Cohomology of groups and splendid equivalences of derived categories

By ALEXANDER ZIMMERMANN

Faculté de Mathématiques; Université de Picardie; 33 rue St Leu; 80039 Amiens Cedex; France

                      e-mail: Alexander.Zimmermann@u-picardie.fr

To Professor Klaus W. Roggenkamp on the occasion of his sixtieth birthday

Abstract

In an earlier paper we studied the impact of equivalences between derived categories of group rings on their cohomology rings. Especially the group of auto-equivalences T rP ic(RG) of the derived category of a group ring RG as introduced by Raphaël Rouquier and the author defines an action on the cohomology ring of this group. We study this action with respect to the restriction map, transfer, conjugation and the local structure of the group G.

Introduction

Let R be a complete discrete valuation domain or a field. The cohomology ring of a finite group G with values in the trivial module R has a very rich structure. One of these structures is the structure of a Mackey functor (see e.g. [14, (53.3)] or [1]) with structure mappings being the transfer, the restriction and the conjugation. Let T rP ic(B_{0}(RG)) be the group of isomorphism classes of auto-equivalences of standard type of the derived category of the principal block of RG as defined and studied by Raphaël Rouquier and the author in [13]. Let HD_{R}(G) be the subgroup of those auto-equivalences which fix the trivial module up to isomorphism. In [16] an action of HD_{R}(G) on H^{∗}(G, R) is defined so that the cohomology ring is an R (HD_{R}(G))-module. We study in the present paper the connection between the functor ‘cohomology of a finite group G’ and, for fixed G, the action of HD_{R}(G).

A major problem is that derived equivalences are not well suited for restriction to subrings or quotient rings. But, if one restricts to splendid equivalences introduced by J. Rickard [9], then, at least for centralizers C_{G}(Q) of p-subgroups Q and in case R is a field, restriction maps may be defined using the Brauer construction [3].

In the present paper we define the respective structures and show compatibility of the Mackey functor structure with respect to local subgroups in a certain sense and under some additional hypothesis. We show that this action of the group of splendid auto-equivalences may be interpreted as action of the sheaf HSplen_{k}(−) of splendid auto-equivalences fixing the trivial module on the Mackey functor H^{∗}(−, 𝔽_{p}), “mod p group cohomology”.

It should be noted that it is in general not possible to modify an element in T rP ic(RG) in the natural way so that the modified element fixes the trivial module. An example is given in section 1.

The paper is organized as follows. In Section 1 the most basic definitions concerning derived equivalences are given. Section 2 defines the subgroup of splendid auto-equivalences and in Section 3 we prove the compatibility of the Mackey functor structure with the structure as modules for the group of splendid auto-equivalences of the derived category. Finally Section 4 states the main theorem. The notations and definitions used in the paper follow the conventions of [7].

1 Recall some facts

1.1 Equivalences and auto-equivalences of derived categories

In [13] Raphaël Rouquier and the author defined and studied the group of auto-equivalences of a derived category.

Recall that, following Bernhard Keller [6], a two-sided tilting complex X for two R-algebras Λ and Γ so that Λ is flat as an R-module is an element X in D^{b}(Λ ⊗_{R} Γ^{op}) so that X ⊗_{Γ}^{𝕃} − is an equivalence of categories.

We denote by [X] the isomorphism class of a complex X in the derived category.

In case one of Λ or Γ is a symmetric R-algebra, then by [15] the other is as well. So, if Λ is a classical symmetric order over a Dedekind domain, then Γ is a classical symmetric order as well. In this case the inverse to X ⊗_{Γ}^{𝕃} − is Hom_{R}(X, R) ⊗_{Γ}^{𝕃} −. If R is a field, analogous statements hold as was already observed by Rickard.

Definition 1.1 [13] Suppose Λ is an R-projective R-algebra. Then,

T rP ic_{R}(RG) := {isomorphism classes [X] | X ∈ D^{b}(Λ⊗_{R} Λ^{op}) is a two-sided tilting complex}

is a group under − ⊗_{Λ}^{𝕃} −.

Let G be a group and R a commutative ring. In [16] it is shown that the group

HD_{R}(G) := Stab_{T rP ic_{R}(RG)}(R) := {[X] ∈ T rP ic(RG) | X ⊗_{RG}^{𝕃} R ≃ R}

acts on the cohomology ring H^{∗}(G, R). This action is functorial in R.

1.2 Definition of splendid equivalences

We recall Rickard’s definition of a splendid equivalence [9]. Let p be a rational prime and let R be either a field of characteristic p or a complete discrete valuation domain of characteristic 0 with field of fractions of characteristic p. Let G and H be finite groups with a common Sylow p subgroup P. Let B_{0}(RG) be the principal block of the group ring RG and B_{0}(RH) likewise. A p-permutation module is a direct factor of a permutation module. Let ΔP be the diagonal embedding of P in G × H.

A bounded complex X of finitely generated B_{0}(RG) ⊗_{R} B_{0}(RH)^{op}-modules is said to be a splendid tilting complex if

• the complex Hom_{B_{0}(G)}(X, X) is isomorphic to B_{0}(H) in the category K^{b}(B_{0}(H)) and Hom_{B_{0}(H)}(X, X) is isomorphic to B_{0}(G) in the category K^{b}(B_{0}(G))

• each homogeneous component of X is projective as B_{0}(RG) as well as B_{0}(RH)-module

• all homogeneous components are relatively ΔP -projective p-permutation modules.

