AND TOBY GEE

Abstract. We show that the category of smooth representations of GL2 (Qp )on p-power torsion modules localizes over a certain projective scheme, and givesome applications.

Contents
1. Introduction
2. Preliminaries on smooth GL2 (Qp )-representations
3. Localization of smooth GL2 (Qp )-representations
4. Applications and examples
Appendix A. Category-theoretic background
References

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1. Introduction
In this paper we establish a localization theory for the category of smooth representations of GL2 (Qp ) on Zp -modules on which p is locally nilpotent, for any primep ≥ 5.
1.1. Initial statement of results. In order to make a precise statement, we introduce some notation. Let O denote the ring of integers in a finite extension Eof Qp and let A denote the category of smooth representations of GL2 (Qp ) on
O-modules on which p acts locally nilpotently, and which have a central character×equal to some fixed character ζ : Q×p → O . Let F denote the residue field of O.
Let X denote a chain of projective lines over F with ordinary double points,of length (p ± 1)/2 (where the sign is equal to −ζ(−1)). Our definition of X is
motivated by the mod p semisimple local Langlands correspondence for GL2 (Qp )due to Breuil [Bre03b, Defn. 1.1]. Indeed, one can think of X as a moduli space ofsemisimple representations ρ : GQp → GL2 (Fp ), with the intersection points of the
AD was supported in various stages of this project by the James D. Wolfensohn Fund at
the Institute for Advanced Study, as well as by the Engineering and Physical Sciences ResearchCouncil [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number
Theory (The London School of Geometry and Number Theory), University College London, andImperial College London. ME was supported in part by the NSF grants DMS-1601871, DMS1902307, and DMS-1952705. TG was supported in part by an ERC Advanced grant, EPSRC grant
EP/L025485/1 and a Royal Society Wolfson Research Merit Award. 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).
1

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A. DOTTO, M. EMERTON, AND T. GEE

P1 s corresponding to irreducible representations. It then follows from Paškūnas’
results [Paš13] that the blocks in the subcategory Al.adm of locally admissible representations with central character ζ are in bijection with the closed points of X.
Given a closed subset Y of X with open complement U := X \ Y , we let AY denote the subcategory of A consisting of those representations all of whose irreducible
subquotients lie in blocks corresponding to closed points of Y . The subcategory AY
is localizing (in the usual sense, recalled in Appendix A.2), and is in particular a
Serre subcategory of A, and we set AU := A/AY .
Our main result shows that the category A can be localized over X, in thefollowing precise sense (see Theorem 3.3.1 and Remark 3.3.2).
Theorem 1.1.1. The collection {AU } forms a stack (of abelian categories) overthe Zariski site of X.
1.2. Applications. In Section 4 we demonstrate that Theorem 1.1.1 (and the results that go into its proof) has many concrete consequences for the representation
theory of GL2 (Qp ). In particular we compute various Ext1 groups between compact
inductions of Serre weights, and between such compact inductions and irreducible
representations. For example, in Proposition 4.2.2 we show that if σ is a Serreweight and π is absolutely irreducible, thendim Ext1 (π, c-IndG
KZ (σ)) ≤ 1;
in addition, we explicitly describe all of the non-split extensions that arise. Similarly, in Proposition 4.2.4 we compute the dimensions of the Ext1 groups for extensions in the opposite direction (with a genericity assumption), and in Section 4.3
we compute the Ext1 groups between full compact inductions of Serre weights.
In addition, in Section 4.1 we make precise the fashion in which the categoryof finitely generated smooth representations is built out of finite length representations (which admit finite filtrations by irreducible representations), together withcompact inductions of Serre weights. In particular we prove Proposition 4.1.3,
which shows that every finitely generated representation π admits a maximal subobject πfl which is of finite length, and the quotient π/πfl is a successive extensionof submodules of representations c-IndG
KZ σ, for σ a Serre weight. (The structureof these submodules is not mysterious; indeed, as recalled in Remark 4.1.4, mostsuch submodules are isomorphic to c-IndG
KZ σ.)
1.3. The relationship to the Bernstein centre. Our results can be understoodby comparison with other, more classical contexts, in the representation theory of
p-adic reductive groups. Note that the particular choice of coefficient ring O does
not affect things much, as long as its residue field F has characteristic p. Thus itmakes sense to compare our results with the classical case of smooth representationsof a p-adic reductive group G over a field F of characteristic zero.
The first thing to note is that, in this classical case just as in our case, there aremany smooth representations, indeed naturally occuring ones, that are not admissible. For example, if G = GL2 (Qp ) and V is a finite dimensional representation
Gof KZ = GL2 (Zp )Q×
p , then c-IndKZ V is admissible if and only if V has vanishing“Jacquet module” (see e.g. [Bus90, Thm. 1 supp.] for a proof of a more general
result), a condition which certainly need not hold in characteristic zero (e.g. if V
is the trivial representation). In characteristic p, c-IndG
KZ V is never admissible,

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )3provided V itself is non-zero. At least morally, this is because in characteristic p,
the Jacquet module of a non-zero V will never vanish.
However, if F is a field of characteristic zero, if G is any p-adic reductive group,
and if Z denotes the Bernstein centre of the category of smooth G-representations on
F-vector spaces, then Bernstein’s results [Ber84] show that any finitely generated
smooth G-representation is admissible over Z. (This is the p-adic analogue ofa theorem of Harish-Chandra, to the effect that if G is a real reductive groupwith Lie algebra g, then any finitely generated U (g), K -module is admissible
over the centre Z of U (g).) Thus we may think of a not-necessarily-admissible
smooth G-representation in characteristic zero as being a “family” of admissible
representations parameterized by some subscheme of Spec(Z). An analogous result
has also recently been proved by Dat–Helm–Kurinczuk–Moss [DHKM22] for the
category of smooth representations of G over a field of characteristic ` 6= p, or moregenerally over a Noetherian Z` -algebra.
On the other hand, the analogous result is not true for our category A. Indeed,
independent results of A.D. and Ardakov–Schneider [AS21, Dot21] show that theBernstein centre of A is trivial. Thus any finitely generated non-admissible representation (such as the c-IndG
KZ V introduced above) provides a counterexample tothe analogue of Bernstein’s result.
The localization theory of this paper was developed in part to rectify this absenceof an interesting Bernstein centre for A: rather than simply regarding objects of A
as lying over the Spec of its Bernstein centre, we localize them over the non-affine
variety X. Then, by forming the Bernstein centres of the various categories AU ,we may endow X with the structure of a topologically ringed space, and we expect
to prove (in forthcoming work) that this endows X with the structure of a formalb — such that X itself is the underlying
scheme over O — which we denote by X
b The triviality of the Bernstein centre of A would thenreduced scheme over F of X.
correspond to the fact that the only global sections of the structure sheaf OXb are
the constant functions. In turn, this statement itself would be an extension to theb of the fact that the only globally defined functions on the connected
thickening X
projective variety X are the constant functions — thus the fact that X is projective,
rather than affine, is closely related to the fact that the Bernstein centre of A istrivial.
1.4. The relationship with local Langlands. Our definition of X suggests that
there is a connection between our results and the mod p or p-adic local Langlands
correspondence, and indeed, this is the case. To explain and motivate this, we firstrecall the analogous result in the ` 6= p context. Namely, if F is a field of characteristic zero or ` 6= p, or more generally if O is a complete DVR whose residue field
is such a field F, then, for any p-adic field F and any n ≥ 1, we may consider themoduli stack X parameterizing n-dimensional representations of the Weil–Delignegroup of F over O-algebras, and the local Langlands correspondence over O identifies Z — the Bernstein centre of O[GLn (F )] — with the ring of functions on X ,by a theorem of Helm–Moss [HM18]. Since X is a reductive quotient of an affine
scheme, we may rephrase this as saying that Spec Z is the moduli space associatedto X , or more precisely an adequate moduli space in the sense of Alper [Alp14].
In forthcoming work we expect to prove a p-adic analogue of this result. Namely,b — the thickening of X induced by the localized Bernstein cenwe will show that X
tre of A discussed above — is a formal moduli space associated to the stack XQp

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A. DOTTO, M. EMERTON, AND T. GEE

parameterizing (ϕ, Γ)-modules arising from continuous rank two p-adic representations of GalQp of appropriately fixed determinant.
Remark 1.4.1. In the ` 6= p case, the identification of the Bernstein centre with thering of functions on X is an outward manifestation of a deeper phenomenon, namely
the categorical local Langlands correspondence considered in [BZCHN20, FS21,
Hel20, Zhu20]. In the forthcoming paper [DEG], we establish (using the resultsof this paper as one our tools) an analogous categorical p-adic local Langlands
correspondence for GL2 (Qp ), in the form of a fully faithful functor from the derived
category of A to an appropriate derived category of coherent sheaves on the stackof (ϕ, Γ)-modules X2,Qp mentioned above.
However, while we anticipate that fully faithful functors of this kind exist in
great generality (in particular, for the representations of GLn (F ) for any n and any
p-adic local field F ), we don’t expect the results of the present paper to generalisein any obvious way, even to GL2 (F ) with F 6= Qp .
Relatedly, on the Galois side of the Langlands correspondence, we do not expect
the stacks of (ϕ, Γ)-modules to admit interesting associated moduli spaces beyond
the cases of GL2 (Qp ) and GL1 (F ), F being any finite extension of Qp . For example,
if X2,Qp2 denotes the analogous stack for GL2 (Qp2 ), then the various specialisation
relations in reducible families cause any morphism X2,Qp2 → Z with Z being a
locally separated formal algebraic space to factor through the structure morphism
X2,Qp2 → Spf O.
1.5. Methods of proof. The subcategory Aadm of A consisting of admissible
representations is well-understood, thanks to the results of Paškūnas [Paš13]. As
already mentioned above, it factors into a product of blocks, which are labelled by
the closed points of X. In our perspective, the objects of Al.adm are supported on
finite closed subsets of X, so that our localization theory, for these objects, is justa restatement of Paškūnas’s results.
The novel aspects of our theory are seen when the objects under consideration
are not locally admissible, since then they will localize over subsets of X with
non-empty interior. A basic consequence of the localization theory is that, if two
representations have disjoint support, then all Exti groups between them vanish.
Conversely, proving such statements is the key to proving Theorem 1.1.1. The key
result is Lemma 3.1.6, which (at least morally) shows that if σ is a Serre weight, andπ is a supported on some closed subset Y , then the Exti between π and c-IndG
KZ σ
are also supported on Y . It is proved by exploiting the fact that these Exti groups
are typically not finite dimensional over F, but they are of countable dimension.
Since they are also modules over the Hecke algebra H(σ) ∼= F[T ], this makes themamenable to an application of a well-known technique of Dixmier [Dix63], at least
when F is uncountable; and we can always arrange that F is uncountable via an
appropriate base-change. With this result in hand, we are able to show that for
any open cover of X, the corresponding “Čech resolution” of any π is acyclic, andwe deduce Theorem 1.1.1 from this acyclicity.
Remark 1.5.1. We do not explicitly use the p-adic local Langlands correspondence
in our arguments. However, we do use the results of Paškūnas [Paš13], whose
proofs use p-adic local Langlands. In particular we frequently use the classificationof blocks in the locally admissible category Al.adm , and for some of the results inLOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )5Section 3.8, we furthermore make use of Paškūnas’s description of the Bernsteincentres of these blocks.
Remark 1.5.2. We expect that the results of this paper extend to the cases of p = 2,
3, with minor modifications to the statements (e.g. to the definition of X in thecase p = 2: in this case X should consist of a single P1 ). Our arguments would
necessarily become more complicated in the cases p = 2, 3, so that we could nolonger argue uniformly in p. Note that this already happens for the description ofblocks in [Paš13], which is not carried out in loc. cit. for p = 2, 3, but is rather
in [Paš14, PT21]. Thus we have not attempted to make the modifications of ourarguments that would be required to treat these cases.
1.6. A guide to the paper. In Section 2 we recall some basic results on thesmooth p-adic representation theory of GL2 (Qp ), and explain our interpretation ofPaškūnas’ classification [Paš13] of the blocks of locally admissible representationsin terms of the closed points of X. In Section 2.5 we prove some general results on
Ext groups, including finiteness properties, compatibility with extension of scalars,
and compatibility with colimits. Finally in Section 2.6 we prove some technicalresults on the Bernstein centres of these blocks.
We establish our localization theory in Section 3. We begin with the basic definitions of our localizing categories, and then prove the crucial Lemma 3.1.6 mentioned
above, and (with some work) deduce Theorem 1.1.1. The rest of Section 3 is devoted to a discussion of the analogue on the representation theory side of completion
along a closed subset of X. In particular, we prove Theorem 3.8.1, which is theanalogue of Beauville–Laszlo gluing in this setting. As well as being of independentinterest, this is used crucially in our forthcoming paper [DEG].
In Section 4 we give a variety of examples and application of our localization
theory, showing in particular how it can be used to compute extensions between
compact inductions c-IndG
KZ σ, and extensions between such compact inductions
and admissible representations. Finally in Appendix A we recall some backgroundmaterial in category theory that we use in the body of the paper.
1.7. Notation and conventions. We fix throughout the paper a prime p ≥5. Fix an algebraic closure Qp of Qp , and write GQp for the absolute Galois
group Gal(Qp /Qp ). Let O denote the ring of integers in a fixed finite extension
E of Qp , and let F be the residue field of O; all representations considered in thispaper will be on O-modules.
We write G = GL2 (Qp ), K = GL2 (Zp ), and Z = Q×
p for the centre of G.
×
We fix a continuous character ζ : Q×
throughout
the paper. We write
→
O
p
×
×
ω : Q×→F
forthereduction
mod
p
of
the
character
Q
→
Z×
p
p
p
p , x 7→ x|x|.
We always let A denote a complete Noetherian local O-algebra, where “local” isunderstood in the strong sense that A is a local ring, and the morphism O → A isa local morphism; given the first assumption, this second hypothesis is equivalent
to requiring that the residue field k be of characteristic p. We then let AA denotethe category of smooth G-representations with central character ζ on locally mA torsion A-modules. In more detail, mA denotes the maximal ideal of A, a module
is locally mA -torsion if each element is annihilated by some power of mA , and a
representation is smooth if each element is fixed by some open subgroup of G; so
we are considering precisely those representations over A which are called smooth in[Eme10a] (and which have the required central character). In the case that A = O

6

A. DOTTO, M. EMERTON, AND T. GEE

we write A for AA . If V is a finite length object of A we will write JH(V ) for themultiset of Jordan–Hölder factors of V .
We will write Iζ for the two-sided ideal of A[G] or A[KZ] (or other group algebrasof groups containing Z) generated by [z] − ζ(z) for z ∈ Z.
We write CA for the category of smooth KZ-representations with central character ζ on A-modules on which p is locally nilpotent. In the case that A = O wewrite C for CA . A Serre weight is an irreducible object of CA , or equivalently of Ck .
Since every irreducible k-representation of K is defined over Fp , and Z is actingby ζ, the set of isomorphism classes of Serre weights is independent of A.
More explicitly, the isomorphism classes of Serre weights are represented by the
representations Symb F2p ⊗ deta of GL2 (Fp ), where 0 ≤ a < p − 1 and 0 ≤ b ≤ p − 1.
It is sometimes convenient to view a as an element of Z/pZ, and we will do so
without further comment.
We make some use of tame types, by which we mean the principal series andcuspidal representations of GL2 (Fp ). For any pair of characters χ1 6= χ2 : F×p →GL (F )

O× , we have the principal series representation I(χ1 , χ2 ) = IndB(F2 p ) p χ1 ⊗ χ2 ,×and for any character χ : F×which does not factor through the norm, thep2 → O
corresponding cuspidal representation Θ(χ). (Up to sign, these are the Deligne–
Lusztig inductions of regular characters of the nonsplit maximal torus in GL2 (Fp ).)These representations are irreducible of dimensions p+1, p−1 respectively, and theonly isomorphisms between them are that I(χ1 , χ2 ) ∼= I(χ2 , χ1 ) and Θ(χ) ∼= Θ(χp ).
The Jordan–Hölder factors of their reductions modulo p are as follows. Firstly, if
we write χi (x) = xni , then the Jordan–Hölder factors of the reduction of I(χ1 , χ2 )are Sym[n1 −n2 ] ⊗ detn2 and Sym[n2 −n1 ] ⊗ detn1 , where [ni − nj ] denotes the uniqueinteger in (0, p − 1) congruent to ni − nj (mod p − 1).
Secondly, if χ(x) = xi+(p+1)j with 1 ≤ i ≤ p, then if 1 < i < p then the reductionof Θ(χ) has two Jordan–Hölder factors, namely Symi−2 ⊗ det1+j and Symp−1−i ⊗ deti+j ;while if i = 1 or p, then it is irreducible and isomorphic to Symp−2 ⊗ det1+j .
1.8. Acknowledgements. We would like to thank Pierre Colmez, Gabriel Dospinescu,
and Vytautas Paškūnas for helpful correspondence and conversations, and comments on an earlier draft of this paper.
2. Preliminaries on smooth GL2 (Qp )-representations
2.1. Irreducible representations. If A = k is an algebraically closed field of
characteristic p, the irreducible objects of Ak have been classified in [BL94, Bre03a].
As we will often need to extend scalars to transcendental extensions of k, in thissection we will extend their results to arbitrary and possibly imperfect extensions
k/Fp .
To state the classification, recall more generally that the restriction functorfrom AA to CA has an exact left adjoint given by compact induction and denoted
V 7→ c-IndG
KZ (V ).
The case when V = σ is an irreducible k[KZ]-module (a “Serre weight”) isof particular importance. We begin by recalling the description, due to Bartheland Livné [BL94], of their endomorphism rings, which we denote by H(σ) :=EndG (c-IndG
KZ σ).

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )7

Lemma 2.1.1. We have an isomorphism(2.1.2)H(σ) ∼= k[T ],
generated by a specific choice of Hecke operator T . Furthermore, c-IndG
KZ σ is freeover k[T ].
In what follows we will denote T by Tp in case of a notational clash, for instance
if T ∈ k[T ] is a formal variable.
We also recall (again from [BL94]) the classification of G-equivariant morphisms
between the compact inductions of Serre weights.
G
0
Lemma 2.1.3. If σ and σ 0 are distinct Serre weights then HomG (c-IndG
KZ σ , c-IndKZ σ) =
p−1
0
s
s
0 except in the case when {σ, σ } = {det , Sym⊗ det } for some s, in whichcase we have the following:(1) All elements of

G
p−1
0
s
⊗ det sHomG c-IndG
KZ Sym ⊗ det , c-IndKZ Sym

and of0p−1
s⊗ det s , c-IndG
HomG c-IndG
KZ Sym ⊗ det
KZ Sym

are k[T ]-equivariant, where k[T ] acts on each of source and target in thenatural way, i.e. with T acting by the appropriate Tp .
(2) Each of

G
0
p−1
s
HomG c-IndG⊗ det sKZ Sym ⊗ det , c-IndKZ Sym
and
0
p−1
s
⊗ det s , c-IndG
HomG c-IndG
KZ Sym ⊗ det
KZ Symis free of rank 1 over k[T ].
(3) We can choose k[T ]-generators
G
p−1
0
s
⊗ det sα ∈ HomG c-IndG
KZ Sym ⊗ det , c-IndKZ Sym



0
p−1
s
⊗ det s , c-IndG
β ∈ HomG c-IndG
KZ Sym ⊗ det
KZ Sym

andsuch thatα ◦ β = β ◦ α = T 2 − 1.
The cokernels of α and β have length 2.
This lemma has the following useful corollary.
Corollary 2.1.4.
(1) If σ is not a twist of Sym0 or Symp−1 , then any subobject of c-IndG
KZ σ isof the form f (T )c-IndG
σ
for
some
f
(T
)
∈
k[T
].
KZ
(2) If σ is a twist of Sym0 or Symp−1 , then any subobject π of c-IndG
KZ σsatisfies inclusions
G(T 2 − 1)f (T )c-IndG
KZ σ ⊆ π ⊆ f (T )c-IndKZ σ

for some f (T ) ∈ k[T ]. Furthermore, π is a k[T ]-submodule of c-IndG
KZ σ.
Proof. If π is non-zero, then it contains a Serre weight σ 0 , and the inclusionGG0
π → c-IndG
KZ σ induces a G-equivariant morphism c-IndKZ σ → c-IndKZ σ. The
corollary now follows easily from Lemma 2.1.3.
8

A. DOTTO, M. EMERTON, AND T. GEEThis corollary is the beginning of the classification of irreducible G-representations.
Namely, we see that if π is irreducible, then by choosing a Serre weight σ in π, we
obtain a surjection
G
c-IndGKZ σ/f (T )c-IndKZ σ → π,
for some irreducible polynomial f (T ) ∈ k[T ]. Classifying irreducibles then amountsGto analyzing the structure of such quotients c-IndG
KZ σ/f (T )c-IndKZ σ (with f (T )irreducible). In the case when k is algebraically closed, so that f (T ) = T − λ forsome λ ∈ k, this was done by Barthel and Livné when λ 6= 0, and by Breuil [Bre03a]when λ = 0. Theorem 2.1.5 and Lemma 2.1.6 below summarize their results.
Theorem 2.1.5. Let k be an algebraically closed field of characteristic p. Every
irreducible object of Ak is isomorphic to a representation in the following list:×(1) η ◦ det for some smooth character η : Q×p →k ;
×(2) (η ◦ det) ⊗ St for some smooth character η : Q×p → k , where St is the
Steinberg representation of G; or(3) c-IndG
KZ (σ)/(f (T )), where σ is an irreducible representation of k[KZ] and f =T − λ for some λ ∈ k, and(σ, λ) 6∈ {(Sym0 ⊗ dets , ±1), (Symp−1 ⊗ dets , ±1)}.
Conversely, all these representations are irreducible and pairwise nonisomorphic, with the following exceptions:
s
G
p−1
0
∼
⊗ dets )/(T − λ) if λ 6= ±1.
(c-IndG
KZ Sym ⊗ det )/(T − λ) = (c-IndKZ Sym(c-IndG Symr ⊗ dets )/T ∼= (c-IndG Symp−1−r ⊗ detr+s )/T.
KZ

KZ

We recall what happens in the excluded cases of part (3) of the preceding theorem.
Lemma 2.1.6. There are non-split short exact sequences
p−1
0 → (nr±1 ◦det)⊗det s → (c-IndG⊗ det s )/(T ±1) → (nr±1 ◦det)⊗StG ⊗det s → 0
KZ Symand0ss0 → (nr±1 ◦det)⊗StG ⊗det s → (c-IndG
KZ Sym ⊗ det )/(T ±1) → (nr±1 ◦det)⊗det → 0.

These short exact sequences are induced by the morphisms of Lemma 2.1.3(3).
It is a standard result that they are not split; we will give a proof in the course ofproving Lemma 2.1.12 below.
The next result generalizes Theorem 2.1.5 to an arbitrary coefficient field. See [Paš13,
Section 5.3] for similar results under the assumption that k is perfect.
Theorem 2.1.7. Assume A is a complete Noetherian local O -algebra, and let π
be an irreducible object of AA . Then π is isomorphic to one of the following:×(1) η ◦ det for some character η : Q×p →k ;
×(2) (η ◦ det) ⊗ StG for some character η : Q×p → k ; orGG(3) c-IndKZ σ /f (T ) c-IndKZ σ for some irreducible k[KZ]-representation σ
and some irreducible polynomial f (T ) ∈ k[T ], with f (T ) 6= T ± 1 if σ is a
twist of Sym0 or Symp−1 .

Conversely, each of these representations is irreducible.

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )9

Proof. If π is an irreducible object of AA then it is a k[G]-module. As noted above,
there exists a surjectionΠ := (c-IndG
KZ σ)/f (T ) → π
for some irreducible k[KZ]-representation σ and some irreducible polynomial f (T ) ∈k[T ]. Bearing in mind Theorem 2.1.5, it suffices to prove that Π is irreducible whenever deg(f ) > 1. To do so, observe first that
Homk[G] (c-IndG
KZ σ, Π)is deg(f )-dimensional over k. Indeed, since both representations are finitely generated over k[G], by [Paš13, Lemma 5.1] this can be checked after extending scalars
to an algebraic closure of k. Then it follows from the fact that Π ⊗k k has deg(f )Jordan–Hölder factors, each of which contains σ as a KZ-subrepresentation withmultiplicity one.
Now let Π0 ⊂ Π be a nonzero G-stable subspace. We claim that Π0 contains theimage of a nonzero element of Homk[G] (c-IndG
KZ σ, Π). To see that this implies thetheorem, notice that this space contains the field k[T ]/f (T ), which has dimension
deg(f ) over k, and so coincides with it. It follows that every nonzero element isbijective, and so Π0 = Π.
To prove the claim, notice that since Π0 is nonzero it contains the image of anonzero morphism
0c-IndGKZ (σ ) → Π
for some irreducible k[KZ]-module σ 0 . Extending scalars to k and using Theorem 2.1.5 we see that σ ∼= σ 0 , except possibly when σ and σ 0 are twists of Sym0
p−1
or Sym . However, in this case we have thatp−1
02c-IndG)[1/(T 2 − 1)] ∼= c-IndG
KZ (Sym
KZ (Sym )[1/(T − 1)],and we are done since our assumption on f (T ) implies that Π has no Jordan–Hölder
factors isomorphic to characters or Steinberg twists.

