Duals and admissibility in natural characteristic

By Peter Schneider and Claus Sorensen

Abstract

In this article we introduce a derived smooth duality functor on the unbounded derived category of smooth -representations of a -adic Lie group . Here is a field of characteristic . Using this functor we relate various subcategories of admissible complexes in .

1. Introduction

Let be a -adic Lie group of dimension , and let be a field of characteristic . We denote by the abelian category of smooth -representations in -vector spaces.

In this paper we endow the unbounded derived category with a tensor product plus internal hom functor , and begin exploring the resulting closed symmetric monoidal category. The duality functor is of particular interest to us. It gives a derived approach to the higher smooth duality functors introduced by Kohlhaase in Reference Koh, realizing them as cohomological functors .

Our first result (Proposition 2.7) shows that the functors are compatible with duals on the Hecke side. If denotes the Hecke algebra of a torsion free open pro- subgroup , we give an -equivariant spectral sequence with -page converging to the twisted dual Hecke modules . Here the character turns out to coincide with the duality character in Reference Koh. This is a non-trivial fact and we give a proof. In particular if is an open subgroup of the -points of a connected reductive group over a -adic field .

Motivated by Reference DGA, which gives a differential graded version of the Hecke algebra along with an equivalence between and the derived category of differential graded modules over , we turn to studying the functor in the derived setting.

We first observe that is involutive on the subcategory of complexes with admissible cohomology representations for all . We then introduce a possibly larger subcategory

consisting of globally admissible complexes, by which we mean is finite-dimensional for all . As we show in Theorem 4.5, a complex belongs to precisely when the natural biduality morphism

is a quasi-isomorphism. As a result, the notion of being globally admissible is independent of the choice of . Finally we show that a globally admissible satisfying various boundedness conditions actually lies in the subcategory . For instance, Corollary 4.12 tells us contains exactly those complexes whose total cohomology is finite-dimensional.

To orient the reader we point out that is equivalent to the category of differential graded -modules with finite-dimensional cohomology spaces in each degree. We have work in progress aiming at an intrinsic description of the duality functor on corresponding to .

2. Higher smooth duality

For any compact open subgroup we have the completed group ring of over . This is a noetherian ring (cf. Reference pLG, Theorem 33.4). We let denote the abelian category of left -modules. However is also a pseudocompact ring (cf. Reference pLG, IV §19). We therefore also have the abelian category of pseudocompact left -modules together with the obvious forgetful functor . Both categories have enough projective objects. Any finitely generated -module is pseudocompact in a natural way. Moreover, such an is projective in if and only if it is projective in . This leads to the natural isomorphism

for any finitely generated module in and any pseudocompact module in .

Pontrjagin duality gives rise to the equivalence of categories

where, of course, in order to make a left module we use the inversion map on . See Reference Koh, Th. 1.5 for instance. In particular, we have the natural isomorphisms

If we apply this with the trivial -representation and use Equation 1 we obtain the natural isomorphism

for in .

If is another open subgroup then in Equation 2 we have on both sides the obvious restriction maps. On the left-hand side this follows from the fact that the restriction functor preserves injective objects (as follows from Frobenius reciprocity and the exactness of compact induction ). On the right-hand side the functor preserves projective objects since is free over .

Hence we may pass to the inductive limit

Note that, for in , the right-hand side is Kohlhaase’s higher smooth dual functors

in Reference Koh. We use the left-hand side to understand these as derived functors. For any in we introduce

Via the -action defined by , for , this is again an object in . Since the functors

are left exact we have the corresponding right derived functors

It is well-known (and easy to show) that has enough injectives. On the contrary does not in general have enough projectives.

Lemma 2.1.
(i)

If is injective in then is -acyclic for any compact open subgroup .

(ii)

.

Proof.

By definition . Note that any injective object in remains injective when viewed in , as explained right after Equation 2. Therefore the lemma follows from Proposition 2.2 in the appendix by Verdier in Reference CG.

We see that, in particular, we can rewrite Kohlhaase’s functors as the derived functors

We first note that in the range . More generally we have the following.

Lemma 2.2.

for any .

Proof.

By Reference Bru, Theorem 4.1 the global dimension of as a pseudocompact ring is equal to the cohomological dimension of . By Lazard (cf. Reference CG, I-47) the latter is equal to provided is pro- and torsion free. Since contains arbitrarily small open pro- subgroups without torsion we conclude from Lemma 2.1(ii) combined with the isomorphism that indeed for any .

Proposition 2.3.

For any compact open subgroup we have the -spectral sequence

In particular,

Proof.

This is the composed functor spectral sequence which exists by Lemma 2.1(i).