It is a consequence of Rickard’s theory of equivalences between derived categories [8] that

X ⊗_{B_{0}(RG)} − : D^{b}(B_{0}(RH)) ⟶ D^{b}(B_{0}(RH))

is an equivalence of triangulated categories, and hence X is a two-sided tilting complex.

1.3 The Brauer construction

Let R be a discrete valuation ring with residue field k := R/rad R of characteristic p. Let Q be a p-subgroup of G. Then, (see [9]) the map

−(Q) : kG − mod ⟶ k N_{G}(Q) − mod
M ⟶ M^{Q}/( ∑_{R<Q} T r_{R}^{Q}(M^{R}))

extends to a functor. For a permutation module M = kΩ one has M(Q) ≃ k(Ω^{Q}). Moreover

B_{0}(kG)(Q) ≃ B_{0}(kC_{G}(Q)).

We have the following

Lemma 1.2 (Rickard [9]) (−^{G})(Q) is isomorphic to (−(Δ(Q))^{C_{G}(Q)} as functors from the category of relatively Δ(Q)-projective p-permutation k(G × Q) modules to the category of kvector spaces.

2 Splendid auto-equivalences

Suppose that R is a complete discrete valuation ring of characteristic 0 and residue field of characteristic p or a field of characteristic p. Let B_{0}(RG) be the principal block of RG.

We denote by (X) the homotopy equivalence class of a complex X.

Definition 2.1

SplenP ic_{R}(G) := {(X) | X is splendid and [X] ∈ T rP ic_{R}(B_{0}(RG))}

Then, Rickard proves

Theorem 1 (Rickard [9]) For any (X) ∈ SplenP ic_{k}(B_{0}(kG)), applying the Brauer functor one has (X(ΔQ)) ∈ SplenP ic_{k}(B_{0}(C_{G}(Q))).

Lemma 2.2 SplenP ic_{R}(G) is a group which maps to T rP ic_{R}(B_{0}(RG)).

Proof. If X is splendid, then Hom_{R}(X, R) is splendid as well. In fact, Hom_{R}(X, R) is B_{0}(RG)-projective on either side as the homogeneous components of this complex are the R-duals of the homogeneous components of X. These are projective as the group ring is a symmetric algebra.

Moreover,

Hom_{B_{0}(RG)}(Hom_{R}(X, R), Hom_{R}(X, R)) ≃ Hom_{R}(Hom_{R}(X, R) ⊗_{RG} X, R)
≃ Hom_{R}(B_{0}(RG), R)
≃ B_{0}(RG)

is a chain of isomorphisms in the homotopy category of B_{0}(RG) bimodules. Similarly,

Hom_{B_{0}(RG)^{op}}(Hom_{R}(X, R), Hom_{R}(X, R)) ≃ Hom_{R}(X ⊗_{RG} Hom_{R}(X, R), R)
≃ Hom_{R}(B_{0}(RG), R)
≃ B_{0}(RG)

Finally, all homogeneous terms of Hom_{R}(X, R) are Δ(P)-projective permutation modules, as they are just the duals of the terms of X.

We shall have to prove that if X and Y are splendid, also Z := X ⊗_{B_{0}(RG)} Y is splendid. In fact, it is clear that Z is a complex of left and right projective modules since these are direct sums of tensor products of left and right projective modules. Moreover,

Hom_{R}(Z, R) ⊗_{B_{0}(RG)} Z ≃ Hom_{R}(Y, R) ⊗_{B_{0}(RG)} Hom_{R}(X, R) ⊗_{B_{0}(RG)} X ⊗_{B_{0}(RG)} Y
≃ Hom_{R}(Y, R) ⊗_{B_{0}(RG)} B_{0}(RG) ⊗_{B_{0}(RG)} Y
≃ B_{0}(RG)

and likewise Z ⊗_{B_{0}(RG)} Hom_{R}(Z, R) ≃ B_{0}(RG) in the homotopy category of B_{0}(RG)- bimodules.

Since the homogeneous terms of Z are direct sums of tensor products of Δ(P)-projective p-permutation modules, they are Δ(P)-projective p-permutation modules as well.

This concludes the proof.

Definition 2.3 We define

HSplen_{R}(G) := Stab_{SplenP ic_{R}(G)}(R) = {(X) ∈ SplenP ic_{R}(G) | X ⊗_{RG} R ≃ R}

The group HSplen_{k}(G) contains those outer automorphisms of G which is the identity on a fixed Sylow subgroup. Denote by Inn(RG) the group of inner automorphisms of RG, denote by Aut^{P} (G) the group of automorphisms of G whose restriction to P is the identity on P and denote by Inn(G) the group of inner automorphisms of G.

Lemma 2.4 Out_{R}^{P}(G) := Aut^{P} (G)/(Inn(RG) ∩ Aut^{P} (G)) ≤ HSplen_{R}(G).