We note the following easy corollaries of these classification results.
Corollary 2.1.8. Let σ1 , σ2 be Serre weights and let fi ∈ H(σi ) be irreducible
G∼
monic polynomials of degree greater than one. Then c-IndG
KZ (σ1 )/f1 = c-IndKZ (σ2 )/f2
if and only if σ1 = σ2 and f1 = f2 .
Proof. Passing to a finite extension l/k where both f1 and f2 split, this is animmediate consequence of Theorem 2.1.5, Theorem 2.1.7, and [Paš13, Lemma 5.1].
Corollary 2.1.9. Let σ be a Serre weight. If π 0 is a non-zero subobject of c-IndG
KZ σ,
G
0
then (c-IndKZ σ)/π is of finite length.
Proof. Corollary 2.1.4 allows us to reduce to the case π 0 = f (T )c-IndG
KZ σ forsome non-zero f (T ) ∈ k[T ], and then by factoring f (T ), we reduce to the casethat f (T ) is irreducible, in which case the corollary follows from Theorem 2.1.7
and Lemma 2.1.6.

We will use the following result in the proof of Lemma 3.6.6.

10

A. DOTTO, M. EMERTON, AND T. GEE

Lemma 2.1.10. Let σ be a Serre weight, and g be a non-zero irreducible element ofG
H(σ). Then (c-IndG
KZ σ)[1/g]/(c-IndKZ σ) is Artinian, and is in fact an essential
GG1extension of g c-IndKZ σ/c-IndKZ σ.
Proof. Write Π := c-IndG
KZ σ. It is enough to show that for each n ≥ 1,
1
1
Π → nΠ
gg
is an essential extension. Let Θ be a sub-G-representation of g1n Π/Π; we can and1
do assume that Θ is not contained in gn−1
Π/Π. Let Θ be the image of Θ under
the isomorphism1∼Π/Π −→ Π/g n Πgngiven by multiplication by g n ; we need to show Θ has non-trivial intersection withg n−1 Π/g n Π. By Corollary 2.1.4, Θ is stable under the action of H(σ), and inparticular under the action of g. We have g n−1 Θ 6= 0 by our assumption that Θ is1
not contained in gn−1
Π, and g n−1 Θ ⊆ Θ ∩ (g n−1 Π/g n Π), as required.

In fact, the Artinian property for locally admissible representations follows fromfiniteness of the socle, by Paškūnas’s results on the structure of blocks. We prove
this in the following proposition.
Proposition 2.1.11. Let J be an injective object of Al.adm . Then J is Artinian if
and only if socG (J) has finite length.
Proof. One direction is immediate, so we assume that socG (J) is finite-dimensional
and prove that J is Artinian. After a finite extension of scalars, J is isomorphic to a
direct sum of injective envelopes of absolutely irreducible objects, hence by [Paš13,Corollary 5.18] it suffices to prove that indecomposable injective objects of any
absolutely irreducible block of Al.adm are Artinian. This is a consequence of one of
the main results of [Paš13], which shows that the endomorphism rings of projective
generators of blocks are right and left Noetherian: in fact, they are finite modules
over their centres, which are Noetherian rings.

We will also find the following refinement of Lemma 2.1.10 useful.
Lemma 2.1.12. Let σ be a Serre weight, and g = T − λ ∈ H(σ) for some λ ∈ k.
G
Then (c-IndG
KZ σ)[1/g]/(c-IndKZ σ) is uniserial.
Proof. Except in the case when σ is a twist of Sym0 or Symp−1 and g = T ± 1, weGknow by Theorem 2.1.7 that c-IndG
KZ σ/g(c-IndKZ σ) is irreducible, in which thelemma is an easy consequence of Lemma 2.1.10. If we are in one of the remainingcases, we may twist so that σ = Sym0 or Symp−1 and g = T − 1, and so we assume
this from now on.
0
Since we are working modulo powers of T − 1, we localize each of c-IndG
KZ σ
G
p−1
and c-IndKZ Sym
at the prime ideal (T − 1) of k[T ], so as to obtain a chain ofinclusions
G0p−1(c-IndG)(T −1)KZ Sym )(T −1) ⊃ (c-IndKZ Sym
G0p−1⊃ (T − 1)(c-IndG)(T −1)KZ Sym )(T −1) ⊃ (T − 1)(c-IndKZ Sym

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )11with successive quotients equal to 1G (the trivial representation of G), StG , and0
1G again. Since any non-isomorphic map of (c-IndG
KZ Sym )(T −1) to itself factors0
p−1
through the quotient by (T − 1)(c-IndG, we
KZ Sym )(T −1) , and similarly for Sym
see that the induced extensions of 1 by StG and of StG by 1 are non-split. Thiseasily proves the uniseriality in the cases of Sym0 and Symp−1 for g = T − 1 (andthus completes the proof of the lemma).

2.2. Blocks, and a chain of P1 s. We now discuss the full subcategory Al.adm
of A consisting of locally admissible representations. Recall [Eme10a, Defn. 2.2.15]
that a representation π ∈ A is locally admissible if every vector v ∈ π is smoothand generates an admissible representation, and is locally finite if for every v therepresentation generated by v is of finite length. In this setting locally admissible
representations are locally finite by [Eme10a, Thm. 2.2.17]. Hence the category
Al.adm is locally finite, unlike A, and so admits a decomposition into blocks. These
blocks are studied intensively in [Paš13].
Recall that, by definition, a block of Al.adm is an equivalence class of (isomorphism classes of) irreducible objects1 under the equivalence relation generated by
π1 ∼ π2 if Ext1A (π1 , π2 ) 6= 0 or Ext1A (π2 , π1 ) 6= 0.
As in the proof of [Paš13, Proposition 5.34], or equivalently using the spectral
sequence (2.5.5), we get the same equivalence relation if we work instead with
Ext1AF , which we will do in what follows. The blocks B that contain absolutely
irreducible representations are as follows:(1) B = {π} for an irreducible supersingular representation π,−1−1χ1 )} for characters χ1 , χ2 : Q×
χ2 ), IndG
(2) B = {IndGp →B (χ2 ⊗ ω
B (χ1 ⊗ ω−1
×±1
F such that χ1 χ2 6= 1, ω ,×−1(3) B = {IndGχ)} for a character χ : Q×p → F , andB (χ ⊗ ωG×−1(4) B = {χ, χ ⊗ StG , IndB (ωχ ⊗ ω χ)} for a character χ : Q×
p →F .
The following Proposition completes the classification of the blocks; the proof
makes use of Theorem 2.1.7 and [Paš13, Proposition 5.33], asserting that Exti commutes with finite extensions of F for representations of finite length (see Proposition 2.5.13 for a generalization to Afg ).
Proposition 2.2.1. Let σ = Symr ⊗ dets be a Serre weight, and let f ∈ H(σ) be0an irreducible polynomial of degree n > 1. Let π = c-IndG
KZ (σ)/f (T ), and let π
0
be an irreducible object of A not isomorphic to π. Then π and π are in the same
block of Al.adm if and only if
π0 ∼= c-IndG (σ 0 )/f ∗ (T )KZ

for σ 0 = Symp−3−r ⊗ detr+s+1 and f ∗ (T ) = T n f (1/T ).
Remark 2.2.2. By Lemma 2.1.3 we can assume without loss of generality that 0 ≤r ≤ p − 2. If r = p − 2, the exponent p − 3 − r denotes p − 2.
Proof of Proposition 2.2.1. Assume that π ∼ π 0 , and let π0 = π, π1 , . . . , πm = π 0
be irreducible objects of A such that Ext1 (πj , πj+1 ) 6= 0 or Ext1 (πj+1 , πj ) 6= 0 for
all j. Since the map (σ, f ) 7→ (σ 0 , f ∗ ) is an involution, by induction on m it suffices
1We note that it follows from the results of [BL94] and [Bre03a] that the irreducible objects of
A are automatically admissible, and hence lie in Al.adm . Thus we can equally well regard this asan equivalence relation on the irreducible objects of A.

12

A. DOTTO, M. EMERTON, AND T. GEE0∗to prove that π1 ∼= π or π1 ∼= c-IndG
KZ (σ )/f (T ). Passing to a finite extensionof F and using [Paš13, Proposition 5.33] this is immediate from the classificationof blocks containing principal series representations.
0∗Conversely, assume that π 0 ∼= c-IndG
KZ (σ )/f (T ). Again by [Paš13, Proposi1tion 5.33] it suffices to prove that ExtAF0 (π ⊗F F0 , π 0 ⊗F F0 ) 6= 0, where F0 is a
splitting field of f over F. Since
G
0−1
Ext1AF0 (c-IndG)) 6= 0KZ (σ)/(T − λ), c-IndKZ (σ )/(T − λfor all roots λ of f in F0 , this is immediate since F0 /F is separable.



Remark 2.2.3. In the more general case of Ak with coefficients in a field k ofcharacteristic p, it could be the case that the splitting field l of f is inseparable
over k. This situation can be treated using the fact that for all i, j > 0 we have
G
i
0
−1 j
Ext1Al (c-IndG) ) 6= 0,
KZ (σ)/(T − λ) , c-IndKZ (σ )/(T − λwhich follows from the long exact sequences in Ext associated to
0
−1
0
−1 j
0
−1 j−1)→0) → c-IndG)→ c-IndG
0 → c-IndG
KZ (σ )/(T −λKZ (σ )/(T −λKZ (σ )/(T −λ

and
G
G
j
j−1
0 → c-IndG→ 0.
KZ (σ)/(T − λ) → c-IndKZ (σ)/(T − λ) → c-IndKZ (σ)/(T − λ)Now we let X denote a chain of P1 ’s over F with ordinary double points, oflength (p ± 1)/2, where the sign is positive if and only if ζ is odd. We choose
coordinates on each irreducible component of X in such a way that each singularpoint corresponds to 0 on one intersecting component and ∞ on the other. We
will refer to the points 0 and ∞ as marked points. We are going to label thecomponents of X by cuspidal types Θ(χ) with central character ζ = χp+1 ; notethat since Θ(χ) ∼= Θ(χ0 ) if and only if χ0 = χ or χ0 = χp , there are as many
cuspidal types with central character ζ as irreducible components of X. In order
to construct the labelling, it will be useful to have the following definition.
Definition 2.2.4. We will say that two cuspidal types τ1 , τ2 are adjacent if there
exist σi ∈ JH(τ i ) such that {σ1 , σ2 } is the set of Jordan–Hölder factors of a principal
series representation of F[GL2 (Fp )].
Recall from [BP12, Corollary 5.6] that the extensions amongst irreducible F[GL2 (Fp )]representations are classified by the following proposition.
Proposition 2.2.5. Let σ = Symb F2p ⊗ deta . Then one of the following is true:(1) there exist exactly two Serre weights σ1 , σ2 such that Ext1 (σ, σi ) is nonzero.
One of these extensions is the mod p reduction of a lattice in a cuspidal type,
and the other is the mod p reduction of a lattice in a principal series type.
(2) b = p − 2, there exists a cuspidal type τ with τ = {σ}, and there exists a
unique Serre weight σ1 such that Ext1 (σ, σ1 ) 6= 0. This nonsplit extension
is the mod p reduction of a lattice in a principal series type.
(3) b = 0, and there exists a unique Serre weight σ1 such that Ext1 (σ, σ1 ) 6= 0.
This nonsplit extension is the mod p reduction of a lattice in a cuspidal
type.
(4) b = p − 1, and σ is a projective F[GL2 (Fp )]-module.
Corollary 2.2.6. Let τ be a cuspidal type. Then one of the following is true:

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )13(1) there exist exactly two cuspidal types adjacent to τ , or(2) τ contains a twist of Sym0 or Symp−2 , and there exists precisely one cuspidal type adjacent to τ .
Proof. This is immediate from Proposition 2.2.5.

Proposition 2.2.7. There exist exactly two bijections τ 7→ X(τ ), from the set of
cuspidal types with central character ζ to the set of irreducible components of X,
with the following property: X(τ1 ) ∩ X(τ2 ) is not empty if and only if τ1 and τ2 areadjacent.
Proof. Assume first that ζ is odd. Twisting by the determinant, we may assume
that ζ|µp−1 (Qp ) = ω −1 . Then there exists a cuspidal type τ with central character ζsuch that τ ∼= Symp−2 , which therefore needs to be sent to one of the components
with only one singular point. By Corollary 2.2.6, the bijection is determined bychoosing which one.
Similarly, if ζ is even we can twist and assume that ζ|µp−1 (Qp ) = 1. In thiscase, the bijection is determined by the image of the cuspidal type whose reduction
contains the trivial F[GL2 (Fp )]-representation.

In what follows we will make an arbitrary choice amongst the two bijections
constructed in Proposition 2.2.7.
Remark 2.2.8. This ambiguity is related to the fact that the group Z/2 × Z/2
acts on the category A by autoequivalences arising from twisting by the quadratic×
characters Q×
p → F , since twisting by ramified characters changes the K-socle ofan irreducible object of A. On the other hand, twisting by the unramified quadratic
character gives rise to an isomorphism(c-IndG Symr ⊗ dets )/(T + λ) ∼= (nr−1 ◦ det) ⊗ (c-IndG Symr ⊗ dets )/(T − λ).
KZ

KZWe record an additional property of the map τ 7→ X(τ ).
Lemma 2.2.9. If τ1 , τ2 are adjacent cuspidal types, the pair (σ1 , σ2 ) in Definition 2.2.4 is unique.
Proof. Let σ10 , σ20 be the other Jordan–Hölder factors of τ i , if any exist. Let x =dim σ1 . Then dim σ2 = p + 1 − x, dim σ10 = p − 1 − x, and dim σ20 = x − 2. Since
σ2 is not isomorphic to σ20 , and there is a principal series extension between σ1
and σ2 , there is no principal series extension between σ1 and σ20 . Similarly, there is
no principal series extension between σ10 and σ2 . Finally, dim σ10 + dim σ20 = p − 3,so there is no principal series extension between σ10 and σ20 .

The following corollary is immediate (recalling that by definition, a principal
series type is irreducible).
Corollary 2.2.10. The map τ 7→ X(τ ) induces a bijection from singular pointsof X to isomorphism classes of principal series types.
2.3. Coordinates on X. Choose a Serre weight σ (with compatible central character). Recall that we have chosen coordinates on each irreducible componentof X in such a way that each singular point corresponds to 0 on one intersecting component and ∞ on the other. We will use the isomorphism (2.1.2) and themap τ 7→ X(τ ) to construct morphisms from the spectra of various Hecke algebras H(σ) to X. Thus we will be able to regard c-IndG
KZ σ as “lying over” a copy of

14

A. DOTTO, M. EMERTON, AND T. GEE

A1 = Spec F[T ] = Spec H(σ), which will form the basis of our localization theory
for AA .
Definition 2.3.1. Let σ be a Serre weight, and assume σ is not a twist of Symp−1 .
Then σ is contained in a unique cuspidal type τ . Define a map fσ : Spec H(σ) → X
in the following way.
(1) If σ is not a twist of Sym0 , Symp−2 , then it is contained in a unique principal
series type τ 0 . Then fσ is the inclusion of Spec H(σ) ∼= A1 in X(τ ) preserving the given coordinates and sending 0 to the point of X(τ ) corresponding
to τ 0 under Corollary 2.2.10.
(2) If σ ∼= (χ ◦ det) ⊗ Sym0 , then fσ is the inclusion of Spec H(σ) ∼= A1 in X(τ )preserving the given coordinate and sending 0 to the nonsingular marked
point.
(3) If σ ∼= (χ ◦ det) ⊗ Symp−2 , then fσ is the degree-two map A1 → X(τ ), (x 7→x + x−1 )±1 . (The sign is chosen so that 0 is sent to the singular point.)
We extend this definition to the case σ ∼= (χ ◦ det) ⊗ Symp−1 by letting fσ be the∼same map as in Case (2) for σ = (χ ◦ det) ⊗ Sym0 .
Remark 2.3.2. The definition in the case of twists of Symp−2 is motivated by the existence of extensions between nonisomorphic irreducible quotients of c-IndG
KZ (σ)/(T −λ), which does not happen for any other weights.
The definition for twists of Symp−1 and Sym0 is motivated by the isomorphism




1
1
G
0
p−1
∼
(Sym
)
c-Ind
c-IndG
(Sym
)
=
KZ
KZ
T2 − 1T2 − 1constructed in Lemma 2.1.3. Furthermore, by Lemma 2.1.6 if T = ±1 we still have
an isomorphism after semisimplification: more precisely,




G
p−1
0
JH c-IndG
(Sym
)/(T
±
1)
=
JH
c-Ind
(Sym
)/(T
±
1)
.
KZ
KZ
The key property of the maps fσ is given in the following proposition.
Proposition 2.3.3. Let x be a closed point of X. Define


[
(σ)
⊗
x
.
Bx =
JH c-IndG
H(σ)
K
σ s.t. x∈fσ (A1 )Then Bx is a block of AF , and the map x 7→ Bx is a bijection from the set of closed
points of X to the set of blocks of AF .
Proof. We begin with the case of F-rational points x ∈ X(F). Assume first that ζis even and x is not a marked point. There are two possibilities for the set ofweights σ such that fσ (A1 ) contains x: up to twist, it has the form
{Symi , Symp−3−i ⊗ deti+1 }for i 6= 0 (which are the factors of a cuspidal type), or{Sym0 , Symp−1 , Symp−3 ⊗ det}
(which are the factors of a cuspidal type, together with a Steinberg weight). By
Remark 2.3.2 it follows that in either case there exists λ ∈ k × such that




G
i
p−3−i
Bx = JH c-IndG⊗ deti+1 )/(T − λ−1 ,KZ (Sym )/(T − λ) ∪JH c-IndKZ (Sym

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )15which in the terminology of Section 2.2 is a block of type (2) if i 6= 0 and a blockof type (4) if i = 0.
Now assume that x is a marked point. Up to twist, the set of σ such that fσ (A1 )contains x has the form
{Symi , Symp−1−i ⊗ deti }which are the factors of a principal series type. We have




G
i
i
p−1−i
Bx = JH c-IndG
(Sym
)/T
∪
JH
c-Ind
(Sym
⊗
det
)/T
,
KZ
KZ
but these are two isomorphic supersingular irreducible representations, so we get a
block of type (1).
The analysis in the case that ζ is odd is similar. The only new case is when x isa point of a component indexed by a cuspidal type whose reduction is isomorphic
to a twist of σ = Symp−1 In thi case, if x is a singular point of X then fσ (x)−1 = 0,
so again Bx is a block of type (1). On the other hand, if x is a regular point of X
then there exists λ ∈ F× such that fσ−1 (x) = {λ, λ−1 }. The corresponding set
o
n
G
−1
)
Bx = c-IndG
KZ (σ)/(T − λ), c-IndKZ (σ)/(T − λis a block of type (2) if λ 6=∈ {±1} and a block of type (3) if λ ∈ {±1}.
This defines a map from X(F) to the set of blocks of Al.adm containing absolutely irreducible representations, and the classification in Section 2.2 implies that
it is a bijection. There remains to treat the case where x is not F-rational. Then xcorresponds under fσ to a maximal ideal of H(σ) generated by an irreducible polynomial f of degree n > 1, andG
c-IndGKZ (σ) ⊗H(σ) x = c-IndKZ (σ)/(f )is irreducible. Proposition 2.2.1 shows that Bx is a block consisting of two nonisomorphic irreducible representations, neither of which is absolutely irreducible.
Again by Proposition 2.2.1, the map x 7→ Bx defines a bijection from the set ofclosed points of X not defined over F to the set of blocks of AF not containing
absolutely irreducible representations. This completes the proof.

2.4. Categories of representations. Recall that we assume throughout the paper that A is a complete local Noetherian O-algebra with residue field of characteristic p. We say that a representation of G on an A-module is finitely generated
if it is finitely generated as an A[G]-module. We note that a smooth representationof G on an A-module, admitting a central character, is finitely generated if andonly if it is a quotient of a compactly induced representation c-IndG
KZ V , for somefinite length A-module V endowed with a smooth action of KZ. Any object of
AA is thus a colimit of objects of the form c-IndG
KZ V , where V is a finite length
A-module endowed with a smooth KZ-action.
We frequently use the following result:
Theorem 2.4.1. If A is a complete Noetherian local O-algebra, then any subrepresentation of a finitely generated smooth representation of G over A with fixed
central character ζ : Z → A× is again finitely generated. Equivalently, any finitely
generated smooth representation of G over A with central character ζ is Noetherian.
Proof. Let π be a finitely generated smooth A[G]-representation. By definition,
π is a quotient of c-IndG
KZ (V ) for some smooth A[KZ]-module V of finite length
over A[KZ]. Since quotients, subobjects and extensions of Noetherian objects are

16

A. DOTTO, M. EMERTON, AND T. GEE

Noetherian, it suffices to prove the theorem when π = c-IndG
KZ (V ), and it further
suffices to prove it in the case that V has length one, hence is an irreducible k[KZ]module. In this case the result follows from Corollary 2.1.9.

Remark 2.4.2. Theorem 2.4.1 does not hold in general for non-abelian p-adic reductive groups. For example, it fails for GL2 (Qp2 ), and presumably any time we’reoutside the case of GL2 (Qp ), or some essentially equivalent context.
The following Lemma (and its proof) goes through unchanged for general p-adic
reductive groups.
Lemma 2.4.3. If A is a complete Noetherian local O-algebra, then AA is a Grothendieck
category.
Proof. Since the category of A-modules is a Grothendieck category, it is enoughto show that AA has a generator. To form a generator of AA , let W run over a
set of isomorphism class representatives of smooth finite length A[KZ]-modules.
Since K has a cofinal countable system of open normal subgroups with finite quotient,
L these representatives form a countable set. We can take the generator to
be W c-IndGKZ W.
Corollary 2.4.4. If A is a complete Noetherian local O-algebra, the abelian category AA is locally Noetherian. The Noetherian objects of AA are precisely thefinitely generated objects.
Proof. By Lemma 2.4.3, filtered colimits are exact in AA ; so in order to see that AA
is locally Noetherian, we only need to exhibit a set of Noetherian generators of AA .
By Theorem 2.4.1, the compact inductions c-IndG
KZ (V ) are Noetherian whenever V
is a finite length smooth A[KZ]-module, since the finite length quotients of A are
Artinian rings. To see that these are generators, let f : π1 → π2 be a nonzero
morphism between objects of AA . We need to find a morphism c-IndG
KZ (V ) → π1
whose composition with f is nonzero. To do so, it suffices to choose v ∈ π1 suchthat f (v) 6= 0, and then notice that it generates a finite length A[KZ]-module,
since π1 is a smooth representation. This concludes the proof of the first assertion.
By Theorem 2.4.1, the finitely generated objects of AA are Noetherian, and theconverse is immediate from the definition.

2.5. Generalities about Ext’s in AA . The inclusion of AA into the category of
all A[G]-modules with central character ζ is exact and fully faithful, and admits
a right adjoint, given by passing to the submodule consisting of smooth vectors
annihilated by some power of mA . This right adjoint preserves injectives, and sosince the latter category admits enough injectives (being the category of modules
over a ring), so does the category AA . Of course, since AA is a Grothendieck
category, it also admits enough injectives for abstract reasons.
Recall that the functor c-IndG
KZ is an exact functor from CA to AA . It is left
adjoint to the forgetful functor (i.e. restriction) from AA to CA , and so this latterfunctor preserves injectives. Thus we have natural isomorphisms(2.5.1)∼i
ExtiAA (c-IndG
KZ V, π) −→ ExtCA (V, π)

whenever V is a object of CA and π is an object of AA . We will use (2.5.1) toreduce some questions about Ext groups in AA to Ext groups in CA . We begin
with the following basic statement about these Ext groups, whose statement andproof extend in an obvious way to arbitrary compact p-adic analytic groups.

