Duals and admissibility in natural characteristic

Abstract

In this article we introduce a derived smooth duality functor $R\underline{\operatorname {Hom}}(-,k)$ on the unbounded derived category $D(G)$ of smooth $k$-representations of a $p$-adic Lie group $G$. Here $k$ is a field of characteristic $p$. Using this functor we relate various subcategories of admissible complexes in $D(G)$.

1. Introduction

Let $G$ be a $p$-adic Lie group of dimension $d$, and let $k$ be a field of characteristic $p$. We denote by $\operatorname {Mod}(G)$ the abelian category of smooth $G$-representations in $k$-vector spaces.

In this paper we endow the unbounded derived category $D(G)=D(\operatorname {Mod}(G))$ with a tensor product $\otimes _k$ plus internal hom functor $R\underline{\operatorname {Hom}}$, and begin exploring the resulting closed symmetric monoidal category. The duality functor $R\underline{\operatorname {Hom}}(-,k)$ is of particular interest to us. It gives a derived approach to the higher smooth duality functors $S^j$ introduced by Kohlhaase in Koh, realizing them as cohomological functors $h^j(R\underline{\operatorname {Hom}}(-,k))=\underline{\operatorname {Ext}}^j(-,k)$.

Our first result (Proposition 2.7) shows that the functors $S^j$ are compatible with duals on the Hecke side. If $H_U$ denotes the Hecke algebra of a torsion free open pro-$p$ subgroup $U \subseteq G$, we give an $H_U$-equivariant spectral sequence with $E_2$-page $H^i(U,S^j(V))$ converging to the twisted dual Hecke modules $H^{d-i-j}(U,V)^{\vee }(\chi _G)$. Here the character $\chi _G: G \rightarrow k^{\times }$ turns out to coincide with the duality character in Koh. This is a non-trivial fact and we give a proof. In particular $\chi _G=1$ if $G$ is an open subgroup of the $\mathfrak{F}$-points of a connected reductive group over a $p$-adic field $\mathfrak{F}$.

Motivated by DGA, which gives a differential graded version of the Hecke algebra $H_U^{\bullet }$ along with an equivalence between $D(G)$ and the derived category $D(H_U^{\bullet })$ of differential graded modules over $H_U^{\bullet }$, we turn to studying the functor $R\underline{\operatorname {Hom}}(-,k)$ in the derived setting.

We first observe that $R\underline{\operatorname {Hom}}(-,k)$ is involutive on the subcategory $D_{adm}(G)$ of complexes $V^{\bullet }$ with admissible cohomology representations $h^i(V^{\bullet })$ for all $i\in \mathbb{Z}$. We then introduce a possibly larger subcategory

$$\begin{equation*} D(G)^a \supseteq D_{adm}(G) \end{equation*}$$

consisting of globally admissible complexes, by which we mean $H^i(U,V^{\bullet })$ is finite-dimensional for all $i\in \mathbb{Z}$. As we show in Theorem 4.5, a complex $V^{\bullet }$ belongs to $D(G)^a$ precisely when the natural biduality morphism

$$\begin{equation*} \eta _{V^{\bullet }}: V^{\bullet } \longrightarrow R\underline{\operatorname {Hom}}(R\underline{\operatorname {Hom}}(V^{\bullet },k),k) \end{equation*}$$

is a quasi-isomorphism. As a result, the notion of being globally admissible is independent of the choice of $U$. Finally we show that a globally admissible $V^{\bullet }$ satisfying various boundedness conditions actually lies in the subcategory $D_{adm}(G)$. For instance, Corollary 4.12 tells us $D_{adm}^b(G)$ contains exactly those complexes $V^{\bullet }$ whose total cohomology $H^*(U,V^{\bullet })$ is finite-dimensional.

To orient the reader we point out that $D(G)^a$ is equivalent to the category $D_{fin}(H_U^{\bullet })$ of differential graded $H_U^{\bullet }$-modules with finite-dimensional cohomology spaces in each degree. We have work in progress aiming at an intrinsic description of the duality functor on $D(H_U^{\bullet })$ corresponding to $R\underline{\operatorname {Hom}}(-,k)$.

2. Higher smooth duality

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

$$\begin{equation} \operatorname {Ext}_{\operatorname {Mod}_{pc}(\Omega (K))}^*(M,N) \cong \operatorname {Ext}_{\operatorname {Mod}(\Omega (K))}^*(M,N) {\tag{1}} \end{equation}$$

for any finitely generated module $M$ in $\operatorname {Mod}(\Omega (K))$ and any pseudocompact module $N$ in $\operatorname {Mod}_{pc}(\Omega (K))$.

Pontrjagin duality gives rise to the equivalence of categories

$$\begin{align*} \operatorname {Mod}(K)^{op} & \xrightarrow {\;\simeq \;} \operatorname {Mod}_{pc}(\Omega (K)), \\ V & \longmapsto V^\vee \coloneq \operatorname {Hom}_k(V,k), \end{align*}$$

where, of course, in order to make $V^\vee$ a left module we use the inversion map $g \mapsto g^{-1}$ on $K$. See Koh, Th. 1.5 for instance. In particular, we have the natural isomorphisms

$$\begin{equation*} \operatorname {Ext}_{\operatorname {Mod}(K)}^*(V_1,V_2) \cong \operatorname {Ext}_{\operatorname {Mod}_{pc}(\Omega (K))}^*(V_2^\vee ,V_1^\vee ) \ . \end{equation*}$$

If we apply this with the trivial $K$-representation $V_2 \coloneq k$ and use 1 we obtain the natural isomorphism

$$\begin{equation} \operatorname {Ext}_{\operatorname {Mod}(K)}^*(V,k) \cong \operatorname {Ext}_{\operatorname {Mod}(\Omega (K))}^*(k,V^\vee ) {\tag{2}} \end{equation}$$

for $V$ in $\operatorname {Mod}(K)$.

If $K' \subseteq K$ is another open subgroup then in 2 we have on both sides the obvious restriction maps. On the left-hand side this follows from the fact that the restriction functor $\operatorname {Mod}(K) \rightarrow \operatorname {Mod}(K')$ preserves injective objects (as follows from Frobenius reciprocity and the exactness of compact induction $\text{ind}_{K'}^K$). On the right-hand side the functor $\operatorname {Mod}(\Omega (K))\rightarrow \operatorname {Mod}(\Omega (K'))$ preserves projective objects since $\Omega (K)$ is free over $\Omega (K')$.

Hence we may pass to the inductive limit

$$\begin{equation} \varinjlim _K \operatorname {Ext}_{\operatorname {Mod}(K)}^*(V,k) \cong \varinjlim _K \operatorname {Ext}_{\operatorname {Mod}(\Omega (K))}^*(k,V^\vee ) \ . {\tag{3}} \end{equation}$$

Note that, for $V$ in $\operatorname {Mod}(G)$, the right-hand side is Kohlhaase’s higher smooth dual functors

$$\begin{equation*} S^*(V) \coloneq \varinjlim _K \operatorname {Ext}_{\operatorname {Mod}(\Omega (K))}^*(k,V^\vee ) \end{equation*}$$

in Koh. We use the left-hand side to understand these as derived functors. For any $V_1, V_2$ in $\operatorname {Mod}(G)$ we introduce

$$\begin{align*} \underline{\operatorname {Hom}}(V_1,V_2) & \coloneq \{f \in \operatorname {Hom}_k(V_1,V_2) : f \ \text{is $K$-equivariant} \\ & \qquad \qquad \qquad \qquad \text{for some compact open subgroup $K \subseteq G$} \}. \end{align*}$$

Via the $G$-action defined by ${^g f} \coloneq g f (g^{-1} -)$, for $g \in G$, this is again an object in $\operatorname {Mod}(G)$. Since the functors

$$\begin{align*} \underline{\operatorname {Hom}}(V_1,-): \operatorname {Mod}(G) & \longrightarrow \operatorname {Mod}(G), \\ V_2 & \longmapsto \underline{\operatorname {Hom}}(V_1, V_2) \end{align*}$$

are left exact we have the corresponding right derived functors

$$\begin{equation*} \underline{\operatorname {Ext}}^i(V_1, V_2) \qquad \text{for $i \geq 0$}. \end{equation*}$$

It is well-known (and easy to show) that $\operatorname {Mod}(G)$ has enough injectives. On the contrary $\operatorname {Mod}(G)$ does not in general have enough projectives.

Lemma 2.1.

(i)

If $V_2$ is injective in $\operatorname {Mod}(G)$ then $\underline{\operatorname {Hom}}(V_1,V_2)$ is $H^0(U,-)$-acyclic for any compact open subgroup $U \subseteq G$.

(ii)

$\underline{\operatorname {Ext}}^*(V_1,V_2) = \varinjlim _K \operatorname {Ext}_{\operatorname {Mod}(K)}^*(V_1,V_2)$.

Proof.

By definition $\underline{\operatorname {Hom}}(V_1,V_2) = \varinjlim _K \operatorname {Hom}_{\operatorname {Mod}(K)}(V_1,V_2)$. Note that any injective object in $\operatorname {Mod}(G)$ remains injective when viewed in $\operatorname {Mod}(U)$, as explained right after 2. Therefore the lemma follows from Proposition 2.2 in the appendix by Verdier in CG.

■

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

$$\begin{equation*} S^*(V) = \underline{\operatorname {Ext}}^*(V,k) \ . \end{equation*}$$

We first note that $S^j(V)=0$ in the range $j>d$. More generally we have the following.

Lemma 2.2.

$\underline{\operatorname {Ext}}^i(V_1, V_2) = 0$ for any $i > d$.

Proof.

By Bru, Theorem 4.1 the global dimension of $\Omega (K)$ as a pseudocompact ring is equal to the cohomological dimension of $K$. By Lazard (cf. CG, I-47) the latter is equal to $d$ provided $K$ is pro-$p$ and torsion free. Since $G$ contains arbitrarily small open pro-$p$ subgroups without torsion we conclude from Lemma 2.1(ii) combined with the isomorphism $\operatorname {Ext}_{\operatorname {Mod}(K)}^*(V_1,V_2) \cong \operatorname {Ext}_{\operatorname {Mod}_{pc}(\Omega (K))}^*(V_2^\vee ,V_1^\vee )$ that indeed $\underline{\operatorname {Ext}}^i(V_1, V_2) = 0$ for any $i > d$.