Proof. The observation that an automorphism of G is trivial in the Picard group if and only if it is inner in RG rather than in G is a known fact. What we have to show is therefore the following: Let α be an automorphism of G fixing a Sylow p subgroup P and let [α] be its image in P ic_{R}(G). Then, [α] ∈ HSplen_{R}(G).

It is clear that _{α}RG_{1} is projective from either side.

Let P be a Sylow p-subgroup of G. Since α|_{P} = id_{P}, the bimodule _{α}RG_{1} is ΔP -projective. Hence, we proved the statement.

Remark 2.5 In general Inn(RG)∩Aut(G) ≠ Inn(G). See [11] for a finite group G where this strict inequality holds for any semi-local coefficient domain R. Martin Hertweck recently gave an example where one can remove the condition of R being semi-local [5]. For cohomology of finite groups one may always assume that R is local though.

2.1 The trivial module may not be preserved

Let A be an algebra over a field k.

Lemma 2.6 Let Y and Z be right bounded complexes of A-modules. Then,

Hom_{k}(Y, k) ⊗_{k} Z ≃ RHom_{k}(Y, Z)

as A ⊗_{k} A^{op}-modules.

Proof. Let M and N be A-modules. Then,

Hom_{k}(M, k) ⊗_{k} N ˜ ⟶ Hom_{k}(M, N)
ϕ ⊗_{k} n ↦ m ↦ ϕ(m)n

is an isomorphism of A ⊗_{k} A^{op}-modules.

Moreover, the degree (i, j) homogeneous component of the bicomplex Hom_{k}(Y, Z) is Hom_{k}(Y_{−i}, Z_{j}) while the degree i homogeneous component of Hom_{k}(Y, k) is Hom_{k}(Y_{−i}, k). Therefore, Hom_{k}(Y_{−i}, Z_{j}) has degree (i, j) in the bicomplex Hom_{k}(Y, k) ⊗_{k} Z. It is moreover clear that the bi-differentials of the bicomplexes Hom_{k}(Y, Z) and of Hom_{k}(Y, k) ⊗_{k} Z correspond under the above isomorphism at the level of modules. Therefore, using again the above isomorphisms on the level of modules, the total complexes of Hom_{k}(Y, Z) and of Hom_{k}(Y, k) ⊗_{k} Z are isomorphic. This completes the proof.

Example: Denoting by T (G) the épaisse sub-categorie of the stable module categorie which contains the trivial module, Carlson and Rouquier prove [4] that any functor F : T (G) ⟶ T (H) of triangulated categories has the property that the functor F̃ := (Hom_{k}(F (k), k) ⊗_{k} −) ∘ F sends the trivial module to the trivial module (see also Roggenkamp [10]).

We shall show with an example that the analogous statement is not true for splendid equivalences and by consequence for derived equivalences neither.

Let k = 𝔽_{3} be the field with three elements and let G = H = 𝔖_{3} be the symmetric group on three letters.

Then there are two indecomposable projective kG-modules P_{+} and P_{−} up to isomorphism and the complex X_{+} defined by

… ⟶ 0 ⟶ P_{+} ⊗_{k} Hom_{k}(P_{+}, k) ⟶ kG ⟶ 0 ⟶ …

with homology concentrated in degree 0 and 1 is a splendid tilting complex with isomorphism class in SplenP ic_{k}(G). If P_{+} denotes the projective cover of the trivial module, then

X_{+} ⊗_{kG} k ≃ (… ⟶ 0 ⟶ P_{+} ⊗_{k} k ⟶ k ⟶ 0 ⟶ …) ≃ Ω^{1}(k)[1]

where Ω is the syzygy operator. This module is two-dimensional and represents the non trivial element in Ext^{1}(k_{−}, k) where k_{−} is the sign representation.

Denoting the k-dual of an object Y by Y^{}ˇ , we shall prove

(X ⊗_{B} k)^{}ˇ ≃ k^{}ˇ ⊗_{B} X^{}ˇ .

Hence, by adjointness of tensor products and covariant homomorphism functor

Hom_{k}(X ⊗_{B} k, k) ≃ Hom_{B}(X, Hom_{k}(k, k))
≃ Hom_{B}(X, k^{}ˇ )

Since Hom_{B}(X, −) is an inverse to the functor − ⊗_{A} X, as well as − ⊗_{B} Hom_{k}(X, k), we get that

Hom_{B}(X, k^{}ˇ ) ≃ k^{}ˇ ⊗_{B} Hom_{k}(X, k) .

Hence,

(X ⊗_{kG} k)^{}ˇ ⊗_{k} (X ⊗_{kG} k) ≃ End_{k}(X ⊗_{kG} k)
≃ End_{k}(Ω^{1}(k))

and the latter is a four-dimensional, non trivial module.

3 Group cohomology as module and as Mackey functor

As is well known, H^{∗}(G, k) together with restriction, transfer and conjugation is a Mackey functor. We shall discuss the compatibility of this structure with respect to the action of splendid auto-equivalences of the derived category.

For the convenience of the reader we shall first recall the definition of a Mackey functor and define what we understand by an action of a group sheaf on a Mackey functor.

3.1 Action on Mackey functors

Let G be a group and let R be a commutative ring. We recall the definition of a Mackey functor (see e.g. [1, 14]). A Mackey functor is a pair of functors (M^{∗}, M_{∗}), the first contravariant and the second covariant, from the category of G-sets to the category of R-modules so that M_{∗} and M^{∗} are identical on objects subject to the following conditions.