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )17

Lemma 2.5.2. Assume that (A, m, k) is a complete Noetherian local O-algebra andthat V, W are objects of CA which are finitely generated over A. Then
ExtiCA (V, W )is finitely generated over A.
Proof. By induction on lengthK (V ), it suffices to prove the lemma when V is irreducible. We begin by proving that ExtiCk (V, W ) is finite-dimensional over k whenever W is an object of Ck of finite dimension over k. By induction on lengthK (W ),it suffices to prove this claim when W is irreducible. Let K1 be the first congruencesubgroup of K, which acts trivially on V and W . Since
HomCk (V, −) = Homk[KZ/K1 ]/Iζ (V, (−)K1 )there is a spectral sequence(2.5.3)

Extik[KZ/K1 ]/Iζ (V, H j (K1 , W )) ⇒ Exti+j
Ck (V, W ).

The group K1 acts trivially on W , which is a finite-dimensional k-vector space,and so H j (K1 , W ) is also finite-dimensional for all j: see for example [SW00, Cor.
4.2.5, Thm. 5.1.2] for a discussion of this fact, which follows from Lazard’s resultson the structure of k[[K]]. Finally, since k[KZ/K1 ]/Iζ is a finite k-algebra, we seethat the groups on the E2 -page are also finite-dimensional, completing the proof ofthe claim.
Now let W be an A-finite object of CA . Since A is Noetherian we know that
ExtjA (k, W ) is a finite k-vector space for all j, hence the lemma follows from theclaim proved before and the spectral sequence (2.5.5) to follow.
Lemma 2.5.4. Let (A, m, k) be a complete Noetherian local O-algebra, and let V, W
be objects of Ck . Then there is a spectral sequence(2.5.5)ExtiCk (V, ExtjA (k, W )) ⇒ Exti+j
CA (V, W ).

The same is true with Ak replacing Ck and AA replacing CA .
Proof. Notice that
HomCA (V, −) = HomCk (V, (−)[m])as functors from CA to k-vector spaces.
If I is an injective object of CA , then I[m] is injective in Ck , since (–)[m] isright adjoint to the inclusion of Ck into CA , which is exact. Furthermore, I is also
injective as an A-module, so that the right derived functors of (–)[m] on CA coincidewith (the composite of the forgetful functor to A-modules and) the right derived
functors of (–)[m] on the category of A-modules. To see this A-module injectivityof I, notice that if a → A is the inclusion of an ideal and ϕ : a → I is A-linear
then ϕ(a) is contained in I K0 for some open normal subgroup K0 ⊂ K, since A is
Noetherian. Viewing a and A as trivial K0 -modules, it follows that ϕ extends to amap A → I, since I is also injective as a K0 -representation.
Putting together the observations of the preceding paragraphs gives a Grothendieck
spectral sequence as in the statement of the lemma. The same proof works in thecase of AA . (To see that the forgetful functor from AA to A-modules preserves
injectives, we can factor this functor through the forgetful functor to CA , and then
combine what was proved above with the fact that restriction to a compact open
subgroup preserves injectives.)

18

A. DOTTO, M. EMERTON, AND T. GEE

Lemma 2.5.6. Assume A is a complete Noetherian local O-algebra. If π is a
finitely generated object of AA , then each ExtiAA (π, –) commutes with filtered colimits.
Proof. This follows from Corollary 2.4.4 and Proposition A.1.1 (3).

Lemma 2.5.7. Assume that A is a complete Noetherian local O-algebra. If π
and π 0 are objects of AA , with π being finitely generated and π 0 being countably
generated, then each ExtiAA (π, π 0 ) is a countably generated A-module.
Proof. Since A is Noetherian, every submodule of a countably generated A-module
is countably generated. Since π is finitely generated, and thus finitely presentedby Theorem 2.4.1, a standard dimension-shifting argument reduces us to checking
the claim in the case when π = c-IndG
KZ V, for some finitely generated A-module V
endowed with a smooth KZ-action. We then consider the isomorphisms
∼

∼

i
0
0
ExtiAA (c-IndG
ExtiCA (V, W ),KZ V, π ) −→ ExtCA (V, π ) −→ lim−→
Wwhere W runs over the finitely generated A[KZ]-submodules of π 0 . (The firstisomorphism is by (2.5.1), and the last isomorphism is proved in the same way as
Lemma 2.5.6.) The directed set of such W contains a countable cofinal subset,
since π 0 is countably generated, and thus the lemma follows from the fact that each
ExtiCA (V, W ) is finitely generated, by Lemma 2.5.2.

We will sometimes make use of the following comparison (due to Paškūnas)
between Ext groups in the locally admissible and smooth categories.
Lemma 2.5.8. If π, π 0 are objects of Al.adm , then ExtiAl.adm (π, π 0 ) = ExtiA (π, π 0 ).
Proof. This is immediate from [Paš13, Cor. 5.18].



A different (more elementary) comparison of Ext groups occurs if we consider
Afg inside A. The former category does not have enough injectives, but (as notedin A.1.2) we can define Exti in Afg via Yoneda extensions. We then have thefollowing result.
∼Lemma 2.5.9. If π and π 0 are objects of Afg , then ExtiAfg (π, π 0 ) −→ ExtiA (π, π 0 ).
Proof. This follows from Corollary 2.4.4 and Lemma A.1.3.



We next establish some base-change results about Ext groups.
Lemma 2.5.10. If A is a finite-dimensional associative algebra over a field k, if I
is an injective A-module, and if l is any extension of k, then l ⊗k I is an injective
l ⊗k A-module.
Proof. We begin with the case that I is finite-dimensional over k. It follows from
Baer’s criterion that I is injective if and only if I ∨ is projective over Aop . Similarlyl ⊗k I is injective over l ⊗k A if and only if (l ⊗k I)∨ ∼= l ⊗k I ∨ is projective overopl⊗k A . Thus it suffices to prove the analogue of the lemma for projective modules,
and thus for free modules (using the characterization of projective modules as direct
summands of free modules). The case of free modules is clear, and thus the lemmais proved in this case.
Since A and l⊗k A are Noetherian, the property of being injective over these rings
is preserved under the formation of filtered colimits. Since Aop is finite dimensional,

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )19

we know that every simple Aop -module has a finite-dimensional projective envelope.
Thus any simple A-module has a finite-dimensional injective envelope. It follows
that any injective A-module is a filtered colimit of finite dimensional injective Amodules, since it contains an injective envelope of any finite-dimensional submodule.
Hence the lemma follows from the finite-dimensional case.

Corollary 2.5.11. If k ⊆ l is an extension of fields, and if V is a finite dimensional
object of Ck , then for any object W of Ck , the base-change mapl ⊗k ExtiCk (V, W ) → ExtiCl (l ⊗k V, l ⊗k W )is a natural isomorphism.
Proof. The case i = 0 is true since V is finitely generated over a Noetherian quotientof k[KZ]. By dimension shifting, it then suffices to prove that l ⊗k I is injective
if I is injective in Ck .
The usual argument for proving Baer’s criterion shows that a smooth KZrepresentation I over k, with fixed central character ζ, is injective if and only
Ext1Ck (V, I) = 0 for each finite-dimensional object V of Ck . Since some open subgroup H of K acts trivially on V , we deduce that I is injective if and only if its
submodule of invariants I H is injective as a k-representation of KZ/H, with fixed
central character, for each open subgroup H of K.
Assume that this holds. Then (l ⊗k I)H = l ⊗k I H , by the case i = 0, andLemma 2.5.10 shows that l ⊗k I H is injective as an l-representation of KZ/H,
with central character ζ. We conclude that l ⊗k I is an injective object of Cl , asclaimed.

Corollary 2.5.12. If O ⊂ O0 is an unramified extension, and if V is a finite
dimensional object of CF , then for any object W of CF , the base-change map
F0 ⊗F ExtiCO (V, W ) → ExtiCO0 (F0 ⊗F V, F0 ⊗F W )is a natural isomorphism.
Proof. For any i and j, there is a base-change isomorphism ∼F0 ⊗F ExtiCF V, ExtjO (F, W ) −→ ExtiCF0 F0 ⊗F V, F0 ⊗O Extj (F, W )= ExtiCF0 F0 ⊗F V, ExtjO0 (F0 , F0 ⊗F W ) ,the first isomorphism following from Corollary 2.5.11, and the equality holding
because O0 is unramified over O, so that the derived functors RHomO0 (F0 , O0 ⊗O –)and O0 ⊗O RHomO (F, –) coincide on the category of O-modules; both are computedby the complex$·
O0 ⊗O (–) −→ O0 ⊗O (–)(where $ is a uniformizer of O).
If we now consider the spectral sequence (2.5.5) for each of O and O0 , we find thatthese base-change isomorphisms abut to the base-change morphism in the statementof the present corollary, showing that this morphism is also an isomorphism, as
claimed.
Proposition 2.5.13. If l is an extension of k, and if π and π 0 are objects of Ak ,with π being finitely generated, then there is a natural isomorphism∼l ⊗k ExtiAk (π, π 0 ) −→ ExtiAl (l ⊗k π, l ⊗k π 0 ),

20

A. DOTTO, M. EMERTON, AND T. GEE

for each i ≥ 0.
Similarly, if π and π 0 are objects of AF with π finitely generated, and O ⊂ O0 is
an unramified extension, then there is a natural isomorphism∼

F0 ⊗F ExtiAO (π, π 0 ) −→ ExtiAO0 (F0 ⊗F π, F0 ⊗F π 0 )Remark 2.5.14. If we were working in the category of all k[G]-modules (with fixed
central character ζ), then this result would be straightforward, since we would
be able to compute the Ext’s using a resolution of π by finite rank free k[G]/Iζ modules. It is the fact that we are working in the category Ak , i.e. that we have
imposed smoothness, which makes the result less obvious.
We also remark that in the case when π is assumed to be of finite length, theresult is proved by Paškūnas [Paš13, Proposition 5.33]
Proof of Prop. 2.5.13. Using Theorem 2.4.1 and dimension shifting, one easily reduces to the case when π = c-IndG
KZ V for some finitely generated smooth KZrepresentation V . Since compact induction is compatible with extension of scalars,
the proposition follows from Corollary 2.5.11, Corollary 2.5.12 and (2.5.1).
Remark 2.5.15. Our proof of Proposition 2.5.13 uses as input the fact that finitely
generated objects of Ak and AO are Noetherian (i.e. Theorem 2.4.1). Bearing in
mind Remark 2.4.2, we see that the argument, and perhaps also the result, won’t extend to more general p-adic Lie groups as stated. It seems plausible that it should atleast hold in general with “finitely generated” replaced by “finitely presented”, andit’s likely that Shotton’s results [Sho20] about smooth finitely presented GL2 (F )representations can be used to show this for GL2 (F ), for arbitrary p-adic fields F .
(Note that for GL2 (Qp ) all finitely generated representations are finitely presentedby Theorem 2.4.1.)
2.6. Bernstein centres of blocks. The results in this section might be of independent interest, but they are quite technical in nature and will only be applied in
Section 3.8. Let x denote a block of absolutely irreducible representations of Al.adm ,and write Ax for the category AY with Y = {x}. The paper [Paš13] describes an
ex defined as follows.
equivalenceLof Ax with the category of modules over a ring E∨
Let πx =πandchooseaninjective
envelopeπ→Jxx , so that Px = Jxπ∈xex be the endomorphism ring of Jx , which is
is a projective envelope of πx∨ . Let E
naturally a topological ring. Then the functor HomG (−, Jx ) is an equivalence of Ax
ex -modules, and so it defines an isomorphismwith the category of compact left E
ex .
between the Bernstein centre of Ax and the centre Zx of E
Remark 2.6.1. If τ ∈ Ax and J is an injective object in Ax , and E = EndG (J),then E is a compact ring and HomG (τ, J) is a compact E-module: this follows
from [Gab62, Section IV.4]. Hence its Pontrjagin dual HomG (τ, J)∨ is a discrete
E-module.
On the other hand, HomG (τ, J) still makes sense for general τ ∈ A. If τ =c-IndGKZ λ for a finite length KZ-representation λ then the action of E on HomG (τ, J)
is continuous for the discrete topology on this module. This follows from [Paš15,
Lemma 2.1]. Hence HomG (τ, J)∨ is a compact E-module.
If π ∈ x is an irreducible object we will write π → Jπ for an injective envelopeof π in Ax . It is often (but not always) the case that Zx is isomorphic to End(Jπ )LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )21

for an irreducible object π ∈ x. More precisely, we have the following result ofPaškūnas.
Theorem 2.6.2. Let x be a block of absolutely irreducible objects of A, and choose π ∈x. Then the natural map Zx → EndG (Jπ ) is an isomorphism if x is a block oftype (1), (2) or (4).
Proof. This is [Paš13, Proposition 6.3] for type (1), [Paš13, Corollary 8.7] fortype (2), and [Paš13, Corollary 10.78, Theorem 10.87] for type (4).

We will often apply this result together with the following proposition.
Proposition 2.6.3. Let π be an absolutely irreducible object of A. Let x be theblock of A containing π, and choose an injective envelope π → Jπ in Ax . If σ is a
Serre weight, then∨
HomA (c-IndG
KZ σ, Jπ )is a cyclic module over the endomorphism ring Eπ = EndAx (Jπ ).
Proof. This is proved in most cases in [HT15, Proposition 2.9]. We give a different,
uniform proof as follows. Let mπ be the Jacobson radical of Eπ . By Nakayama’s∨
lemma for compact modules, it suffices to prove that HomA (c-IndG
KZ σ, Jπ ) /mπ is
at most one-dimensional. We know thatG∨∨∼
HomA (c-IndG
KZ σ, Jπ ) /mπ = HomA (c-IndKZ σ, Jπ [mπ ])and the representation Jπ [mπ ] contains π as a subquotient with multiplicity one.
Then the proposition follows from the fact that c-IndG
KZ σ has at most one locallyadmissible quotient X with socle π appearing with multiplicity one in X.
To see this, observe that X is necessarily a quotient of c-IndG
KZ σ/f (T ) forsome Hecke operator f (T ), and since the socle of X is absolutely irreducible wehave f (T ) = (T − λ)n for some n. Since X contains π as a subquotient with multiplicity one, we have n = 1. Hence X is irreducible except possibly in the casethat σ is a twist of Sym0 or Symp−1 and λ = ±1. However, in this case there are
only two possibilities for X, and they are distinguished by their G-socle.
Now let σ be a Serre weight and choose λ ∈ F. In Section 3.8 we will be working
with the inverse system {τn } of finite length G-representations defined bynτn = c-IndG
KZ σ/(T − λ) .

These are all contained in the same block, which we will denote by x. There isan action of the Hecke algebra H(σ) on τn for all n, and the transition maps areequivariant for this action. In fact, the action extends to the completion of H(σ)
at the maximal ideal generated by (T − λ), which is isomorphic to a power series
ring in one variable. We will show that in most cases this action is induced by theBernstein centre of the block Ax .
Proposition 2.6.4. In the notation of the previous paragraph, if x has type (1),(2) or (4) then there exists an element of the Bernstein centre Zx of Ax inducing
the Hecke operator T on τn for all n.
Proof. Assume first that π = τ1 is irreducible, so that every composition factorof τi is isomorphic to π. If π 0 is another irreducible object of Ax and π 0 → Jπ0 is

22

A. DOTTO, M. EMERTON, AND T. GEE

an injective envelope in Ax then the space HomA (τi , Jπ0 ) vanishes for all i, since
socG (Jπ0 ) ∼= τi is not a composition factor of τi . Hence the functor
Mπ (−) = HomA (−, Jπ )∨is fully faithful on the full subcategory of Ax generated by the representations {τn }.
By Proposition 2.6.3 the Eπ -module Mπ (c-IndG
KZ σ) is cyclic. Hence there exists Tπ ∈ Eπ such that T = Tπ as endomorphisms of Mπ (c-IndG
KZ σ). By Theorem 2.6.2 the natural map Zx → Eπ is an isomorphism. Since Mπ (c-IndG
KZ σ) surjects onto Mπ (τi ) equivariantly for T and Tπ , we deduce that there exists Tπ ∈ Zx
such that T = Tπ as endomorphisms of Mπ (τi ). Since the natural map
EndAx (τi ) → EndEπ (Mπ (τi ))is an isomorphism, the proposition follows in the case that τ1 is irreducible.
After twisting there remains to prove the proposition in the case that σ =Sym0 , Symp−1 and λ = 1. Hence τ1 is a nonsplit extension of 1 by St and the block−1). We will follow [Paš13,
containing τ1 is x = {1, St, π}, where π = IndG
B (ω ⊗ ω
Section 10.4] and work in a quotient category of Ax . By [Paš13, Lemma 10.84] thekernel of the functor
HomAx (−, Jπ ⊕ JSt )∨is the category T(k) of modules with trivial G-action. By [Gab62, IV.4, Théorème 4]
this functor defines an equivalence of the quotient category Q(k) = Ax /T(k)with the category of left discrete EndG (Jπ ⊕ JSt )-modules. (Compare the proofof [Paš13, Corollary 10.85], and note that our categories are opposite to those denoted in [Paš13] by the same symbols.) More precisely, the functor
HomQ(k) (−, Jπ ⊕ JSt )is an equivalence, but by the equivalence of (b) and (c) in [Gab62, III.2, Lemme 1]
the natural map
HomAx (X, Jπ ⊕ JSt ) → HomQ(k) (X, Jπ ⊕ JSt )is an isomorphism for every object X of Ax .
Since HomA (τi , Jπ ) = 0 for all i, the functor MSt (−) = HomA (−, JSt )∨ induces
a fully faithful functor on the full subcategory of Q(k) generated by the τi . By
Proposition 2.6.3 we know that MSt (c-IndG
KZ σ) is cyclic over the endomorphismring ESt of JSt . Arguing as before, it follows from Theorem 2.6.2 that there exists Tx ∈ ESt ∼= Zx such that T = Tx as endomorphisms of HomAx (τi , JSt ). The
faithfulness of MSt (−) implies that T = Tx as elements of EndQ(k) (τi ). This implies
that the image in Ax of(T − Tx ) : τi → τi
has trivial G-action. This is true for all i, and the only submodule of τi with trivial
G-action has length at most one: this is an immediate consequence of Lemma 2.1.12.
Using this for τi+1 together with the fact that T − Tx commutes with the projection
τi+1 → τi it follows that T − Tx = 0 as elements of EndG (πi ).
Blocks of type (3) are an exception to Theorem 2.6.2 and Proposition 2.6.4 because the rings EndG (Jπ ) are no longer commutative. The following weaker result
will suffice for our purposes; by a more detailed analysis of the endomorphism
rings Ex , taking into account their relationship with Galois pseudodeformation

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )23rings, it should actually be possible to compute explicit integral dependence equations for Hecke operators over the centre of the block.
Proposition 2.6.5. Let σ be a twist of Symp−2 , let λ = ±1, and letiτi = c-IndG
KZ σ/(T − λ) .

Let x be the block of type (3) containing the irreducible representation π = τ1 . Then
there exist a positive integer n and central elements z0 , z1 , . . . , zn ∈ Zx containedin the unique maximal ideal mx ⊂ Zx such that(T − λ)n+1 + zn (T − λ)n + · · · + z1 (T − λ) + z0 = 0
holds in EndG (τi ) for all i.
Proof. Fix an injective envelope π → Jπ in Ax and let Eπ = EndG (Jπ ). The
functorMπ (−) = HomG (−, Jπ )∨is defined on A, and it restricts to an equivalence of Ax with the category of discrete
left Eπ -modules. Hence the natural map Zx → Eπ induces an isomorphism Zx →Z(Eπ ). By [Paš13, Corollary 9.25], the ring Eπ is a free module of rank 4 over its
centre, which is a local ring. By Proposition 2.6.3, the module
M = Mπ (c-IndG
KZ σ)
is cyclic over Eπ . It follows that M is finite over Zx . (See the proof of [San16,
Lemma 2.6] for more details.)
The Hecke algebra H(σ) acts on M by Zx -linear endomorphisms. The endomorphism induced by (T − λ) on M ∨ is locally nilpotent, because every G-linear
map c-IndG
KZ σ → Jπ factors through a representation of finite length, in fact
through one of the τi . The quotient M/mx M is a finite-dimensional vector spaceover the residue field of Zx , and it is dual to M ∨ [mx ], hence (T − λ) induces
a nilpotent endomorphism of M/mx M . It follows that there exists n > 0 suchthat (T − λ)n (M ) ⊂ mx M . This implies that there exist z0 , z1 , . . . , zn ∈ mx suchthat
(T − λ)n+1 + zn (T − λ)n + · · · + z1 (T − λ) + z0 = 0
as endomorphisms of M . Hence the same relation holds as endomorphisms of Mπ (τi )for all i, since they are quotients of M . Since the functor Mπ (−) is fully faithfulon Ax , this implies the proposition.

3. Localization of smooth GL2 (Qp )-representations
3.1. Localization. In this subsection we begin to explain how we localize A over X.
The fundamental input is that the closed points of X are in bijection with the blocks
l.adm
of AO
.
Definition 3.1.1. If Y is a closed subset of X, then we let AY denote the subcategory of A consisting of those representations all of whose irreducible subquotientslie in a block corresponding to a closed point of Y .
We refer to Appendix A for the basic definitions and properties of localizing
categories.
Lemma 3.1.2. The subcategory AY is localizing.