The above spectral sequence has an additional equivariance property which we now describe. We fix a compact open subgroup and consider the compact induction in . We then have the endomorphism ring so that becomes a right -module. Frobenius reciprocity gives a natural isomorphism of functors on . By using injective resolutions it extends to a natural isomorphism of cohomological functors

Through its right action on the right-hand side becomes a left -module. In this way is equipped with a left -module structure. In particular, carries a left -module structure which is functorial in and . By derivation we obtain a functorial left -module structure on . Up to isomorphism the latter -module is independent of the choice of injective resolution of .

Lemma 2.4.

The spectral sequence in Proposition 2.3 is -equivariant.

Proof.

This is straightforward from the way the composed functor spectral sequence is constructed.

We now suppose in addition that is pro- and torsion free. Then is a Poincaré group of dimension (Reference CG I-47 Ex. (3)). A straightforward variant of the appendix by Verdier in Reference CG (or Tate’s Appendix 1 in the 1997 English translation) therefore gives the following: In we have the dualizing object

which actually is isomorphic to the trivial representation in , together with an isomorphism

which is natural in in ; this latter isomorphism is induced by the Yoneda product

(Definition 4.5, Proposition 3.1.5, and first displayed formula on p. V-20). In the following we will keep writing and view it as a trivial -representation. From now on we assume that comes from a given -representation (by restriction to ) and we will see that then all terms in the above Yoneda pairing carry a natural left -action.

(A)

From the proof of Proposition 8.4.i in Reference OS we know a formula for the action on . Viewing as the convolution algebra of -bi-invariant functions with compact support on we denote by , for , the characteristic function of the double coset in . The diagram

is commutative.

(B)

By Reference CG I Proposition 18 the same is also a dualizing object in for any open subgroup .

(C)

As introduced above, we have a natural left -action on . To give an explicit formula we let be any other object in and we first recall that, for any open subgroup and any , we have the following natural maps:

The restriction map which derives the obvious forgetful map on homomorphisms. (Recall that restriction preserves injective objects.)

The corestriction map which derives the map which sends a -equivariant homomorphism to the -equivariant homomorphism .

The conjugation map which derives the map which sends a -equivariant homomorphism to the -equivariant homomorphism .

(D)

As for (A) it is straightforward to verify that, for any , the diagram

is commutative.

(E)

It is easily checked that the map

is an anti-involution of the -algebra , again viewed as a convolution algebra as in part (A) It sends to .

Lemma 2.5.

For any and any the diagram of Yoneda pairings

is commutative.

Proof.

We fix injective resolutions and in , which as noted earlier remain injective resolutions after restriction to any given open subgroup of .

The upper rectangle: Let be a -equivariant and a -equivariant homomorphism of complexes representing classes and , respectively. Then also represents whereas is represented by . We compute

The middle rectangle: Let be a -equivariant and a -equivariant homomorphism of complexes representing classes and , respectively. Then and are represented by and . We compute

The lower rectangle: This is entirely analogous to the computation for the upper rectangle.

By Reference CG I-50(4) the two corestriction maps in the rightmost column of the diagram in Lemma 2.5 are isomorphisms between one-dimensional vector spaces. The composition

is therefore multiplication by a scalar , which happens to be independent of .

Lemma 2.6.

The map is a character which is independent of and trivial on any pro- subgroup of .

Proof.

We first show the independence of . Suppose is another open torsion free pro- subgroup of and consider subgroups . Again by Reference CG I-50(4) corestriction gives isomorphisms

This gives a canonical isomorphism between the dualizing objects which in turn gives a canonical isomorphism . Altogether this shows is independent of , and it is obviously trivial on .

Suppose that we have checked the multiplicativity of already and let be any pro- subgroup of . Note that, as a -adic Lie group, always has an open torsion free pro- subgroup; see Reference pLG, Theorem 27.1 for instance which even shows the existence of a -valuable subgroup. Hence factorizes through a finite quotient which is a -group. Since any finite subgroup of has order prime to it follows that is trivial on . To establish multiplicativity let . Since conjugation commutes with corestriction we have the following three commutative diagrams, which together show our claim:

and

The map

is an algebra homomorphism. Pulling back an -module along this homomorphism defines the twisted -module . More explicitly acts on by the rule .

Also note that we may use the anti-involution in (E) to make the -linear dual of a left -module again into a left -module. More explicitly for and .

Using (A) and (C) we may rewrite the diagram in Lemma 2.5 as the commutative diagram

Then this says that the duality isomorphism Equation 4 in fact is an isomorphism of -modules

Altogether this yields the following spectral sequence alluded to in Section 1.

Proposition 2.7.

For any compact open subgroup which is pro- and torsion free and any in we have an -equivariant -spectral sequence

Proof.

The spectral sequence arises by combining the second spectral sequence in Proposition 2.3 (observe Lemma 2.4) with the duality isomorphism Equation 5.

Remark 2.8.