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Proposition 2.3.

For any compact open subgroup $U \subseteq G$ we have the $E_2$-spectral sequence

$$\begin{equation*} H^i(U,\underline{\operatorname {Ext}}^j(V_1,V_2)) \Longrightarrow \operatorname {Ext}_{\operatorname {Mod}(U)}^{i+j}(V_1,V_2) \ . \end{equation*}$$

In particular,

$$\begin{equation*} H^i(U,S^j(V)) \Longrightarrow \operatorname {Ext}_{\operatorname {Mod}(U)}^{i+j}(V,k) \ . \end{equation*}$$

Proof.

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

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The above spectral sequence has an additional equivariance property which we now describe. We fix a compact open subgroup $U \subseteq G$ and consider the compact induction $\mathbf{X}_U \coloneq \operatorname {ind}_U^G(k)$ in $\operatorname {Mod}(G)$. We then have the endomorphism ring $H_U \coloneq \operatorname {End}_{\operatorname {Mod}(G)}(\mathbf{X}_U)^{op}$ so that $\mathbf{X}_U$ becomes a right $H_U$-module. Frobenius reciprocity gives a natural isomorphism of functors $H^0(U,-) \cong \operatorname {Hom}_{\operatorname {Mod}(G)}(\mathbf{X}_U,-)$ on $\operatorname {Mod}(G)$. By using injective resolutions it extends to a natural isomorphism of cohomological functors

$$\begin{equation*} H^*(U,-) \cong \operatorname {Ext}_{\operatorname {Mod}(G)}^*(\mathbf{X}_U,-) \ . \end{equation*}$$

Through its right action on $\mathbf{X}_U$ the right-hand side becomes a left $H_U$-module. In this way $H^*(U,-)$ is equipped with a left $H_U$-module structure. In particular, $\operatorname {Hom}_{\operatorname {Mod}(U)}(V_1,V_2) = H^0(U, \underline{\operatorname {Hom}}(V_1,V_2)) \cong \operatorname {Hom}_{\operatorname {Mod}(G)}(\mathbf{X}_U,\underline{\operatorname {Hom}}(V_1,V_2))$ carries a left $H_U$-module structure which is functorial in $V_1$ and $V_2$. By derivation we obtain a functorial left $H_U$-module structure on $\operatorname {Ext}_{\operatorname {Mod}(U)}^*(V_1,V_2)$. Up to isomorphism the latter $H_U$-module is independent of the choice of injective resolution of $V_2$.

Lemma 2.4.

The spectral sequence in Proposition 2.3 is $H_U$-equivariant.

Proof.

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

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

$$\begin{equation*} \hat{I} \coloneq \varinjlim _{K \subseteq U, \operatorname {cores}} \operatorname {Hom}_k(H^d(K,k),k) \ , \end{equation*}$$

which actually is isomorphic to the trivial representation $k$ in $\operatorname {Mod}(U)$, together with an isomorphism

$$\begin{equation} \operatorname {Hom}_k(H^i(U,V),k) \cong \operatorname {Ext}^{d-i}_{\operatorname {Mod}(U)}(V,\hat{I}) \cong \operatorname {Ext}^{d-i}_{\operatorname {Mod}(U)}(V,k) \qquad \text{for any $i \geq 0$} {\tag{4}} \end{equation}$$

which is natural in $V$ in $\operatorname {Mod}(U)$; this latter isomorphism is induced by the Yoneda product

$$\begin{equation*} \operatorname {Ext}^{d-i}_{\operatorname {Mod}(U)}(V,\hat{I}) \times H^i(U,V) \longrightarrow H^d(U,\hat{I}) \end{equation*}$$

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

(A)

From the proof of Proposition 8.4.i in OS we know a formula for the $H_U$ action on $H^*(U,V)$. Viewing $H_U$ as the convolution algebra of $U$-bi-invariant functions with compact support on $G$ we denote by $\tau _h \in H_U$, for $h \in G$, the characteristic function of the double coset $UhU$ in $G$. The diagram$$\begin{equation*} \vcenter{\img[][189pt][53pt][{$\xymatrix{ H^*(U,V) \ar[d]_{\res} \ar[r]^{\tau_h \cdot} & H^*(U,V) \\ H^*(U \cap h^{-1}Uh,V) \ar[r]^{h_\ast} & H^*(U \cap hUh^{-1},V) \ar[u]_{\cores} }$}]{Images/img5c6efd6c5286aef72a3b95b049dfa045.svg}} \end{equation*}$$

is commutative.

(B)

By CG I Proposition 18 the same $\hat{I}$ is also a dualizing object in $\operatorname {Mod}(U')$ for any open subgroup $U' \subseteq U$.

(C)

As introduced above, we have a natural left $H_U$-action on $\operatorname {Ext}^*_{\operatorname {Mod}(U)}(V,\hat{I})$. To give an explicit formula we let $V'$ be any other object in $\operatorname {Mod}(G)$ and we first recall that, for any open subgroup $U' \subseteq U$ and any $h \in G$, we have the following natural maps:

–

The restriction map $\operatorname {Ext}^*_{\operatorname {Mod}(U)}(V,V') \xrightarrow {\operatorname {res}} \operatorname {Ext}^*_{\operatorname {Mod}(U')}(V,V')$ which derives the obvious forgetful map on homomorphisms. (Recall that restriction $\operatorname {Mod}(U) \rightarrow \operatorname {Mod}(U')$ preserves injective objects.)

–

The corestriction map $\operatorname {Ext}^*_{\operatorname {Mod}(U')}(V,V') \xrightarrow {\operatorname {cores}} \operatorname {Ext}^*_{\operatorname {Mod}(U)}(V, V')$ which derives the map which sends a $U'$-equivariant homomorphism $f : V \rightarrow V'$ to the $U$-equivariant homomorphism $\sum _{g \in U/U'} g f (g^{-1}-) : V \rightarrow V'$.

–

The conjugation map $\operatorname {Ext}^*_{\operatorname {Mod}(U)}(V,V') \!\xrightarrow {h_*}\! \operatorname {Ext}^*_{\operatorname {Mod}(hUh^{-1})}(V,\! V')$ which derives the map which sends a $U$-equivariant homomorphism $f : V \rightarrow V'$ to the $hUh^{-1}$-equivariant homomorphism $h f(h^{-1} -) : V \rightarrow V'$.

(D)

As for (A) it is straightforward to verify that, for any $h \in G$, the diagram$$\begin{equation*} \vcenter{\img[][229pt][57pt][{$\xymatrix{ \Ext^*_{\Mod(U)}(V,V') \ar[d]_{\res} \ar[r]^{\tau_h \cdot} & \Ext^*_{\Mod(U)}(V,V') \\ \Ext^*_{\Mod(U \cap h^{-1}Uh)}(V,V') \ar[r]^{h_\ast} & \Ext^*_{\Mod(U \cap hUh^{-1})}(V,V') \ar[u]_{\cores} }$}]{Images/img6efc6cf07eb1d84725334a11ef27569b.svg}} \end{equation*}$$

is commutative.

(E)

It is easily checked that the map$$\begin{align*} H_U & \longrightarrow H_U, \\ \tau & \longmapsto \tau (-^{-1}) \end{align*}$$

is an anti-involution of the $k$-algebra $H_U$, again viewed as a convolution algebra as in part (A) It sends $\tau _h$ to $\tau _{h^{-1}}$.

Lemma 2.5.

For any $0 \leq i \leq d$ and any $h \in G$ the diagram of Yoneda pairings

$$\begin{equation*} \vcenter{\img[][337pt][145pt][{$\xymatrix{ \Ext^{d-i}_{\Mod(U)}(V,\hat{I}) \ar[d]_{\res} &\!\! \times\!\! & H^i(U,V) \ar[r] & H^d(U,\hat{I}) \\ \Ext^{d-i}_{\Mod(U \cap h^{-1}Uh)}(V,\hat{I}) \ar[d]_{h_*} &\!\! \times\!\! & H^i(U \cap h^{-1}Uh,V) \ar[u]_{\cores} \ar[r] & H^d(U \cap h^{-1}Uh,\hat{I}) \ar[u]_{\cores} \ar[d]^{h_*} \\ \Ext^{d-i}_{\Mod(U \cap hUh^{-1})}(V,\hat{I}) \ar[d]_{\cores} &\!\! \times\!\! & H^i(U \cap hUh^{-1},V) \ar[u]_{h_*^{-1}} \ar[r] & H^d(U \cap hUh^{-1},\hat{I}) \ar[d]^{\cores} \\ \Ext^{d-i}_{\Mod(U)}(V,\hat{I}) &\!\! \times\!\! & H^i(U,V) \ar[u]_{\res} \ar[r] & H^d(U,\hat{I}) }$}]{Images/imgbcda489921ca7436523b2c32cbf3ddf2.svg}} \end{equation*}$$

is commutative.

Proof.

We fix injective resolutions $V \xrightarrow {\simeq } \mathcal{J}^\bullet$ and $\hat{I} \xrightarrow {\simeq } \mathcal{I}^\bullet$ in $\operatorname {Mod}(G)$, which as noted earlier remain injective resolutions after restriction to any given open subgroup of $G$.