24

A. DOTTO, M. EMERTON, AND T. GEE

Proof. To see that AY is a Serre subcategory, it suffices to check that given an
exact sequence
0 → π1 → π → π2 → 0
in A the set JH(π) of irreducible subquotients of π coincides with JH(π1 ) ∪ JH(π2 ).
Since it is immediate that JH(π1 ) ∪ JH(π2 ) ⊆ π, let V be an irreducible subquotientof π. Then there are subspaces W1 ⊂ W2 ⊂ π such that V ∼= W2 /W1 , and V is asubquotient of π2 unless W1 + π1 = W2 + π1 . Similarly, V is a subquotient of π1
unless W1 ∩ π1 = W2 ∩ π1 . Since W1 6= W2 , these two equalities cannot hold at thesame time.
To see that AY is localizing it thus suffices to check that it is closed under directsums.
L Assume that πi ∈ AY and V is an irreducible submodule of a quotientofi∈I πi . Since V is cyclic over O[G],L there exists a finite subset J ⊂ I suchthat V is a submodule of a quotient of i∈J πi . By the previous discussion, V ∈∪i∈J JH(πi ) ⊂ AY , implying the claim.
Lemma 3.1.3. If Y = {y1 , . . . , yn } is a finite closed subset of X, then AY is the
product of the finitely many blocks corresponding to the points yi . In particular, AY
is contained in Al.adm , and is an Artinian category.
Proof. If we write Y = {y1 , . . . , yn }, then AY consists of objects whose subquotientslie in the blocks corresponding to one of the points yi . If π is such an object, let σGbe a Serre weight, and let c-IndG
KZ σ → π be a morphism; then since c-IndKZ σ has
subquotients lying in infinitely many different blocks, this morphism must factor
through a non-trivial, and hence finite length, quotient of c-IndG
KZ (σ). Thus any
finitely generated subrepresentation of π is of finite length, and hence locally admissible, so that AY ⊆ Al.adm . The lemma then follows from the theory of blocks
for Al.adm .
Definition 3.1.4. If U is an open subset of X, then we write AU := A/AY , whereY := X \ U .
Since AY is localizing there is a fully faithful right adjoint (jU )∗ : AU → A to thecanonical quotient functor (jU )∗ : A → AU . Where U is understood, we write j∗for (jU )∗ , and j ∗ for (jU )∗ .
Lemma 3.1.5.
(1) The right adjoint j∗ commutes with filtered colimits.
(2) The category AU is locally Noetherian.
fg(3) If we let Afg
U denote the essential image of A under the localization functor,fgthen AU is precisely the subcategory of Noetherian objects in AU .
Proof. This follows from Lemma A.2.8 and the fact that A and AY are both generated by finitely generated objects, which are Noetherian objects in A by Corollary 2.4.4.
Lemma 3.1.6. Let σ be a Serre weight, let Y be a closed subset of X, and chooseg ∈ H(σ) such that fσ−1 (Y ) = V (g) (a closed subset
 of Spec H(σ)). Then for any
object π of AY , we have ExtiA π, (c-IndGσ)[1/g]= 0 for all i.
KZProof. Applying Lemma A.2.7, we see that it suffices to prove the claimed vanishingin the case when π is finitely generated, which we assume from now on. By dévissage
we can furthermore assume that the maximal ideal of O acts trivially on π. Using

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )25the spectral sequence (2.5.5) andto prove the fact that O is a DVR, it sufficesiσ)[1/g]
=
0
for
all
i.
The
formation
of
this
Ext
-moduleisthat ExtiAF π, (c-IndG
KZ
compatible with extension of scalars, by Proposition 2.5.13, and so we may and do
replace F by some uncountable extension. We thus assume that F is uncountable
for the remainder of the argument. Also, if g = 0 then (c-IndG
KZ σ)[1/g] = 0, andthe claimed vanishing follows immediately. Thus we may and do assume for theremainder of the proof that g 6= 0.
We first suppose that Y is a finite union of closed points. In this case, Lemma 3.1.3
shows that any object π of AY decomposes as a direct sum of objects lying in thevarious blocks corresponding to the points of Y . It will be useful in this case toprove a slightly stronger statement, namely that ExtiA π, (c-IndG
KZ σ)[1/h] = 0 for
any non-zero multiple h of g. To this end, we note that if f ∈ H(σ) is coprimeGto h (and so in particular coprime to g), then (c-IndG
KZ σ)/f (c-IndKZ σ) is of finite length, and has no Jordan–Hölder factor lying in AY (this last claim following
from the relationship between the morphisms fσ and the blocksof Al.adm ). ThusGGi
by Lemma 2.5.8 we have ExtA π, (c-IndKZ σ)/f (c-IndKZ σ) = 0 for all i, and aconsideration of the long exact Ext sequence arising from the short exact sequence
f·

G
G
G
0 → (c-IndG
KZ σ)[1/h] −→ (c-IndKZ σ)[1/h] → (c-IndKZ σ)/f (c-IndKZ σ) → 0shows that f acts invertibly on ExtiA π, (c-IndG
KZ σ)[1/h] . Certainly h also acts
invertibly, and so this Ext module is a vector space over the fraction field of H(σ),
which is isomorphic to F(T ), a field of uncountable dimension over F. On the other
hand, Lemma 2.5.7 shows that this Ext module is of countable dimension over F.
Thus it must vanish, as claimed.
We now consider the general case. Since we are assuming that π is finitely
generated, it is a quotient of a representation of the form c-IndG
KZ V , for somefinite dimensional KZ-representation V . A dévissage using the long exact Ext
sequence reduces to the case when π is a quotient of c-IndG
KZ τ , for some Serreweight τ . We consider two cases: (i) π is equal to c-IndG
KZ τ , or (ii) π is a properquotient of c-IndG
KZ τ . In this latter case, π is of finite length, and so lies in AY0
for some finite closed subset Y0 of Y . The claimed vanishing then follows from thespecial case already proved, and so we now put ourselves in case (i).
The assumption that c-IndG
KZ τ is an object of AY implies that for each nonGzero element q ∈ H(τ ), the (finite length) quotient (c-IndG
KZ τ )/q(c-IndKZ τ ) liesin AY0 for some finite closed subset Y0 of Y . The result we already proved thenGGshows that ExtiA (c-IndG
KZ τ )/q(c-IndKZ τ ), (c-IndKZ σ)[1/g] = 0 for each i, and aconsideration of the long exact Ext sequence arising from the short exact sequence
q·

G
G
G
0 → c-IndG
KZ τ −→ c-IndKZ τ → (c-IndKZ τ )/q(c-IndKZ ) → 0Gshows that q acts invertibly on ExtiA c-IndG
KZ τ, (c-IndKZ σ)[1/g] . This Ext module is thus a vector space over the fraction field of H(τ ), while also being a countabledimensional F-vector space, by Lemma 2.5.7. Since the fraction field of H(τ ) is isomorphic to F(T ), which is of uncountable F-dimension when F itself is uncountable,
we find that this Ext module vanishes, as claimed.
Proposition 3.1.7. Let Y be a closed subset of X, and write U := X \ Y .

26

A. DOTTO, M. EMERTON, AND T. GEE(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-IndGKZ σ → (c-IndKZ σ)[1/g]can be identified with the unit morphismG∗c-IndGKZ σ → jU ∗ jU c-IndKZ σ.

(2) The functor jU ∗ is exact.
Proof. We first put ourselves in the situation of (1). If g = 0 then c-IndG
KZ σ is anG∗
object of AY , and both (c-IndGσ)[1/g]andjj(c-Indσ)vanish,sothat (1)∗
KZ
KZ
follows immediately. If g 6= 0, then the embedding(3.1.8)Gc-IndGKZ σ ,→ (c-IndKZ σ)[1/g]is the colimit of the embeddings
c-IndGKZ σ →

1(c-IndG
KZ σ).
gn n
GThe cokernel of this latter morphism is isomorphic to c-IndG
KZ σ /g c-IndKZ σ ,which is of finite length, and lies in AY . Thus each of these inclusions induces
an isomorphism after applying j∗ j ∗ (–), and hence the colimiting inclusion (3.1.8)
also induces an isomorphism after applying this functor. Thus it suffices to showthat (c-IndG
KZ σ)[1/g] is in the image of j∗ , or equivalently, by Lemma A.2.2, thatσ)[1/g]= 0 for i = 0, 1, whenever π is an object of AY . This
ExtiA π, (c-IndG
KZ
follows from Lemma 3.1.6.
To prove (2), it suffices, by Corollary A.2.6, along with Lemmas 2.5.6 and 3.1.5,
to show that Ext2A (π, j∗ j ∗ π 0 ) = 0, as π runs over the objects of AY and π 0 runs overa collection of generators of A. We let π 0 range over the representations c-IndG
KZ V,
where V is a finite length smooth O[KZ]-representation. A dévissage then reduces
us to checking the required vanishing when V = σ is a Serre weight. In this case,the required vanishing again follows from Lemma 3.1.6.
Corollary 3.1.9. Let Y be a closed subset of X, and write U := X \ Y . If π is anobject of AY , and π 0 is any object of A, then Exti (π, jU ∗ jU∗ π 0 ) = 0 for all i.
Proof. This follows directly from Proposition 3.1.7 (2), together with Corollary A.2.6.
In fact, it was essentially proved directly in the course of proving Proposition 3.1.7 (2).
Lemma 3.1.10. If σ is a Serre weight, and if c-IndG
KZ σ ,→ π is an essential
embedding whose cokernel is of finite length, then there exists a non-zero g ∈ H(σ)
such that π (thought of as an overmodule of c-IndKZ σ) is contained in g1 ·c-IndG
KZ σ.
0
Proof. Consider the short exact sequence 0 → c-IndG
KZ σ → π → π → 0. Since0
π is of finite length, it is supported on some finite closed subset Y of X. Write∼
U := X \Y . Then jU ∗ jU∗ π = 0, and so jU ∗ c-IndG
KZ σ −→ jU ∗ π. Proposition 3.1.7 (1)Gshows that jU ∗ c-IndG
KZ σ = (c-IndKZ σ)[1/g] for some non-zero element g of H(σ).

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )27

Consider the commutative squarec-IndGKZ σ/π(c-IndG
KZ σ)[1/g]
jU ∗ jU∗ π

Since the left-hand vertical arrow is injective, and since the upper horizontal arrow isan essential embedding, we find that the right-hand vertical arrow is injective. ThusG
we find that π embeds as a submodule of (c-IndG
KZ σ)[1/g] containing c-IndKZ σ.
Since π is finitely generated (being an extension of a finite length module by afinitely generated module), we see that if we replace g by a sufficiently large power,
then in fact π ⊆ g1 · c-IndG
KZ σ, as claimed.
Remark 3.1.11. An immediate consequence of Lemma 3.1.10 is that if c-IndG
KZ σ →π is an essential embedding with cokernel of finite length then the maximal idealof O acts trivially on π. This can also be seen directly: if πE ∈ O is a uniformizer,
the image of πE : π → π has finite length, and so intersects c-IndG
KZ σ trivially.
G
Since the embedding c-IndKZ σ → π is essential, this implies that πE : π → π isthe zero map.
Remark 3.1.12. Note that if σ is not isomorphic to a twist of Sym0 or Symp−1 ,Gthen (c-IndG
KZ σ)/h(c-IndKZ σ) is irreducible, for each irreducible h ∈ H(σ). In
this case, by choosing g in Lemma 3.1.10 appropriately, we may even assume thatπ = g1 · c-IndG
KZ σ. In the exceptional cases, we get a counterexample from theexact sequence (3.2.2) below.
Remark 3.1.13. Another way to prove Lemma 3.1.10 is to use the isomorphism
∼

G
G
1 0
Hom(π 0 , (c-IndG
KZ σ)[1/g]/c-IndKZ σ) −→ Ext (π , c-IndKZ σ)of Section 3.6 below.
3.2. Čech resolutions. If {U0 , . . . , Un } is any finite open cover of X, then for any
object π of A, we obtain a functorial Čech resolutionY(3.2.1)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 ,and the differentials are given by the usual formulas (see e.g. [Sta, Tag 01FG]).
(In the terminology of the Stacks Project, we are working with the ordered Čechcomplex.) We are going to prove that the complex (3.2.1) is always exact. To
do so we will use another resolution of smooth GL2 (Qp )-representations, arisingfrom the Bruhat–Tits tree of PGL2 (Qp ). Let N be the normalizer of the Iwahori
subgroup Iw. Write δ : N → O× for the orientation character, which is trivial
0 1
on IwZ and takes value −1 at. We identify the representation space of δ
p 0with O. Then we have an exact sequence of O[G]-representations(3.2.2)∂sum

0 → c-IndG→ c-IndG
N (δ) −KZ (triv) −−→ triv → 0,

see for example [Dot, Section 2.4.14].

28

A. DOTTO, M. EMERTON, AND T. GEE

Proposition 3.2.3. For any object π of A, and any finite open cover {Ui } of X,
the resolution (3.2.1) is acyclic.
Proof. By Frobenius reciprocity there is a short exact sequenceG
0 → c-IndG
N δπ → c-IndKZ π → π → 0

where δ is the nontrivial quadratic character of N/IwZ.
G
GKZ
Since p > 2, c-IndG
N δπ is a direct summand of c-IndIwZ π = c-IndKZ (c-IndIwZ π).
Since the formation of each of the terms in (3.2.1) is exact, we reduce to verifying
the claim of the proposition in the case when π is of the form c-IndG
KZ V for somesmooth O[KZ]-representation V . Since both compact induction, and the formation of the terms in (3.2.1), are compatible with passing to filtered colimits (useLemma A.2.8 (1)), we then reduce to the case when V is finitely generated. Finally, since compact induction is exact, we reduce to the case when π is of the formc-IndGKZ σ for some Serre weight σ.
The open cover {Ui } pulls back via fσ to an open cover D(gi ) of Spec H(σ).
Proposition 3.1.7 then implies that (3.2.1) may be identified with the tensor product(3.2.4)•(c-IndG
KZ σ) ⊗H(σ) K (g0 , . . . , gn ),where K • (g0 , . . . , gn ) denotes the usual Čech complex for a finite open cover ofSpec of a ring by distinguished opens associated to the sequence {g0 , . . . , gn }. Since{D(gi )} is an open cover of the affine scheme Spec H(σ), the complex K • (g0 , . . . , gn )is acyclic, and remains acyclic after tensoring with any H(σ)-module. We consequently find that (3.2.4) is acyclic, and thus that (3.2.1) is acyclic, as claimed. 
Corollary 3.2.5. If Y ⊂ U is an inclusion of a closed subset of X in an open subsetof X, and if π is an object of AY , then the natural morphism π → (jU )∗ (jU )∗ π is
an isomorphism; in other words, π lies in the essential image of (jU )∗ .
Proof. This follows from the acyclicity of the Čech resolution of π with respect tothe open cover {U, V := X \ Y } of X, together with the fact that, by definition,(jV )∗ π = (jU ∩V )∗ π = 0.
Corollary 3.2.6. Let Y and W be disjoint closed subset of X. If π is an object of
AY , and π 0 is an object of AW , then Exti (π, π 0 ) = 0 for all i.
Proof. This follows immediately from Corollaries 3.1.9 and 3.2.5 (the latter being
applied to the inclusion W ⊆ X \ Y ).

Lemma 3.2.7. Let Y be a closed subset of X, let σ be a Serre weight, and write
fσ−1 (Y ) = V (g) (a closed subset of Spec H(σ)) for some g ∈ H(σ). Then for any
object π of AY , each element of ExtiA (c-IndG
KZ σ, π) is annihilated by some powerof g.
Proof. By Lemma 2.5.6 we can assume that π is finitely generated, and by dévissage
we can assume that the maximal ideal of O acts trivially on π. Using (2.5.5) it
suffices to prove the claim for ExtiAF instead of ExtiA . By Proposition 2.5.13, wemay verify this after extending scalars to an uncountable extension of F. Thus we
may and do assume that F is uncountable.
G
If h ∈ H(σ) is coprime to g, then (c-IndG
KZ σ)/h(c-IndKZ σ) is of finite length,
and lies in AW , for some finite closed subset W of X \ Y . If we set U :=X \ W , then π = (jU )∗ (jU )∗ π, by Corollary 3.2.5 (note that Y ⊆ U ). Thus

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )29

GExtiA (c-IndG
KZ σ)/h(c-IndKZ σ), π vanishes for every i, by Corollary 3.1.9, and so
a consideration of the long exact sequence of Ext’s arising from the short exact
sequenceh·GGG
0 → c-IndG
KZ σ −→ c-IndKZ σ → (c-IndKZ σ)/h(c-IndKZ σ) → 0

shows that h acts invertibly on ExtiA (c-IndG
KZ σ, π); thus the H(σ)-module structure on ExtiA (c-IndGσ,π)extendstoaH(σ)
S -module structure, where S deKZ
notes the multiplicative subset of elements coprime to g. Since F is uncountable, the ring H(σ)S has uncountable F-dimension, while Lemma 2.5.7 shows that
Gi
ExtiA (c-IndG
KZ σ, π) has countable F-dimension. Thus ExtA (c-IndKZ σ, π) cannot
contain any H(σ)S torsion-free submodule. The lemma follows.

Remark 3.2.8. We explain an alternate proof of Lemma 3.2.7 in the case when Y
is finite. In this case Lemma 3.1.3 shows that AY is a product of blocks in Al.adm ,and it follows from [Paš13, Cor. 5.18] that any injective resolution of π in AY also
provides an injective resolution in A. But if I • is such a resolution in AY , thenGG•nany element of HomA (c-IndG
KZ σ, I ) factors through (c-IndKZ σ)/g (c-IndKZ σ),
for some n.
3.3. A stack of abelian categories. For each open subset U of X we have thelocalized category AU , and for each open subset V ⊂ U we have the natural localization functor jU∗ V : AU → AV .
Theorem 3.3.1. The collection {AU } together with the localization functors jU∗ V :
AU → AV for V ⊂ U forms a stack (of abelian categories) over the Zariski siteof X.
Remark 3.3.2. By a stack in abelian categories over X we simply mean a stackover the Zariski site of X whose fiber categories are abelian. There are further
properties one could ask of such an object, such as exactness of all the pullback
functors, and indeed the stack determined by the {AU } has a lot more structure,such as pushforward functors and acyclic Čech resolutions. This is very similar
to the formalism underlying cohomological descent [Sta, Tag 0D8D], as might be
expected taking into account the connection between the categories AU and sheaves
over a stack of (ϕ, Γ)-modules explained in the introduction.
Proof of Theorem 3.3.1. Since the localization functors jU∗ V : AU → AV are theidentity on objects, it is straightforward to check that U 7→ AU defines a pseudofunctor in the sense of [Vis05, Definition 3.10]. More precisely, the universal property of localization specifies a natural isomorphism jU∗ W ∼= jV∗ W jU∗ V associated to
any composite inclusion W ⊂ V ⊂ U of subsets of V , such that jU∗ U is the identity
and Properties (a) and (b) in [Vis05, Definition 3.10] are satisfied. By the procedure
explained in [Vis05, Section 3.1.3] we obtain a fibered category A• → X.
Following [Vis05, Definition 4.6] we need to prove that for any covering {Ui →U }i∈I the functor from AU to the category A• ({Ui → U }) of objects with descentdata is an equivalence. Since X is quasicompact, we can assume without loss ofgenerality that the indexing set is equal to {0, 1 . . . , n}. We will deduce the result
from Proposition 3.2.3. In what follows we will write Uij = Ui ∩ Uj and abbreviate∗
the pullback and pushforward associated to the inclusion Uij ⊂ Ui by ji,ij
and ji,ij∗ ,∗and those associated to the inclusion Uij ⊂ U by jij and jij∗ .

30

A. DOTTO, M. EMERTON, AND T. GEE

Choose objects πi ∈ AUi and isomorphisms
∼

∗
∗
ji,ij
πi −
→ jj,ij
πj(3.3.3)in AUij for all i, j, satisfying the cocycle condition after pullback to Uijk . Fix∗
indices i < j and define πij = ji,ij
πi . The adjunction between pullback and pushforward defines a map uij : πi → ji,ij∗ πij , and pushing forward to U we find amap
uij : ji∗ πi → jij∗ πij .
Similarly, the isomorphism (3.3.3) yields a mapu+ij : jj∗ πj → jij∗ πij .
Putting these together we obtain(3.3.4)u+ij − uij :Yiji∗ πi →Y

jij∗ πij .

i<j

If we are in the special case that Ui0 = U for some index i0 , the cocycle condition∗
implies that the maps ji∗0 ,i0 i πi0 → πi = ji,i
π define an isomorphism in A• ({Ui →0i iU }). Since the formation of (3.3.4) defines a functor on A• ({Ui → U }), it follows
that the complex (3.3.4) is isomorphic to the first truncation of the Čech resolution of πi0 . Furthermore, the formation of (3.3.4) is compatible with pullback forany open inclusion V ⊂ U . It follows from this discussion together with Proposition 3.2.3 and the exactness of pullback functors that if we define π by the exact
sequenceY(3.3.4) Y
0→π→ji∗ πi −−−−→
jij∗ πijii<jthen ji∗ (π) ∼= πi , proving essential surjectivity of AU → A• ({Ui → U }).
Now let α : π1 → π2 be a morphism in AU . It follows immediately from Proposition 3.2.3 and functoriality of the Čech resolutions that if ji∗ (α) = 0 for all ithen α = 0, which proves that AU → A• ({Ui → U }) is faithful. To prove that it isfull we begin with representations π, τ ∈ AU together with morphisms of descentdata αi : ji∗ π → ji∗ τ . The argument for essential surjectivity implies that the αi
induce a morphism on the first-truncated Čech resolutions of π and τ , and so amorphism α : π → τ . Since the formation of the Čech resolution commutes with ji∗
we deduce that that j ∗ (α) = αi , which concludes the proof.

3.4. Further results about j∗ . As we see from part (1) of Proposition 3.1.7, it is
not typically the case that j∗ j ∗ π is finitely generated, even if π is. Indeed we havethe following result.
Lemma 3.4.1. If U is an open subset of X, and if jU ∗ jU∗ π is finitely generated forsome object π of A, then the natural morphism π → jU ∗ jU∗ π is surjective.
Proof. We may write π as the filtered colimit of its finitely generated submodules πi ,and then Lemma 3.1.5 shows that j∗ j ∗ π is the filtered colimit of the j∗ j ∗ (πi ).
Since j∗ j ∗ π is finitely generated, and so Noetherian, by assumption, we find thatj∗ j ∗ (πi ) = j∗ j ∗ π for some value of i. Replacing π by πi , we may thus assume that
π is finitely generated.
Choose a surjection c-IndG
KZ V → π, for some finite length O[KZ]-representation V .
We proceed by induction on the length of V (the case V = 0 being trivial). Let σ
be an irreducible subrepresentation of V , and let π 0 denote the image of c-IndG
KZ σ

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )31

in π. If π 0 = 0 then we replace V by V /π 0 , and we are done by induction. Otherwise, since j∗ is exact, we see that j∗ j ∗ (π/π 0 ) is a quotient of j∗ j ∗ π, and so isfinitely generated. Thus our inductive hypothesis shows that (π/π 0 ) → j∗ j ∗ (π/π 0 )is surjective. Also, since j∗ j ∗ π 0 is a subobject of j∗ j ∗ π, and since finitely generated
G-representations are Noetherian, we see that j∗ j ∗ π 0 is finitely generated.
∼G00
Now either c-IndG
KZ σ −→ π , or else π is a proper quotient of c-IndKZ σ, in
0which case it is of finite length. In the latter case, it is automatic that π → j∗ j ∗ π 0 issurjective. In the former case, a consideration of the formula of Proposition 3.1.7 (1)shows that since j∗ j ∗ π 0 = j∗ j ∗ (c-IndG
KZ σ) is finitely generated, the natural morphism π 0 → j∗ j ∗ π 0 is an isomorphism, and so in particular is surjective. Again
using the fact that j∗ is exact, we see that we have a morphism of short exact
sequences/π/ (π/π 0 )
/0
/ π0
0

0


/ j∗ j ∗ π 0
/ j∗ j ∗ π/ j∗ j ∗ (π/π 0 )/0in which the outer two vertical arrows are surjections. The middle vertical arrowis thus a surjection as well.

Corollary 3.4.2. If π is finitely generated and lies in the essential image of j∗ ,then the same is true for any subquotient of π.
∼

Proof. The assumption on π is equivalent to asking that π −→ j∗ j ∗ π and thatj∗ j ∗ π be finitely generated (see the proof of Lemma A.2.2). Bearing this in mind,
since j∗ is exact, it suffices to prove the statement of the lemma for subobjectsof π, since it then follows for quotients, and so also for subquotients. If π 0 is any
subobject of π, we see that j∗ j ∗ π 0 is a subobject of j∗ j ∗ π, and so finitely generated;thus Lemma 3.4.1 implies that the natural morphism(3.4.3)π 0 → j∗ j ∗ π 0

is surjective. The left-exactness of j∗ , and the fact that the corresponding map for
π is an isomorphism, implies that (3.4.3) is also injective. Thus in fact (3.4.3) is an
isomorphism, and so π 0 also lies in the essential image of j∗ .
Remark 3.4.4. The preceding corollary is not true in general without the assumptionof finite generation. Indeed, if π is finitely generated while j∗ j ∗ π is not finitely
generated (Proposition 3.1.7 provides plenty of examples of such π), then if we let
π 0 denote the image of π in j∗ j ∗ π, we have that π 0 ( j∗ j ∗ π, while j∗ j ∗ π 0 = j∗ j ∗ π.
Thus j∗ j ∗ π is an object in the essential image of j∗ , while its subobject π 0 is not inthis essential image.
Corollary 3.4.5. Let Y be a closed subset of X, and write U := X \ Y . If π is a
finitely generated object lying in the essential image of jU ∗ , and if π 0 is an object
of AY , then ExtiA (π, π 0 ) = 0 for all i.
Proof. Arguing by dévissage, and taking into account Corollary 3.4.2, we may assume that π is a quotient of c-IndG
KZ σ, for some Serre weight σ. If it is a properquotient, then it is of finite length, and so lies in AW , for some finite closed subset of W of U , while π 0 = (jV )∗ (jV )∗ π, if V = X \ W (by Corollary 3.2.5; notethat Y ⊆ V ). In this case the lemma follows from Corollary 3.1.9. Otherwise, we32

A. DOTTO, M. EMERTON, AND T. GEE

may assume that π = c-IndG
KZ σ, in which case the claimed vanishing follows from
Lemma 3.2.7.

We may use the preceding result to strengthen Lemma 3.4.1.
Corollary 3.4.6. Let Y be a closed subset of X, and write U := X \ Y . If jU ∗ jU∗ π
is finitely generated, then the natural morphism π → jU ∗ jU∗ π is a split surjection.
Proof. Lemma 3.4.1 ensures that the morphism π → j∗ j ∗ π is surjective. If wedenote its kernel by π 0 , then π 0 lies in AY . The short exact sequence
0 → π 0 → π → j∗ j ∗ π → 0
represents an element of Ext1A (j∗ j ∗ π, π 0 ), and Corollary 3.4.5 shows that this Ext
module vanishes. Thus the short exact sequence splits, as claimed.