Suppose that where is a finite extension and is a connected reductive -split group over . Assuming that a pro- Iwahori subgroup of is torsion free it is shown in Reference OS Proposition 7.16 that . Under additional assumptions this was proved before in Reference Koz. In the preprint Reference KS21 Koziol and Schwein give an alternate proof of the triviality of the orientation character via Moy-Prasad groups (still assuming pro- Iwahori is torsion free). We extend this result in Lemma 2.10.

The spectral sequence in Proposition 2.7 was obtained by different means in Reference Ko, Theorem 1.3 in the generality of a -adic reductive group and a torsion free pro- Iwahori subgroup .

We will show that in fact coincides with the duality character introduced by Kohlhaase in Reference Koh after Definition 3.12 and which we temporarily denote by .

Proposition 2.9.

We have .

Proof.

The character describes the -action on a certain one dimensional -vector space the original definition of which we do not need. Instead we use Reference Koh Proposition 3.2 which says that, for any compact open subgroup , there is a natural -equivariant isomorphism such that:

(1)

For any the diagram

is commutative, where is the conjugation isomorphism (compare with the argument in the third paragraph of the proof of Reference Koh Proposition 3.13).

(2)

For any open subgroup the diagram

is commutative. Moreover is the composite of the restriction map

and the map

which is induced by the Pontrjagin dual of the extension by zero map .

The Pontrjagin dual of being we have, using Equation 2, the isomorphism

Combining it with the above two diagrams we arrive at the commutative diagrams

and

On the other hand, taking now we note that the duality isomorphism Equation 4 for and is given by

Let denote the map which sends to the constant function with value on . Then the above isomorphism is equivalent to the isomorphism

The first isomorphism being natural in conjugation by and this conjugation sending to we see that we have the commutative diagram

Furthermore, if is any open subgroup, then we have the commutative diagram of duality pairings

Here the top, resp. bottom, rectangle is commutative by the top rectangle in Lemma 2.5, resp. the functoriality of the Yoneda pairing. Note that the middle column maps to . Hence we obtain the commutative diagram

By combining it with the diagram Equation 7 we deduce the left-hand triangle of the commutative diagram

where the right-hand oblique arrows are our standard identifications. This means that the isomorphism does not depend on the subgroup . With this information we consider the commutative diagram

whose left-hand rectangle arises by combining Equation 6 and Equation 8. Since the horizontal arrows coincide we conclude that .

One immediately infers the triviality of for open subgroups of -adic reductive groups:

Lemma 2.10.

Suppose that is a connected reductive group over a finite extension of ; if is an open subgroup of then .

Proof.

Proposition 2.9 together with Reference Koh, Corollary 5.2 shows the assertion in the case . In general let denote the Weil restriction of to . It is shown in Reference Oes App. 3 that again is a connected linear algebraic group with the property that as -adic Lie groups. Since our field extension is separable it follows from loc. cit. A.3.4 that with also is reductive. This reduces the general case to the case .

3. Derived smooth duality

We begin by recalling some general nonsense about the adjunction between tensor product and Hom-functor which for three -vector spaces , , and is given by the linear isomorphism

Suppose that all three vector spaces carry a left -action. Then and are equipped with the -action defined by

and

respectively. The above adjunction is equivariant for these two actions. If we restrict to the diagonal -action, and take -invariants, then the above adjunction induces the adjunction isomorphism

If the -action on the is smooth then this also can be written as an isomorphism

Let denote the unbounded derived category of . The tensor product functor

where the -action on the tensor product is the diagonal one, is exact in both variables. Therefore it extends directly (i.e., without derivation) to the functor

which we usually denote simply by .⁠Footnote1 On the other hand, since is a Grothendieck category, we have for any in the total derived functor

1

This uses the fact that for any two complexes of vector spaces one of which is acyclic their tensor product is acyclic as well. Indeed by the Künneth formula. Recall that is a field.

such that for any in and . We want to extend this to a bifunctor . First we recall that has arbitrary direct products (but which are not exact); we will denote these by to avoid confusion with the cartesian direct product. Hence, for any two complexes and in we may define the complex

in in the usual way. By construction we have that

is the inductive limit over all compact open subgroups of the usual Hom-complexes for the abelian categories .

The adjunction Equation 10 shows that the assumptions of Reference KS Theorem 14.4.8 are satisfied (with , the tensor product functor, and ). Hence we obtain the following result.

Proposition 3.1.

The total derived functor exists and can be computed by where is a homotopically injective resolution. Moreover, there are the natural adjunctions

and

for any in .

Remark 3.2.

For future reference we mention that the local version of the above adjunction also holds. That is

for all . To see this pick a homotopically injective resolution in . Note that remains homotopically injective upon restriction to any compact open subgroup (by Frobenius reciprocity and exactness of ). Furthermore is homotopically injective by adjunction and the previous footnote. By Proposition 3.1 for we have

Taking the limit over and invoking the description Equation 11 gives the result.