(M1) Let X and Y be G-sets and i_{X} resp. i_{Y} be the injections of X or Y in X ∐ Y. Then they are supposed to satisfy M^{∗}(i_{X}) ⊕ M^{∗}(i_{Y}) and M_{∗}(i_{X}) ⊕ M_{∗}(i_{Y}) are mutually inverse isomorphisms between M_{∗}(X ∐ Y) and M_{∗}(X) ∐ M_{∗}(Y).

(M2) Moreover, they satisfy the following condition on pullbacks: If

T ⟶^{γ} Y
δ ↓  ↓ α
Z ⟶^{β} X

is a pullback of G-sets, then M^{∗}(β)M_{∗}(α) = M_{∗}(δ)M^{∗}(γ).

A morphism of Mackey functors θ : (M^{∗}, M_{∗}) ⟶ (N^{∗}, N_{∗}) is a natural transformation θ : M_{∗} ⟶ N_{∗} and θ : M^{∗} ⟶ N^{∗} (the same for M^{∗} and for M_{∗}).

A group sheaf on a partially ordered set ℭ of subgroups of G is a contravariant functor G : ℭ ⟶ 𝔊roup where 𝔊roup is the category of groups with group homomorphisms.

Definition 3.1 An operation of a group sheaf on a Mackey functor M is an operation of G(S) on M(S) for any S ∈ ℭ by morphisms of Mackey functors.

3.2 Elementary properties

Lemma 3.2 We have a homomorphism of groups

−(ΔQ) : SplenP ic_{k}(G) ⟶ SplenP ic_{k}(C_{G}(Q))
(X) ⟶ (X(ΔQ))

Proof. Let (X), (Y) ∈ HSplen_{k}(G). By Theorem 1 one has (X(ΔQ)) ∈ SplenP ic_{k}(B_{0}(C_{G}(Q))). To simplify notation denote for a moment H := C_{G}(Q).

One has the following isomorphism of functors.

Hom_{B_{0}(kG)}(X, −) ≃ X^{−1} ⊗_{B_{0}(kG)} −

Moreover, using Lemma 1.2,

X^{−1} ⊗_{B_{0}(kG)} − ≃ (Hom_{B_{0}(kG)}(X, −))(ΔQ)
≃ Hom_{B_{0}(kH)}(X(ΔQ), −(ΔQ))

which may be applied to kG since kG is ΔQ-projective. Then, since group rings are symmetric and therefore the k-dual is isomorphic to the B_{0}(kH)-dual on the principal block,

X^{−1}(ΔQ) ≃ X(ΔQ)^{−1} .

Moreover,

(Hom_{B_{0}(kG)}(X, Y))(ΔQ) ≃ Hom_{B_{0}(kH)}(X(ΔQ), Y (ΔQ))

as kH ⊗_{k} kH^{op}-bimodules again by Lemma 1.2. But this may be interpreted as

((X^{−1} ⊗_{B_{0}(kG)} Y) ⊗_{B_{0}(kG)} kG)(ΔQ) ≃ ((X^{−1}(ΔQ) ⊗_{B_{0}(kH)} Y (ΔQ)) ⊗_{B_{0}(kH)} kH)
≃ ((X(ΔQ)^{−1} ⊗_{B_{0}(kH)} Y (ΔQ)) ⊗_{B_{0}(kH)} kH)

and therefore

(X^{−1} ⊗_{B_{0}(kG)} Y)(ΔQ) ≃ (X(ΔQ)^{−1} ⊗_{B_{0}(kH)} Y (ΔQ))

This concludes the proof.

Lemma 3.3 If X is a splendid tilting complex, then

X(ΔQ) ⊗_{kC(Q)} k ≃ X^{G}(Q)

Proof. Let X ∈ SplenP ic_{k}(G). An inverse to the auto-equivalence X ⊗_{kG}− of D^{b}(B_{0}(kG)) is Hom_{k}(X, k) ⊗_{kG} − which is isomorphic to Hom_{kG}(X, −). It is clear that if X is splendid, also Y := Hom_{k}(X, k) is splendid. So, Lemma 1.2 implies that

(Y (ΔQ))^{C_{G}(Q)} ≃ Y^{G}(Q)

But, Raphaël Rouquier proves in [12, Lemma 2.2] that for p-permutation modules taking k-duals commutes up to functorial isomorphism with the Brauer functor. Hence,

Hom_{kC_{G}(Q)}(X(ΔQ), k) ≃ (Hom_{kG}(X, k))(Q)

Since Y (ΔQ)⊗_{kC_{G}(Q)} − is an inverse to Hom_{kC_{G}(Q)}(Y (ΔQ), −) which in turn is an inverse to Hom_{kC_{G}(Q)}(X(ΔQ), −), we get

Y^{G}(Q) ≃ Y (ΔQ) ⊗_{kC_{G}(Q)} k

Replacing X by Y taking double duals, the lemma is proved.