The upper rectangle: Let $\beta ^\bullet : \mathcal{J}^\bullet \rightarrow \mathcal{I}^\bullet [d-i]$ be a $U$-equivariant and $\alpha ^\bullet : k \rightarrow \mathcal{J}^\bullet [i]$ a $U \cap h^{-1}Uh$-equivariant homomorphism of complexes representing classes $[\beta ^\bullet ] \in \operatorname {Ext}^{d-i}_{\operatorname {Mod}(U)}(V,\hat{I})$ and $[\alpha ^\bullet ] \in H^i(U \cap h^{-1}Uh,V)$, respectively. Then $\beta ^\bullet$ also represents $\operatorname {res}[\beta ^\bullet ]$ whereas $\operatorname {cores}[\alpha ^\bullet ]$ is represented by $\sum _{g \in U / U \cap h^{-1}Uh} {^g \alpha }^\bullet$. We compute

$$\begin{align*} [\beta ^\bullet [i]] \circ \operatorname {cores}[\alpha ^\bullet ] & = [\beta ^\bullet [i] \circ \sum _{g \in U / U \cap h^{-1}Uh} {^g \alpha }^\bullet ] = [\beta ^\bullet [i] \circ \sum _{g \in U / U \cap h^{-1}Uh} g \alpha ^\bullet (g^{-1} -)] \\ & = [\sum _{g \in U / U \cap h^{-1}Uh} g(\beta ^\bullet [i] \circ \alpha ^\bullet )(g^{-1} -)]\! =\! [\sum _{g \in U / U \cap h^{-1}Uh} {^g (}\beta ^\bullet [i] \circ \alpha ^\bullet )] \\ & = \operatorname {cores}(\operatorname {res}[\beta ^\bullet [i]] \circ [\alpha ^\bullet ]) \ . \end{align*}$$

The middle rectangle: Let $\beta ^\bullet : \mathcal{J}^\bullet \rightarrow \mathcal{I}^\bullet [d-i]$ be a $U \cap h^{-1}Uh$-equivariant and $\alpha ^\bullet : k \rightarrow \mathcal{J}^\bullet [i]$ a $U \cap hUh^{-1}$-equivariant homomorphism of complexes representing classes $[\beta ^\bullet ] \in \operatorname {Ext}^{d-i}_{\operatorname {Mod}(U \cap h^{-1}Uh)}(V,\hat{I})$ and $[\alpha ^\bullet ] \in H^i(U \cap hUh^{-1},V)$, respectively. Then $h_*[\beta ^\bullet ]$ and $h_*^{-1}[\alpha ^\bullet ]$ are represented by ${^h \beta }^\bullet$ and ${^{h^{-1}} \alpha }^\bullet$. We compute

$$\begin{align*} h_*[\beta ^\bullet [i]] \circ [\alpha ^\bullet ] & = [{^h \beta }^\bullet [i] \circ \alpha ^\bullet ] = [h\beta ^\bullet [i](h^{-1}\alpha ^\bullet (-))] = [h\beta ^\bullet [i](h^{-1}\alpha ^\bullet (h h^{-1}-))] \\ & = [h\beta ^\bullet [i]({^{h^{-1}} \alpha ^\bullet }(h^{-1}-))] = [h (\beta ^\bullet [i] \circ {^{h^{-1}} \alpha ^\bullet })(h^{-1}-)]\\ & = [{^h (}\beta ^\bullet [i] \circ {^{h^{-1}} \alpha ^\bullet })] = h_* ([\beta ^\bullet [i]] \circ h_*^{-1}[\alpha ^\bullet ]) \ . \end{align*}$$

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

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By 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

$$\begin{equation*} H^d(U,\hat{I}) \overset{\sim }{\longleftarrow } H^d(U \cap h^{-1}Uh,\hat{I}) \overset{h_*}{\longrightarrow } H^d(U \cap hUh^{-1},\hat{I}) \overset{\sim }{\longrightarrow } H^d(U,\hat{I}) \end{equation*}$$

is therefore multiplication by a scalar $\chi _G(h) \in k^\times$, which happens to be independent of $U$.

Lemma 2.6.

The map $\chi _G : G \rightarrow k^\times$ is a character which is independent of $U$ and trivial on any pro-$p$ subgroup of $G$.

Proof.

We first show the independence of $U$. Suppose $U'$ is another open torsion free pro-$p$ subgroup of $G$ and consider subgroups $U'' \subset U \cap U'$. Again by CG I-50(4) corestriction gives isomorphisms

$$\begin{equation*} H^d(U,k) \overset{\sim }{\longleftarrow } H^d(U'',k) \overset{\sim }{\longrightarrow } H^d(U',k). \end{equation*}$$

This gives a canonical isomorphism between the dualizing objects $\hat{I}_U \simeq \hat{I}_{U'}$ which in turn gives a canonical isomorphism $H^d(U,\hat{I}_U) \simeq H^d(U',\hat{I}_{U'})$. Altogether this shows $\chi _G(h) \in k^\times$ is independent of $U$, and it is obviously trivial on $U$.

Suppose that we have checked the multiplicativity of $\chi _G$ already and let $U_0$ be any pro-$p$ subgroup of $G$. Note that, as a $p$-adic Lie group, $G$ always has an open torsion free pro-$p$ subgroup; see pLG, Theorem 27.1 for instance which even shows the existence of a $p$-valuable subgroup. Hence $\chi _G | U_0$ factorizes through a finite quotient which is a $p$-group. Since any finite subgroup of $k^\times$ has order prime to $p$ it follows that $\chi _G$ is trivial on $U_0$. To establish multiplicativity let $g, h \in G$. Since conjugation commutes with corestriction we have the following three commutative diagrams, which together show our claim:

$$\begin{equation*} \vcenter{\img[][237pt][55pt][{$\xymatrix{ H^d(U \cap gUg^{-1}, \hat{I}) & H^d(U \cap gUg^{-1} \cap ghU(gh)^{-1}, \hat{I}) \ar[l]_-{\cores}^-{\cong} \\ H^d(U \cap g^{-1}Ug, \hat{I}) \ar[u]_{g_*} & H^d(U \cap g^{-1}Ug \cap hUh^{-1}, \hat{I}) \ar[l]_-{\cores}^-{\cong} \ar[u]_{g_*} , }$}]{Images/img0e1fcbbb8fc5acfe432ca045114a699f.svg}} \end{equation*}$$

$$\begin{equation*} \vcenter{\img[][243pt][55pt][{$\xymatrix{ H^d(U \cap hUh^{-1}, \hat{I}) & H^d(U \cap g^{-1}Ug \cap hUh^{-1}, \hat{I}) \ar[l]_-{\cores}^-{\cong} \\ H^d(U \cap h^{-1}Uh, \hat{I}) \ar[u]_{h_*} & H^d(U \cap(gh)^{-1}Ugh \cap h^{-1}Uh, \hat{I}) \ar[l]_-{\cores}^-{\cong} \ar[u]_{h_*} , }$}]{Images/img7bd60129cbd72eef1d337bd7f774a32a.svg}} \end{equation*}$$ and

$$\begin{equation*} \vcenter{\img[][304pt][96pt][{$\xymatrix{ H^d(U \cap gUg^{-1} \cap ghU(gh)^{-1}, \hat{I}) \ar[r]^-{\cores}_-{\cong} & H^d(U \cap ghU(gh)^{-1}, \hat{I}) \\ H^d(U \cap g^{-1}Ug \cap hUh^{-1}, \hat{I}) \ar[r]^-{\cores}_-{\cong} \ar[u]_{g_*} & H^d(g^{-1}Ug \cap hUh^{-1}, \hat{I}) \ar[u]_{g_*} \\ H^d(U \cap(gh)^{-1}Ugh \cap h^{-1}Uh, \hat{I}) \ar[u]_{h_*} \ar[r]^-{\cores}_-{\cong} & H^d(U \cap(gh)^{-1}Ugh, \hat{I}) \ar[u]_{h_*} \ar@/_15ex/[uu]_{(gh)_*} . }$}]{Images/imgd82d874ad290dc632ea0503232a524f7.svg}} \end{equation*}$$ ■

The map

$$\begin{align*} H_U & \longrightarrow H_U, \\ \tau & \longmapsto \chi _G \tau \ \text{(pointwise product of functions)} \end{align*}$$

is an algebra homomorphism. Pulling back an $H_U$-module $M$ along this homomorphism defines the twisted $H_U$-module $M(\chi _G)$. More explicitly $\tau _h \in H_U$ acts on $m \in M(\chi _G)$ by the rule $\tau _h \star m=\chi _G(h)\tau _h(m)$.

Also note that we may use the anti-involution in (E) to make the $k$-linear dual $M^\vee \coloneq \operatorname {Hom}_k(M,k)$ of a left $H_U$-module $M$ again into a left $H_U$-module. More explicitly $(\tau _h f)(m)=f(\tau _{h^{-1}}m)$ for $f \in M^{\vee }$ and $h \in G$.

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

$$\begin{equation*} \vcenter{\img[][229pt][58pt][{$\xymatrix{ \Ext^{d-i}_{\Mod(U)}(V,\hat{I}) \ar[d]_{\tau_h \cdot} & \times& H^i(U,V) \ar[r] & k \ar[d]^{\chi_G(h) \cdot} \\ \Ext^{d-i}_{\Mod(U)}(V,\hat{I}) & \times& H^i(U,V) \ar[u]_{\tau_{h^{-1}} \cdot} \ar[r] & k . }$}]{Images/img5e2263ae4f27c8afc2cf29fea1da734a.svg}} \end{equation*}$$

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

$$\begin{equation} \operatorname {Ext}^{d-i}_{\operatorname {Mod}(U)}(V,k) \xrightarrow {\:\cong \;} H^i(U,V)^\vee (\chi _G) \ . {\tag{5}} \end{equation}$$

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

Proposition 2.7.

For any compact open subgroup $U \subseteq G$ which is pro-$p$ and torsion free and any $V$ in $\operatorname {Mod}(G)$ we have an $H_U$-equivariant $E_2$-spectral sequence

$$\begin{equation*} H^i(U,S^j(V)) \Longrightarrow H^{d-i-j}(U,V)^\vee (\chi _G) \ . \end{equation*}$$

Proof.

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

■

Remark 2.8.

Suppose that $G = \mathbf{G}(\mathfrak{F})$ where $\mathfrak{F}/\mathbb{Q}_p$ is a finite extension and $\mathbf{G}$ is a connected reductive $\mathfrak{F}$-split group over $\mathfrak{F}$. Assuming that a pro-$p$ Iwahori subgroup $U$ of $G$ is torsion free it is shown in OS Proposition 7.16 that $\chi _G = 1$. Under additional assumptions this was proved before in Koz. In the preprint KS21 Koziol and Schwein give an alternate proof of the triviality of the orientation character $\chi _G$ via Moy-Prasad groups (still assuming pro-$p$ 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 Ko, Theorem 1.3 in the generality of a $p$-adic reductive group $G$ and a torsion free pro-$p$ Iwahori subgroup $U$.