Corollary 3.3.

is a closed symmetric monoidal category.

For viewed as complex concentrated in degree zero we, in particular, obtain the total derived duality functor

such that for any in and any . In order to see in which way is a dualizing object for we have to introduce two finiteness conditions.

First we make the following observation.

Lemma 3.4.

The functor is way-out in both directions, and in particular respects .

Proof.

We refer to Reference Har, p. 68 for what it means to be way-out, but the actual definition is not important here. By Reference Har, Proposition I.7.6 is way-out (in both directions) if and only if there is an such that for all and . By (the proof of) Lemma 2.2 we may take when is sufficiently small. Finally by Equation 11 we conclude that itself is way-out.

Remark 3.5.

In general the trivial -representation does not have finite injective dimension in . Nevertheless, as the previous proof shows, we have

for all in .

Next we recall that a representation in is called admissible if, for any open subgroup , the vector space of -fixed vectors is finite dimensional. In fact, it suffices to check the defining condition for a single compact open subgroup (apply the Nakayama lemma to the dual -module or see Reference Koh Lemma 1.7). The full subcategory of admissible representations in is a Serre subcategory (cf. Reference Em1 Proposition 2.2.13). Hence we have the strictly full triangulated subcategories and of those complexes whose cohomology representations are admissible.

Lemma 3.6.

The derived duality functor respects both subcategories and .

Proof.

It is shown in Reference Koh Corollary 3.15 that for an admissible representation in the representations are admissible as well. Hence for an admissible the complex lies in . On the other hand we have observed already that our functor is way-out in both directions in the sense of Reference Har §7. Therefore our assertion follows from loc. cit. Proposition I.7.3.

Let be any complex in and fix an injective resolution . We construct a natural transformation

as follows. Inserting the definitions we have to produce, for any , a natural -equivariant map

compatible with the differentials. It is straightforward to check that the maps

have these properties.

Proposition 3.7.

If the complex has admissible cohomology then the natural transformation is a quasi-isomorphism.

Proof.

Since we have a natural transformation between way-out functors the lemma on way-out-functors (Reference Har Prop. I.7.1(iii)) tells us that we need to establish the assertion only in the case where our complex is a single admissible representation (viewed as a complex concentrated in degree zero). In fact, by loc. cit. Prop. I.7.1(iv) we can go one step further. Suppose given a class of admissible representations such that every admissible representation is embeddable into a finite direct sum of representations in this class. Then it suffices to check the assertion for representations in . We cannot apply this directly, though. First let us fix a compact open subgroup in . Then we observe:

Any admissible -representation is also admissible as a -representation;

is also an injective resolution in ;

the natural transformation remains the same if constructed for considered only as a -representation.

This means that, for the purposes of our proof, we may assume that our group is compact. Let denote, as before, the vector space of -valued locally constant functions on . Equipped with the left translation action it is an admissible smooth -representation. We have . Let be any admissible representation in . Then is a finitely generated (pseudocompact) -module (Reference Koh Proposition 1.9(i)). Hence we find a surjection in for some integer . It is the dual of an injective map in . Therefore we can take the single object for the class . By Reference Koh Proposition 3.13 we have, for any integer , that

where is Kohlhaase’s duality character. Hence and then . One checks from the proof in loc. cit. that the latter quasi-isomorphism is induced by the natural transformation .

In other words:

Corollary 3.8.

On the functor is involutive.

Next we extend the involutivity of to a potentially larger category.

4. Globally admissible complexes

In this section we will generalize some of the results in Section 3 to a subcategory of which is potentially larger than . The possible drawback is that the defining condition for this subcategory is a “global” finiteness condition.

We let denote the abelian category of -vector spaces and its unbounded derived category. In the following we fix an open subgroup which is pro- and torsion free. As recalled in the proof of Lemma 2.2 the functor

has finite cohomological dimension . Hence its total derived functor exists (cf. Reference Har Corollary I.5.3)). It is given by composing

with the restriction functor .

On the other hand the functor on of taking the -linear dual is exact and therefore passes directly to a functor form to which, for simplicity, we also denote by .

Theorem 4.1.

The diagram

is commutative (up to a natural isomorphism). More precisely, there is a natural isomorphism of functors

Proof.

The upper rectangle is commutative since restriction from to preserves homotopically injective resolutions. For the lower triangle we first observe that the second adjunction formula in Proposition 3.1 tells us that the composed functor is naturally isomorphic to the functor . Hence it remains to exhibit a natural isomorphism

For this we start with the Yoneda pairing

By our assumption on the group the natural homomorphism

is an isomorphism and the upper truncation at degree (cf. Reference Har p. 69/70) maps to its cohomology in degree . (The latter identification is given by the trace map in Verdier’s appendix to Reference CG.)