3.5. Completion. If Y is a closed subset of X, then AY is an abelian subcategory(indeed a localizing subcategory) of A. Thus the discussion of Section A.3.17 applies, and shows that the inclusion Pro(AY ) ,→ Pro(A) admits a left adjoint, which
we denote by π 7→ π
bY . Being a left adjoint, this functor preserves colimits when
they exist. In particular, it is right exact. We sometimes write simply πb if Y isunderstood from the context.
The following lemma describes π
bY explicitly when π ∈ A.
Lemma 3.5.1. If π is an object of A, there is a natural isomorphism∼πbY −→ “ lim ”π 0 ,←−where π 0 runs over the cofiltered directed set of quotients of π lying in AY .
Proof. This is a particular instance of Lemma A.3.18.



Recall that Afg denotes the full subcategory of A consisting of finitely generated
objects. Similarly, for any closed subset Y of X, we write Afg
Y to denote the fullsubcategory of AY consisting of finitely generated objects. If Y is a finite closed
subset of X, then Lemma 3.1.3 shows that Afg
Y may equally well be described as
the subcategory of AY consisting of finite length objects.
Corollary 3.5.2. If π is an object of Afg , then πbY is an object of Pro(AfgY ).
Proof. This follows immediately from Lemma 3.5.1.

Proposition 3.5.3. Let Y be a closed subset of X, and suppose that0 → π 0 → π → π 00 → 0
is a short exact sequence in Afg , with π 0 being an object of Afg
Y . Then we may finda commutative diagram
0

/ π0

/π

/ π 00

/0

0

/ π0


/π
e


/π
e00

/0

in which the right-hand (and hence also the middle) vertical arrow is surjective, andin which πe00 (and hence also πe) are objects of Afg
Y.

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )3300
Proof. By assumption we may find a surjection c-IndG
KZ τ → π , for some finite
length representation τ of O[KZ], and we argue by induction on the KZ-lengthof τ . Thus, to begin with, we assume that τ = σ is a Serre weight. If the surjection
00
00
c-IndGKZ σ → π is not an isomorphism, then π is of finite length, and so it isan object of AW , for some finite closed subset W of X. Write W 0 = W ∩ Y and
W 00 = W \ Y , so that W is the disjoint union of its closed subsets W 0 and W 00 .
Correspondingly, we may write π 00 = π100 ⊕ π200 , where π100 is an object of AW 0 ⊆AY , and π200 is an object of AW 00 . Since Y and W 00 are disjoint, it follows from
Corollary 3.2.6 that Ext1A (π200 , π 0 ) = 0. Thus the natural map

Ext1A (π100 , π 0 ) → Ext1A (π 00 , π 0 )given by pullback along the surjection π 00 → π100 , is an isomorphism, and so π ispulled back from an extension π1 of π100 by π 0 .
∼00−1Suppose, then, that c-IndG
KZ σ −→ π . Write fσ (Y ) = V (g), for some g ∈fg
00
H(σ). If g = 0, then π itself is an object of AY , and there is nothing to prove.
0
Otherwise, g is non-zero. The extension π represents a class of Ext1A (c-IndG
KZ σ, π ),nand Lemma 3.2.7 shows that some power g of g annihilates this class. A consideration of the long exact Ext sequence associated to the short exact sequence
gn ·

G
G
G
n
0 → c-IndG
KZ σ −→ c-IndKZ σ → (c-IndKZ σ)/g (c-IndKZ ) → 0

then shows that the class of π arises by pullback along the surjection c-IndG
KZ σ →
Gn),
whosetargetisanobjectofA.
σ)/g(c-Ind(c-IndGY
KZ
KZ
We now turn to the general case, when τ is not necessarily irreducible. Our
inductive hypothesis allows us to choose a subrepresentation π000 of π 00 such that thestatement of the proposition holds for each of π000 and π 00 /π000 in place of π 00 . Pullingback the extension given by π along the inclusion π000 ⊆ π 00 , we obtain a short exact
sequence
0 → π 0 → π0 → π000 → 0,
and by assumption we may find a subrepresentation π2 ⊆ π0 such that π 0 ∩ π2 = 0,
and such that π0 /π2 is an object of Afg
Y . Next consider the short exact sequence∼

0 → π0 /π2 → π/π2 → π/π0 (−→ π 00 /π000 ) → 0.
Again by assumption, we may find a subrepresentation π1 of π containing π2 suchthat π0 ∩ π1 = π2 , and such that π/π1 (= (π/π2 )/(π1 /π2 )) is an object of Afg
Y.
Then π 0 ∩ π1 = π 0 ∩ π0 ∩ π1 = π 0 ∩ π2 = 0, and so setting π
e = π/π1 does the job. 
Corollary 3.5.4. The restriction to Afg of the functor π 7→ πbY is exact.
Proof. If π 0 ⊆ π is an inclusion of finitely generated objects of A, we must provethat the induced morphism πb0 Y → π
bY is a monomorphism. By Lemma A.3.9 andcthe definition of the functor (–)Y as an adjoint, it suffices to show any morphism
π 0 → π 0 whose target lies in AY can be placed in a commutative diagram
π0

/π


π0


/ π,

where π also lies in AY and in which the bottom arrow is a monomorphism. In
fact, it suffices to do this with π 0 replaced by the image π
e0 of π 0 in π 0 . (Indeed, if

34

A. DOTTO, M. EMERTON, AND T. GEE

we then find a requisite monomorphism embedding π
e0 ,→ πe, we can take π to be00
the coproduct of πe and π over πe .)If we pushout the short exact sequence
0 → π 0 → π → π/π 0 → 0
along the surjection π 0 → π
e0 , and then apply Proposition 3.5.3, we obtain therequired quotient πe of π.

If π is an object of A, and π 0 is an object of AY , and if we write πbY = “ limI ”πi ,←−then the adjunction property of completion can be expressed as∼∼

HomPro(AY ) (bπY , π 0 ) −→ lim HomAY (πi , π 0 ) −→ HomAY (π, π 0 ).
−→I

(The first isomorphism is an instance of (A.3.1), and the second isomorphism is
induced by the maps π → πi induced by the unit of adjunction π → πbY .) Thefollowing lemma extends this result to higher Ext.
Lemma 3.5.5. Let π be an object of Afg , and write πbY = “ lim ”πi , where (by←−I

0Corollary 3.5.2) the πi are objects of Afg
Y . If π is an object of AY , then, for any
value of n, the natural morphism∼

πY , π 0 ) −→ lim ExtnAY (πi , π 0 ) → ExtnA (π, π 0 )ExtnPro(AY ) (b−→Iis an isomorphism.
Proof. We note that the first isomorphism in the displayed chain of isomorphisms
is an instance of Lemma A.3.13.
We may write π 0 as the filtered colimit of its finitely generated subobjects.
Lemma 2.5.6, along with its analogue for AY , shows that each side of the natural morphism under consideration commutes with the passage to filtered colimitsin π 0 , and thus we may and do assume for the remainder of the proof that π 0 is
finitely generated.
Lemma 2.5.9 shows that any object of ExtnA (π, π 0 ) may be represented as a lengthn Yoneda extension involving finitely generated objects of A. Taking completions
along Y , and taking into account Corollary 3.5.4, we obtain a length n Yoneda Ext
in Pro(AY ) between πbY and π 0 . Thus we find that the morphism
ExtnPro(AY ) (bπY , π 0 ) → ExtnA (π, π 0 )is surjective, for every n. It now follows by a formal argument that this map is infact an isomorphism for every n.
In more detail, since the formation of filtered colimits is exact, we obtain amorphism
Ext•Pro(AY ) (bπY , –) → Ext•A (π, –)of δ-functors on AY . This morphism is an isomorphism in degree zero, is surjectivein every degree, and its source is an effaceable δ-functor. (If π 0 is an object of
AY , we can embed it into an injective object of AY , which by Lemma A.3.11
remains injective in Pro(AY ) and so effaces the higher ExtnPro(AY ) (bπY , –).) An easy
dimension-shifting argument shows that this morphism is then an isomorphism, as
claimed.
LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )35

We have the following corollary, which in the case of finite Y also follows directly from Lemma 3.1.3 together with [Paš13, Cor. 5.18] (as was already noted in
Remark 3.2.8).
Corollary 3.5.6. The inclusion AY ,→ A preserves injectives. Consequently, if π
and π 0 are objects of AY , then the natural morphism
ExtnAY (π, π 0 ) → ExtnA (π, π 0 )is an isomorphism, for every value of n.
Proof. To show that an object π 0 of A is injective, it suffices to show that ExtiA (π, π 0 ) =0 for all i > 0, for finitely generated objects π. If π 0 is an injective object of AY ,and if π is finitely generated, it follows from Lemma 3.5.5 that indeed these higher
Ext do vanish. Thus π 0 is an injective object of A. The claimed isomorphism of
Ext groups follows immediately.
c is particularly well-behaved
3.5.7. Ind-completion. Corollary 3.5.4 shows that (–)
Y
fg
on A , and this suggests the following modification of its definition. Namely, we∼recall (see e.g. Section A.1.2) the equivalence Ind(Afg ) −→ A, and then extend
c : Afg → Pro(Afg ) formally to an exact
the exact (by Corollary 3.5.4) functor (–)
YYfg
colimit-preserving functor Ind(A ) → Ind Pro(AfgY ). We continue to denote thisb Y , relying on context to distinguish it from the functor defined on A
functor by (–)
earlier.
fg
There is a canonical functor Ind Pro(AfgY ) → Pro Ind(AY ), while the equivalence∼fg
fg
Ind(AY ) → AY induces an equivalence Pro Ind(AY ) −→ Pro(AY ); composing these
gives a functor(3.5.8)

Ind Pro(AfgY ) → Pro(AY ).

c
We then have the following commutative diagram comparing our “redefined” (–)
Y
to our original definition:
Ind(Afg )

c
(–)
Y

Y

∼


A

/ Ind Pro(Afg )(3.5.8)c(–)
Y/ Pro(AY )3.5.9. Further results for finite Y . We present some additional results that hold in
the case when Y is finite.
Lemma 3.5.10. Assume that Y is finite. If π is an object of Afg”π
Y , and if “ lim←−m mfg
is a countably indexed object of Pro(AY ), then the natural map
ExtnPro(Afg ) (π, “ lim ”πm ) → lim ExtnC (π, πm )
←
−
←
−
Y
m
m
is an isomorphism for every n.
Proof. This follows from Lemma A.3.14 and the fact that all Hom modules in AfgYare of finite length.
36

A. DOTTO, M. EMERTON, AND T. GEE

Our next result has some formal similarity to Lemma 3.5.5, but seems to lie
deeper: its proof relies on Propositions 2.6.5 and 2.6.4 above, and these results
require Paškūnas’ work [Paš13] on the p-adic local Langlands correspondence for
their proof.
0
Proposition 3.5.11. Let Y be a finite subset of X. If π is an object of Afg
Y and πfg
is an object of A , then passing to the completion along Y induces isomorphisms∼

ExtiAfg (π, π 0 ) −→ ExtiPro(Afg ) (π, πbY0 ).
Y

fg

Proof. Although the category A does not have enough injectives or projectives,
the map in the statement of the proposition is well-defined by Corollary 3.5.4,
viewing the Exti -groups as morphisms in the derived category, or equivalently asgroups of Yoneda extensions.
Since Y is finite, Afg
Y is a finite length category, and so it decomposes as a direct
product of blocks. Writing π as a direct sum according to this decomposition we
see that it suffices to prove the proposition when Y = {x} consists of a single closed
points of X, possibly not defined over F.
On the other hand, every object of Afg has a finite filtration whose graded pieces
are either finite length representations or isomorphic to c-IndG
KZ (σ) for a Serreweight σ. Since π 0 7→ π
bY0 is exact, it follows by dévissage that it suffices to prove0
the claim when π is irreducible, and either π 0 = c-IndG
KZ (σ) or π has finite length.
0
When π has finite length the result is immediate from Lemma A.3.12.
Assume now that π 0 = c-IndG
KZ (σ). By the discussion in Section 2.1, there exists
a polynomial fx (T ) ∈ HG (σ) ∼= F[T ] such that the inverse system
n


o
n
πn0 = c-IndG
KZ σ /fx
is cofinal in the diagram defining π
bY0 . By construction, π10 is the largest multiplicityG
free quotient of c-IndKZ σ contained in Afg
Y.
It follows from the above and Lemma 3.5.10 that it suffices to prove that themapExti (π, c-IndG
Exti (π, πn0 )KZ (σ)) → lim←−
n0
induced by the projections c-IndG
KZ (σ) → πn is an isomorphism.

Notice furthermore that fx is regular on the compact induction, and so we have short exact
sequences00
0 → πm→ πn+m→ πn0 → 0

for all positive integers m, n.
As explained at the end of Section 3.6, it follows from Corollary 3.1.9 that wehave an isomorphism∼G
lim ExtiA (π, πn0 ) −→ Exti+1
A (π, c-IndKZ (σ))−→
ninduced by connecting homomorphism of the exact sequence(3.5.12)G
0 → c-IndG
π0 → 0KZ (σ) → c-IndKZ (σ)[1/fx ] → lim−→ nand the fact (proved in Lemma 2.5.6) that Exti commutes with filtered colimits.

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )370Pulling back (3.5.12) along πm→ lim πn0 , and then pushing forward along c-IndG
KZ (σ) →−→0πn , we rephrase our goal as proving that the map
00lim ExtiA (π, πm) → lim Exti+1
A (π, πm )←−−→induced by the connecting homomorphisms of the exact sequences000 → πm→ πn+m→ πn0 → 0

is an isomorphism.
Making a finite unramified extension O0 /O and applying Proposition 2.5.13, andnoting that all terms of the inverse limit have finite O-length, we can furthermore
assume that Y = {x} is a block of absolutely irreducible representations. Hencethe polynomial fx can be taken to be linear in almost all cases: the only exception
is for blocks of type (2) consisting of quotients of c-IndG
KZ (σ) when σ is a twistp−2−1)
of Sym . Indeed, when λ 6= ±1 the representation c-IndG
KZ (σ)/(T − λ)(T − λis multiplicity-free and contained in a single block, and so we need to take fx =0
is a(T − λ)(T − λ−1 ), which is quadratic in T . However, the representation πm
direct sum
G
−1 m
m
0
) ,
πm
= c-IndG
KZ (σ)/(T − λ) ⊕ c-IndKZ (σ)/(T − λ00→ πn0 → 0 is also a direct sum of exact→ πn+mand the exact sequence 0 → πmsequences. Hence the proposition follows from the next result.
Proposition 3.5.13. Let σ be a Serre weight and λ ∈ F. Definenτn = c-IndG
KZ (σ)/(T − λ)and let π be an irreducible object of the block x containing τn . Then the connecting
homomorphisms of the exact sequence(T −λ)n0 → τm −−−−−→ τn+m → τn → 0
induces an isomorphism
lim ExtiA (π, τm ) → lim Exti+1
A (π, τn )←−−→for all i ≥ 0.
L
Proof. Let Jx = π∈x Jπ be the direct sum of injective envelopes of all the irreducible objects of Ax . Then the functor
Mx (−) = HomA (−, Jx )∨is an equivalence of Ax with the category of discrete left modules over Ex =EndA (Jx ). Hence it suffices to prove the proposition after applying Mx (−).
Define M = limi Mx (τi ), which is a module over Ex (in fact, since the Mx (τi )←−have finite length, this is actually a compact Ex -module). The Hecke operator (T −λ) induces compatible endomorphisms of all the M (τi ), hence also of M . There isan exact sequence
0 → M → M [1/(T − λ)] → lim Mx (τi ) → 0.
−→i

Since the groups
ExtiEx (Mx (π), Mx (τj )) ∼= ExtiAx (π, τj )

38

A. DOTTO, M. EMERTON, AND T. GEE

have finite dimension over k, the same argument as in Lemma A.3.14 implies thatthe natural map∼

Exti+1→ lim Exti+1
Ex (Mx (π), M ) −Ex (Mx (π), Mx (τj ))←−j

is an isomorphism. So we obtain a connecting homomorphism
lim ExtiEx (Mx (π), Mx (τk )) → lim Exti+1
Ex (Mx (π), Mx (τj ))
−→
←−
j

k

which can be checked to coincide with the map induced by the connecting homomorphisms of the short exact sequences
0 → Mx (τm ) → Mx (τn+m ) → Mx (τn ) → 0
So it suffices to prove that
(3.5.14)E i = ExtiEx (Mx (π), M [1/(T − λ)]) = 0

for all i. We are going to deduce this from Propositions 2.6.4 and 2.6.5, using thefact that this Exti -group is a module over the centre Zx of Ex in a unique way
under the actions on either factor.
Assume first that x is a block of type (1), (2) or (4). Then it follows from
Proposition 2.6.4 that there exists an element Tx −λ ∈ Zx inducing the operator T −λ on Mx (τi ) for all i, and so on M too. Since π is an absolutely irreducible objectof Ax and Tx − λ = 0 on τ1 , we also have Tx − λ = 0 on π. Hence the vanishingof (3.5.14) follows from the fact that Tx − λ is simultaneously invertible and zeroon E i .
Assume now that x has type (3). Then E i has an invertible endomorphism T −λarising from the second factor. Furthermore, by Proposition 2.6.5 there are central
elements z0 , . . . , zn ∈ Zx such that(T − λ)n+1 + zn (T − λ)n + · · · + z1 (T − λ) + z0 = 0
as endomorphisms of Mx (τi ), hence as endomorphisms of M and E i . In addition,
the zi are contained in the maximal ideal mx , and so they are zero on Mx (π),
which is the only irreducible object of the block. So they are zero on E i . It follows
that (T − λ)n+1 is simultaneously invertible and zero on E i , and so E i = 0.

3.5.15. From formal to literal completions. When Y is finite, since the objects ofAfg
Y are Artinian when regarded as objects of A, it follows that the functor
lim : Pro(AfgY ) → O[G]-Mod,←−
given by mapping a pro-object “ limI ”πi to its literal projective limit limi πi in the←−←−
category of O[G]-modules, is faithful and exact [Jen72, Cor. 7.2].
If we compose this with the exact (by Corollary 3.5.4) completion functor Afg →Pro(AfgY ), we obtain the exact “literal completion along Y ” functor
c : Afg → O[G]-Mod,lim (–)←− Y
which we denote by π 7→ lim πb ,←− YLemma 3.5.16. For any object π of Afg , there is a natural morphism of O[G]modules π → lim π
b .
←− Y

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )39

Proof. As discussed in A.3.17, the unit of adjunction induces a morphism π → πbY
in Pro(A). Regarding this as a morphism in the larger category Pro(O[G])-Mod,and them passing to literal projective limits in O[G]-Mod (and recalling that π isconstant as a pro-object), we obtain the desired morphism.
The discussion at end of A.3.17 shows that, if we apply Lemma 3.5.1 to write∼πbY −→ “ lim ”π 0 , where π 0 runs over the cofiltered directed set of quotients of π 0 , so←−that lim πbY = lim π 0 , then the morphism of the preceding lemma is the one induced←−←−by the collection of quotient morphisms π → π 0 .
If we pass to Ind-categories, then “literal projective limit” formally extends toan exact and faithful functor lim : Ind Pro(AfgY ) → Ind O[G]-Mod. Thus if π is an←−∼object of A −→ Ind Afg , we may form the “literal completion” lim πb, which is an←−object of Ind O[G]-Mod. There is again a natural morphism π → lim πb.
←−3.5.17. Some analogues for AU . Suppose that Y ⊆ U for some open subset U of X.
Write Z := X \ U. If π and π 0 are objects of AZ and AY respectively, then
HomAY (bπY , π 0 ) = HomA (π, π 0 ) = 0
d vanishes on AZ , and so the functor π
by Corollary 3.2.6. Thus (−)
bY factors
Ythrough AU , and then extends canonically to a functor Pro(AU ) → Pro(AY ) prec for this induced functor,
serving cofiltered limits. We use the same notation (–)
Y
so that by definition we have
∗
jd
bY ,
U πY = π(3.5.18)
for objects π of Pro(A).

Lemma 3.5.19. If πU is an object of Pro(AU ), then there is a natural isomorphism∼dj\∗ (πU )Y −→ (πU )Y .
c and j∗ are extended from AU to Pro AU via the
Proof. Since the functors (–)
Y
c is extended from A to Pro(A) via
universal property of the latter, and similarly (–)
Y
the universal property of the latter, it suffices to construct the natural isomorphism
for objects πU of AU .
∼Now the counit of adjunction gives an isomorphism jU∗ jU ∗ πU −→ πU , and therequired isomorphism follows from (3.5.18).

Recall from Corollary 3.2.5 that the unit of adjunction id → jU ∗ jU∗ restricts to
an isomorphism on AY , so that jU∗ embeds AY fully faithfully into AU . Thus, if
πU is an object of Pro(AU ) and π 0 is an object of Pro(AY ), we find that
∼
0
0
[
\
HomPro(AY ) ((πU )Y , π ) −→ HomPro(AY ) ((j∗ πU )Y , π )∼= HomPro(A) (j∗ πU , π 0 ) −→ HomPro(A) (j∗ πU , j∗ j ∗ π 0 )= HomPro(AU ) (πU , j ∗ π 0 ).
(The first isomorphism is given by Lemma 3.5.19, while, as already noted, the second isomorphism follows from (3.5.18). The equalities are instances of the various
c on Pro(AU ) satisfies an analogous
adjunctions that we’ve established.) Thus (–)
Ycadjunction to the one satisfied by (–)Y on Pro(A).

40

A. DOTTO, M. EMERTON, AND T. GEE

c on AU satisfies analogues of the results we proved for
We now show that (–)
Y
c on A.
(–)
Y
Corollary 3.5.20. If π is an object of Afg
bY is an object of Pro(AfgU , then π
Y ).
Proof. This follows directly from Lemma 3.1.5(3) and Corollary 3.5.2.

c to Afg is exact.
Lemma 3.5.21. The restriction of (–)
UY∗
Proof. Any exact sequence in Afg
U may be written as the image under jU of an exact
fg
sequence in A . The claim then follows from Corollary 3.5.4 and (3.5.18).
Lemma 3.5.22. Let πU be an object of Afg
U , and (via Corollary (3.5.20))write
0
d
(π
)
=
“
lim
”π
,
where
the
π
are
objects
of
Afg
U Y
i
Y . If π is an object of AY , then,←− iIfor any value of n, there are natural isomorphisms
∼
n
n
0
0
∗ 0
d
ExtnPro(AY ) ((πU )Y , π ) −→ lim ExtAY (πi , π ) → ExtAU (πU , j π ).
−→I∗

Proof. If we write πU = j π for some object π of A, take into account (3.5.18) andapply Lemma 3.5.5, then the lemma follows from the chain of isomorphisms andadjunctions∼

ExtnA (π, π 0 ) −→ ExtnA (π, j∗ j ∗ π 0 ) = ExtnAU (j ∗ π, j ∗ π 0 ) = ExtnAU (πU , j ∗ π 0 )(the isomorphism being the consequence of Corollary 3.2.5 that we’ve alreadynoted).

3.6. The functors ΓY and R1 ΓY . If Y is a closed subset of X, then we define
the functor ΓY : A → AY as follows: ΓY (π) is the maximal subobject of π lying in
AY . Equivalently, ΓY (π) is the kernel of the natural morphism π → j∗ j ∗ π (where,
as usual, we write U := X \ Y ). The functor ΓY is right adjoint to the inclusion of
AY in A; since this latter functor is exact, we see that ΓY takes injectives in A to
injectives in AY . Corollary 3.5.6 then shows that ΓY in fact takes injectives in A
to injectives in A.
We may consider the derived functors Ri ΓY of ΓY . Their computation is facilitated by the following result.
Lemma 3.6.1. If I is an injective object of A, then the natural map I → j∗ j ∗ I issurjective.
Proof. As already noted, it follows from Corollary 3.5.6 that ΓY (I) is again an
injective object of A. Thus the inclusion ΓY (I) ,→ I is split, and so the image J of
I in j∗ j ∗ I is yet again injective. The inclusion J ,→ j∗ j ∗ I is thus also split. The
characterization given by Lemma A.2.2 then shows that J lies in the essential imageof j∗ , and thus that J = j∗ j ∗ J. Thus the surjection I → J induces an identificationj∗ j ∗ I → j∗ j ∗ J = J, and the lemma is proved.

Corollary 3.6.2. For any object π of A, the derived functors of ΓY are computedby the complex
π → j∗ j ∗ π.
In particular, the only non-trivial higher derived functor is R1 ΓY .