We will show that $\chi _G$ in fact coincides with the duality character introduced by Kohlhaase in Koh after Definition 3.12 and which we temporarily denote by $\chi _G^\mathrm{Koh}$.

Proposition 2.9.

We have $\chi _G = \chi _G^\mathrm{Koh}$.

Proof.

The character $\chi _G^\mathrm{Koh}$ describes the $G$-action on a certain one dimensional $k$-vector space $E^d(k)$ the original definition of which we do not need. Instead we use Koh Proposition 3.2 which says that, for any compact open subgroup $G_0\! \subseteq \! G$, there is a natural $G_0$-equivariant isomorphism $\ell _{G,G_0}\! :\! E^d(k) \!\xrightarrow {\cong }\! \operatorname {Ext}^d_{\operatorname {Mod}(\Omega (G_0))}(k, \Omega (G_0))$ such that:

(1)

For any $g \in G$ the diagram$$\begin{equation*} \vcenter{\img[][242pt][62pt][{$\xymatrix{ E^d(k) \ar[d]_{\chi_G^\mathrm{Koh}(g) \cdot} \ar[rr]^-{\ell_{G,G_0}} && \Ext^d_{\Mod(\Omega(G_0))}(k, \Omega(G_0)) \ar[d]^{g_*} \\ E^d(k) \ar[rr]^-{\ell_{G,g G_0 g^{-1}}} && \Ext^d_{\Mod(\Omega(g G_0 g^{-1}))}(k, \Omega(gG_0 g^{-1})) }$}]{Images/imge93ca829b9fd58a583c7b32321523952.svg}} \end{equation*}$$

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

(2)

For any open subgroup $G_1 \subseteq G_0$ the diagram$$\begin{equation*} \vcenter{\img[][155pt][79pt][{$\xymatrix@R=0.5cm{ & \Ext^d_{\Mod(\Omega(G_0))}(k, \Omega(G_0)) \ar[dd]^{\ell_{G_0,G_1}} \\ E^d(k) \ar[ur]^-{\ell_{G,G_0}} \ar[dr]_-{\ell_{G,G_1}} \\ & \Ext^d_{\Mod(\Omega(G_1))}(k, \Omega(G_1)) }$}]{Images/img79ac9dfcb1afa099ca343504d34b5e00.svg}} \end{equation*}$$

is commutative. Moreover $\ell _{G_0,G_1}$ is the composite of the restriction map$$\begin{equation*} \operatorname {Ext}^d_{\operatorname {Mod}(\Omega (G_0))}(k, \Omega (G_0)) \xrightarrow {\operatorname {res}} \operatorname {Ext}^d_{\operatorname {Mod}(\Omega (G_1))}(k, \Omega (G_0)) \end{equation*}$$

and the map$$\begin{equation*} \operatorname {Ext}^d(k, j_{G_1.G_0}^\vee ) : \operatorname {Ext}^d_{\operatorname {Mod}(\Omega (G_1))}(k, \Omega (G_0)) \rightarrow \operatorname {Ext}^d_{\operatorname {Mod}(\Omega (G_1))}(k, \Omega (G_1)) \end{equation*}$$

which is induced by the Pontrjagin dual $j_{G_1,G_0}^\vee$ of the extension by zero map $j_{G_1,G_0} : C^\infty (G_1,k) \rightarrow C^\infty (G_0,k)$.

The Pontrjagin dual of $C^\infty (G_0,k)$ being $\Omega (G_0)$ we have, using 2, the isomorphism

$$\begin{equation*} P_{G_0} : \operatorname {Ext}^d_{\operatorname {Mod}(\Omega (G_0))}(k,\Omega (G_0)) \xrightarrow {\cong } \operatorname {Ext}^d_{\operatorname {Mod}(G_0)}(C^\infty (G_0,k),k) \ . \end{equation*}$$

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

$$\begin{equation} \vcenter{\img[][275pt][63pt][{$\xymatrix{ E^d(k) \ar[d]_{\chi_G^\mathrm{Koh}(g) \cdot} \ar[rrr]^-{P_{G_0} \circ\ell_{G,G_0}}_-{\cong} &&& \Ext^d_{\Mod(G_0)}(C^\infty(G_0,k),k) \ar[d]^{g_*} \\ E^d(k) \ar[rrr]^-{P_{g G_0 g^{-1}} \circ\ell_{G,g G_0 g^{-1}}}_-{\cong} &&& \Ext^d_{\Mod(g G_0 g^{-1})}(C^\infty(g G_0 g^{-1},k),k) }$}]{Images/imgd1d71390b04fc0697b10943fbd659095.svg}} {\tag{6}} \end{equation}$$

and

$$\begin{equation} \vcenter{\img[][294pt][82pt][{$\xymatrix@R=0.5cm{ & \Ext^d_{\Mod(G_0)}(C^\infty(G_0,k),k) \ar[dr]^{\res} \\ E^d(k) \ar[ur]^-{P_{G_0} \circ\ell_{G,G_0}\qquad}_{\cong} \ar[dr]_-{P_{G_1} \circ\ell_{G,G_1}\qquad}^{\cong} & & \Ext^d_{\Mod(G_1)}(C^\infty(G_0,k),k) \ar[dl]^{\qquad\Ext^d(j_{G_1,G_0},k)} \\ & \Ext^d_{\Mod(G_1)}(C^\infty(G_1,k),k) . }$}]{Images/img6e5b7a247523aaf18f192da7358bc58d.svg}} {\tag{7}} \end{equation}$$

On the other hand, taking now $G_0=U$ we note that the duality isomorphism 4 for $V = C^\infty (U,k)$ and $i=0$ is given by

$$\begin{align*} \operatorname {Ext}^d_{\operatorname {Mod}(U)}(C^\infty (U,k),k) & \xrightarrow {\;\cong \;} \operatorname {Hom}_k(\operatorname {Hom}_{\operatorname {Mod}(U)}(k,C^\infty (U,k)), H^d(U,k)), \\ e & \longmapsto \big [\phi \mapsto \phi ^*(e)\big ] \ . \end{align*}$$

Let $\operatorname {con}_U : k \rightarrow C^\infty (U,k)$ denote the map which sends $1 \in k$ to the constant function with value $1$ on $U$. Then the above isomorphism is equivalent to the isomorphism

$$\begin{align*} \operatorname {Ext}^d_{\operatorname {Mod}(U)}(C^\infty (U,k),k) & \xrightarrow {\;\cong \;} H^d(U,k), \\ e & \longmapsto \operatorname {con}_U^*(e) \ . \end{align*}$$

The first isomorphism being natural in conjugation by $g \in G$ and this conjugation sending $\operatorname {con}_U$ to $\operatorname {con}_{gUg^{-1}}$ we see that we have the commutative diagram

$$\begin{align} \vcenter{\img[][253pt][62pt][{$\xymatrix{ \Ext^d_{\Mod(U)}(C^\infty(U,k),k) \ar[d]_{g_*} \ar[rr]^-{\con_U^*} && H^d(U,k) \ar[d]^{g_*} \\ \Ext^d_{\Mod(gUg^{-1})}(C^\infty(gUg^{-1},k),k) \ar[rr]^-{\con_{gUg^{-1}}^*} && H^d(gUg^{-1},k). }$}]{Images/img502115bd70d1a57cb2b436a8b461761a.svg}} {\tag{8}} \end{align}$$

Furthermore, if $U' \subseteq U$ is any open subgroup, then we have the commutative diagram of duality pairings

$$\begin{equation*} \vcenter{\img[][319pt][101pt][{$\xymatrix{ \Ext^d_{\Mod(U)}(C^\infty(U,k),k) \ar[d]_{\res} & \times& H^0(U,C^\infty(U,k)) \ar[r] & H^d(U,k) \\ \Ext^d_{\Mod(U')}(C^\infty(U,k),k) \ar[d]_{\Ext^d(j_{U',U},k)} & \times& H^0(U',C^\infty(U,k)) \ar[u]_{\cores} \ar[r] & H^d(U',k) \ar[u]_{\cores} \ar@{=}[d] \\ \Ext^d_{\Mod(U')}(C^\infty(U',k),k) & \times& H^0(U',C^\infty(U',k)) \ar[u]_{H^0(U',j_{U',U})} \ar[r] & H^d(U',k) . \\ }$}]{Images/img15dcf955ba2099606a6d8d7de0e2cea7.svg}} \end{equation*}$$

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 $\operatorname {con}_{U'}$ to $\operatorname {con}_U$. Hence we obtain the commutative diagram

$$\begin{align*} \vcenter{\img[][189pt][62pt][{$\xymatrix{ \Ext^d_{\Mod(U)}(C^\infty(U,k),k) \ar[d]_{\Ext^d(j_{U',U},k) \circ\res} \ar[r]^-{\con_U^*} & H^d(U,k) \\ \Ext^d_{\Mod(U')}(C^\infty(U',k),k) \ar[r]^-{\con_{U'}^*} & H^d(U',k) \ar[u]^{\cores} . }$}]{Images/img63f303752ecf2f5ae2560a1d3dd6711c.svg}} \end{align*}$$

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

$$\begin{equation*} \vcenter{\img[][151pt][73pt][{$\xymatrix@R=0.5cm{ & H^d(U,k) \ar[dr]_{\cong} \\ E^d(k) \ar[ur]^{\con_U^* \circ P_{U} \circ\ell_{G,U}\qquad}_{\cong} \ar[dr]_{\con_{U'}^* \circ P_{U'} \circ\ell_{G,U'}\qquad}^{\cong} && k \\ & H^d(U',k) \ar[uu]^{\cores} \ar[ur]^{\cong} , }$}]{Images/img8ed2a5251fb1cb87227fa0d7b65e697e.svg}} \end{equation*}$$

where the right-hand oblique arrows are our standard identifications. This means that the isomorphism $\operatorname {con}_U^* \circ P_{U} \circ \ell _{G,U} : E^d(k) \xrightarrow {\cong } k$ does not depend on the subgroup $U$. With this information we consider the commutative diagram

$$\begin{equation*} \vcenter{\img[][311pt][60pt][{$\xymatrix{ E^d(k) \ar[d]_{\chi_G^\mathrm{Koh}(g) \cdot} \ar[rrrrr]^{\con_U^* \circ P_{U} \circ\ell_{G,U}}_{\cong} &&&&& H^d(U,k) \ar[d]_{g_*} \ar[r]^-{\cong} & k \ar[d]^{\chi_G(g) \cdot} \\ E^d(k) \ar[rrrrr]^{\con_{gUg^{-1}}^* \circ P_{gUg^{-1}} \circ\ell_{G,gUg^{-1}}}_{\cong} &&&&& H^d(gUg^{-1},k) \ar[r]^-{\cong} & k }$}]{Images/img480924253f29e2cb219c03b69385c476.svg}} \end{equation*}$$

whose left-hand rectangle arises by combining 6 and 8. Since the horizontal arrows coincide we conclude that $\chi _G^\mathrm{Koh}(g) = \chi _G(g)$.