The Yoneda pairing therefore induces a pairing

and hence a natural homomorphism

To show that it is an isomorphism we need to check that the map induced on cohomology

is bijective. If is a single representation in degree zero then we have seen this already in Equation 4. By Example 1 on p. 68 in Reference Har the functor and hence also the functor are way-out in both directions. Similarly, by Lemma 2.2 and Reference Har Proposition I.7.6 the functor is way-out in both directions as well. Hence it follows from Reference Har Proposition I.7.1(iii) that Equation 13 is always bijective.

Definition 4.2.

A complex in is globally admissible if its cohomology groups , for any , are finite dimensional vector spaces. Let denote the strictly full triangulated subcategory of all globally admissible complexes.

We will see only later in Corollary 4.6 that Definition 4.2, indeed, does not depend on the choice of . To rephrase Definition 4.2 let denote the strictly full triangulated subcategory of all objects all of whose cohomology vector spaces are finite dimensional. Then is the full preimage in of under the functor .

Corollary 4.3.

The duality functor respects the subcategory .

Proof.

This is immediate from Theorem 4.1 since the functor on respects the subcategory .

In Equation 12 we introduced the biduality morphism . Our further analysis of it will be based upon the following general observation.

Lemma 4.4.

A homomorphism in is an isomorphism if and only if the induced map , for any , is bijective.

Proof.

This is an immediate consequence of the equivalence between and the derived category of a certain differential graded algebra in Reference DGA Theorem 9. By construction the functor has the property that .

Theorem 4.5.

The biduality morphism , for any in , is an isomorphism if and only if lies in .

Proof.

According to Lemma 4.4 we have to check that the maps

are bijective for any if and only if lies in . By Proposition 4.1 we have natural isomorphisms

For the remainder of this proof we fix an isomorphism . The trace map then yields an isomorphism . We will just write instead of in what follows.

We now claim that the diagram

where denotes the natural map from a -vector space into its double dual, is commutative up to the sign . This immediately shows that is bijective if and only if is bijective which, of course, is the case if and only if the vector space is finite dimensional.

To establish this claim we compute by using an injective resolution of in and hence in . Then by Proposition 3.1. Moreover the adjunction property Equation 10 implies that always is homotopically injective. Finally we may also assume that is homotopically injective. Our diagram therefore becomes

where denotes as usual the unbounded homotopy category of complexes in . We first recall that, under our identification , the map is explicitly given by

Now let . By definition of its image under the top horizontal arrow in the above diagram is the homotopy class of the homomorphism of complexes

induced by . Under the right vertical arrow it is further mapped to the linear map

But corresponds under to the linear map in sending to . Hence the preimage of Equation 14 under the bottom horizontal map in the diagram is equal to as claimed.

Corollary 4.6.

The subcategory in is independent of the choice of the subgroup .

What is the relation between the subcategories and ? We had observed earlier that a representation in is admissible if and only if the vector space is finite dimensional. Moreover, by Reference Em2 Lemma 3.3.4, we have the following fact.

Lemma 4.7.

If in is admissible then all the vector spaces , for , are finite dimensional.

Lemma 4.7 says that, for an admissible , the complex lies in . By Example 1 on p. 68 in Reference Har the functor is way-out in both directions. Therefore Reference Har Proposition I.7.3(iii) implies that the functor maps to . This proves the following.

Proposition 4.8.

.

Alternatively this can be seen by combining Proposition 3.7 and Proposition 4.5.

On the full subcategories of complexes bounded below or above we have stronger results.

Proposition 4.9.
(i)

A complex in lies in if and only if is finite dimensional for any . I.e., we have

Similarly for .

(ii)

More generally, a globally admissible complex with some vanishing differential lies in the subcategory .

Proof.

First of all, in part (i) it suffices to show the -version. For if lies in then its dual lies in . Furthermore belongs to if does by Corollary 4.3. In that case, once we show the -version, we conclude that is an object of . However, by Lemma 3.6 the functor preserves . Since the functor is involutive on by Proposition 4.5 we conclude that indeed belongs to .

We proceed to show the -version in part (i). The direct implication holds true by Proposition 4.8. For the reverse implication we now assume that all the are finite dimensional, and is bounded below.

Choose an integer such that for any . In this situation it is a standard fact (cf. Reference KS Exercise 13.3) that we have . Hence is finite dimensional. As recalled before Lemma 4.7 this implies that is admissible. Moreover, Lemma 4.7 then says that is finite dimensional for any . We now use the distinguished triangles

in (cf. Reference KS Proposition 13.1.15(i)). Since in the left triangle implies that is finite dimensional for any . Using this as an input for the long exact cohomology sequence associated with the right triangle we conclude that is finite dimensional for any as well. This proves the case of the following statement :

Proceeding inductively, to show for we may repeat our initial reasoning for the complex . We obtain in particular that is admissible for any .

