Noname manuscript No. (will be inserted by the editor)

Restriction between Cohomology Algebras of Blocks of Finite Groups

Constantin Cosmin Todea

Abstract Let k be an algebraically closed field of characteristic p and G be a finite group. Let N be a normal subgroup of G and c be a G-stable block of kN. We shall discuss the cohomology algebra of the block c, defined by M. Linckelmann and, in this case a generalized block cohomology which can be defined using some generalized Brauer pairs, denoted (c,G)-Brauer pairs, which were are introduced by R. Kessar and R. Stancu. We also analyze the restriction map between these two cohomology algebras associated to the block c through transfer maps between the Hochschild cohomology algebras of kGc and of the block c.

1 Introduction

Let G be a finite group, N be a normal subgroup, k be an algebraically closed field of prime characteristic p and let c be a G-stable block of kN. We denote by A_{1} = kN as G-algebra and consider kN as interior N-algebra. Thus N_{{c}} and G_{{c}} are both pointed groups on A_{1}.

Using the approach from [4], in [8, Section 3] R. Kessar and R. Stancu gave the definition of the generalized Brauer pair and the generalized Brauer category. In the next lines we restate some needed results and give some properties which already has been pointed out in a more general case of twisted group algebras by A. Marcus in [16, Section 3]. We also include the approach of pointed groups, which is not used in [8].

Since A_{1} is a p-permutation G-algebra the results of [4, Section 3] apply to our situation. For any Q, a p-subgroup of G the canonical projection from kN to kC_{N}(Q) induces an algebra homomorphism from (kN)^{Q} onto kC_{N}(Q), the Brauer homomorphism denoted by Br_{Q}^{N}. Explicitly Br_{Q}^{N}(x) = x if x ∈ C_{N}(Q) and Br_{Q}^{N}(x) = 0 if x ∉ C_{N}(Q). A (c,G)-Brauer pair is a pair (Q,e_{Q}) where Q is a p-subgroup of G such that Br_{Q}^{N}(c) ≠ 0 and e_{Q} is a block of kC_{N}(Q) such that Br_{Q}^{N}(c)e_{Q} ≠ 0. When G = N a (c,G)-Brauer pair is also known as a c-Brauer pair.

If (R,e_{R}) and (Q,e_{Q}) are two (c,G)-Brauer pairs, we say that (Q,e_{Q}) is contained in (R,e_{R}) and we write (Q,e_{Q}) ≤ (R,e_{R}), if Q ≤ R and for any primitive idempotent i ∈ (kN)^{R} such that Br_{R}^{N}(i)e_{R} ≠ 0 we have that Br_{Q}^{N}(i)e_{Q} ≠ 0. This is an order relation on the set of (c,G)-Brauer pairs, compatible with the conjugation of G. If (R,e_{R}) is a given (c,G)-Brauer pair, for any Q ≤ R there is a unique (c,G)-Brauer pair such that (Q,e_{Q}) ≤ (R,e_{R})(see [4, Theorem 1.14]).

By [4, Theorem 1.14] we know that G acts transitively on maximal (c,G)-Brauer pairs, equivalently all maximal (c,G)-Brauer pairs are G-conjugate. If (P,e_{P}) is a maximal (c,G)-Brauer pair then P is called a (c,G)-defect group. If N = G then P is a defect group of c in the usual sense.

C.C. Todea Department of Mathematics, Technical University, Cluj Napoca, Romania, Str. G.Baritiu, no.25, 400027 E-mail: Constantin.Todea@math.utcluj.ro

We remind that P_{γ} is a defect pointed group of G_{{c}} if P_{γ} is a maximal local pointed group on A_{1} included in G_{{c}}. This is equivalent to P being a maximal p-subgroup of G such that Br_{P}^{N}(c) ≠ 0. We apply [9, Proposition 5.3] in our particular case with N_{{c}} ≤ G_{{c}}, where c is a G-stable block of kN and obtain the following remark. Notice that N_{{c}} is also a pointed group on kN as N-algebra.