Remark 3.4 Let α ∈ Aut^{P} (G). Then,

( _{α}kG_{1})(ΔQ) ⊗_{C_{G}(Q)} k ≃ ( _{α}kG_{1})^{G}(Q) = k(Q) = k

3.3 The transfer with respect to local subgroups

We will prove in this section that the transfer in group cohomology is compatible with the action of HSplen_{k}(G). For a subgroup H of a group G denote by tr_{H}^{G} the transfer map.

Proposition 3.5 Let G be a finite group and let Q be a p-subgroup. Let k be a field of characteristic p. For any X with [X] ∈ HSplen_{k}(G) we denote by F_{X} the action of [X] on H^{∗}(G, k). Suppose that X(ΔQ) ∈ HSplen_{k}(C_{G}(Q)). Then the diagram

H^{∗}(G, k)  ⟶^{F_{X}}  H^{∗}(G, k)
tr_{C_{G}(Q)}^{G} ↑  ↑ tr_{C_{G}(Q)}^{G}
H^{∗}(C_{G}(Q), k) ⟶^{F_{X(ΔQ)}} H^{∗}(C_{G}(Q), k)

is commutative.

Proof. The transfer map tr_{C_{G}(Q)}^{G} can be defined as composite

Hom_{kC_{G}(Q)}(k, k[n])≃Hom_{kG}(k, kG⊗_{kC_{G}(Q)}k[n]) → Hom_{kG}(k, kG⊗_{kG}k[n]) ≃ Hom_{kG}(k, k[n])

and so we have to show that two diagrams are commutative. The first one is

Hom_{kG}(k, kG ⊗_{kC} k[n]) ⟶ Hom_{kG}(k, kG ⊗_{kG} k[n])
F_{X} ↓  ↓ F_{X}
Hom_{kG}(X^{G}, X ⊗_{kC} k[n]) ⟶ Hom_{kG}(X^{G}, X ⊗_{kG} k[n])

where the commutativity is clear. The second diagram is

Hom_{kC_{G}(Q)}(k, k[n])  ˜ ⟶  Hom_{kG}(k, kG ⊗_{kC} k[n])
F_{X(ΔQ)} ↓  ↓ F_{X}
Hom_{kC_{G}(Q)}(X^{G}(Q), X^{G}(Q)[n]) ⟶^{α} Hom_{kG}(X^{G}, X ⊗_{kC_{G}(Q)} k[n])

where commutativity is less clear. But, all mappings beside α are defined as isomorphisms, so there is only one α possible which makes the diagram commutative. We have to show that this α is the mapping coming from the natural isomorphism from Frobenius reciprocity.

Since X^{G} ≃ k in the derived category of G-modules, the complex X^{G} is exact in any degree besides in degree 0, where its homology is the trivial module. Hence, the restriction of X^{G} to C_{G}(Q) is exact as well, besides in degree 0, where its homology is the trivial module k. Since X^{G}(Q) ≃ k in the derived category of C_{G}(Q)-modules, the complex X^{G}(Q) is exact in any degree besides in degree 0, where its homology is the trivial module. So, in the derived category of complexes of C_{G}(Q)-modules, we get the following isomorphism:

res_{C_{G}(Q)}^{G}(X^{G}) ≃^{φ} X^{G}(Q) ≃ k

Now, α decomposes as follows:

Hom_{kC_{G}(Q)}(k, k[n]) ⟶^{F_{X(ΔQ)}} Hom_{kC_{G}(Q)}(X^{G}(Q), X^{G}(Q)[n])
⟵^{φ^{∗}(φ_{∗})^{−1}} Hom_{kC_{G}(Q)}(X^{G}, X^{G}[n])
≃  Hom_{kG}(X^{G}, X^{G}[n] ↑_{C_{G}(Q)}^{G})  by Frobenius reciprocity

Now, we see that

X ⊗_{kG} Hom_{kC_{G}(Q)}(kG, k) ≃ Hom_{kC_{G}(Q)}(kG, X ⊗_{kG} k)
x ⊗ ϕ ↦ (m ↦ x ⊗ ϕ(m))

are naturally isomorphic. Using again Lemma 3.3 we get that F_{X}^{−1}∘α∘F_{X(ΔQ)} equals Frobenius reciprocity Hom_{kC_{G}(Q)}(k, k[n]) ≃ Hom_{kG}(k, k ↑_{C_{G}(Q)}^{G}).

This proves Proposition 3.5.

3.4 The restriction with respect to local subgroups

We shall prove sensibly the same thing as for transfer now for restriction.

Proposition 3.6 Let G be a finite group and let Q be a p-subgroup. Let k be a field of characteristic p. For any X with [X] ∈ HSplen_{k}(G) we denote by F_{X} the action of [X] on H^{∗}(G, k). Suppose that X(ΔQ) ∈ HSplen_{k}(C_{G}(Q)). Then the diagram

H^{∗}(G, k)  ⟶^{F_{X}}  H^{∗}(G, k)
res_{C_{G}(Q)}^{G} ↓  ↓ res_{C_{G}(Q)}^{G}
H^{∗}(C_{G}(Q), k) ⟶^{F_{X(ΔQ)}}  H^{∗}(C_{G}(Q), k)

is commutative.