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )41

Proof. If I • is an injective resolution of π, then the various R• ΓY (π) are computed as the cohomology of the kernel of the morphism of complexes I • → j∗ j ∗ I • .
Lemma 3.6.1 shows that this kernel coincides with the cone (up to a shift). Now
since j∗ is exact, the complex j∗ j ∗ I • resolves j∗ j ∗ π, and so this shifted cone isquasi-isomorphic to the complex π → j∗ j ∗ π, proving the lemma.

Since ΓY preserves injectives, we obtain, for any objects π of AY and π 0 of A, aspectral sequence
0(3.6.3)E2p,q := ExtpAY π, Rq ΓY (π 0 ) =⇒ Extp+q
A (π, π ).
In the case when π 0 → j∗ j ∗ π 0 is injective, i.e. when ΓY (π 0 ) = 0, this simplifies tothe formula0
ExtpAY π, R1 ΓY (π 0 ) = Extp+1
A (π, π ),which can be obtained directly by computing the long exact sequence of Ext’s
arising from the short exact sequence
0 → π 0 → j∗ j ∗ π 0 → R1 ΓY (π 0 ) → 0,
and taking into account Corollary 3.1.9.
Lemma 3.4.1 shows that if π and R1 ΓY (π) are both finitely generated, then infact R1 ΓY (π) = 0. Thus R1 ΓY (π) is typically not finitely generated. Nevertheless,
we can gain some control over it, using the following results. (By contrast, if π isfinitely generated then so is ΓY (π), by Theorem 2.4.1, since it is a submodule of π.)
Lemma 3.6.4. Suppose that Y is a closed subset of X, and that σ is a Serre weight.
As in the statement of Proposition 3.1.7, write fσ−1 (Y ) = V (g) for some g ∈ H(σ).
GG1
If g = 0, then ΓY (c-IndG
KZ σ) = c-IndKZ σ and R ΓY (c-IndKZ σ) = 0, while if g 6=GG
G0, then ΓY (c-IndKZ σ) = 0, while R1 ΓY (c-IndKZ σ) = (c-IndG
KZ σ)[1/g]/(c-IndKZ σ).
Proof. Proposition 3.1.7 shows that the exact sequenceGG∗1
0 → ΓY (c-IndG
KZ σ) → c-IndKZ σ → j∗ j c-IndKZ σ → R ΓY (σ) → 0

reduces to eitherG
0 → c-IndG
KZ σ → c-IndKZ σ → 0 → 0 → 0

or
G
G
G
0 → 0 → c-IndG
KZ σ → (c-IndKZ σ)[1/g] → (c-IndKZ σ)[1/g]/(c-IndKZ σ) → 0,

depending on whether g = 0 or not.



Lemma 3.6.5. If Y is a closed subset of X and π is of finite length, then RΓ1Y (π) =0.
Proof. Since π is of finite length, it is an `object of AZ for some finite closed set Z
of X, which we write in the form Z := Z1 Z2 , where Z1 = Y ∩ Z and Z2 = Z \ Z1 .
Correspondingly we may write π = π1 ⊕π2 with πi an object of AflYi . The complexes
0 → ΓY (πi ) → πi → j∗ j ∗ πi → R1 ΓY (πi ) → 0
then reduce to
0 → π1 → π1 → 0 → 0 → 0,
respectively
0 → 0 → π2 → π2 → 0 → 0,

42

A. DOTTO, M. EMERTON, AND T. GEEin the case i = 1, respectively 2. In particular, in either case we have R1 ΓY (πi ) = 0,
and so R1 ΓY (π) = 0.

We may now prove the following result.
Lemma 3.6.6. If Y is a closed subset of X and π is finitely generated, then RΓ1Y (π)
is an Artinian object of AY .
Proof. By the usual dévissage, we reduce to the case when either π is of finite
length, or π is of the form c-IndG
KZ σ. In the finite length case, Lemma 3.6.5 showsthat R1 ΓY even vanishes. In the case of c-IndG
KZ σ, Lemma 3.6.4 reduces us toGshowing that if g is a non-zero element of H(σ), then (c-IndG
KZ σ)[1/g]/(c-IndKZ σ)
is Artinian, which is Lemma 2.1.10.
Definition 3.6.7. If π is any object of A, then we let πf.l. denote the maximal
subobject of π which is locally of finite length; equivalently, this is the maximal
subobject of π which is locally admissible.
We may write πf.l. = limY ΓY (π), where Y runs over all finite closed subsets−→of X. We then see that this functor has a single non-vanishing higher derived
1functor, namely limY R ΓY .
−→3.7. More on Pro(AfgY ). We now extend some of the preceding constructions to thecontext of Pro(Afg),
sincewe will need them for the Beauville–Laszlo-type resultsYof the following section.
fg
fg
We may form the quotient category Pro(AfgY )U := Pro(AY )/AY , and just as
fg
in the case of A considered above, we let j ∗ : Pro(AfgY ) → Pro(AY )U denote thecanonical functor. Since this functor is exact, its Ind-extension admits a rightadjoint (applying the Ind-analogue of the discussion of A.3.7), which we denotefg
by j∗ : Ind Pro(AfgY )U → Ind Pro(AY ). We will only consider the restriction offg
this functor to Pro(AY )U . (But note that the target of this restriction is still
Ind Pro(AfgY ).)c : Afg → Pro(Afg ) induces a corresponding functor
The completion functor (–)
Y
fg
fg
fg
fg
fg
Afg
U := A /AY → Pro(AY )/AY =: Pro(AY )U ,c By construction, then, the diagram
which we again denote by (–).
Afg

(3.7.1)jc(–)/ Pro(Afg )
Y

∗


Afg
U

j∗


c
(–)
/ Pro(Afg )UYcommutes up to natural transformation.
We also have a diagram(3.7.2)

c
(–)

Afg
U

/ Pro(Afg )U
Y

j∗

j∗


A

∼

/ Ind Afgc(–)


/ Ind Pro(Afg )Y

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )43c : Afg → Pro(Afg ) to the corresponding Ind-categories,(where we have extended (–)
Y
as in 3.5.7 above), but the commutativity of this diagram is less obvious, and infact we only establish it under the additional hypothesis that Y is finite. In thatcase, it is more-or-less the content of the following proposition.
Proposition 3.7.3. Assume that Y is finite. If π is an object of Afg , then there∼∗∗
is a natural isomorphism j\b.
∗ j π −→ j∗ j π
Proof. From (3.7.1) (and its formal extension to Ind-objects), we obtain the first
and third of the following sequence of isomorphisms
∼

∼

∼

∗
∗ j j ∗ π −→ j
∗ π −→ j ∗ πdj ∗ j\b,∗ j π −→ j \∗the second being clear. Thus, by adjunction, we obtain a morphism
∗
∗
j\
b,
∗ j π → j∗ j π(3.7.4)

which we must show is an isomorphism.
Completing the natural (unit to the adjunction) morphism π → j∗ j ∗ π induces a∗morphism π
b → j\∗ j π. There is also the natural (unit to the adjunction) morphism∗πb → j∗ j π
b. The resulting diagram(3.7.5)

π
b

∗
/ j\
∗j π

π
b


/ j∗ j ∗ πb(3.7.4)is easily checked to commute. (By its construction, it suffices to check commutativ∗ π and all the arrows
ity after applying j ∗ , in which case all the vertices become jd∗
become the identity morphism of jd
π to itself.)
Our next step is to embed (3.7.5) in a larger commutative diagram. To begin
with, we consider the tautological exact sequence
0 → ΓY (π) → π → j∗ j ∗ π → R1 ΓY (π) → 0,
∼

fg
∗
fg
with ΓY (π) an object of Afg
Y , π an object of A , j∗ j π an object of A −→ Ind A ,∼fg
1
and R ΓY (π) an object of AY −→ Ind AY . Applying the completion functor (in its
Ind-category incarnation following 3.5.7 above), which is exact by Corollary 3.5.4,
we obtain the exact sequence1∗
0 → ΓY (π) → π
b → j\∗ j π → R ΓY (π) → 0,

where we still have ΓY (π) being an object of Afg
b being an object of
Y , we have π
fg
fg
∗
\
Pro(AY ), we have j∗ j π being an object of Ind Pro(AY ), and we have R1 ΓY (π)
being an object of Ind Afg
Y.
We next define V and W to be the kernel and cokernel respectively of the naturalmorphism π
b → j∗ j ∗ πb. The diagram (3.7.5) can then be placed in the commutative
diagram with exact rows
(3.7.6)0/ ΓY (π)

0


/V

/π
b

∗
/ j\
∗j π

/π
b


/ j∗ j ∗ πb/ R1 ΓY (π)

/0


/W

/0

(3.7.4)

44

A. DOTTO, M. EMERTON, AND T. GEE

Now, by its construction, ΓY (π) can be characterized as the object of Afg
Y thatrepresents the functor HomAfg (–, π) on Afg.
Similarly,Vcanbecharacterized
as theYfg
fg
0
object of AY that represents the functor HomPro(Afg ) (–, πb) on AY . The Ext case ofY
Proposition 3.5.11 then implies that the morphism ΓY (π) → V is an isomorphism.
A similar argument shows that the morphism R1 ΓY (π) → W is an isomorphism.
Indeed, the long exact Ext sequences attached to the short exact sequences
0 → π/ΓY (π) → j∗ j ∗ π → R1 ΓY (π) → 0
and0→πb/V → j∗ j ∗ πb→W →0give rise to the horizontal arrows of the commutative square of functors (on AfgY)/ Ext1A –, π/ΓY (π)
HomA –, R1 ΓY (π)

HomInd Pro(Afg ) (–, W )Y/ Ext1

Ind Pro(Afg
Y )

(–, π
b/V )

in which the horizontal arrows are monomorphisms. In fact, the discussion following
Corollary 3.6.2 shows that the upper horizontal arrow is an isomorphism, and theExt1 case of Proposition 3.5.11 shows that the right-hand vertical arrow is also
an isomorphism. It follows that the other two arrows — in particular, the lefthand vertical arrow — are isomorphisms. This implies that R1 ΓY (π) → W is an
isomorphism.
We have now seen that the outer vertical arrows in (3.7.6) are isomorphisms.
The five lemma then implies that (3.7.4) is an isomorphism, as required.

We finish this section by stating a result related to the proof of the preceding
proposition. In order to make the statement, we first recall (continuing to assume
that Y is finite) from 3.5.15 that we have the “literal projective limit” functor
lim : Ind Pro(Afgπb.
Y ) → Ind O[G]-Mod, and there is a natural morphism π → lim←−←−
Lemma 3.7.7. Assume Y is finite. If π is an object of Afg , then there is a natural
short exact sequence∗0→π→π
b × j∗ j ∗ π → j\∗j π → 0

Proof. As in the proof of Proposition 3.7.3, we may complete the tautological exact
sequence
0 → ΓY (π) → π → j∗ j ∗ π → R1 ΓY (π) → 0
to obtain an exact sequence1∗
0 → ΓY (π) → π
b → j\∗ j π → R ΓY (π) → 0.

Now formation of literal projective limits is exact; also both ΓY (π) and R1 ΓY (π)
are in Ind Afg
acts identically on each
Y (i.e. are trivial in the Pro-aspect), and so lim←−of them. Thus we obtain an exact sequence in Ind O[G]-Mod
0 → ΓY (π) → lim π
b → lim j\j ∗ π → R1 ΓY (π) → 0.
←−←− ∗

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )45

Now consider the squareπ/ j∗ j ∗ π


π
b


∗
/ j\
j
∗ π

If we regard this as a morphism (in the vertical direction) of complexes (in thehorizontal direction), then the discussion of the preceding paragraph shows thatthe vertical morphisms induce a quasi-isomorphism of complexes. Passing to theassociated total complex yields the required exact sequence.

3.8. Beauville–Laszlo-type gluing. The results of this section are not used elsewhere in this paper, but they are crucially applied in [DEG]. The categorical p-adic
Langlands correspondence developed in [DEG] relates smooth representations of
GL2 (Qp ) to sheaves on stacks, and the arguments made in [DEG] rely on being
able to glue representations in the same way that we can glue sheaves. In fact,
in order to make use of the results of [Paš13], we need an analogue of Beauville–
Laszlo gluing of sheaves (i.e. of gluing sheaves over formal completions at closed
loci to sheaves on the complementary open substack). Theorem 3.8.1 provides sucha result.
Theorem 3.8.1. Suppose that Y is a finite closed subset of X, with open complement U . The canonical functorfg
Afg −→ Pro(AfgY ) ×Pro(Afg )U AU ,Y

induced by completion along Y and by localization over U , is an equivalence of
categories.
Proof. As usual, it suffices to show that the functor is fully faithful and essentiallysurjective. We begin by showing faithfulness. To this end, suppose that f, g :π1 ⇒ π2 are morphisms in Afg , for which the induced morphisms π
c1 → πc2 andj ∗ π1 → j ∗ π2 coincide. We then find that the morphisms
lim π
c × j∗ j ∗ π1 → lim π
c × j∗ j ∗ π2
←− 1
←− 2
coincide. The injectivity aspect of Lemma 3.7.7 shows that f and g in fact coincide.
We turn to proving fullness. Thus we suppose given a pair π1 , π2 of objects offg
A , together with morphisms g : π
b1 → πb2 and h : j ∗ π1 → j ∗ π2 , such that j ∗ g = bhas morphisms∗π = j∗π∗π .
jdc1 → j ∗ π
b2 = jd1
2
Applying j∗ , we then find that j∗ j ∗ g and j∗ bh coincide as morphisms∗∗∗j\c1 → j∗ j ∗ π
b2 = j\∗ j π1 = j∗ j π∗ j π2(where we are using the identifications given by Proposition 3.7.3).
“Literal projective limit” formally extends to an exact and faithful functor lim :←−∼fg
Ind Pro(Afg)→
Ind
O[G]-Mod.
Thus
if
π
is
an
object
of
A
−→
Ind
A
,
we
may
Y
form the “literal completion” lim πb, which is an object of Ind O[G]-Mod. There is←−a natural morphism π → lim π
b.
←−

46

A. DOTTO, M. EMERTON, AND T. GEE

We now make use of the “literal projective limit” of 3.5.15, and the naturalmorphism π → lim πb, where π is an object of A. We will apply this with π = j∗ j ∗ πi←−(i = 1, 2), and so obtain a commutative square(3.8.2)j∗ j ∗ π1/ lim j\j ∗ π1
←− ∗

j∗ h

←
− ∗

/ lim j\j ∗ π2←− ∗∗j∗ j π2

lim j j ∗ g=lim jd
∗h

←
−

∗
∗
We may also apply lim to the natural morphisms π
bi → j\bi , to obtain∗ j π i = j∗ j π←−
a commutative square(3.8.3)

lim π
b
←− 1
lim g

←
−


lim π
b
←− 2

/ lim j\j ∗ π1←− ∗lim j∗ j ∗ g=lim jd
∗h

←
−

/ lim j\j ∗ π2←− ∗←−We may then “glue” the two squares (3.8.2) and (3.8.3) into the following commutative square, in which the horizontal arrows are given by taking the differenceof the horizontal arrows in those preceding two squares:lim π
b × j∗ j ∗ π1←− 1
lim g×j∗ h

←
−


lim π
b2 × j∗ j ∗ π2←−

/ lim j\j ∗ π1←− ∗lim j j ∗ g=lim jd
∗h

←
− ∗

/ lim
\
j j ∗ π2←− ∗←−Now Lemma 3.7.7 shows that the vertical morphisms in this square induce a morphism f : π1 → π2 , which in turn gives rise to the morphisms lim g and j∗ h after←−
literally completing, respectively applying j∗ j ∗ . Since the formation of literal projective limits is faithful, we see that f gives rise to g after completion. We also seethatj ∗ f = j ∗ j∗ j ∗ f = j ∗ j∗ h = h.
Thus f : π1 → π2 induces (g, h) : (bπ1 , j∗ π1 ) → (bπ2 , j∗ π2 ), and fullness is proved.
It remains to prove
essential
surjectivity.
To
this end, suppose that we are givenfg
a pair π
b1 , (π2 )U , where π
b1 is an object of Pro(AfgY ) and π2 is an object of A ,fgwith image (π2 )U := j ∗ π2 in the quotient category AU . We are furthermore given
an isomorphism of these objects in the quotient category Pro(AfgY )U , which we canrepresent by a “roof”, i.e. a pair of morphisms π
b1 , πb2 ⇒ πb3 , where πb2 denotes
the completion of π2 along Y , and π
b3 is another object of Pro(Afg).
Thispair ofYfgmorphisms has the property that their kernels and cokernels lie in AY . (This isfg
why they give rise to an isomorphism in Pro(A Y )U .)In order to show that the pair π
b1 , (π2 )U is in the image of the functor, we needin particular to find some object π1 of A whose completion along Y is πb1 . We will
do this by repeatedly applying Proposition 3.5.11, and it will turn out that ourconstruction also guarantees that (π1 )U = (π2 )U .

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )47

The kernel π 0 of π
b2 → πb3 is supported on Y by assumption, and so lies in ΓY (bπ2 ),which by Proposition 3.5.11 is equal to ΓY (π2 ). In other words, π 0 is actually asubobject of π2 , and the exactness of completion ensures that (π2 /π 0 )b = π
b2 /π 0 .
00
Again by assumption, the cokernel π of the morphism π
b2 → πb3 is supported
on Y . Thus πb3 represents an element of Ext1Pro(Afg ) (π 00 , π
b2 /π 0 ). By Proposition 3.5.11,
Ythis corresponds to an element of Ext1Afg (π 00 , π2 /π 0 ), i.e. we can find an object π3of Afg that contains π2 /π 0 , with (π3 /(π2 /π 0 ) = π 00 , and whose completion along Y
coincides with πb3 .
Now let π 000 denote the cokernel of the morphism π
b1 → πb3 , which is yet again
supported on Y . By Lemma 3.5.5, the surjection π
b3 → π 000 then induces and is
induced by a surjection π3 → π 000 . We denote the kernel by π4 . If we let π (4) denotethe kernel of π
b1 → πb3 , then we get a short exact sequence
0 → π (4) → π
b1 → πb4 → 0,which (by an application of Proposition 3.5.11 as in the preceding paragraph) arises
by completing a short exact sequence
0 → π (4) → π1 → π4 → 0
of objects in Afg . In other words, π
b1 is the completion of an object π1 , admitting
a morphism to π3 , which induces the given morphism π
b1 → πb3 after completion. We claim that the functor of the proposition takes π1 to the pair π
b1 , (π2 )U .
Indeed the completion of π1 is πb1 by construction, while chasing through the construction above we find in turn that
(π1 )U = (π4 )U = (π3 )U = (π2 )U ,as required.



Remark 3.8.4. It is likely that Theorem 3.8.1 holds for arbitrary closed subsets
Y of X, and indeed a proof in this case should reduce to the finite case via an
appropriate application of the gluing result for the (finite!) closed subset U ∩ Y .
Since the case of finite Y suffices for our envisaged applications, we haven’t tried
to write up the details of this reduction.
4. Applications and examples
4.1. The structure of smooth representations. In many of our arguments wehave used dévissage to reduce to the cases of irreducible representations, and representations of the form c-IndG
KZ σ. We can use our results to make more precise theway in which these representations interact. We begin with the following structural
results.
Proposition 4.1.1. Every object of A/Al.adm is locally of finite length.
Proof. Since Al.adm is localizing, the quotient functor A → Al.adm preserves colimits. It follows that it suffices to prove that π = c-IndG
KZ (σ) is irreducible in thequotient category for all Serre weights σ. In order to do this, it suffices to provethat every nonzero arrow X → π in the quotient category is surjective. Writing X
as a colimit of finitely generated representations, it suffices to prove this when X
is finitely generated.
We can represent the arrow by a “roof diagram” in A, i.e. a pair of arrows
X → Z, π → Z for some object Z, such that π → Z has locally admissible kernel

48

A. DOTTO, M. EMERTON, AND T. GEE

and cokernel. Replacing Z with the subrepresentation generated by the imagesof X and π, we can assume that Z is finitely generated. Hence π → Z has kerneland cokernel of finite length.
Since π has no subobjects of finite length, it follows that π → Z is injective.
Quotienting out by Zf.l. , we can assume that π → Z is furthermore an essential
embedding. It follows from Lemma 3.1.10 that Z is isomorphic to a submoduleof π. Since every nonzero submodule of π has cofinite length in π, it follows thatevery nonzero submodule of Z has cofinite length in Z. Hence the map X → Z has
cokernel of finite length, and so it is surjective in the quotient category A/Al.adm ,which was to be proved.

Remark 4.1.2. By definition (see e.g. [Kan21, §2.4]), Proposition 4.1.1 says that the
Krull–Gabriel dimension of A is 1.
We have the following more concrete variant of Proposition 4.1.1. Recall from
Definition 3.6.7 that we write πf.l. for the maximal subobject of π which is locallyof finite length (equivalently, locally admissible).
Proposition 4.1.3. If π ∈ A is finitely generated, then π/πf.l. is a successive
extension of representations isomorphic to submodules of c-IndG
KZ σ, for various
Serre weights σ.
Proof. We suppose that πf.l. = 0, and show that π is a successive extension of
submodules of representations of the form c-IndG
KZ σ. We write π as the quotientof c-IndG
V
for
some
smoothO[KZ]-representation
V of finite O[KZ]-length, andKZargue by induction on the length of V . Let σ be an irreducible subrepresentation ofV , and consider the induced morphism c-IndG
KZ σ → π. If this morphism is zero, we
may replace V by V /σ, and thus we may assume it is non-zero. Since πf.l. = 0 by
assumption, it is then furthermore injective. Now consider the short exact sequenceG
0 → c-IndG
KZ σ → π → π/c-IndKZ σ → 0

Pulling back along the finite length part of the target, we obtain a short exact
sequenceG
0
0 → c-IndG
KZ σ → π → (π/c-IndKZ σ)f.l. → 00which is furthermore essential, since πf.l.
⊆ πf.l. = 0. Lemma 3.1.10 and Re00mark 3.1.12 then show that π is isomorphic to a submodule of c-IndG
KZ σ forG000some σ . By construction, (π/π )f.l. = 0, and π/π is a quotient of c-IndKZ (V /σ).
The result follows by induction.
Remark 4.1.4. If σ is not isomorphic to a twist of Sym0 or Symp−1 , it follows fromG
Lemma 2.1.3 that every submodule of c-IndG
KZ (σ) is isomorphic to c-IndKZ (σ), andin fact coincides with the image of a Hecke operator in H(σ).
4.2. Extensions between irreducible representations and compact inductions. It is possible to use our results to compute Ext groups between irreducible
representations and compact inductions in complete generality. We begin withthe following qualitative statement, whose proof depends upon the main resultsof [Paš13].
Lemma 4.2.1. If π is of finite length, and π 0 is finitely generated, then the Omodule ExtiA (π, π 0 ) is of finite length for all i ≥ 0.