Remark 1.1 P_{γ} is a defect pointed group of G_{\{c\}} on A_{1} if and only if \overline{P}= PN/N is a Sylow p-subgroup of \overline{G}= G/N and there is Q_{δ} a defect pointed group of N_{\{c\}} on kN as N-algebra such that Q_{δ} ≤ P_{γ}. In this case Q = P ∩ N.

By Remark 1.1 and Proposition 2.3, which we shall prove in Section 2, in the following sections we will work under the situation:

Situation (∗). Let G be a finite group, N be a normal subgroup of G and let c be a G-stable block of kN. Let P_{γ} a defect pointed group of G_{{c}} on A_{1} and Q_{δ} = (P ∩ N)_{δ} ≤ P_{γ} the corresponding defect pointed group of N_{{c}}. Then there is a unique, maximal (c,G)-Brauer pair (P,e_{P}) such that Br_{P}^{N}(i)e_{P} ≠ 0, for any i ∈ γ, and a unique, maximal c-Brauer pair (Q,e_{Q}) such that Br_{Q}^{N}(j)e_{Q} ≠ 0, for any j ∈ δ. Moreover we have that (Q,e_{Q}) ≤ (P,e_{P}).

Let A and B be symmetric R-algebras over a commutative ring R. Let X be a bounded complex of finite generated A − B-bimodules projective as left A-modules and right B-modules. In [11] M. Linckelmann defined the transfer map associated to X denoted t_{X} : HH^{∗}(B) −→ HH^{∗}(A), between the Hochschild cohomology algebras. The element π_{X} = t_{X}^{0}(1_{B}) is called the relatively X-projective element of Z(A). If π_{X} is invertible the normalized transfer map associated with X is the following:

T_{X}: HH^{∗}(B) −→ HH^{∗}(A), T_{X}(τ) = π_{X}^{−1} t_{X}(τ).

He also introduced the notion of X-stable elements. We denote by HH_{X}^{∗}(A) the graded subalgebra of HH^{∗}(A) consisting of all X-stable elements. From [11] we know that if π_{X} is invertible the normalized transfer T_{X} induces a graded surjective of k-algebras homomorphism

R_{X}: HH_{X^{∗}}^{∗}(B) −→ HH_{X}^{∗}(A).

In Section 3 under the situation (∗) similar to block cohomology from [11, Definition 5.1] we will give the definition of the generalized block cohomology H^{∗}(G,N,c,P_{γ}) and we shall prove that there is a generalized restriction

res_{N,c}^{G,N,c}: H^{∗}(G,N,c,P_{γ}) −→ H^{∗}(N,c,Q_{δ}).

From [11] we know that the cohomology algebra H^{∗}(N,c,Q_{δ}) is embedded into the jkN-stable subalgebra of HH^{∗}(kQ) by the diagonal approximation δ_{Q} : H^{∗}(Q,k) −→ HH^{∗}(kQ), where j ∈ δ is a source idempotent. The composite

\xymatrix{H^{∗}(N,c,Q_{δ}) \ar[r]^{δ_{Q}} & HH_{jkN}^{∗}(kQ) \ar[r]^{R_{kNj}} & HH_{kNj}^{∗}(kNc)},

of δ_{Q} and the normalized transfer map R_{kNj} is an injective graded k-algebra homomorphism. M. Linckelmann introduced in [12] the notion of varieties for bounded complexes of finite generated modules associated to block cohomology, which is a theoretic analogue of the cohomology variety defined by Carlson [5]. In our case we consider the variety of finite generated kNc-modules as the maximal ideal spectrum of the cohomology algebra H^{∗}(N,c,Q_{δ}).

There is a similar theorem to [11, Theorem 5.6] which we shall prove in Section 4.