Proof. Again using Lemma 3.3 and the hypothesis, res_{C_{G}(Q)}^{G}(X^{G}) ≃ k ≃ X^{G}(Q) in the derived category of bounded complexes of res_{C_{G}(Q)}^{G}-modules. Let φ : X^{G} ≃ X^{G}(Q) be this isomorphism. Let β : X^{G} ≃ k and α : X^{G}(Q) ⟶ k so that φ = α ∘ β^{−1}. Then F_{X} acts on ξ ∈ H^{∗}(G, k) as (β^{∗})^{−1}β_{∗}(X ⊗ χ) and F_{X(ΔQ)} acts on η ∈ H^{∗}(C_{G}(Q), k) as (α^{∗})^{−1}α_{∗}(X(ΔQ) ⊗ η). Therefore, the following diagram is commutative:

Hom_{D^{b}(kG)}(k, k[n]) ⟶  Hom_{D^{b}(kG)}(X^{G}, X^{G}[n])
res_{C_{G}(Q)}^{G} ↓  ↓ res_{C_{G}(Q)}^{G}
Hom_{D^{b}(kC_{G}(Q))}(k, k[n])  Hom_{D^{b}(kC_{G}(Q))}(res_{C_{G}(Q)}^{G}X^{G}, res_{C_{G}(Q)}^{G}X^{G}[n])
∥  ↓ (φ^{∗})(φ_{∗})^{−1}
Hom_{D^{b}(kC_{G}(Q))}(k, k[n]) ⟶  Hom_{D^{b}(kG)}(X^{G}(Q), X^{G}(Q)[n])

Now, since φ = α ∘ β^{−1},

Hom_{D^{b}(kG)}(X^{G}, X^{G}[n])  ⟶^{(β^{∗})^{−1}β_{∗}} Hom_{D^{b}(kG)}(k, k[n])
↓ res_{C_{G}(Q)}^{G}  ↓ res_{C_{G}(Q)}^{G}
Hom_{D^{b}(kC_{G}(Q))}(X^{G}, X^{G}[n]) ⟶^{(β^{∗})^{−1}β_{∗}} Hom_{D^{b}(kC_{G}(Q))}(k, k[n])
φ^{∗}(φ_{∗})^{−1} ↓  ∥
Hom_{D^{b}(kG)}(X^{G}(Q), X^{G}(Q)[n]) ⟶^{(α^{∗})^{−1}α_{∗}} Hom_{D^{b}(kC_{G}(Q))}(k, k[n])

is commutative. Combining these two commutative diagrams we get a commutative diagram

Hom_{D^{b}(kG)}(k, k[n])  ⟶^{F_{X}}  Hom_{D^{b}(kG)}(k, k[n])
↓ res_{C_{G}(Q)}^{G}  ↓ res_{C_{G}(Q)}^{G}
Hom_{D^{b}(kC_{G}(Q))}(k, k[n]) ⟶^{F_{X(ΔQ)}} Hom_{D^{b}(kC_{G}(Q))}(k, k[n])

This concludes the proof.

3.5 An example and an application

Remark 3.7 Proposition 3.5 implies that for R = 𝔽_{p} and G = 𝔖_{p} the symmetric group on p letters, any element in HSplen_{R}(G) acts trivially on H^{∗}(G, R). This fact was proven in [16] by explicit calculation.

In fact,

tr_{C_{p}}^{𝔖_{p}} ∘ res_{C_{p}}^{𝔖_{p}} : H^{∗}(𝔖_{p}, 𝔽_{p}) ⟶ H^{∗}(𝔖_{p}, 𝔽_{p})

is just multiplication by [𝔖_{p} : C_{p}] = (p − 1)! and is hence invertible since we are working in characteristic p. So, res_{C_{p}}^{𝔖_{p}} is linear with respect to the action of splendid auto-equivalences

[X] ∈ HSplen_{k}(𝔖_{p}) with (X(ΔC_{p})) ∈ HSplen_{k}(C_{p}) if and only if tr_{C_{p}}^{𝔖_{p}} is.

Denoting C_{p} :=< (1, 2, 3, …, p) > a p-Sylow subgroup of 𝔖_{p}, it is well known that

res_{C_{p}}^{G} : H^{∗}(𝔖_{p}, R) ⟶ H^{∗}(C_{p}, R)

is injective since the p-Sylow subgroup of 𝔖_{p} is abelian. Hence, the fact that the restriction to the centralizer of C_{p} is compatible with the action of HSplen_{k}(𝔖_{p}) proves that the action of HSplen_{R}(G) on H^{∗}(𝔖_{p}, R) is trivial if the action of HSplen_{R}(C_{p}) on H^{∗}(C_{p}, R) is trivial and one may prove that the trivial module is fixed also when one takes the Brauer functor of the auto-equivalence in question. This can be checked individually for the auto-equivalences considered in [16]. Moreover,

HSplen_{R}(C_{p}) ⊆ {ϕ ∈ Aut_{R}(RC_{p})| ^{ϕ}R ≃ R}

as C_{p} =< c > is a p-group. Any such automorphism α_{x} is fixed by α_{x}(c) = 1 + (c − 1)x for an x ∈ F C_{p}. Since H^{∗}(C_{p}, R) is generated by an element ζ in degree 2, and an element χ in degree 1 we have to evaluate the action in degree 1 and 2 only. Ones sees readily that α_{x}(ζ) = x · ζ and α_{x}(χ) = x · χ. Now, the image of res_{C_{p}}^{𝔖_{p}} in H^{∗}(C_{p}, R) is the fix point ring H^{∗}(C_{p}, R)^{C_{p−1}} under the normalizer action. This action is multiplication by a primitive p − 1-th root of unity in R/pR and H^{∗}(C_{p}, R)^{C_{p−1}} is generated in degree 2(p − 1) and 2p − 3. The operation of α_{x} is by multiplication by x^{p−1}, but the unit group of R is cyclic of order p − 1.