Finally part (ii) is a combination of the -versions. If the differential vanishes one can decompose as a sum of the two naive truncations . If is globally admissible so are the direct summands and . Therefore they both lie in by part (i), which immediately implies also lies in as claimed.

Remark 4.10.

One can relax the condition in part (ii) of Proposition 4.9 slightly. If is split somewhere, meaning at some there is a morphism such that , then the map gives rise to a quasi-isomorphism . The direct sum is a complex with a vanishing differential at . Applying (ii) shows that lies in provided it is globally admissible. For the definition of a split complex we refer the reader to Reference Wei, Df. 1.4.1.

Unfortunately we do not have an example showing the inclusion in Proposition 4.8 could be strict for certain .

Proposition 4.11.

For any in and any particular we have

In particular, if in satisfies for all (resp. ) then belongs to (resp. ).

Proof.

The proof of the first claim is almost literally the same argument as the one for the reverse implication in Reference DGA Proposition 5, but for a single . Now invoke Proposition 4.9.

We finish with a characterization of .

Corollary 4.12.

The subcategory consists of all complexes in whose total cohomology is finite dimensional.

Proof.

This is an immediate consequence of Proposition 4.11, Lemma 4.7, and the hypercohomology spectral sequence.

Remark 4.13.

If is compact then the natural functor

is an equivalence. Similarly for . This follows from Reference Em2 Proposition 2.1.9, and Reference Har, Proposition I.4.8 (which is also an easy consequence of Reference KS, Theorem 13.2.8).

Acknowledgment

We would like to thank the referee for thoughtful feedback which substantially improved our presentation.

Mathematical Fragments

Equation (1)
Equation (2)
Lemma 2.1.
(i)

If is injective in then is -acyclic for any compact open subgroup .

(ii)

.

Lemma 2.2.

for any .

Proposition 2.3.

For any compact open subgroup we have the -spectral sequence

In particular,

Lemma 2.4.

The spectral sequence in Proposition 2.3 is -equivariant.

Equation (4)
Lemma 2.5.

For any and any the diagram of Yoneda pairings

is commutative.

Equation (5)
Proposition 2.7.

For any compact open subgroup which is pro- and torsion free and any in we have an -equivariant -spectral sequence

Proposition 2.9.

We have .

Equation (6)
Equation (7)
Equation (8)
Lemma 2.10.

Suppose that is a connected reductive group over a finite extension of ; if is an open subgroup of then .

Equation (10)
Equation (11)
Proposition 3.1.

The total derived functor exists and can be computed by where is a homotopically injective resolution. Moreover, there are the natural adjunctions

and

for any in .

Lemma 3.6.

The derived duality functor respects both subcategories and .

Equation (12)
Proposition 3.7.

If the complex has admissible cohomology then the natural transformation is a quasi-isomorphism.

Theorem 4.1.

The diagram

is commutative (up to a natural isomorphism). More precisely, there is a natural isomorphism of functors

Equation (13)
Definition 4.2.

A complex in is globally admissible if its cohomology groups , for any , are finite dimensional vector spaces. Let denote the strictly full triangulated subcategory of all globally admissible complexes.

Corollary 4.3.

The duality functor respects the subcategory .

Lemma 4.4.

A homomorphism in is an isomorphism if and only if the induced map , for any , is bijective.

Theorem 4.5.

The biduality morphism , for any in , is an isomorphism if and only if lies in .

Equation (14)
Corollary 4.6.

The subcategory in is independent of the choice of the subgroup .

Lemma 4.7.

If in is admissible then all the vector spaces , for , are finite dimensional.

Proposition 4.8.

.

Proposition 4.9.
(i)

A complex in lies in if and only if is finite dimensional for any . I.e., we have

Similarly for .

(ii)

More generally, a globally admissible complex with some vanishing differential lies in the subcategory .

Proposition 4.11.

For any in and any particular we have

In particular, if in satisfies for all (resp. ) then belongs to (resp. ).

Corollary 4.12.

The subcategory consists of all complexes in whose total cohomology is finite dimensional.