■

One immediately infers the triviality of $\chi _G$ for open subgroups of $p$-adic reductive groups:

Lemma 2.10.

Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $\mathfrak{F}$ of $\mathbb{Q}_p$; if $G$ is an open subgroup of $\mathbf{G}(\mathfrak{F})$ then $\chi _G = 1$.

Proof.

Proposition 2.9 together with Koh, Corollary 5.2 shows the assertion in the case $\mathfrak{F} = \mathbb{Q}_p$. In general let $\mathbf{G}'$ denote the Weil restriction of $\mathbf{G}$ to $\mathbb{Q}_p$. It is shown in Oes App. 3 that $\mathbf{G}'$ again is a connected linear algebraic group with the property that $\mathbf{G}(\mathfrak{F}) = \mathbf{G}'(\mathbf{Q}_p)$ as $p$-adic Lie groups. Since our field extension is separable it follows from loc. cit. A.3.4 that with $\mathbf{G}$ also $\mathbf{G}'$ is reductive. This reduces the general case to the case $\mathfrak{F} = \mathbb{Q}_p$.

■

3. Derived smooth duality

We begin by recalling some general nonsense about the adjunction between tensor product and Hom-functor which for three $k$-vector spaces $V_1$, $V_2$, and $V_3$ is given by the linear isomorphism

$$\begin{align} \operatorname {Hom}_k(V_1 \otimes _k V_2, V_3) & \xrightarrow {\;\cong \;} \operatorname {Hom}_k(V_1, \operatorname {Hom}_k(V_2,V_3)), {\tag{9}}\\ A & \longmapsto \lambda _A(v_1)(v_2) \coloneq A (v_1 \otimes v_2) \ . \end{align}$$

Suppose that all three vector spaces carry a left $G$-action. Then $\operatorname {Hom}_k(V_1 \otimes _k V_2, V_3)$ and $\operatorname {Hom}_k(V_1, \operatorname {Hom}_k(V_2,V_3))$ are equipped with the $G \times G \times G$-action defined by

$$\begin{equation*} {^{(g_1,g_2,g_3)} A}(v_1 \otimes v_2) \coloneq g_3 A(g_1^{-1} v_1 \otimes g_2^{-1} v_2) \end{equation*}$$

and

$$\begin{equation*} {^{(g_1,g_2,g_3)} \lambda }(v_1)(v_2) \coloneq g_3 (\lambda (g_1^{-1} v_1)(g_2^{-1} v_2)) , \end{equation*}$$

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

$$\begin{equation*} \operatorname {Hom}_{k[G]}(V_1 \otimes _k V_2, V_3) \xrightarrow {\;\cong \;} \operatorname {Hom}_{k[G]}(V_1, \operatorname {Hom}_k(V_2,V_3)) \ . \end{equation*}$$

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

$$\begin{equation} \operatorname {Hom}_{\operatorname {Mod}(G)}(V_1 \otimes _k V_2, V_3) \cong \operatorname {Hom}_{\operatorname {Mod}(G)}(V_1, \underline{\operatorname {Hom}}(V_2,V_3)) \ . {\tag{10}} \end{equation}$$

Let $D(G)$ denote the unbounded derived category of $\operatorname {Mod}(G)$. The tensor product functor

$$\begin{align*} \operatorname {Mod}(G) \times \operatorname {Mod}(G) & \longrightarrow \operatorname {Mod}(G), \\ (V_1,V_2) & \longmapsto V_1 \otimes _k V_2 \ , \end{align*}$$

where the $G$-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

$$\begin{align*} D(G) \times D(G) & \longrightarrow D(G), \\ (V_1^\bullet ,V_2^\bullet ) & \longmapsto \mathrm{tot}_\oplus (V_1^\bullet \otimes _k V_2^\bullet ) \ , \end{align*}$$

which we usually denote simply by $V_1^\bullet \otimes _k V_2^\bullet$.⁠¹ On the other hand, since $\operatorname {Mod}(G)$ is a Grothendieck category, we have for any $V_0$ in $\operatorname {Mod}(G)$ 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 $h^*(V_1^\bullet \otimes _k V_2^\bullet )\simeq h^*(V_1^{\bullet })\otimes _k h^*(V_2^{\bullet })$ by the Künneth formula. Recall that $k$ is a field.

✖

$$\begin{equation*} R\underline{\operatorname {Hom}}(V_0,-) : D(G) \longrightarrow D(G) \end{equation*}$$

such that $R^j\underline{\operatorname {Hom}}(V_0,V) = \underline{\operatorname {Ext}}^j(V_0,V)$ for any $V$ in $\operatorname {Mod}(G)$ and $j \geq 0$. We want to extend this to a bifunctor $D(G)^{op} \times D(G) \rightarrow D(G)$. First we recall that $\operatorname {Mod}(G)$ has arbitrary direct products (but which are not exact); we will denote these by $\prod ^\infty$ to avoid confusion with the cartesian direct product. Hence, for any two complexes $V_1^\bullet$ and $V_2^\bullet$ in $\operatorname {Mod}(G)$ we may define the complex

$$\begin{equation*} \underline{\operatorname {Hom}}^\bullet (V_1^\bullet , V_2^\bullet ) \coloneq {\prod _{j \in \mathbb{Z}}}^\infty \underline{\operatorname {Hom}}(V_1^j,V_2^{j+ \bullet }) \end{equation*}$$

in $\operatorname {Mod}(G)$ in the usual way. By construction we have that

$$\begin{align} \underline{\operatorname {Hom}}^\bullet (V_1^\bullet , V_2^\bullet ) & = \varinjlim _K \big ( \prod _{j \in \mathbb{Z}} \underline{\operatorname {Hom}}(V_1^j,V_2^{j+ \bullet }) \big )^K = \varinjlim _K \prod _{j \in \mathbb{Z}} \underline{\operatorname {Hom}}(V_1^j,V_2^{j+ \bullet })^K \\ & = \varinjlim _K \prod _{j \in \mathbb{Z}} \operatorname {Hom}_{\operatorname {Mod}(K)}(V_1^j,V_2^{j+ \bullet }) {\tag{11}}\\ & = \varinjlim _K \operatorname {Hom}_{\operatorname {Mod}(K)}^\bullet (V_1^\bullet , V_2^\bullet ) \end{align}$$

is the inductive limit over all compact open subgroups $K \subseteq G$ of the usual Hom-complexes for the abelian categories $\operatorname {Mod}(K)$.

The adjunction 10 shows that the assumptions of KS Theorem 14.4.8 are satisfied (with $\mathcal{P}_i = \mathcal{C}_i = \operatorname {Mod}(G)$, $G$ the tensor product functor, and $F_1 = F_2 = \underline{\operatorname {Hom}}$). Hence we obtain the following result.

Proposition 3.1.

The total derived functor $R\underline{\operatorname {Hom}}(-,-) : D(G)^{op} \times D(G) \longrightarrow D(G)$ exists and can be computed by $R\underline{\operatorname {Hom}}(V_1^\bullet ,V_2^\bullet ) = \underline{\operatorname {Hom}}^\bullet (V_1^\bullet ,J^\bullet )$ where $V_2^\bullet \xrightarrow {\simeq } J^\bullet$ is a homotopically injective resolution. Moreover, there are the natural adjunctions

$$\begin{equation*} \operatorname {Hom}_{D(G)}(V_1^\bullet \otimes _k V_2^\bullet ,V_3^\bullet ) = \operatorname {Hom}_{D(G)}(V_1^\bullet , R\underline{\operatorname {Hom}}(V_2^\bullet ,V_3^\bullet )) \end{equation*}$$

and

$$\begin{equation*} R\operatorname {Hom}_{\operatorname {Mod}(G)}(V_1^\bullet \otimes _k V_2^\bullet ,V_3^\bullet ) = R\operatorname {Hom}_{\operatorname {Mod}(G)}(V_1^\bullet , R\underline{\operatorname {Hom}}(V_2^\bullet ,V_3^\bullet )) \end{equation*}$$

for any $V_i^\bullet$ in $D(G)$.

Remark 3.2.

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

$$\begin{equation*} R\underline{\operatorname {Hom}}(V_1^\bullet \otimes _k V_2^\bullet ,V_3^\bullet ) = R\underline{\operatorname {Hom}}(V_1^\bullet , R\underline{\operatorname {Hom}}(V_2^\bullet ,V_3^\bullet )) \end{equation*}$$

for all $V_i^{\bullet }$. To see this pick a homotopically injective resolution $V_3^\bullet \xrightarrow {\simeq } J^\bullet$ in $\operatorname {Mod}(G)$. Note that $J^{\bullet }$ remains homotopically injective upon restriction to any compact open subgroup $K \subseteq G$ (by Frobenius reciprocity and exactness of $\operatorname {ind}_K^G$). Furthermore $\underline{\operatorname {Hom}}^{\bullet }(V_2^{\bullet },J^{\bullet })$ is homotopically injective by adjunction and the previous footnote. By Proposition 3.1 for $K$ we have

$$\begin{equation*} \operatorname {Hom}_{\operatorname {Mod}(K)}^{\bullet }(V_1^\bullet \otimes _k V_2^\bullet ,J^\bullet ) = \operatorname {Hom}_{\operatorname {Mod}(K)}^{\bullet }(V_1^\bullet , \underline{\operatorname {Hom}}^{\bullet }(V_2^\bullet ,J^\bullet )). \end{equation*}$$

Taking the limit over $K$ and invoking the description 11 gives the result.