One would like to get the same result for R being the p-adic integers. But, using the Brauer construction one is forced to work over a field as ring of coefficients (but see [17], where the lifting question to a complete discrete valuation domain is studied and where parts of the present results are announced). Moreover, what we proved is valid only for splendid auto-equivalences. We plan in the near future to give a handy criterion for deciding whether an equivalence is splendid.

3.6 Conjugation

Proposition 3.8 Let x ∈ G and Q be a p-subgroup of G. Then, denoting by c_{x} the conjugation morphism, for any X with (X) ∈ HSplen_{R}(G) the following diagram is commutative:

H^{∗}(C(Q), R)  ⟶^{c_{x}}  H^{∗}(C(Q)^{x}, R)
X(ΔQ) ⊗_{RG} − ↓  ↓ X(ΔQ^{x}) ⊗_{RG} −
H^{∗}(C(Q), R)  ⟶^{c_{x}}  H^{∗}(C(Q)^{x}, R)

Proof. Lemma 3.3 implies that we have to prove that the diagram

Hom_{RC_{G}(Q)}(R, R[n])  ⟶^{c_{x}}  Hom_{RC_{G}(Q)^{x}}(xR, xR[n])
X(ΔQ) ⊗_{RG} − ↓  ↓ X(ΔQ^{x}) ⊗_{RG} −
Hom_{RC_{G}(Q)}(X^{G}(Q), X^{G}(Q)[n]) ⟶^{c_{x}} Hom_{RC_{G}(Q)}(X^{G}(Q^{x}), X^{G}(Q^{x})[n])

is commutative. But this is clear.

4 Group sheaf action on the Mackey functor H^{∗}(−, k)

We are ready to state and to prove our main theorem.

Denote by 𝔊roup the category of groups with morphisms being group homomorphisms and denote by k − 𝔄lg the category of k-algebras with morphisms being algebra homomorphisms. Let k be a field of characteristic p and let G be a finite group. Denote for any p-subgroup Q of G by HSplen_{k}(C_{G}(Q)) the group of splendid auto-equivalences of the bounded derived category D^{b}(kC_{G}(Q)) which fix the trivial module.

Fix a set 𝔓_{p,G} of p-subgroups of G which may contain all p-subgroups of G or not. Suppose that for any Q_{1}, Q_{2} ∈ 𝔓_{p,G} we have Q_{1} < Q_{2} ⇒ Q_{2} ≤ C_{G}(Q_{1}). This is the case for example if all elements of 𝔓_{p,G} are abelian. In fact this is also necessary for all but the maximal elements of 𝔓_{p,G}. This gives a category whose objects is 𝔓_{p,G} and whose morphism sets are inclusion (that is Hom_{𝔓_{p,G}}(Q_{1}, Q_{2}) contains one element, the inclusion, if Q_{1} ≤ Q_{2} and Hom_{𝔓_{p,G}}(Q_{1}, Q_{2}) = ∅ if Q_{1} ⊈ Q_{2}). Define

ℭ_{p,G} := {C_{G}(Q)| Q ∈ 𝔓_{p,G}} ∪ {{1}}

with morphisms being again induced by inclusion. Taking centralizers C_{G}(−) : 𝔓_{p,G} ⟶ ℭ_{p,G} is a contravariant functor: Q_{1} ⊆ Q_{2} ⇒ C_{G}(Q_{1}) ⊇ C_{G}(Q_{2}). Define a contravariant functor by

ℭ_{p,G} ⟶ 𝔊roup
C_{G}(Q) ↦ HSplen_{k}(C_{G}(Q))

and which maps an inclusion C_{G}(Q_{1}) < C_{G}(Q_{2}) to the Brauer construction:

HSplen_{k}(C_{G}(Q_{2}))) ∋ (X) ↦ (X(ΔQ_{1})) ∈ HSplen_{k}(C_{G}(Q_{1}))) .

Then, the functor ℭ_{p,G} ⟶ 𝔊roup which is HSplen_{k}(−) on objects and HSplen_{k}(C_{G}(Q_{1}) ≤ C_{G}(Q_{2}))(X) := (X(ΔQ_{1})) on morphisms is a sheaf of groups.

Furthermore,

H^{∗}(−, k) : ℭ_{p,G} ⟶ k − 𝔄lg

is a contravariant functor of k-algebras, a ringed space. Moreover, (H^{∗}(−, k), res, T r, c) is a Mackey functor on the set of all subgroups of G. The Mackey functor structure may be restricted to ℭ_{p,G} if this set is stable under intersection. It is clear that ℭ_{p,G} is stable under conjugation. So, the condition to be stable under intersection implies that the Mackey formula makes sense for functors on ℭ_{p,G}.