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )49

Proof. By dévissage it suffices to prove the claim when π 0 has finite length or π 0 is
the compact induction of a weight. The first case is a consequence of [Paš13], so we
assume that π 0 = c-IndG
KZ (σ). The representation π is an object of AY for somefinite Y , and the spectral sequence (3.6.3) implies that
∼ Exti (π, R1 ΓY (π 0 )).
Exti+1 (π, π 0 ) =AAYBy Lemma 3.6.4 and Lemma 2.1.10 we know that R1 ΓY (π 0 ) is Artinian, and so it
suffices to prove that ExtiAY (π, τ ) has finite O-length whenever τ is an Artinian
object of AY . However, the socle of τ has finite length, hence by Proposition 2.1.11
there exists an exact sequence
0 → τ → J → τ0 → 0
where J is an injective Artinian object of AY . It follows that τ 0 is also Artinian, andso by dimension shifting and induction it suffices to prove that HomAY (π, τ 0 ) has
finite O-length whenever τ 0 is Artinian. This follows again from Proposition 2.1.11.
For extensions of absolutely irreducible representations and compact inductionsof Serre weights we can be a lot more precise, as in the following proposition.
Proposition 4.2.2. Let σ be a Serre weight and π an absolutely irreducible objectof A. Then dim Ext1A (π, c-IndG
KZ (σ)) ≤ 1, and if it is not zero then one of thefollowing is true:(1) π is a quotient of c-IndG
KZ (σ) by some T −λ, and the extension is isomorphic
toT −λ
0 → c-IndG−−→ c-IndG
KZ (σ) −KZ (σ) → π → 0.
p−1∼(2) π = χ ◦ det, σ = Sym⊗(χ ◦ det)|KZ , and the extension is isomorphic toG
0 → c-IndG
KZ (σ) → c-IndN (χ ◦ det) → π → 0.
∼ (χ ◦ det)|KZ , and the extension is isomorphic to(3) π = (χ ◦ det) ⊗ StG , σ =the preimage of π underG
p−1⊗(χ ◦ det)|KZ ) → π ⊕ (π ⊗ (nr−1 ◦ det)) → 0.
0 → c-IndG
KZ (σ) → c-IndKZ (Sym

Proof. Let x be the block of π and write Y = {x}. Since ΓY (c-IndG
KZ (σ)) = 0 the
spectral sequence (3.6.3) yields an isomorphism∼

Hom(π, R1 ΓY (c-IndG→ Ext1 (π, c-IndG
KZ (σ))) −KZ (σ)).
By Lemma 2.1.10 we know that the socle of R1 ΓY (c-IndG
KZ (σ)) has multiplicity
one, which implies the assertion on dimensions.
Assume that π is not a character or a Steinberg twist. Then Lemma 2.1.10 imG
plies that the socle of R1 ΓY (c-IndG
KZ (σ)) is isomorphic to a quotient of c-IndKZ (σ),
Gand so if Ext1 (π, c-IndG
KZ (σ)) does not vanish then π is a quotient of c-IndKZ (σ).
Since we have already proved that there is only one isomorphism class of nonsplit
extensions, it follows that it must be the one given in the proposition.
Assume now that π is a character; the case of Steinberg twists is treated similarly.
Twisting by χ−1 ◦ det we can assume that π = 1 is trivial and then we deduce from
Lemma 2.1.10 thatsoc(R1 ΓY (c-IndG (Symp−1 ))) = soc(c-IndG (Symp−1 )/(T − 1)) ∼
=1
KZ

KZ
G
G
soc(R ΓY (c-IndKZ (1))) = soc(c-IndKZ (1)/(T − 1)) ∼
= StG
1

50

A. DOTTO, M. EMERTON, AND T. GEE

p−1
hence if Ext1 (π, c-IndG. To find a repreKZ (σ)) does not vanish then σ = Sym
sentative for the extension class it suffices to start from the exact sequencesβp−1
0 → c-IndG−→ c-IndG
KZ SymKZ 1 → 1 ⊕ nr−1 → 0
G
0 → c-IndG
N (δ) → c-IndKZ 1 → 1 → 0

defined in Lemma 2.1.3 and (3.2.2), and to twist the second one by nr−1 .

Remark 4.2.3. Let Y be the block of the trivial representation of G. Using Lemma 2.1.12
we can refine the conclusion of Proposition 4.2.2: if τ is an extension of 1 by π which
is not a quotient of c-IndG
KZ (1) then
HomG (τ, R1 ΓY (c-IndG
KZ (1))) = 0
and so there are no extensions of τ by c-IndG
KZ (1).
Using our results in Section 2.6 it is also possible to compute the Ext1 -groups
in the other direction. For simplicity, we will only deal with compact inductions ofgeneric Serre weights.
Proposition 4.2.4. Let σ be a Serre weight, let π be an absolutely irreducible
object of A, and assume that σ is not a twist of Sym0 , Symp−2 or Symp−1 . ThenG
Ext1A (c-IndG
KZ σ, π) is not zero if and only if the completion of c-IndKZ (σ) at theblock of π is not zero. If this is the case thendim Ext1A (c-IndG
KZ σ, π) = 2 if π is supersingular= 1 otherwise.
Proof. If the completion is zero then
ExtiA (c-IndG
KZ (σ), π) = 0
for all i by Lemma 3.5.5. Otherwise, let τ = c-IndG
KZ (σ)/(T − λ) be the only irre(σ)
in
the
same
block
of
π, and let τi = c-IndG
ducible quotient of c-IndG
KZ (σ)/(T −KZiλ) . By Lemma 3.5.5 there is an isomorphism∼

lim Ext1AY (τi , π) −→ Ext1A (c-IndG
KZ (σ), π)−→iwhich is equivariant for the action of (T − λ) on the first factor. However, by
Proposition 2.6.4 the Hecke operator (T − λ) is induced by an element of theBernstein centre of AY , which must act by zero on π since it does so on τ . It
follows that (T − λ) is zero on Ext1A (c-IndG
KZ (σ), π), and so we have an exact
sequenceG
0 → HomA (τ, π) → Hom(c-IndG
KZ (σ), π) → Hom(c-IndKZ (σ), π) →→ Ext1A (τ, π) → Ext1 (c-IndG
KZ (σ), π) → 0.
The proposition follows from this together with the computation of Ext1 -groups between absolutely irreducible objects of A. More specifically, we know that Ext1A (π, π)
has dimension 3 if π is supersingular [Paš10, Theorem 10.13] and dimension 2 otherwise [Paš13, Section 8]. On the other hand, if τ is in the same block of π but is notisomorphic to π then π is not supersingular, and dim Ext1A (τ, π) = 1, see [Paš13,
Section 8].
LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )51

4.3. Extensions between compact inductions. We can also consider the Ext
groups between full compact inductions. We have the following general result.
Lemma 4.3.1. Let σ0 and σ1 be Serre weights, and assume they are not twists
Gof Symp−1 . If Exti (c-IndG
KZ σ1 , c-IndKZ σ0 ) is non-zero for some i, then either σ0and σ1 are isomorphic, or they are the Jordan–Hölder factors of the reduction of atame type.
Proof. For i = 0, 1 let τi be the unique cuspidal type containing σi in its semisimplified mod p reduction. If τ0 = τ1 then σ0 = σ1 or, by definition, there exists atame type whose reduction has Jordan–Hölder factors equal to σ0 and σ1 .
Assume that τ0 6= τ1 . If τ0 and τ1 are not adjacent then the intersection X(τ0 ) ∩X(τ1 ) is empty, by Proposition 2.2.7. Let ji be the inclusion of the complementof X(τi ) in X. By Proposition 3.1.7 (1), the natural maps
(4.3.1)G∗c-IndGKZ (σ0 ) → j1∗ j1 c-IndKZ (σ0 )(4.3.1)G∗c-IndGKZ (σ1 ) → j0∗ j0 c-IndKZ (σ1 )are isomorphisms. On the other hand, c-IndG
KZ (σi ) is an object of AX(τi ) by Definition 3.1.1. Hence
G
Exti (c-IndG
KZ σ1 , c-IndKZ σ0 ) = 0
by Corollary 3.1.9 and Corollary 3.4.5.
Finally, if τ0 and τ1 are adjacent but σ0 and σ1 are not constituents of a principalG
series type then the same argument as above shows that Exti (c-IndG
KZ σ1 , c-IndKZ σ0 )vanishes. Indeed, the image of fσ0 does not intersect X(τ1 ), and similarly the imageof fσ1 does not intersect X(τ0 ). Hence the localization morphisms (4.3.1) are still
isomorphisms.
p−1, c-IndG
Remark 4.3.2. The same argument proves that if Exti (c-IndG
KZ σ)
KZ Sym0
p−1
p−3
is non-zero for some i then σ ∈ {Sym , Sym , Sym⊗ det}.

The computation of Ext0 (i.e. of Hom) is due to Barthel and Livné [BL94] andis treated in Lemma 2.1.3. In the remainder of this section we explicitly computeGsome instances of the H(σ0 ) ⊗k H(σ1 )-module Ext1 (c-IndG
KZ σ1 , c-IndKZ σ0 ). If wewrite H(σ0 ) = F[S] and H(σ1 ) = F[T ], then H(σ0 ) ⊗F H(σ1 ) = F[S, T ]. (Notethat even if σ0 and σ1 are isomorphic, the Hecke operators S and T give distinct
actions on Exti , at least a priori.)
Lemma 4.3.3. Suppose that σ0 and σ1 are the Jordan–Hölder factors of the reG
duction of an irreducible principal series type. Then Ext1 (c-IndG
KZ σ1 , c-IndKZ σ0 )Gis one-dimensional, spanned by the class of c-IndKZ τ , where τ denotes the nonsplit extension of σ1 by σ0 . As an H(σ0 ) ⊗k H(σ1 )-module it is isomorphic to
F[S, T ]/(S, T ).

G
Proof. Corollary 3.1.9 shows that Exti c-IndG
KZ σ1 , (c-IndKZ σ0 )[1/S] = 0 for all i,while Lemma 2.5.6 shows that

G
G
G
i
Exti c-IndG
KZ σ1 , (c-IndKZ σ0 )[1/S] = Ext (c-IndKZ σ1 , c-IndKZ σ0 )[1/S];
G
thus we see that Exti (c-IndG
KZ σ1 , c-IndKZ σ0 ) consists of S-torsion elements.
The long exact Ext sequence associated to the short exact sequence
Sn ·

G
G
G
n
0 → c-IndG
KZ σ0 −→ c-IndKZ σ0 → (c-IndKZ σ0 )/S (c-IndKZ σ0 ) → 0

52

A. DOTTO, M. EMERTON, AND T. GEE

yields an isomorphismGGn
Hom c-IndG
KZ σ1 , (c-IndKZ σ0 )/S (c-IndKZ σ0 )∼G−→ S n -torsion in Ext1 (c-IndG
KZ σ1 , c-IndKZ σ0 ).
GGGWhen n = 1, the quotients (c-IndG
KZ σ1 )/T (c-IndKZ σ1 ) and (c-IndKZ σ0 )/S(c-IndKZ σ0 )are isomorphic to the same supersingular irreducible representation π, but the thickenings
Gn(c-IndG
KZ σ1 )/T (c-IndKZ σ1 )andGn(c-IndGKZ σ0 )/S (c-IndKZ σ0 )don’t coincide to any higher order. More precisely, the elements of Ext1 (π, π) classifying these extensions for n = 2 are linearly independent: see [AB15, Theorem 1.2,
Section 7.3] for a proof of this fact.
Hence

G
G
n
Hom c-IndG
KZ σ1 , (c-IndKZ σ0 )/S (c-IndKZ σ0 )is one-dimensional, no matter what the value of n is, because these morphisms haveGGGnto factor through c-IndG
KZ σ1 /T (c-IndKZ σ1 ), and so through c-IndKZ σ1 /T (c-IndKZ σ1 )Gby the discussion above. We find that Ext1 (c-IndG
KZ σ1 , c-IndKZ σ0 ) is one-dimensional,
and annihilated by S.
If g ∈ H(σ1 ) is coprime to T (i.e. does not vanish at T = 0), then Corollary 3.1.9
GG
shows that Exti (c-IndG
KZ σ1 )/g(c-IndKZ σ1 ), c-IndKZ σ0 ) = 0 for all i. A consideration of the long exact Ext sequence associated to the short exact sequence
g·

G
G
G
0 → c-IndG
KZ σ1 −→ c-IndKZ σ1 → (c-IndKZ σ1 )/g(c-IndKZ σ1 ) → 0
Gthen shows that g acts invertibly on Ext1 (c-IndG
KZ σ1 , c-IndKZ σ0 ). Thus this Ext
module is annihilated by T as well, since it has dimension one over k.
It remains to show that c-IndG
KZ τ is a non-split extension. Suppose otherwise.
Then the surjectionGG
c-IndGKZ σ0 → (c-IndKZ σ0 )/T (c-IndKZ σ0 )would extend to a surjectionGG
c-IndGKZ τ → (c-IndKZ σ0 )/T (c-IndKZ σ0 ),G
inducing a KZ-equivariant embedding τ ,→ (c-IndG
KZ σ0 )/T (c-IndKZ σ0 ). However,there is no such embedding, so c-IndG
KZ τ is indeed non-split.

Remark 4.3.4. Here is another viewpoint on this computation. If we let τ 0 denote∼the non-split extension of σ0 by σ1 , then there is an isomorphism c-IndG
KZ τ −→GG
0c-IndKZ τ : indeed, the source is isomorphic to c-IndIZ χ (for some character χ) ands
the target is isomorphic to c-IndG
IZ χ , and these are well-known to be isomorphic.
Furthermore, the composite
∼

G
G
G
0
0 → c-IndG
KZ σ0 → c-IndKZ τ −→ c-IndKZ τ → c-IndKZ σ0 → 0

is equal to the Hecke operator S. Thus we see that the element of the extension
G
G
group Ext1 (c-IndG
KZ σ1 , c-IndKZ σ0 ) classified by c-IndKZ τ is annihilated by S.

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )53

Remark 4.3.5. Let πi = c-IndG
KZ (σi ) where σ0 , σ1 are as in Lemma 4.3.3. If wepush out the extension c-IndG
KZ τ along the inclusion
π0 ,→ π0 [1/T ],then the resulting extension does split. Indeed, if ρ is the cuspidal type containing σ1 , then π1 lies in AX(ρ) . On the other hand, Proposition 3.1.7 shows that
π0 [1/T ] = jU ∗ jU∗ (π0 ), where U denotes the complement in X of X(ρ). So we seethat Ext1 (π1 , π0 [1/T ]) = 0.
Concretely, this means that there is a KZ-equivariant embedding τ ,→ π0 [1/T ]extending the canonical embedding σ0 ,→ π0 ,→ π0 [1/T ]. This corresponds to thefact that π0 [1/T ] is a family of principal series representations of Serre weight σ0 .
Suppose now that σ0 , σ1 , are the constituents of a tame cuspidal type, and again
write τ for the non-split extension of σ1 by σ0 .
Lemma 4.3.6. If σ0 , σ1 , are the constituents of a tame cuspidal type, then there
are isomorphisms
G
Ext1 (c-IndG
KZ σ1 , c-IndKZ σ0 )
∼



∼



G
−→ Ext1 (c-IndG
KZ σ1 )[1/T ], c-IndKZ σ0
G−→ Ext1 c-IndG
KZ σ1 , (c-IndKZ σ0 )[1/S]∼G−→ Ext1 (c-IndG
KZ σ1 )[1/T ], (c-IndKZ σ0 )[1/S] .
GGProof. The usual argument shows that ExtiA (c-IndG
KZ σ1 )/T (c-IndKZ σ1 ), c-IndKZ σ0 =G
0 for all i, so that multiplication by T induces an isomorphism from Ext1A (c-IndG
KZ σ1 , c-IndKZ σ0 )GG
to itself. Writing (c-IndKZ σ1 )[1/T ] as the colimit of c-IndKZ σ1 under multiplication by T , and using the spectral sequenceGE2p,q := lim (p) Extq (c-IndG
KZ σ1 , c-IndKZ σ0 )←−T·

G=⇒ Extp+q (c-IndG
KZ σ1 )[1/T ], c-IndKZ σ0(and recalling that for an inverse system of isomorphisms, the higher derived inverse
limits all vanish), we obtain the first claimed isomorphism of the lemma. The
remaining isomorphisms are proved similarly.

Notice that the localizations (c-IndG
KZ σi )[1/T ] are algebraic families of principal
series over the multiplicative group Gm , a fact that goes back to [BL94]. It follows
that the Ext groups in Lemma 4.3.6 can be computed using the techniques of [BP12,§6-7] to relate these Ext1 of compact inductions to certain Ext1 of Hecke modules.
The result is the following analogue of Lemma 4.3.3, which would also follow by
the method of [Eme10b, Section 4] if the δ-functor of derived ordinary parts had
an extension to non-admissible representations. However, a more natural way toprove Lemma 4.3.7 (at least from the optic of this paper) is as a consequence of
our expected results on Bernstein centres explained in Section 1.3, so we don’t givea proof here.
Lemma 4.3.7. If σ0 , σ1 , are the constituents of a tame cuspidal type, then as an
G
F[S, T ]-module, Ext1 (c-IndG
KZ σ1 , c-IndKZ σ0 ) is free of rank one over F[S, T ]/(ST −G1), generated by the class of c-IndKZ τ .

54

A. DOTTO, M. EMERTON, AND T. GEE

Remark 4.3.8. In the context of Lemma 4.3.7 both compact inductions are objects of AX(τ ) for the same cuspidal type τ . The compact induction c-IndG
KZ τ issupported on the entirety of X(τ ) ∼= P1 , in the sense that it lies in AX(τ ) andhas irreducible quotients lying in every block parameterized by the closed points ofX(τ ). Indeed, its quotient c-IndG
KZ σ1 has irreducible quotients lying over the wholeof fσ1 (A1 ); the remaining point X(τ ) \ fσ1 (A1 ) corresponds to the supersingularG
quotient (c-IndG
KZ σ0 )/T (c-IndKZ σ0 ), which contains a copy of τ and is thus alsoGa quotient of c-IndKZ τ .
If we let U ∼= Gm denote the complement of the marked points in X(τ ), thenjU ∗ jU∗ (c-IndG
KZ τ ) “lies over” U in an intuitive sense, and gives a family of atomes
automorphes. It is an extension ofG
jU ∗ jU∗ (c-IndG
KZ σ1 ) = (c-IndKZ σ1 )[1/T ]byG
jU ∗ jU∗ (c-IndG
KZ σ0 ) = (c-IndKZ σ0 )[1/S].
(Both the equalities follow from Proposition 3.1.7.)
To see the relationship between Lemma 4.3.7 and the Bernstein centre, noticethat the equality ST = 1 in the endomorphism ring ofG
Ext1 (c-IndG
KZ σ1 [1/T ], c-IndKZ σ0 [1/S])−1
implies that there exists an automorphism T + of jU ∗ jU∗ (c-IndG
KZ τ ) which induces S+
on the subobject and T on the quotient. We expect the automorphism T to be
induced by the Bernstein centre of AU . Taking the cokernel of T + − λ for λ ∈ k ×
corresponds to taking the fibre at λ ∈ Gm , and yields the corresponding atome
automorphe.

Appendix A. Category-theoretic backgroundIn this appendix we recall various results about abelian categories and theirlocalizations. Much of this material goes back to Gabriel’s thesis [Gab62], andmost of the rest of it can be found in [KS06]. Many of the results that we needfrom the previous two references are collected in [Kan21, §2], which we frequently
refer to for convenience.
In order to be able to use these references we need to fix a Grothendieck universe,
which we do without further comment. Sets are small if they belong to this fixed
universe, and all limits and colimits are assumed to be small, i.e. can be written as(co)limits over small indexing sets.
A.1. Grothendieck and locally Noetherian categories. We recall that an
abelian category A is a Grothendieck category if it satisfies (AB5) (which is to saythat A is cocomplete and that the formation of filtered colimits in A is exact), andit furthermore admits a set of generators (i.e. a small set of objects Gi with theproperty that for any nonzero morphism f : X → Y in A, there is a morphismg : Gi → X for some Gi such that f g 6= 0). Every object in a Grothendieck
category admits an injective envelope [Gab62, §II.6 Thm. 2].
Recall that an object X of an abelian category is Noetherian if it satisfies theascending chain condition on subobjects, and is compact if Hom(X, −) commuteswith filtered colimits.
An abelian category is called locally Noetherian if it is Grothendieck, and furthermore admits a set of generators that are Noetherian.

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )55

Proposition A.1.1. Suppose that A is a locally Noetherian category. Then
(1) An object of A is compact if and only if it is Noetherian.
(2) A filtered colimit of injective objects of A is injective.
(3) If X is a Noetherian object of A, then for each n ≥ 0 the functor ExtnA (X, −)commutes with filtered colimits.
Proof. For (1), note that any Noetherian object of A is compact by [Gab62, §2.4
Cor. 1]. Conversely if X is compact then by writing X = limi Xi as a filtered−→colimit of Noetherian objects, we see that we can factor the identity morphism
1X through some Xi ; so X is a retract of a Noetherian object and is thus itself
Noetherian. Part (2) is part of [Gab62, §2.4 Cor. 1]. Part (3) is [Kan21, Prop.
2.7].

A.1.2. A vs. Afg . If A is a locally Noetherian abelian category, we write Afg for thefull subcategory of compact (or equivalently, by the preceding result, Noetherian)objects. The canonical functor Ind(Afg ) → A given by evaluating inductive limits
in A is an equivalence.
We can’t expect the category Afg to have enough injectives, but can compute
Exts in it via Yoneda Exts. We then have the following result.
Lemma A.1.3. Suppose that A is locally Noetherian. If X and Y are objects of∼
Afg , then ExtiAfg (X, Y ) −→ ExtiA (X, Y ) for all i.
Proof. If i = 1, this reduces to the statement that Afg is closed under extensions
in A. If i > 1, then we may write a class in ExtiA (X, Y ) as a Yoneda product c ∪ d1000for c ∈ Exti−1
A (X, X ) and d ∈ ExtA (X , Y ) for some object X of A.
0
Writing X as the filtered colimit of its compact subobjects, and taking into account Proposition A.1.1 3, we find an inclusion i : X 00 ,→ X of a compact object X 00
00
0
0
0
∗
and a class c0 ∈ Exti−1
A (X, X ) such that c = i∗ c . Then c ∪ d = i∗ c ∪ d = c ∪ i d,
1∗00
where now i d ∈ ExtA (X , Y ). The surjectivity of the morphism in the lemma
follows by induction on i. The fact that it is an isomorphism follows by an easy
δ-functor argument.

A.2. Localizing categories. Suppose that A is a Grothendieck category. If B is a
Serre subcategory of A (i.e. a non-empty full subcategory closed under the formationof subquotients and extensions in A), then we may form the quotient category A/B.
We are interested in the question of when the natural functor A → A/B has a rightadjoint; if such a right adjoint exists, we say that B is a localizing subcategory. This
right adjoint, if it exists, is necessarily left-exact: indeed, being a right adjoint,
it preserves all limits. Furthermore, the counit of the adjunction is necessarily an
isomorphism [Gab62, §III.2 Proposition 3], hence the right adjoint is fully faithful
if it exists.
Since left adjoints necessarily preserve colimits, we see that for such a rightadjoint to exist, the Serre subcategory B must be closed under the formation of(small) colimits in A. (Since B is closed under forming quotients in A, this isequivalent to asking that B be closed under the formation of arbitrary direct sumsin A.) It turns out that this condition is also sufficient for B to be localizing. The
quotient category A/B is then also a Grothendieck category. (See e.g. [Kan21, Rem.
2.21] for these facts.)

56

A. DOTTO, M. EMERTON, AND T. GEE

We note that the construction of the adjoint is not too difficult, modulo settheoretic issues. Namely, if A is an object of A, we consider the category of morphisms A → A0 which project to an isomorphism in A/B (equivalently, whose kerneland cokernel both lie in B). Then taking the colimit of A0 over this category gives
the value of the adjoint on A.
We assume throughout the rest of this appendix that B is a localizing subcategoryof A, let j ∗ : A → A/B denote the natural projection, and let j∗ : A/B → A
denote the right adjoint to j ∗ . As already noted, since j∗ is a right adjoint, it isautomatically left-exact. However, it needn’t be exact in general. Another way tothink of this is that the essential image of j∗ is a full subcategory of A which is
intrinsically an abelian category, but is not necessarily an abelian subcategory of A.
Indeed, this essential image will be an abelian subcategory of A if and only if j∗ isactually exact.
Remark A.2.1. Our notation is motivated by the following example. If j : U ⊆ X is
the inclusion of a retrocompact open subset in a scheme X, then we have the adjoint
pair (j ∗ , j∗ ) : QCoh(U ) → QCoh(X), which realizes QCoh(U ) as a Serre quotientcategory of QCoh(X): it is the quotient of QCoh(X) by the Serre subcategory
QCohZ (X) consisting of quasicoherent sheaves, all of whose sections are supported
on Z := X \ U . (Cf. [Sta, Tag 01PD].)
The essential image of j∗ admits the following characterization:
Lemma A.2.2. An object A of A lies in the essential image of j∗ if and only if
Hom(B, A) = Ext1 (B, A) = 0 for any object B of B.
Proof. The identity j ∗ A = j ∗ A induces a morphism(A.2.3)

A → j∗ j ∗ A

We note two basic facts:(i) applying j ∗ to (A.2.3) recovers the identity morphism j ∗ A = j ∗ A;(ii) the object A is in the essential image of j∗ if and only if (A.2.3) is an
isomorphism.
From (i), we see that the kernel and cokernel of (A.2.3) lie in B. Thus if A satisfies
the two conditions appearing in the statement of the lemma, we see that thismorphism is indeed an isomorphism, so that by (ii), the object A lies in the essential
image of j∗ . (More precisely: the vanishing of the Hom gives that there is no kernel,
and then the vanishing of the Ext1 gives that j∗ j ∗ A = A⊕B, with B an object of B.
But Hom(B, j∗ j ∗ A) = Hom(j ∗ B, j ∗ A) = 0 (because j ∗ B = 0); thus the inclusionof B in j∗ j ∗ A is the zero map, so B = 0.)
Conversely, suppose that A lies in the essential image of j∗ . Consider any morphism B → A whose source is an object of B. We have a commutative diagram
B/Aj∗ j ∗ B/ j∗ j ∗ A

By assumption, the right-hand vertical arrow is an isomorphism, while j ∗ B = 0,
since B is an object of B, and thus also j∗ j ∗ B = 0. Consequently the morphism
B → A is equal to zero.