Theorem 1.2 Let N be a normal subgroup of a finite group G, c be a block of kN which is G-stable, P_{γ} be a defect pointed group of G_{{c}} and i ∈ γ. Let (P,e_{P}) be a maximal (c,G)-Brauer pair and consider kGi and ikG as kGc − kP respectively kP − kGc-bimodule.

i) We have that π_{kGi}= Tr_{P}^{G}(i) ∈ Z(kGc)^{×} and π_{ikG}= \frac{dim_{k}(ikG)}{|G|}1_{kP}∈ k^{∗}1_{kP}. ii) If [ζ] ∈ H^{∗}(G,N,c,P_{γ}) then δ_{P}([ζ]) is ikG-stable in HH^{∗}(kP). iii) The morphism T_{kGi}∘ δ_{P} induces an injective graded k-algebra homomorphism

\xymatrix{H^{∗}(G,N,c,P_{γ}) \ar[r]^{δ_{P}} & HH_{ikG}^{∗}(kP) \ar[r]^{R_{kGi}} & HH_{kGi}^{∗}(kGc)}

Under the situation (∗) since Q_{δ} ≤ P_{γ} we choose i ∈ γ and j ∈ δ such that j = ij = j i. Let X = kGc as kNc − kGc-bimodule with X^{∗} = kGc as kGc − kNc-bimodule. We consider ikGj as kP − kQ-bimodule. In the main theorem of this paper we try to understand generalized restriction via transfer map between Hochschild coohomology algebras of kGc and block ideal kNc.We shall prove this theorem in Section 5.

Theorem 1.3 Under the situation (∗) we choose i ∈ γ and j ∈ δ such that j = ij = j i. We assume that δ_{P}(H^{∗}(G,N,c,P_{γ})) ⊂ HH_{ikGj}^{∗}(kP). Then the following diagram of graded k-algebra homomorphisms is commutative:

\xymatrix{H^{∗}(G,N,c,P_{γ}) \ar[r]^{T_{kGi}∘δ_{P}} \ar[d]^{_{res{G,N,c \atop N,c}}} & HH_{X^{∗}}^{∗}(kGc) \ar[d]^{T_{X}} \\ H^{∗}(N,c,Q_{δ}) \ar[r]^{T_{kNj}∘δ_{Q}} & HH^{∗}(kNc)}.

If the block c is principal block of kN then the assumption δ_{P}(H^{∗}(G,N,c,P_{γ})) ⊂ HH_{ikGj}^{∗}(kP) from Theorem 1.3 holds; in this case the above diagram is commutative. Theorem 1.3 allows us to obtain the following result concerning the varieties of modules, which will be proved in Section 6.

Theorem 1.4 We keep the notations and assumptions of Theorem 1.3.

(a) The generalized restriction res_{N,c}^{G,N,c}: H^{∗}(G,N,c,P_{γ}) −→ H^{∗}(N,c,Q_{δ}) induces a finite map (res_{N,c}^{G,N,c})^{∗}, which we denote by r_{G,N,c}^{∗}: V_{N,c}−→ V_{G,N,c}.

b) For any finite generated kGc-module U, we have that

V_{N,c}(U) = (r_{G,N,c}^{∗})^{−1} (V_{G,N,c}(U)).

For basic facts on block theory and pointed groups we refer to [19].

2 More on generalized Brauer pairs and defect pointed groups

Let b be a block of G and D_{δ} a defect pointed group of G_{{b}}, as a pointed group on the G-algebra kG. Let i ∈ δ be a source idempotent. In this case there exists the Brauer homomorphism Br_{D}^{G} : (kG)^{D} −→ kC_{G}(D). Then Br_{D}^{G}(i) ≠ 0 is a primitive idempotent in kC_{G}(D) and thus there is a unique b-Brauer pair (D,e_{D}), with e_{D} a block of kC_{G}(D) such that Br_{D}^{G}(i)e_{D} ≠ 0). The same result can be obtained working with (c,G)-Brauer pairs.

Proposition 2.1 Let P_{γ} be a defect pointed group of G_{{c}} on A_{1}. Then there is a unique (c,G)-Brauer pair (P,e_{P}) such that Br_{P}^{N}(i)e_{P} ≠ 0 for any i ∈ γ. Moreover (P,e_{p}) is a maximal (c,G)-Brauer pair, thus P is a (c,G)-defect group.