References

Reference [Bru]
Armand Brumer, Pseudocompact algebras, profinite groups and class formations, J. Algebra 4 (1966), 442–470, DOI 10.1016/0021-8693(66)90034-2. MR202790,
Show rawAMSref \bib{Bru}{article}{ label={Bru}, author={Brumer, Armand}, title={Pseudocompact algebras, profinite groups and class formations}, journal={J. Algebra}, volume={4}, date={1966}, pages={442--470}, issn={0021-8693}, review={\MR {202790}}, doi={10.1016/0021-8693(66)90034-2}, }
Reference [Em1]
Matthew Emerton, Ordinary parts of admissible representations of -adic reductive groups I. Definition and first properties (English, with English and French summaries), Astérisque 331 (2010), 355–402. MR2667882,
Show rawAMSref \bib{Em1}{article}{ label={Em1}, author={Emerton, Matthew}, title={Ordinary parts of admissible representations of $p$-adic reductive groups I. Definition and first properties}, language={English, with English and French summaries}, journal={Ast\'{e}risque}, number={331}, date={2010}, pages={355--402}, issn={0303-1179}, isbn={978-2-85629-282-2}, review={\MR {2667882}}, }
Reference [Em2]
Matthew Emerton, Ordinary parts of admissible representations of -adic reductive groups II. Derived functors (English, with English and French summaries), Astérisque 331 (2010), 403–459. MR2667883,
Show rawAMSref \bib{Em2}{article}{ label={Em2}, author={Emerton, Matthew}, title={Ordinary parts of admissible representations of $p$-adic reductive groups II. Derived functors}, language={English, with English and French summaries}, journal={Ast\'{e}risque}, number={331}, date={2010}, pages={403--459}, issn={0303-1179}, isbn={978-2-85629-282-2}, review={\MR {2667883}}, }
Reference [Har]
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR0222093,
Show rawAMSref \bib{Har}{book}{ label={Har}, author={Hartshorne, Robin}, title={Residues and duality}, series={Lecture Notes in Mathematics, No. 20}, note={Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne}, publisher={Springer-Verlag, Berlin-New York}, date={1966}, pages={vii+423}, review={\MR {0222093}}, }
Reference [KS]
Masaki Kashiwara and Pierre Schapira, Categories and sheaves, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332, Springer-Verlag, Berlin, 2006, DOI 10.1007/3-540-27950-4. MR2182076,
Show rawAMSref \bib{KS}{book}{ label={KS}, author={Kashiwara, Masaki}, author={Schapira, Pierre}, title={Categories and sheaves}, series={Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]}, volume={332}, publisher={Springer-Verlag, Berlin}, date={2006}, pages={x+497}, isbn={978-3-540-27949-5}, isbn={3-540-27949-0}, review={\MR {2182076}}, doi={10.1007/3-540-27950-4}, }
Reference [Koh]
Jan Kohlhaase, Smooth duality in natural characteristic, Adv. Math. 317 (2017), 1–49, DOI 10.1016/j.aim.2017.06.038. MR3682662,
Show rawAMSref \bib{Koh}{article}{ label={Koh}, author={Kohlhaase, Jan}, title={Smooth duality in natural characteristic}, journal={Adv. Math.}, volume={317}, date={2017}, pages={1--49}, issn={0001-8708}, review={\MR {3682662}}, doi={10.1016/j.aim.2017.06.038}, }
Reference [Koz]
Karol Kozioł, Hecke module structure on first and top pro--Iwahori cohomology, Acta Arith. 186 (2018), no. 4, 349–376, DOI 10.4064/aa170903-24-3. MR3879398,
Show rawAMSref \bib{Koz}{article}{ label={Koz}, author={Kozio\l , Karol}, title={Hecke module structure on first and top pro-$p$-Iwahori cohomology}, journal={Acta Arith.}, volume={186}, date={2018}, number={4}, pages={349--376}, issn={0065-1036}, review={\MR {3879398}}, doi={10.4064/aa170903-24-3}, }
Reference [Ko]
Karol Kozioł, Functorial properties of pro--Iwahori cohomology, J. Lond. Math. Soc. (2) 104 (2021), no. 4, 1572–1614, DOI 10.1112/jlms.12469. MR4339945,
Show rawAMSref \bib{Ko}{article}{ label={Ko}, author={Kozio\l , Karol}, title={Functorial properties of pro-$p$-Iwahori cohomology}, journal={J. Lond. Math. Soc. (2)}, volume={104}, date={2021}, number={4}, pages={1572--1614}, issn={0024-6107}, review={\MR {4339945}}, doi={10.1112/jlms.12469}, }
Reference [KS21]
K. Koziol and D. Schwein, On mod p orientation characters, Preprint, http://www-personal.umich.edu/~kkoziol/orientation.pdf, 2021.
Reference [Oes]
Joseph Oesterlé, Nombres de Tamagawa et groupes unipotents en caractéristique (French), Invent. Math. 78 (1984), no. 1, 13–88, DOI 10.1007/BF01388714. MR762353,
Show rawAMSref \bib{Oes}{article}{ label={Oes}, author={Oesterl\'{e}, Joseph}, title={Nombres de Tamagawa et groupes unipotents en caract\'{e}ristique $p$}, language={French}, journal={Invent. Math.}, volume={78}, date={1984}, number={1}, pages={13--88}, issn={0020-9910}, review={\MR {762353}}, doi={10.1007/BF01388714}, }
Reference [OS]
Rachel Ollivier and Peter Schneider, The modular pro- Iwahori-Hecke Ext-algebra, Representations of reductive groups, Proc. Sympos. Pure Math., vol. 101, Amer. Math. Soc., Providence, RI, 2019, pp. 255–308. MR3930021,
Show rawAMSref \bib{OS}{article}{ label={OS}, author={Ollivier, Rachel}, author={Schneider, Peter}, title={The modular pro-$p$ Iwahori-Hecke Ext-algebra}, conference={ title={Representations of reductive groups}, }, book={ series={Proc. Sympos. Pure Math.}, volume={101}, publisher={Amer. Math. Soc., Providence, RI}, }, date={2019}, pages={255--308}, review={\MR {3930021}}, }
Reference [pLG]
Peter Schneider, -adic Lie groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 344, Springer, Heidelberg, 2011, DOI 10.1007/978-3-642-21147-8. MR2810332,
Show rawAMSref \bib{pLG}{book}{ label={pLG}, author={Schneider, Peter}, title={$p$-adic Lie groups}, series={Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]}, volume={344}, publisher={Springer, Heidelberg}, date={2011}, pages={xii+254}, isbn={978-3-642-21146-1}, review={\MR {2810332}}, doi={10.1007/978-3-642-21147-8}, }
Reference [DGA]
Peter Schneider, Smooth representations and Hecke modules in characteristic , Pacific J. Math. 279 (2015), no. 1-2, 447–464, DOI 10.2140/pjm.2015.279.447. MR3437786,
Show rawAMSref \bib{DGA}{article}{ label={DGA}, author={Schneider, Peter}, title={Smooth representations and Hecke modules in characteristic $p$}, journal={Pacific J. Math.}, volume={279}, date={2015}, number={1-2}, pages={447--464}, issn={0030-8730}, review={\MR {3437786}}, doi={10.2140/pjm.2015.279.447}, }
Reference [CG]
Jean-Pierre Serre, Cohomologie galoisienne (French), Lecture Notes in Mathematics, No. 5, Springer-Verlag, Berlin-New York, 1965. With a contribution by Jean-Louis Verdier; Troisième édition, 1965. MR0201444,
Show rawAMSref \bib{CG}{book}{ label={CG}, author={Serre, Jean-Pierre}, title={Cohomologie galoisienne}, language={French}, series={Lecture Notes in Mathematics, No. 5}, note={With a contribution by Jean-Louis Verdier; Troisi\`eme \'{e}dition, 1965}, publisher={Springer-Verlag, Berlin-New York}, date={1965}, pages={v+212 pp. (not consecutively paged)}, review={\MR {0201444}}, }
Reference [Wei]
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994, DOI 10.1017/CBO9781139644136. MR1269324,
Show rawAMSref \bib{Wei}{book}{ label={Wei}, author={Weibel, Charles A.}, title={An introduction to homological algebra}, series={Cambridge Studies in Advanced Mathematics}, volume={38}, publisher={Cambridge University Press, Cambridge}, date={1994}, pages={xiv+450}, isbn={0-521-43500-5}, isbn={0-521-55987-1}, review={\MR {1269324}}, doi={10.1017/CBO9781139644136}, }