Corollary 3.3.

$(D(G), \otimes _k, k, R\underline{\operatorname {Hom}})$ is a closed symmetric monoidal category.

For $V_2 = k$ viewed as complex concentrated in degree zero we, in particular, obtain the total derived duality functor

$$\begin{equation*} R\underline{\operatorname {Hom}}(-,k) : D(G)^{op} \longrightarrow D(G) \end{equation*}$$

such that $R^j\underline{\operatorname {Hom}}(V,k) = \underline{\operatorname {Ext}}^j(V,k)=S^j(V)$ for any $V$ in $\operatorname {Mod}(G)$ and any $j \geq 0$. In order to see in which way $k$ is a dualizing object for $\operatorname {Mod}(G)$ we have to introduce two finiteness conditions.

First we make the following observation.

Lemma 3.4.

The functor $R\underline{\operatorname {Hom}}(-,k)$ is way-out in both directions, and in particular respects $D^b(G)$.

Proof.

We refer to Har, p. 68 for what it means to be way-out, but the actual definition is not important here. By Har, Proposition I.7.6 $R\operatorname {Hom}_{\operatorname {Mod}(K)}(-,k)$ is way-out (in both directions) if and only if there is an $n_0$ such that $\operatorname {Ext}_{\operatorname {Mod}(K)}^i(V,k)=0$ for all $V \in \operatorname {Mod}(K)$ and $i>n_0$. By (the proof of) Lemma 2.2 we may take $n_0=d$ when $K$ is sufficiently small. Finally by 11 we conclude that $R\underline{\operatorname {Hom}}(-,k)$ itself is way-out.

■

Remark 3.5.

In general the trivial $G$-representation $k$ does not have finite injective dimension in $\operatorname {Mod}(G)$. Nevertheless, as the previous proof shows, we have

$$\begin{equation*} R\underline{\operatorname {Hom}}(V,k)\in D^{[0,d]}(G) \end{equation*}$$

for all $V$ in $\operatorname {Mod}(G)$.

Next we recall that a representation $V$ in $\operatorname {Mod}(G)$ is called admissible if, for any open subgroup $K \subseteq G$, the vector space of $K$-fixed vectors $V^K$ is finite dimensional. In fact, it suffices to check the defining condition for a single compact open subgroup $K$ (apply the Nakayama lemma to the dual $\Omega (K)$-module $V^\vee$ or see Koh Lemma 1.7). The full subcategory $\operatorname {Mod}_{adm}(G)$ of admissible representations in $\operatorname {Mod}(G)$ is a Serre subcategory (cf. Em1 Proposition 2.2.13). Hence we have the strictly full triangulated subcategories $D^b_{adm}(G) \subseteq D^b(G)$ and $D_{adm}(G) \subseteq D(G)$ of those complexes whose cohomology representations are admissible.

Lemma 3.6.

The derived duality functor $R\underline{\operatorname {Hom}}(-,k)$ respects both subcategories $D^b_{adm}(G)$ and $D_{adm}(G)$.

Proof.

It is shown in Koh Corollary 3.15 that for an admissible representation $V$ in $\operatorname {Mod}(G)$ the representations $S^j(V)$ are admissible as well. Hence for an admissible $V$ the complex $R\underline{\operatorname {Hom}}(V,k)$ lies in $D^b_{adm}(G)$. On the other hand we have observed already that our functor is way-out in both directions in the sense of Har §7. Therefore our assertion follows from loc. cit. Proposition I.7.3.

■

Let $V^\bullet$ be any complex in $\operatorname {Mod}(G)$ and fix an injective resolution $k \xrightarrow {\simeq } \mathcal{J}^\bullet$. We construct a natural transformation

$$\begin{equation} \eta _{V^\bullet } : V^\bullet \longrightarrow \underline{\operatorname {Hom}}^\bullet (\underline{\operatorname {Hom}}^\bullet (V^\bullet ,\mathcal{J}^\bullet ),\mathcal{J}^\bullet ) {\tag{12}} \end{equation}$$

as follows. Inserting the definitions we have to produce, for any $\ell \in \mathbb{Z}$, a natural $G$-equivariant map

$$\begin{equation*} \eta _{V^\ell } : V^\ell \longrightarrow {\prod _{j \in \mathbb{Z}}}^\infty \underline{\operatorname {Hom}}({\prod _{i \in \mathbb{Z}}}^\infty \underline{\operatorname {Hom}}(V^i,\mathcal{J}^{i+j}),\mathcal{J}^{j+\ell }) \end{equation*}$$

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

$$\begin{equation*} \eta _{V^\ell }(v)_j((f_{i,j})_i) \coloneq (-1)^{\ell j}f_{\ell ,j}(v) \end{equation*}$$

have these properties.

Proposition 3.7.

If the complex $V^\bullet$ has admissible cohomology then the natural transformation $\eta _{V^\bullet }$ is a quasi-isomorphism.

Proof.

Since we have a natural transformation between way-out functors the lemma on way-out-functors (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 $\mathcal{P}$ 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 $\mathcal{P}$. We cannot apply this directly, though. First let us fix a compact open subgroup $K$ in $G$. Then we observe:

–

Any admissible $G$-representation $V$ is also admissible as a $K$-representation;

–

$k \xrightarrow {\simeq } \mathcal{J}^\bullet$ is also an injective resolution in $\operatorname {Mod}(K)$;

–

the natural transformation $\eta _V$ remains the same if constructed for $V$ considered only as a $K$-representation.

This means that, for the purposes of our proof, we may assume that our group $G$ is compact. Let $C^\infty (G,k)$ denote, as before, the vector space of $k$-valued locally constant functions on $G$. Equipped with the left translation action it is an admissible smooth $G$-representation. We have $C^\infty (G,k)^\vee = \Omega (G)$. Let $V$ be any admissible representation in $\operatorname {Mod}(G)$. Then $V^\vee$ is a finitely generated (pseudocompact) $\Omega (G)$-module (Koh Proposition 1.9(i)). Hence we find a surjection $\Omega (G)^m \twoheadrightarrow V^\vee$ in $\operatorname {Mod}_{pc}(\Omega (G))$ for some integer $m \geq 0$. It is the dual of an injective map $V \hookrightarrow C^\infty (G,k)^m$ in $\operatorname {Mod}(G)$. Therefore we can take the single object $C^\infty (G,k)$ for the class $\mathcal{P}$. By Koh Proposition 3.13 we have, for any integer $j$, that

$$\begin{equation*} R^j\underline{\operatorname {Hom}}(C^\infty (G,k),k) = S^j(C^\infty (G,k)) \cong \begin{cases} \chi _G \otimes _k C^\infty (G,k) & \text{for $j = d$}, \\ 0 & \text{otherwise}, \end{cases} \end{equation*}$$

where $\chi _G : G \rightarrow k^\times$ is Kohlhaase’s duality character. Hence $R\underline{\operatorname {Hom}}(C^\infty (G,k),k) \simeq (\chi _G \otimes _k C^\infty (G,k))[-d]$ and then $R\underline{\operatorname {Hom}}(R\underline{\operatorname {Hom}}(C^\infty (G,k),k),k) \simeq C^\infty (G,k)$. One checks from the proof in loc. cit. that the latter quasi-isomorphism is induced by the natural transformation $\eta _{C^\infty (G,k)}$.

■

In other words:

Corollary 3.8.

On $D_{adm}(G)$ the functor $R\underline{\operatorname {Hom}}(-,k)$ is involutive.

Next we extend the involutivity of $R\underline{\operatorname {Hom}}(-,k)$ 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 $D(G)$ which is potentially larger than $D_{adm}(G)$. The possible drawback is that the defining condition for this subcategory is a “global” finiteness condition.

We let $\operatorname {Vec}$ denote the abelian category of $k$-vector spaces and $D(k)$ its unbounded derived category. In the following we fix an open subgroup $U \subseteq G$ which is pro-$p$ and torsion free. As recalled in the proof of Lemma 2.2 the functor

$$\begin{align*} \operatorname {Mod}(G) & \longrightarrow \operatorname {Vec}, \\ V & \longmapsto V^U = H^0(U,V) \end{align*}$$

has finite cohomological dimension $d$. Hence its total derived functor $RH^0(U,-) : D(G) \longrightarrow D(k)$ exists (cf. Har Corollary I.5.3)). It is given by composing

$$\begin{equation*} R\operatorname {Hom}_{\operatorname {Mod}(U)}(k,-): D(U) \longrightarrow D(k) \end{equation*}$$

with the restriction functor $\text{forget}: D(G)\longrightarrow D(U)$.

On the other hand the functor $\operatorname {Hom}_k(-,k)$ on $\operatorname {Vec}$ of taking the $k$-linear dual is exact and therefore passes directly to a functor form $D(k)^{op}$ to $D(k)$ which, for simplicity, we also denote by $\operatorname {Hom}_k(-,k)$.

Theorem 4.1.

The diagram

$$\begin{equation*} \vcenter{\img[][226pt][95pt][{$\xymatrix{ D(G)^{op} \ar[d]_{\mathrm{forget}} \ar[rr]^-{R\underline{\Hom}(-,\hat{I})} && D(G) \ar[d]^{\mathrm{forget}} \\ D(U)^{op} \ar[d]_{R\Hom_{\Mod(U)}(k,-)} \ar[rr]^-{R\underline{\Hom}(-,\hat{I})} && D(U) \ar[d]^{R\Hom_{\Mod(U)}(k,-)} \\ D(k)^{op} \ar[rr]^-{\Hom_k(-,k)[-d]} && D(k) }$}]{Images/imgc1d18f6fb4d7a10e9ef347fed35d30bd.svg}} \end{equation*}$$

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

$$\begin{equation*} RH^0(U,R\underline{\operatorname {Hom}}(-,\hat{I})) \overset{\sim }{\longrightarrow } \operatorname {Hom}_k(RH^0(U,-),k)[-d]. \end{equation*}$$

Proof.