Proof For i ∈ γ we have that i ∈ (kNc)^{P} is a primitive idempotent with Br_{P}^{N}(i) ≠ 0. Br_{P}^{N}(i) is a primitive idempotent in kC_{N}(P) since Br_{P}^{N} is surjective. It follows that there is a block e_{P}∈ Z(kC_{N}(P)) such that Br_{P}^{N}(i)e_{P} ≠ 0. This block is unique since, otherwise by contradiction it follows that Br_{P}^{N}(i) is a primitive idempotent in kC_{N}(P), which is in the primitive decompositions in kC_{N}(P) of two blocks.

Since P_{γ} is a defect pointed group we have that Br_{P}^{N}(c) ≠ 0 thus Br_{P}^{N}(c)e_{P} ≠ 0. By contradiction, if Br_{P}^{N}(c)e_{P} = 0 then

Br_{P}^{N}(i)e_{P} = Br_{P}^{N}(ic)e_{P} = Br_{P}^{N}(i)Br_{P}^{N}(c)e_{P} = 0,

which is not true. The last part of the proof is obvious.

Next we recall the definition of generalized Brauer category from [8, Definition 3.3], which by [8, Theorem 3.4] is a fusion system. For the basic definitions, notations and properties of fusion system we follow [11].

Definition 2.2 Let N be a normal subgroup of a finite group G, c be a G-stable block of kN and (P,e_{P}) be a maximal (c,G)-Brauer pair. For a subgroup Q of P let e_{Q} be the unique block of kC_{N}(Q) such that (Q,e_{Q}) ≤ (P,e_{P}).We denote by F_{(P,e_{P})}(G,N,c) the category on P with objects the subgroups of P and morphisms:

Hom_{F_{(P,e_{P})}(G,N,c)}(Q,R) = {φ : Q −→ R | φ(u) = gug^{−1} ,∀u ∈ Q,g ∈ G,^{g}(Q,e_{Q}) ≤ (R,e_{R})}.

If N = G we obtain the usual fusion system associated to a block c of kG denoted F_{(P,e_{P})}(G,c).

In the following proposition we keep the notations from Remark 1.1.

Proposition 2.3 Let P_{γ} be a defect pointed group of G_{{c}} on A_{1} and Q_{δ} = (P ∩ N)_{δ} ≤ P_{γ} the corresponding defect pointed group of N_{{c}}. Then there is a unique, maximal (c,G)-Brauer pair (P,e_{P}) such that Br_{P}^{N}(i)e_{P} ≠ 0, for any i ∈ γ, and a unique, maximal c-Brauer pair (Q,e_{Q}) such that Br_{Q}^{N}(j)e_{Q} ≠ 0, for any j ∈ δ. Moreover we have that (Q,e_{Q}) ≤ (P,e_{P}) and F_{(Q,e_{Q})}(N,c) is a normal subsystem of F_{(P,e_{P})}(G,N,c).

Proof The first part of the proof follows from Proposition 2.1. Let i ∈ (kN)^{P} be a primitive idempotent such that Br_{P}^{N}(i)e_{P} ≠ 0. Since Q_{δ} ≤ P_{γ} we have that there is j ∈ δ such that j = ij = j i. Suppose that Br_{Q}^{N}(i)e_{Q} = 0, then

Br_{Q}^{N}(j)e_{Q} = Br_{Q}^{N}(ji)e_{Q} = Br_{Q}^{N}(j)Br_{Q}^{N}(i)e_{Q} = 0,

which contradicts the fact that (Q,e_{Q}) is a c-Brauer pair. For the last part of the proof, since Q = P ∩ N and N is a normal subgroup of G it is easy to check that F_{(Q,e_{Q})}(N,c) is a normal subsystem of F_{(P,e_{P})}(G,N,c).

Lemma 2.4 Let c be a G-stable block of kN, where N is a normal subgroup of G. Let P_{γ} be a defect pointed group of G_{{c}} and i ∈ γ. Then the kGc − kGc-bimodule homomorphism

kGi ⊗_{kP}ikG −→ kGc

induced by multiplication is split.