Article Information

MSC 2020
Primary: 22E50 (Representations of Lie and linear algebraic groups over local fields)
Secondary: 18G80 (Derived categories, triangulated categories), 46A20 (Duality theory for topological vector spaces)
Author Information
Peter Schneider
Math. Institut, Universität Münster, Einsteinstr, 62, 48149 Münster, Germany
pschnei@wwu.de
MathSciNet
Claus Sorensen
Department of Mathematics, UC San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
csorensen@ucsd.edu
MathSciNet
Journal Information
Representation Theory of the American Mathematical Society, Volume 27, Issue 02, ISSN 1088-4165, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , and published on .
Copyright Information
Copyright 2023 American Mathematical Society
Article References
  • Permalink
  • Permalink (PDF)
  • DOI 10.1090/ert/634
  • MathSciNet Review: 4561086
  • Show rawAMSref \bib{4561086}{article}{ author={Schneider, Peter}, author={Sorensen, Claus}, title={Duals and admissibility in natural characteristic}, journal={Represent. Theory}, volume={27}, number={2}, date={2023}, pages={30-50}, issn={1088-4165}, review={4561086}, doi={10.1090/ert/634}, }

Settings

Change font size
Resize article panel
Enable equation enrichment

Note. To explore an equation, focus it (e.g., by clicking on it) and use the arrow keys to navigate its structure. Screenreader users should be advised that enabling speech synthesis will lead to duplicate aural rendering.

For more information please visit the AMS MathViewer documentation.