Define the sub-functor HSplen_{k,𝔓_{p,G}}(−) of HSplen_{k}(−) by

HSplen_{k,𝔓_{p,G}}(C_{G}(Q))

:= {(X) ∈ HSplen_{k}(C_{G}(Q))| (X(ΔQ̂)) ∈ HSplen_{k}(C_{G}(Q̂)) for all Q̂ ∈ 𝔓_{p,G} and Q̂ ≥ Q}

Remark 4.1 Let P be a p-Sylow subgroup of G. By Remark 3.4 one has

{[ϕ] ∈ Out_{R}(G)| ϕ|_{P} = id_{P} } ≤ HSplen_{k,𝔓_{p,G}}(C_{G}(Q)) for all Q ∈ 𝔓_{p,G}.

We are ready to state our main results.

Theorem 2 Let 𝔓_{p,G} be a partially ordered set of p-subgroups of the finite group G. Suppose that if Q_{1} < Q_{2} and Q_{1}, Q_{2} ∈ 𝔓_{p,G}, then Q_{2} centralizes Q_{1}. Let Res: ℭ_{p,G} ⟶ k−Mod and Trans: ℭ_{p,G} ⟶ (k−Mod)^{op} be two functors which are identical on objects: H^{∗}(−, k) : ℭ_{p,G} ⟶ k − 𝔄lg and which are on morphisms

Res(C_{G}(Q_{1}) ≤ C_{G}(Q_{2})) := res_{C_{G}(Q_{1})}^{C_{G}(Q_{2})}

while

Trans(C_{G}(Q_{1}) ≤ C_{G}(Q_{2})) := tr_{C_{G}(Q_{1})}^{C_{G}(Q_{2})}

Then,

HSplen_{k,𝔓_{p,G}}(−) : ℭ_{p,G} ⟶ 𝔊roup

acts by natural transformations on Res and on Trans. That means that the functor giving the group operation structure on each subgroup

HSplen_{k,𝔓_{p,G}}(−) × H^{∗}(−, k) ⟶ H^{∗}(−, k)

is functorial in the sense that for any Q_{1} < Q_{2} one has

HSplen_{k,𝔓_{p,G}}(C_{G}(Q_{1}))) × H^{∗}(C_{G}(Q_{1})), k) ⟶ H^{∗}(C_{G}(Q_{1})), k)
Br_{C_{G}(Q_{2})}^{C_{G}(Q_{1})} ↓ res_{C_{G}(Q_{2})}^{C_{G}(Q_{1})} ↓ res_{C_{G}(Q_{2})}^{C_{G}(Q_{1})} ↓
HSplen_{k,𝔓_{p,G}}(C_{G}(Q_{2}))) × H^{∗}(C_{G}(Q_{2})), k) ⟶ H^{∗}(C_{G}(Q_{2})), k)

is commutative and similarly for Trans.

Proof. This follows from Proposition 3.5 and Proposition 3.6.

Theorem 3 Suppose ℭ_{p,G} is stable under intersection. Then, the constant sheaf

HSplen_{k,𝔓_{p,G}}(G) : ℭ_{p,G} ⟶ 𝔊roup

acts by morphisms of Mackey functors on

H^{∗}(−, k) : ℭ_{p,G} ⟶ k − 𝔄lg

Proof. This is exactly the statement of Proposition 3.5, Proposition 3.6 and Proposition 3.8.

Remark 4.2 1. It is interesting to note that Serge Bouc used in [2] functors M which satisfy only two third of the Mackey functor axioms to construct resolutions of Mackey functors. More precisely, he used functors F which satisfy all the Mackey functor axioms which may be formulated by use of conjugation and restriction only. Dually one could use transfer instead of restriction. In our situation in Theorem 2 this is exactly the type of functors we consider.

2. It would be useful to know when

(X) ∈ HSplen_{k}(G) implies (X(ΔQ)) ∈ HSplen_{k}(C_{G}(Q))].

This would imply considerable simplifications in the formulation of Theorem 2 and Theorem 3.

3. In case ℭ_{p,G} has only one non-trivial element, Theorem 2 is exactly Proposition 3.6 and Proposition 3.5.

Acknowledgement: I wish to thank Serge Bouc for explaining to me the subtleties of Mackey functors and for communicating to me a gap in an earlier version of the paper.

References

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[2] Serge Bouc, Résolutions de foncteurs de Mackey, Proceedings of Symposia in Pure Mathematics 63 (1998) 31-83.

[3] Michel Broué, On Scott modules and p-permutation modules: An approach through the Brauer morphism, Proc. Amer. Math. Soc. 93 (1985) 401-408.

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[8] Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37-48.

[9] Jeremy Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. 72 (1996), 331-358.

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[12] Raphaël Rouquier, Gluing p-permutation modules, preprint (1998).

http://www.math.jussieu.fr/grpsfinis/˜ rouquier

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[16] Alexander Zimmermann, Auto-equivalences of derived categories acting on group cohomology, preprint (1999).

http://delambre.mathinfo.u-picardie.fr/alex/cohobr2.dvi

[17] Alexander Zimmermann, Cohomology of groups, abelian Sylow subgroups and splendid equivalences, Algebra Montpellier Announcements (electronic) 01-2000, 5 pages.