Proof We denote by m the kGc − kGc-bimodule homomorphism induced by multiplication

m: kGi ⊗_{kP}ikG −→ kGc,

and we will defined a section of m. Since P_{γ} is defect pointed group of G_{{c}}, it follows by [19, Proposition 18.3, Lemma 14.1] that there exist u,v ∈ (kN)^{P} such that c = Tr_{P}^{G}(uiv). Let s be the map

s: kGc −→ kGi ⊗_{kP}ikG s(a) = ∑_{x∈[G/P]}axui ⊗_{kP}ivx^{−1} ,

where a ∈ kGc and [G/P] is a set of representatives of left cosets of P in G. The map s is independent of the choice of [G/P] and is indeed a homomorphism of kGc − kGc-bimodules. We also have that

(m ∘ s)(a) = m( ∑_{x∈[G/P]}axui ⊗_{kP}ivx^{−1} ) = ∑_{x∈[G/P]}axuivx^{−1} = aTr_{P}^{G}(uiv) = ac = a,

for a ∈ kGc.

Next we consider A = kG as interior G-algebra and we analyze source algebras in our case. In hypothesis of Remark1.1 let A_{δ} = jAj, where j ∈ (ckN)^{Q} is a primitive idempotent such that Br_{Q}^{N}(j) ≠ 0. Then A_{δ} is a ksubalgebra of A and the interior Q-algebra jA_{1}j is called the source algebra of the interior N-algebra cA_{1}. A_{γ} is the interior P-algebra ikGi, where i ∈ γ. With this notations we have the following proposition, which can be obtained as a consequence of [16, Proposition 3.2]. But we give a different proof here, since we don’t need the approach of twisted group algebras .

Proposition 2.5 If P_{γ} is a defect pointed group of G_{{c}} on A_{1} then A_{γ}, as interior P-algebra is Morita equivalent with A_{{c}}.

Proof We will prove that kGi as kGc − ikGi-bimodule and his dual ikG induces the Morita equivalences. First notice that the homomorphism φ of kGc − kGc-bimodules

φ : kGi ⊗_{ikGi}ikG −→ kGc,

induced by multiplication is still split surjective (the proof is similar to the proof of Lemma 2.4). Let

ψ : ikG ⊗_{kGc}kGi −→ ikGi

be the homomorphism of ikGi − ikGi-bimodules induced by multiplication. It is easy to check that

ψ′: ikGi −→ ikG ⊗_{kGc}kGi, ψ′(iai) = ia ⊗_{kGc}i,

where a ∈ kG, is the inverse homomorphism of ψ. Thus ψ is an isomorphism of ikGi − ikGi-bimodules. Since ϕ,ψ are induced by multiplication the conditions from [1, Definition 2.2.2] (definition of Morita equivalents rings) are satisfied.

As is already pointed in [9, 6.4] we may take the interior P-algebra A_{γ} as the source algebra of the interior Galgebra A_{{c}}. In [16, 3.5] A. Mărcuşalready noticed that Linckelmann’s results [13, 7.7 and 7.8] generalize for the case of twisted group algebras. In our case we have as follows.

Remark 2.6 Let R,S be subgroups of P, T a subgroup of R, i be a source idempotent of (kNc)^{P} where P_{γ} a defect pointed group of G_{{c}} on A_{1}. If φ : T −→ S is a group homomorphism such that the (kR,kS)-bimodule kR ⊗_{kT}^{φ} (kS) is isomorphic to a direct summmad of ikGi then φ is injective and φ ∈ Hom_{F_{(P,e_{P})}(G,N,c)}(T,S).

Remark 2.7 Following [13, 7.1] by Remark 2.6 we have that all indecomposable direct summands of the source algebra ikGi as (kP,kP)-bimodule are the (kP,kP)- bimodules kP ⊗_{kT}^{φ} (kP) where T is a subgroup of G and φ ∈ Hom_{F_{(P,e_{P})}(G,N,c)}(T,P). Moreover when T = P we have that all these indecomposable direct summands are pairwise non-isomorphic.