The upper rectangle is commutative since restriction from $G$ to $U$ 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 $RH^0(U,R\underline{\operatorname {Hom}}(-,\hat{I}))$ is naturally isomorphic to the functor $R\operatorname {Hom}_{\operatorname {Mod}(U)}(-,\hat{I})$. Hence it remains to exhibit a natural isomorphism

$$\begin{align*} R\operatorname {Hom}_{\operatorname {Mod}(U)}(-,\hat{I})\longrightarrow \operatorname {Hom}_k(RH^0(U,-),k)[-d]. \end{align*}$$

For this we start with the Yoneda pairing

$$\begin{equation*} R\operatorname {Hom}_{\operatorname {Mod}(U)}(V^\bullet ,\hat{I}) \times R\operatorname {Hom}_{\operatorname {Mod}(U)}(k,V^\bullet ) \longrightarrow R\operatorname {Hom}_{\operatorname {Mod}(U)}(k,\hat{I}) \ . \end{equation*}$$

By our assumption on the group $U$ the natural homomorphism

$$\begin{align*} \tau ^{\leq d} R\operatorname {Hom}_{\operatorname {Mod}(U)}(k,\hat{I}) \xrightarrow {\cong } R\operatorname {Hom}_{\operatorname {Mod}(U)}(k,\hat{I}) \end{align*}$$

is an isomorphism and the upper truncation $\tau ^{\leq d} R\operatorname {Hom}_{\operatorname {Mod}(U)}(k,\hat{I})$ at degree $d$ (cf. Har p. 69/70) maps to its cohomology $H^d(U,\hat{I})[-d] \cong k[-d]$ in degree $d$. (The latter identification is given by the trace map $\varrho$ in Verdier’s appendix to CG.)

The Yoneda pairing therefore induces a pairing

$$\begin{equation*} R\operatorname {Hom}_{\operatorname {Mod}(U)}(V^\bullet ,\hat{I}) \times R\operatorname {Hom}_{\operatorname {Mod}(U)}(k,V^\bullet ) \longrightarrow k[-d] \end{equation*}$$

and hence a natural homomorphism

$$\begin{equation*} R\operatorname {Hom}_{\operatorname {Mod}(U)}(V^\bullet ,\hat{I}) \longrightarrow \operatorname {Hom}_k(R\operatorname {Hom}_{\operatorname {Mod}(U)}(k,V^\bullet ),k[-d]) \ . \end{equation*}$$

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

$$\begin{equation} \operatorname {Ext}^*_{\operatorname {Mod}(U)}(V^\bullet ,\hat{I}) \longrightarrow \operatorname {Hom}_k(H^{d-*}(U,V^\bullet ),k) {\tag{13}} \end{equation}$$

is bijective. If $V^\bullet$ is a single representation in degree zero then we have seen this already in 4. By Example 1 on p. 68 in Har the functor $RH^0(U,-)$ and hence also the functor $\operatorname {Hom}_k(R\operatorname {Hom}_{\operatorname {Mod}(U)}(k,-),k[-d])$ are way-out in both directions. Similarly, by Lemma 2.2 and Har Proposition I.7.6 the functor $R\operatorname {Hom}_{\operatorname {Mod}(U)}(-,k)$ is way-out in both directions as well. Hence it follows from Har Proposition I.7.1(iii) that 13 is always bijective.

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Definition 4.2.

A complex $V^\bullet$ in $D(G)$ is globally admissible if its cohomology groups $H^i(U,V^\bullet )$, for any $i \in \mathbb{Z}$, are finite dimensional vector spaces. Let $D(G)^a \subseteq D(G)$ 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 $U$. To rephrase Definition 4.2 let $D_{fin}(k) \subseteq D(k)$ denote the strictly full triangulated subcategory of all objects all of whose cohomology vector spaces are finite dimensional. Then $D(G)^a$ is the full preimage in $D(G)$ of $D_{fin}(k)$ under the functor $RH^0(U,-)$.

Corollary 4.3.

The duality functor $R\underline{\operatorname {Hom}}(-k)$ respects the subcategory $D(G)^a$.

Proof.

This is immediate from Theorem 4.1 since the functor $\operatorname {Hom}_k(-,k)$ on $D(k)$ respects the subcategory $D_{fin}(k)$.

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In 12 we introduced the biduality morphism $\eta _{V^\bullet }\! :\! V^\bullet \!\!\rightarrow \! R\underline{\operatorname {Hom}}(R\underline{\operatorname {Hom}}(V^\bullet \!,k),k)$. Our further analysis of it will be based upon the following general observation.

Lemma 4.4.

A homomorphism $V_1^\bullet \rightarrow V_2^\bullet$ in $D(G)$ is an isomorphism if and only if the induced map $H^i(U,V_1^\bullet ) \rightarrow H^i(U,V_2^\bullet )$, for any $i \in \mathbb{Z}$, is bijective.

Proof.

This is an immediate consequence of the equivalence $H$ between $D(G)$ and the derived category of a certain differential graded algebra in DGA Theorem 9. By construction the functor $H$ has the property that $h^*(H(-)) = H^*(U,-)$.

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Theorem 4.5.

The biduality morphism $\eta _{V^\bullet }$, for any $V^\bullet$ in $D(G)$, is an isomorphism if and only if $V^\bullet$ lies in $D(G)^a$.

Proof.

According to Lemma 4.4 we have to check that the maps

$$\begin{equation*} H^i(U,\eta _{V^\bullet }) : H^i(U,V^\bullet ) \rightarrow H^i(U, R\underline{\operatorname {Hom}}(R\underline{\operatorname {Hom}}(V^\bullet ,k),k)) \end{equation*}$$

are bijective for any $i \in \mathbb{Z}$ if and only if $V^\bullet$ lies in $D(G)^a$. By Proposition 4.1 we have natural isomorphisms

$$\begin{equation*} \xi ^i_{V^\bullet } : H^i(U,R\underline{\operatorname {Hom}}(V^\bullet ,\hat{I})) \xrightarrow {\;\cong \;} \operatorname {Hom}_k(H^{d-i}(U,V^\bullet ),k) \ . \end{equation*}$$

For the remainder of this proof we fix an isomorphism $\hat{I}\simeq k$. The trace map $\varrho : H^d(U,\hat{I}) \rightarrow k$ then yields an isomorphism $H^d(U,k)\simeq k$. We will just write $k$ instead of $\hat{I}$ in what follows.

We now claim that the diagram

$$\begin{equation*} \vcenter{\img[][327pt][62pt][{$\xymatrix{ H^i(U,V^\bullet) \ar[d]_{b} \ar[rr]^-{H^i(U,\eta_{V^\bullet})} && H^i(U, R\underline{\Hom}(R\underline{\Hom}(V^\bullet,k),k)) \ar[d]^{\xi^i_{R\underline{\Hom}(V^\bullet,k)}}_{\cong} \\ \Hom_k(\Hom_k(H^i(U,V^\bullet),k),k) \ar[rr]^-{\Hom_k(\xi^{d-i}_{V^\bullet},k)}_-{\cong} && \Hom_k(H^{d-i}(U,R\underline{\Hom}(V^\bullet,k)),k), }$}]{Images/imgb59849206225a23f6f0a7a15d5321980.svg}} \end{equation*}$$

where $b$ denotes the natural map from a $k$-vector space into its double dual, is commutative up to the sign $(-1)^{i(d-i)}$. This immediately shows that $H^i(U,\eta _{V^\bullet })$ is bijective if and only if $b$ is bijective which, of course, is the case if and only if the vector space $H^i(U,V^\bullet )$ is finite dimensional.

To establish this claim we compute $R\underline{\operatorname {Hom}}(-,k)$ by using an injective resolution $\mathcal{J}^\bullet$ of $k$ in $\operatorname {Mod}(G)$ and hence in $\operatorname {Mod}(U)$. Then $R\underline{\operatorname {Hom}}(V^\bullet ,k) = \underline{\operatorname {Hom}}^\bullet (V^\bullet ,\mathcal{J}^\bullet )$ by Proposition 3.1. Moreover the adjunction property 10 implies that $\underline{\operatorname {Hom}}^\bullet (V^\bullet ,\mathcal{J}^\bullet )$ always is homotopically injective. Finally we may also assume that $V^\bullet$ is homotopically injective. Our diagram therefore becomes

$$\begin{equation*} \vcenter{\img[][346pt][62pt][{$\xymatrix{ h^i((V^\bullet)^U) \ar[d]_{b} \ar[rr]^-{H^i(U,\eta_{V^\bullet})} && \Hom_{K(U)}(\prod_{r \in\mathbb{Z}}^{\infty} \underline{\Hom}(V^r,\mathcal{J}^{r + \bullet}),\mathcal{J}^\bullet[i]) \ar[d]^{\xi^i_{R\underline{\Hom}(V^\bullet,k)}}_{\cong} \\ \Hom_k(\Hom_k(h^i((V^\bullet)^U),k),k) \ar[rr]^-{\Hom_k(\xi^{d-i}_{V^\bullet},k)}_-{\cong} && \Hom_k(\Hom_{K(U)}(V^\bullet,\mathcal{J}^\bullet[d-i]),k), }$}]{Images/img8ee05ee176b80b0a63115e6088cdb0c0.svg}} \end{equation*}$$

where $K(U)$ denotes as usual the unbounded homotopy category of complexes in $\operatorname {Mod}(U)$. We first recall that, under our identification $h^d((\mathcal{J}^\bullet )^U) = H^d(U,k) \cong k$, the map $\xi _{V^\bullet }^i$ is explicitly given by

$$\begin{align*} \xi _{V^\bullet }^i : \operatorname {Hom}_{K(U)}(V^\bullet ,\mathcal{J}^\bullet [i]) & \longrightarrow \operatorname {Hom}_k(h^{d-i}((V^\bullet )^U),k), \\ [\epsilon ^\bullet ] & \longmapsto \Big [ [\delta _{d-i}] \longmapsto [\epsilon ^{d-i}(\delta _{d-i})] \Big ] \ . \end{align*}$$