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )57

Now consider a short exact sequence 0 → A → E → B → 0, with B an objectof B. We have a commutative diagram
0

/A

/E

/B

/0

0


/ j∗ j ∗ A/ j∗ j ∗ E/ j∗ j ∗ B/0Again, the left-hand vertical arrow is an isomorphism, and thus the top extension
is pulled back from the bottom one. But also j∗ j ∗ B = 0, so the bottom extension
is trivial. Hence so is the top extension.

Remark A.2.4. It is not hard to prove this result in a more constructive fashion,using the construction of j∗ given above.
We now have the following result, which relates higher Ext’s to higher direct
images of j∗ .
Lemma A.2.5. Let A0 be an object of A/B, and fix some n ≥ 2. Then the following
are equivalent:(1) Exti (B, j∗ A0 ) = 0 for all objects B of B, and all 0 ≤ i ≤ n.
(2) Ri j∗ A0 = 0 for all 1 ≤ i ≤ n − 1.
Proof. Let A0 ,→ I • be an injective resolution (in the category A/B). Then j∗ I •∼computes the various Ri j∗ A0 . Since j ∗ j∗ I • −→ I • , we find that j ∗ Ri j∗ A0 = 0 for
all i > 0, and thus that Ri j∗ A0 is an object of B for i > 0. In particular, we seethat Ri j∗ A0 = 0 if and only if HomA (B, Ri j∗ A0 ) = 0 for all objects B of B.
Now since j∗ is right adjoint to an exact functor, it preserves injectives, and thus
HomA (B, j∗ I • ) computes RHomA (B, j∗ I • ), for any object B of B. But this Hom
is identically zero, by Lemma A.2.2, and so RHomA (B, j∗ I • ) = 0. On the other
hand, we have the usual E2 spectral sequence computing this RHom, which thus
becomes
E2p,q := ExtpA (B, Rq j∗ A0 ) =⇒ 0.
It is now easy to prove the lemma by analyzing this spectral sequence. In moredetail, to see that the first condition implies the second, note that by the conclusionof the preceding paragraph, it is enough to prove that E20,q vanishes for 1 ≤ q ≤n − 1; and if this holds for some q, then actually E2p,q = 0 for all p. In fact,
if E20,q = 0 for some q and for all objects B of B then Rq j∗ A0 = 0, again by theconclusion of the preceding paragraph, and this implies that E2p,q = 0 for all p. 
As a corollary, we have the following result.
Corollary A.2.6. The following are equivalent:(1) The functor j∗ is exact.
(2) The derived functor R1 j∗ is identically zero.
(3) For all objects A lying in the essential image of j∗ , and for all objects B
of B, we have Ext2 (B, A) = 0.
If these equivalent conditions hold, then in fact all the derived functors Ri j∗ (i ≥ 1)
vanish, and Exti (B, A) = 0 for all i ≥ 0 and objects A lying in the essential imageof j∗ and B lying in B.

58

A. DOTTO, M. EMERTON, AND T. GEE

Proof. The equivalence of (1) and (2) is immediate. Lemma A.2.5 with n = 2 yields
that (2) implies (3), and the converse implication follows from Lemma A.2.5 andLemma A.2.2.

We also note the following result, which in practice can simplify the checking of
the various vanishing conditions introduced in the preceding results.
Lemma A.2.7. If {Bj } is a system of generators of B, and A is an object of A,
then the following are equivalent:(1) Exti (Bj , A) = 0 for all Bj and all i ≤ n.
(2) Exti (B, A) = 0 for all objects B of B, and all i ≤ n.
Proof. Clearly (2) implies (1), and so we focus on proving the converse. To say that{Bj } is a system of generators of B is to say that for any object B of B, we may
L ⊕Ifind an epimorphism j Bj j → B. Since B is a Serre subcategory which is closed
under the formation of colimits in A, the kernel of this epimorphism is again an
object of B. A straightforward dimension-shifting argument reduces us to checking
L ⊕I
the vanishing of (2) in the case when B = j Bj j . In this case, we immediately
Q
compute that Exti (B, A) = j Exti (Bj , A)Ij . The vanishing claimed in (2) thus
follows from the vanishing assumed in (1).
Lemma A.2.8. Suppose that A is locally Noetherian. Then:(1) j∗ commutes with filtered colimits.
(2) Both B and A/B are locally Noetherian.
(3) An object of B is Noetherian if and only if it is Noetherian as an objectof A, while an object of A/B is Noetherian if and only if it is isomorphic
to the image of a Noetherian object of A.
Proof. The first two statements are [Gab62, §III.4 Cor. 1]. The third statement isimmediate from [Kan21, Prop. 2.22].

A.3. Pro-categories (and Ind-categories). We now recall some standard properties of Pro- and Ind-categories. It suffices to develop the theory of Pro-categories,
because for any category C, there is an equivalence Ind(C) ∼= Pro(C op )op ; in the
body of the paper we make far more use of Pro-categories than Ind-categories, so
we do not explicitly state results for Ind-categories in this appendix. (Note alsothat some of our references, in particular [KS06], take the opposite approach, so
the corresponding statements for Ind-categories are often more readily available in
the literature.)
If C is a category, one can define its associated pro-category Pro(C) to be thecategory whose objects are the diagrams F : I → C indexed by a cofiltered small
category I, with the morphisms between two diagrams F : I → C, G : J → C being
defined by the formula
HomPro(C) (F, G) = lim lim HomC (F (i), G(j)).
←− −→
J

I

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 fromthe limit limI F (i) in C itself, if this 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 .
←−LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )59

As a special case of the definition of morphisms, we note that if X ∈ C andF : I → C is a cofiltered diagram, then the definitions imply that
(A.3.1)

HomPro(C) (“ lim ”Xi , X) = lim HomC (Xi , X)
←−
−→
I

I

and
(A.3.2)

HomPro(C) (X, “ lim ”Xi ) = lim HomC (X, Xi ).
←−
←−
I

I

There is an equivalent definition of Pro(C), in which the object “ limI ”F (i) is←−
interpreted as the functor(A.3.3)lim HomC F (i), –)−→i

on C. To give a little more detail: C admits its co-Yoneda embedding C ,→Fun(C, Sets)op via X 7→ HomC (X, –). If F : I → C is a diagram indexed by acofiltered small category, then we can form the actual projective limit limI F (i) in←−
Fun(C, Sets)op ; this yields precisely the functor (A.3.3). In this way we obtain afunctor Pro(C) → Fun(C, Sets)op , which is fully faithful.
This functor Pro(C) → Fun(C, Sets)op is an instance of a more general construction, coming from the universal mapping property that Pro(C) satisfies: namely, if
D is a category that admits all cofiltered limits, then pullback of functors along theembedding C ,→ Pro(C) induces an equivalence of categories of functors(A.3.4)∼

Fun0 Pro(C), D) −→ Fun(C, D),

where the domain denotes the full subcategory of Fun Pro(C), D) consisting of
those functors that preserve cofiltered limits. (See e.g. [Lur09, Prop. 5.3.5.10].)The quasi-inverse generalizes the preceding construction: if F : C → D is a functor
whose target admits cofiltered limits, we extend F to Pro(C) via
F (“ lim ”Xi ) := lim F (Xi ).
←−
←−
I

I

A.3.5. Finite limits. One useful fact [KS06, Cor. 6.1.14] is that any morphism in
Pro(C) can be written as a cofiltered limit, over some small category I, of morphisms Xi → Yi in C. Similarly, any pair of morphisms having the same domain
and codomain may be written as a cofiltered limit, over some small category I, ofmorphism pairs Xi ⇒ Yi in C (see [KS06, Cor. 6.1.15]). Then, if C admits equalizers, the same is true of Pro(C), and furthermore we can compute the equalizer of“ limI ”(Xi ⇒ Yi ) as the limit in Pro(C) of the equalizers of the morphisms Xi ⇒ Yi←−(see [AM69, Appendix, Prop. 4.1]). Likewise, if C admits finite products, then sodoes Pro(C), and these are computed “pointwise” in the same way as equalizers. It
follows that if C admits finite limits then so does Pro(C), and then Pro(C) admits
all small limits (since it admits finite limits and cofiltered limits). Compare [KS06,
Proposition 6.1.18].
A.3.6. Finite colimits. If C admits finite colimits, the same is true of Pro(C), andfinite coproduct and coequalizers can be computed “pointwise”, as for equalizers:
this follows again from an application of [AM69, Appendix, Prop. 4.1]. More precisely, the coequalizer of “ limI ”Xi ⇒ Yi is the limit in Pro(C) of the coequalizers←−

60

A. DOTTO, M. EMERTON, AND T. GEEof the various Xi ⇒ Yi , while“ lim ”Xi←−Ia“ lim ”Yj = “ lim ”Xi
←−
←−
J

a

Yj .

I×J

In particular, the inclusion C → Pro(C) preserves finite colimits. Furthermore, finite
colimits commute with cofiltered limits: compare [KS06, Proposition 6.1.19].
A.3.7. Pro-adjoints. Suppose that F : C1 → C2 is a functor preserving finite limits,
between categories each of which admits finite limits. Then we obtain an induced
functor Pro(F ) : Pro(C1 ) → Pro(C2 ), which, by the discussions of Section A.3.5,
also preserves equalizers and finite products, hence finite limits. The functor Pro(F )can also be regarded as corresponding, under the equivalence (A.3.4), to the composite C1 → C2 → Pro(C2 ), and so is seen to preserve cofiltered limits. Thus Pro(F )preserves arbitrary limits, and hence admits a left adjoint G : Pro(C2 ) → Pro(C1 ).
(This is a consequence of the special adjoint functor theorem, which applies because Pro(Ci ) is complete, by A.3.5, and has a cogenerator, see e.g. [Kan21, Thm.
2.39].)
A.3.8. The abelian case. We assume throughout this subsection that C is an abeliancategory (so that in particular all of the discussions above hold, in particular thoseof Section A.3.5). Then one can alternatively define Pro(C) as the opposite of thecategory of left-exact covariant functors C → Ab, the point being that (A.3.3) is
now abelian group-valued and left exact.
The category Pro(C) is again abelian and has exact cofiltered limits. (See e.g.
[Kan21, Thm. 2.39].)
The following lemma provides a criterion for testing if a morphism in Pro(C) isa monomorphism.
Lemma A.3.9. Suppose given a morphism Y → Z in Pro(C) with the property
that, for any morphism Y → Y 0 with Y 0 an object of C, we may find a commutativesquare(A.3.10)

Y

/Z


Y0


/ Z0

in Pro(C) in which the bottom horizontal arrow is a monomorphism; then the given
morphism Y → Z is a monomorphism.
Proof. If we let X denote the kernel of the morphism Y → Z in Pro(C), and write
Y = “ lim ”Yi with the Yi being objects of C — so that in fact Y = lim Yi in Pro(C)
←−
←−
I
I
— then the exact sequence 0 → X → Y → Z in Pro(C) induces exact sequencesa
0 → Xi → Yi → Zi := ZYi
Yin Pro(C), where Xi is defined to be the image (again, in Pro(C)) of X in Yi . We
have X = lim Xi and Z = lim Zi , the limits being taken in Pro(C). (The second
←−
←−
i
I
isomorphism is an instance of a finite colimit commuting with a cofiltered limit.
The first isomorphism then follows from the second.) Thus we see that X = 0 if
and only each Xi = 0.

LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )61

Now form a diagram of the form (A.3.10) with Y 0 = Yi . The morphism Yi =Y → Z 0 then factors through Zi , and since the former morphism is a monomorphism by hypothesis, the morphism Yi → Zi must also be. Hence we find that
Xi = 0, as required.
0Lemma A.3.11. If X ∈ C is an injective object then it remains injective in Pro(C).
Proof. Let α : F → G be a monomorphism in Pro(C). By [AM69, Appendix, Prop.
4.6] one can represent F and G by diagrams F : I → C, G : I → C from the sameindex category I in such a way that α is represented by a natural transformation
α : F → G such that αi : Fi → Gi is a monomorphism for all i ∈ I.
Assume given a map λ : F → X. By definition, it arises from a map λ : Fi → X
for some i. Since αi is a monomorphism and X is injective, we can extend λ to amap Gi → X, which proves that X is injective in Pro(C).

Lemma A.3.12. Suppose that C has enough injectives. If X, Y ∈ C, then thenatural map ExtiC (X, Y ) → ExtiPro(C) (X, Y ) is an isomorphism for all i ≥ 0.
Proof. This is a direct consequence of Lemma A.3.11 and the fact that C → Pro(C)is fully faithful and exact. Indeed, Lemma A.3.11 and our assumption that C hasenough injectives implies that Y admits an injective resolution Y → I • in C which issimultaneously an injective resolution in Pro(C), and then HomC (X, I • ) computes
both ExtiC (X, Y ) and ExtiPro(C) (X, Y ).
The preceding result admits the following evident extension.
Lemma A.3.13. Assume that C has enough injectives. Let “ limI ”Xi be an object←−of Pro(C), and Y be an object of C. Then the natural map
lim ExtnC (Xi , Y ) → ExtnPro(C) (“ lim ”Xi , Y ),
←−
−→
I

I

obtained by pullback and Lemma A.3.12, is an isomorphism for each n ≥ 0.
Proof. Choose an injective resolution Y → J • in C. By Lemma A.3.11 this is aninjective resolution in Pro(C), and we have∼lim HomPro(C) (Xi , J • ) −→ HomPro(C) (“ lim ”Xi , J • ).
−→
←−
I

I

Since the passage to cohomology commutes with lim, the lemma follows.
−→

In general, it’s harder to say anything about Ext groups computed in the otherdirection, i.e. of the form ExtnPro(C) (X, “ limI ”Yi ), but we have the following result←−
which treats a special case.
Lemma A.3.14. Assume that C is Artinian and O-linear, for some commutative
ring O. If X is an object of C such that the O-modules Extn (X, Y ) are of finite
length for all n and all objects Y of C, and if “ limm ”Ym is a countably indexed←−
object of Pro(C), then the natural map(A.3.15)ExtnPro(C) (X, “ lim ”Ym ) → lim ExtnC (X, Ym )
←
−
←
−
m
m

is an isomorphism for every n.

62

A. DOTTO, M. EMERTON, AND T. GEE

Proof. Replacing each Ym by the “infimum” of the collection of images of transition morphisms Ym0 → Ym for m0 ≥ m (by Artinianness, this descending sequenceof images stabilizes) — which replaces “ limn ”Ym by an isomorphic pro-object —←−we may, and do, assume that the transition morphisms between the Ym are epimorphisms. Since Pro(C) has enough projectives ([Oor64, p. 229]), we can compute ExtnPro(C) (X, “ lim ”Ym ) by taking a projective resolution P• → X.
←−m
The projectivity of the terms of P• , together with the fact that the transition
maps between the Ym are epic, implies that the transition maps in the inverse
system of complexes HomPro(C) (P• , Ym ) are surjective, and hence have vanishingR1 lim . We thus find that←−
(A.3.16)∼∼
HomPro(C) (P• , “ lim ”Ym ) −→ lim HomPro(C) (P• , Ym ) −→ R lim HomPro(C) (P• , Ym )
←
−
←
−
←
−
m
m
m
The composite isomorphism gives rise to a spectral sequenceE2p,q := Rp lim ExtqC (X, Ym ) =⇒ H p+q HomPro(C) (P• , “ lim ”Ym )
←
−
←
−
m
m
= Extp+q
lim ”Ym ).
Pro(C) (X, “ ←−mOf course the Rp limm vanish automatically for p ≥ 2, but they also vanish for←−
p = 1, since the inverse sytems ExtqC (X, Ym ) are inverse systems of finite length
O-modules by assumption, and so satisfy the Mittag–Leffler condition. This proves
the lemma.

A.3.17. Completion along an abelian subcategory. Suppose that C0 ,→ C is anabelian subcategory (i.e. a full subcategory which is also abelian and for which
the inclusion is exact). Then we may apply the discussion of Section A.3.7 to seethat the induced inclusion Pro(C0 ) ,→ Pro(C) (which is evidently fully faithful,and by the discussions of (A.3.5) and (A.3.6) is again exact) admits a left adjoint
Pro(C) → Pro(C0 ), which we refer to as the functor of completion of C along C0 ,b
and which we denote by X 7→ X.
b to itself induces, by the adjunction that defines X,
b a
The identity map from X
b in Pro(C).
canonical morphism (the unit of the adjunction) X → X
b explicitly for objects X of C, in the case when
The following lemma describes X
C0 is furthermore closed under the formation of subobjects in C (e.g. a Serre subcategory of C).
Lemma A.3.18. If C0 is closed under the formation of subobjects in C, then there
is a natural isomorphism∼b −→(A.3.19)X“ lim ”X 0 ,←−where X 0 runs over the cofiltered directed set of quotients of X lying in C0 .
Proof. If X → X 0 is a surjection with X 0 lying in C0 , then by adjunction there isb → X 0 . These induced morphisms collectively give rise to
an induced morphism X
the morphism (A.3.19), which we claim is an isomorphism. To see this, it suffices
to show that the induced morphism of functors∼b –)(A.3.20)
HomPro(C0 ) (“ lim ”X 0 , –) −→ HomPro(C0 ) (X,←−LOCALIZATION OF SMOOTH REPRESENTATIONS OF GL2 (Qp )63

on C0 is an isomorphism, and this is what we will show.
This morphism is injective by construction. To see that it is surjective, writeb = “ lim ”Xi , with Xi an object of C0 , Each of the projection morphisms X
b → XiX←−I(in Pro(C0 )) induces a morphism X → Xi in C. If we let Xi0 denote the image of X
(which is again an object of C0 , by assumption), then the factorization X → Xi0 →b → X 0 → Xi , so that the projection
Xi induces a corresponding factorization X
ib
X → Xi factors through the morphism (A.3.19). This shows that (A.3.20) is
indeed surjective, and thus is an isomorphism.
b =In the context of the preceding lemma, the unit of the adjunction X → X
0“ lim ”X is the morphism induced by the various quotient morphisms X → X 0 .
←−b in Pro(C0 ) is its image under theConcretely, if X is an object of C and if X
b (in Pro(C)) through which any
adjoint, then there is a canonical morphism X → X
morphism X → X0 , with X0 an object of C0 , uniquely factors.

References[AB15]

U. K. Anandavardhanan and Gautam H. Borisagar, Iwahori-Hecke model for supersingular representations of GL2 (Qp ), J. Algebra 423 (2015), 1–27.
[Alp14]
Jarod Alper, Adequate moduli spaces and geometrically reductive group schemes,
Algebr. Geom. 1 (2014), no. 4, 489–531.
[AM69]
M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, No. 100,
Springer-Verlag, Berlin-New York, 1969.
[AS21]
Konstantin Ardakov and Peter Schneider, The Bernstein center in natural characteristic, 2021.
[Ber84]J. N. Bernstein, Le “centre” de Bernstein, Representations of reductive groups over a
local field, Travaux en Cours, Hermann, Paris, 1984, Edited by P. Deligne, pp. 1–32.
[BL94]
L. Barthel and R. Livné, Irreducible modular representations of gl2 of a local field,
Duke Math J. 75 (1994), no. 2, 261–292.
[BP12]
Christophe Breuil and Vytautas Paškūnas, Towards a modulo p Langlands correspondence for GL2 , Mem. Amer. Math. Soc. 216 (2012), no. 1016, vi+114.
[Bre03a]
Christophe Breuil, Sur quelques représentations modulaires et p-adiques de GL2 (Qp ).
I, Compositio Math. 138 (2003), no. 2, 165–188.
, Sur quelques représentations modulaires et p-adiques de GL2 (Qp ). II, J.
[Bre03b]Inst. Math. Jussieu 2 (2003), no. 1, 23–58.
[Bus90]Colin J. Bushnell, Induced representations of locally profinite groups, J. Algebra 134(1990), no. 1, 104–114.
[BZCHN20] David Ben-Zvi, Harrison Chen, David Helm, and David Nadler, Coherent Springer
theory and the categorical Deligne–Langlands correspondence, 2020.
[DEG]
Andrea Dotto, Matthew Emerton, and Toby Gee, A categorical p-adic Langlands
correspondence for GL2 (Qp ).
[DHKM22] Jean-Francois Dat, David Helm, Robert Kurinczuk, and Gilbert Moss, Finiteness for
Hecke algebras of p-adic groups, 2022.
[Dix63]
J. Dixmier, Représentations irréductibles des algèbres de Lie nilpotentes, An. Acad.
Brasil. Ci. 35 (1963), 491–519.
[Dot]
Andrea Dotto, Restriction of p-adic representations of GL2 (Qp ) to parahoric subgroups., submitted, https://arxiv.org/abs/2111.12827.
[Dot21], Mod p Bernstein centres of p-adic groups, Math. Res. Lett. (2021), to appear,https://arxiv.org/abs/2105.09367.
[Eme10a]Matthew Emerton, Ordinary parts of admissible representations of p-adic reductive
groups I. Definition and first properties, Astérisque (2010), no. 331, 355–402.
[Eme10b], Ordinary parts of admissible representations of p-adic reductive groups II.
Derived functors, Astérisque (2010), no. 331, 403–459.

64

A. DOTTO, M. EMERTON, AND T. GEE[FS21]

Laurent Fargues and Peter Scholze, Geometrization of the local Langlands correspondence.
[Gab62]
Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–
448.
[Hel20]Eugen Hellmann, On the derived category of the Iwahori–Hecke algebra, 2020.
[HM18]David Helm and Gilbert Moss, Converse theorems and the local Langlands correspondence in families, Invent. Math. 214 (2018), no. 2, 999–1022.
[HT15]
Yongquan Hu and Fucheng Tan, The Breuil-Mézard conjecture for non-scalar split
residual representations, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 6, 1383–1421.
[Jen72]
C. U. Jensen, Les foncteurs dérivés de lim et leurs applications en théorie des mod←−ules, Lecture Notes in Mathematics, Vol. 254, Springer-Verlag, Berlin-New York,1972.
[Kan21]
Ryo Kanda, Extension groups between atoms in abelian categories, J. Pure Appl.
Algebra 225 (2021), no. 9, Paper No. 106669, 37.
[KS06]
Masaki Kashiwara and Pierre Schapira, Categories and sheaves, Grundlehren dermathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],
vol. 332, Springer-Verlag, Berlin, 2006.
[Lur09]
Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton
University Press, Princeton, NJ, 2009.
[Oor64]
F. Oort, Yoneda extensions in abelian categories, Math. Ann. 153 (1964), 227–235.
[Paš10]
Vytautas Paškūnas, Extensions for supersingular representations of GL2 (Qp ),Astérisque (2010), no. 331, 317–353.
[Paš13]
Vytautas Paškūnas, The image of Colmez’s Montreal functor, Publ. Math. Inst.
Hautes Études Sci. 118 (2013), 1–191.
[Paš14], Blocks for mod p representations of GL2 (Qp ), Automorphic forms and Galois representations. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 415, Cambridge Univ. Press, Cambridge, 2014, pp. 231–247.
[Paš15]
Vytautas Paškūnas, On the Breuil–Mézard conjecture, Duke Math. J. 164 (2015),
no. 2, 297–359.
[PT21]
Vytautas Paškūnas and Shen-Ning Tung, Finiteness properties of the category ofmod p representations of GL2 (Qp ), Forum Math. Sigma 9 (2021), Paper No. e80,39.
[San16]
Fabian Sander, A local proof of the Breuil-Mézard conjecture in the scalar semisimplification case, J. Lond. Math. Soc. (2) 94 (2016), no. 2, 447–461.
[Sho20]Jack Shotton, The category of finitely presented smooth mod p representations of
GL2 (F ), Doc. Math. 25 (2020), 143–157.
[Sta]
The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu.
[SW00]
Peter Symonds and Thomas Weigel, Cohomology of p-adic analytic groups, New
horizons in pro-p groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA,
2000, pp. 349–410.
[Vis05]
Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc.,
Providence, RI, 2005, pp. 1–104.
[Zhu20]
Xinwen Zhu, Coherent sheaves on the stack of Langlands parameters, 2020.
Email address: andreadotto@uchicago.edu
Department of Mathematics, University of Chicago, 5734 S. University Ave.,
Chicago, IL 60637, USA
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