3 Restriction in block cohomology

Under the situation (∗) since Q_{δ} is a defect pointed group of N_{{c}} we remind from [11] the usual definition of block cohomology.

Definition 3.1 The cohomology ring of the block c of N associated with Q_{δ} is the subring

H^{∗}(N,c,Q_{δ})

of H^{∗}(Q,k) which consists of all [ζ] ∈ H^{∗}(Q,k) satisfying the stability condition res_{φ}[ζ] = res_{R}^{Q}[ζ], for any R subgroup of Q and any group homomorphism φ : R −→ Q belonging to the fusion system F_{(Q,e_{Q})}(N,c).

Since all defect pointed groups of N_{{c}} are N-conjugate the above ring is up to isomorphism independent of the choice of Q_{δ}.

Remark 3.2 a) Alperin fusion lemma for c-Brauer pairs implies that in order to check that [ζ] ∈ H^{∗}(Q,k) is in H^{∗}(N,c,Q_{δ}) it suffices to check that res_{R}^{Q}[ζ] = res_{φ}[ζ] for any R ≤ Q such that (R,e_{R}) is self-centralizing (that is e_{R} has Z(R) as defect group).

b) If c the principal block of kN then H^{∗}(N,c,Q_{δ}) ≅ H^{∗}(N,k), by usual description of stable elements in ordinary group cohomology.

Under the situation (∗) similar to block cohomology we give the definition of the generalized block cohomology, which if N = G is the above definition.This is possible since there is the fusion system F_{(P,e_{P})}(G,N,c) associated to the larger defect group P. Next we denote by F the fusion system F_{(P,e_{P})}(G,N,c).

Definition 3.3 The generalized cohomology ring of the block c of N associated to P_{γ} is the subring

H^{∗}(G,N,c,P_{γ})

of H^{∗}(P,k) which consists of the elements [ζ] ∈ H^{∗}(P,k) satisfying the stability condition res_{φ}[ζ] = res_{R}^{P}[ζ], for any subgroup R of P and any group homomorphism φ : R −→ P in F.

Since all defect pointed groups of G_{{c}} are G-conjugate the above ring is, up to isomorphism independent of the choice of P_{γ}.

Next we define a restriction from generalized block cohomology to block cohomology.

Proposition 3.4 In situation (∗) for any [ζ] ∈ H^{∗}(G,N,c,P_{γ}) we have that res_{Q}^{P}([ζ]) ∈ H^{∗}(N,c,Q_{δ}).

Proof We will prove that

res_{R}^{Q}(res_{Q}^{P}([ζ])) = res_{φ}(res_{Q}^{P}([ζ])),

for any subgroup R of Q and φ ∈ Hom_{F_{(Q,e_{Q})}(N,c)}(R,Q). Let \overline{φ} = i∘φ, where i: Q −→ P is the inclusion, which remains a morphism in F. By Proposition 2.3 we have that φ ∈ Hom_{F} (Q,R), thus \overline{φ} ∈ Hom_{F} (R,P). Now we obtain that

res_{R}^{Q}(res_{Q}^{P}([ζ])) = res_{R}^{P}([ζ]) = res\overline{φ}([ζ]),

where the las equality follows since [ζ] ∈ H^{∗}(G,N,c,P_{γ}). The proof is finished now by res_{φ}(res_{Q}^{P}([ζ])) = res\overline{φ}([ζ]).

By Proposition 3.4 we may define a restriction from generalized block cohomology to block cohomology.

Definition 3.5 Under the situation (∗) we define the generalized restriction in block cohomology

res_{N,c}^{G,N,c}: H^{∗}(G,N,c,P_{γ}) −→ H^{∗}(N,c,Q_{δ}),

by res_{N,c}^{G,N,c}([ζ]) = res_{Q}^{P}([ζ]) for any [ζ] ∈ H^{∗}(G,N,c,P_{γ}).

4 Proof of Theorem 1.2

Throughout this section we follow the notations and assumptions of the situation (∗) stated in Section 1 and we will prove Theorem 1.2. For proving the main result of this section we need the following remark, a lemma and two propositions. In situation (∗), since we consider P ∩N ≠ 1 it is not difficult to prove that Z(P)∩N ≠ 1. Moreover we have the next general and well known result.