Now let $[v_i] \in h^i((V^\bullet )^U)$. By definition of $\eta _{V^\bullet }$ its image under the top horizontal arrow in the above diagram is the homotopy class of the homomorphism of complexes

$$\begin{align*} {\prod _{r \in \mathbb{Z}}}^{\infty } \underline{\operatorname {Hom}}(V^r,\mathcal{J}^{r + \bullet }) & \longrightarrow \mathcal{J}^\bullet [i], \\ (f_{r,\bullet })_r & \longmapsto (-1)^{i \bullet } f_{i,\bullet }(v_i) \end{align*}$$

induced by $\eta _{V^i}(v_i)_{\bullet }$. Under the right vertical arrow it is further mapped to the linear map

$$\begin{align} \operatorname {Hom}_{K(U)}(V^\bullet ,\mathcal{J}^\bullet [d-i]) & \longrightarrow k, {\tag{14}}\\ [(f_{r,d-i})_r] & \longmapsto (-1)^{i(d-i)} [f_{i,d-i}(v_i)]. \end{align}$$

But $[(f_{r,d-i})_r]$ corresponds under $\xi ^{d-i}_{V^\bullet }$ to the linear map in $\operatorname {Hom}_k(h^i((V^\bullet )^U),k)$ sending $[\delta _i]$ to $[f_{i,d-i}(\delta _i)]$. Hence the preimage of 14 under the bottom horizontal map in the diagram is equal to $(-1)^{i(d-i)} b([v_i])$ as claimed.

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Corollary 4.6.

The subcategory $D(G)^a$ in $D(G)$ is independent of the choice of the subgroup $U \subseteq G$.

What is the relation between the subcategories $D_{adm}(G)$ and $D(G)^a$? We had observed earlier that a representation $V$ in $\operatorname {Mod}(G)$ is admissible if and only if the vector space $H^0(U,V)$ is finite dimensional. Moreover, by Em2 Lemma 3.3.4, we have the following fact.

Lemma 4.7.

If $V$ in $\operatorname {Mod}(G)$ is admissible then all the vector spaces $H^i(U,V)$, for $i \geq 0$, are finite dimensional.

Lemma 4.7 says that, for an admissible $V$, the complex $RH^0(U,V)$ lies in $D_{fin}(k)$. By Example 1 on p. 68 in Har the functor $RH^0(U,-)$ is way-out in both directions. Therefore Har Proposition I.7.3(iii) implies that the functor $RH^0(U,-)$ maps $D_{adm}(G)$ to $D_{fin}(k)$. This proves the following.

Proposition 4.8.

$D_{adm}(G) \subseteq D(G)^a$.

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

On the full subcategories $D^{\pm }(G)$ of complexes bounded below or above we have stronger results.

Proposition 4.9.

(i)

A complex $V^\bullet$ in $D^+(G)$ lies in $D_{adm}(G)$ if and only if $H^i(U,V^\bullet )$ is finite dimensional for any $i \in \mathbb{Z}$. I.e., we have$$\begin{equation*} D^+(G) \cap D_{adm}(G) = D^+(G) \cap D(G)^a. \end{equation*}$$

Similarly for $D^-(G)$.

(ii)

More generally, a globally admissible complex with some vanishing differential lies in the subcategory $D_{adm}(G)$.

Proof.

First of all, in part (i) it suffices to show the $D^+(G)$-version. For if $V^{\bullet }$ lies in $D^-(G)$ then its dual $R\underline{\operatorname {Hom}}(V^{\bullet },k)=\underline{\operatorname {Hom}}^{\bullet }(V^{\bullet },\mathcal{J}^{\bullet })$ lies in $D^+(G)$. Furthermore $R\underline{\operatorname {Hom}}(V^{\bullet },k)$ belongs to $D(G)^a$ if $V^{\bullet }$ does by Corollary 4.3. In that case, once we show the $D^+(G)$-version, we conclude that $R\underline{\operatorname {Hom}}(V^{\bullet },k)$ is an object of $D_{adm}(G)$. However, by Lemma 3.6 the functor $R\underline{\operatorname {Hom}}(-,k)$ preserves $D_{adm}(G)$. Since the functor is involutive on $D(G)^a$ by Proposition 4.5 we conclude that $V^{\bullet }$ indeed belongs to $D_{adm}(G)$.

We proceed to show the $D^+(G)$-version in part (i). The direct implication holds true by Proposition 4.8. For the reverse implication we now assume that all the $H^i(U,V^\bullet )$ are finite dimensional, and $V^{\bullet }$ is bounded below.

Choose an integer $m$ such that $h^j(V^\bullet ) = 0$ for any $j < m$. In this situation it is a standard fact (cf. KS Exercise 13.3) that we have $H^0(U,h^m(V^\bullet )) = R^mH^0(U,V^\bullet ) = H^m(U,V^\bullet )$. Hence $H^0(U,h^m(V^\bullet ))$ is finite dimensional. As recalled before Lemma 4.7 this implies that $h^m(V^\bullet )$ is admissible. Moreover, Lemma 4.7 then says that $H^i(U,h^m(V^\bullet ))$ is finite dimensional for any $i \in \mathbb{Z}$. We now use the distinguished triangles

$$\begin{equation*} \vcenter{\img[][177pt][47pt][{$\xymatrix{ & h^m(V^\bullet)[-m] \ar[dl]_{+1} \\ \tau^{\leq m-1} V^\bullet\ar[rr] & & \tau^{\leq m} V^\bullet\ar[ul] }$}]{Images/imga23e1d9a7820eeaaa23b7d6d5b5a7338.svg}} \!\!\text{and}\!\! \vcenter{\img[][135pt][47pt][{$\xymatrix{ & \tau^{\geq m+1} V^\bullet\ar[dl]_{+1} \\ \tau^{\leq m} V^\bullet\ar[rr] & & V^\bullet\ar[ul] }$}]{Images/img556e51f150b514ee35c12d83d91d42fd.svg}} \end{equation*}$$

in $D(G)$ (cf. KS Proposition 13.1.15(i)). Since $\tau ^{\leq m-1} V^\bullet \simeq 0$ in $D(G)$ the left triangle implies that $H^i(U, \tau ^{\leq m} V^\bullet ) \cong H^{i-m} (U, h^m(V^\bullet ))$ is finite dimensional for any $i \in \mathbb{Z}$. Using this as an input for the long exact cohomology sequence associated with the right triangle we conclude that $H^i(U, \tau ^{\geq m+1} V^\bullet )$ is finite dimensional for any $i \in \mathbb{Z}$ as well. This proves the $n=0$ case of the following statement $P_n$:

$$\begin{equation*} \text{$h^{m+n}(V^{\bullet })$ is admissible and $\tau ^{\geq m+n+1}V^{\bullet }$ is globally admissible.} \end{equation*}$$

Proceeding inductively, to show $P_{n-1} \Rightarrow P_{n}$ for $n>0$ we may repeat our initial reasoning for the complex $\tau ^{\geq m+n} V^\bullet$. We obtain in particular that $h^j(V^\bullet )$ is admissible for any $j \in \mathbb{Z}$.

Finally part (ii) is a combination of the $D^{\pm }(G)$-versions. If the differential $V^n\rightarrow V^{n+1}$ vanishes one can decompose $V^{\bullet }$ as a sum of the two naive truncations $V^{\bullet }=\sigma ^{\leq n}V^{\bullet }\oplus \sigma ^{\geq n+1}V^{\bullet }$. If $V^{\bullet }$ is globally admissible so are the direct summands $\sigma ^{\leq n}V^{\bullet }$ and $\sigma ^{\geq n+1}V^{\bullet }$. Therefore they both lie in $D_{adm}(G)$ by part (i), which immediately implies $V^{\bullet }$ also lies in $D_{adm}(G)$ as claimed.

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Remark 4.10.

One can relax the condition in part (ii) of Proposition 4.9 slightly. If $V^{\bullet }$ is split somewhere, meaning at some $n$ there is a morphism $s: V^n \rightarrow V^{n-1}$ such that $dsd=d$, then the map $ds: V^n \rightarrow \ker (d)$ gives rise to a quasi-isomorphism $V^{\bullet }\rightarrow \tau ^{\leq n}V^{\bullet }\oplus \tau ^{\geq n+1}V^{\bullet }$. The direct sum is a complex with a vanishing differential at $n$. Applying (ii) shows that $V^{\bullet }$ lies in $D_{adm}(G)$ provided it is globally admissible. For the definition of a split complex we refer the reader to Wei, Df. 1.4.1.

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

Proposition 4.11.

For any $V^\bullet$ in $D(G)$ and any particular $i \in \mathbb{Z}$ we have

$$\begin{equation*} H^i(U,V^\bullet ) = 0 \Longrightarrow h^i(V^\bullet ) = 0. \end{equation*}$$

In particular, if $V^\bullet$ in $D(G)^a$ satisfies $H^i(U,V^\bullet ) = 0$ for all $i<\!\!<0$ (resp. $i>\!\!>0$) then $V^{\bullet }$ belongs to $D_{adm}^+(G)$ (resp. $D_{adm}^-(G)$).

Proof.

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

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We finish with a characterization of $D_{adm}^b(G)=D^b(G) \cap D_{adm}(G)$.

Corollary 4.12.

The subcategory $D_{adm}^b(G)$ consists of all complexes $V^\bullet$ in $D(G)$ whose total cohomology $H^*(U,V^\bullet )$ is finite dimensional.

Proof.

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

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Remark 4.13.

If $G$ is compact then the natural functor

$$\begin{equation*} D^+(\operatorname {Mod}_{adm}(G)) \xrightarrow {\simeq } D_{adm}^+(G) \coloneq D^+(G) \cap D_{adm}(G) \end{equation*}$$

is an equivalence. Similarly for $D_{adm}^b(G)$. This follows from Em2 Proposition 2.1.9, and Har, Proposition I.4.8 (which is also an easy consequence of KS, Theorem 13.2.8).