Remark 4.1 Let N be a normal subgroup of a finite group G and P a p-subgroup of G such that P ∩ N ≠ 1. Then Z(P) ∩ N ≠ 1.

We consider B = kNc the primitive G-algebra, which is the localization of G_{{c}} in A_{1} and P_{γ} a defect of B. We remind that S(γ) = B^{P}/m_{γ} is a simple k-algebra called the multiplicity algebra, where m_{γ} = J(B^{P}) is the unique maximal ideal of B^{P} such that γ ⊈ m_{γ}. Then S(γ) ≃ End_{k}(V (γ)), where V (γ) is the simple B^{P}-module called the multiplicity module.

By [19, Lemma 14.5] we can view, slightly differently the multiplicity algebra S(γ) as a simple quotient of kC_{N}(P) and thus S(γ) isomorphic with the k-endomorphism algebra of a simple kC_{N}(P)-module. Explicitly this module is V (γ) = kC_{N}(P)Br_{P}^{N}(i)/J(kC_{N}(P))Br_{P}^{N}(i). In the next lines we denote by \overline{N} = N_{G}(P_{γ})/P and by Remark 4.1 we may consider \overline{C}= C_{N}(P)/Z(P) ∩ N. Remark that \overline{C}= C_{N}(P)/P ∩ C_{N}(P) ≃ PC_{N}(P)/P which is a subgroup of \overline{N}.

Lemma 4.2 In the above conditions it is true that the multiplicity module V (γ) is simple and projective as k\overline{C}-module. Moreover we have that p does not divide | \overline{N}/\overline{C}|.

Proof V (γ) is a simple kC_{N}(P)-module and Z(P) ∩ N a normal p-subgroup of C_{N}(P). By [19, Corollary 21.2] Z(P) ∩ N acts trivially on every simple kC_{N}(P)-module, thus V (γ) is simple as k\overline{C}-module.

The multiplicity algebra S(γ) has a \overline{N}-algebra structure, which is not necessarily interior on restriction to the subgroup PC_{G}(P)/P but is interior on restriction to the subgroup \overline{C}. By [19, Example 10.9] the multiplicity module V (γ) of P_{γ} is endowed with a k_{♯}ˆ\overline{N}-module structure which extends the structure of V (γ) as k\overline{C}-module. Since B is a primitive G-algebra by [19, Theorem 19.2] we have that V (γ) is projective as k_{♯}ˆ\overline{N}-module. By [19, Corollary 17.8] this is equivalent with the fact that the \overline{N}-algebra S(γ) = End_{k}(V (γ)) is projective algebra relative to {1}. Further this is equivalent to the surjectivity of the relative transfer map

t\overline{N} ; 1 : S(γ) −→ S(γ)\overline{N}.

Since V (γ) is simple on restriction to k\overline{C}it follows that S(γ)\overline{C} ≅ k by Schur’s lemma, and a fortiori S(γ)\overline{N} ≅ k. Therefore the relative trace map t\overline{N} ; 1 factorizes as

\xymatrix{S(γ) \ar[r]^{t\overline{C} ; 1} & k \ar[r]^{t\overline{N} ; \overline{C}} & k}.

\overline{N}/\overline{C}acts trivially on k thus by definition of relative trace map t\overline{N} ; \overline{C} is multiplication by [\overline{N}: \overline{C}], which is either 0 or an isomorphism.

We conclude that t\overline{N} ; 1 is surjective if and only if t\overline{C} ; 1 is surjective and [\overline{N}: \overline{C}]1_{k} ≠ 0. By Higman’s criterion the first condition implies that V (γ) is projective as k\overline{C}-module. The second means that p does not divide | \overline{N}/\overline{C}|.

Proposition 4.3 Let P_{γ} be a defect pointed group of G_{{c}} and (P,e_{P}) the unique maximal (c,G)-Brauer pair with the property that Br_{P}^{N}(i)e_{P} ≠ 0. Then it follows that Z(P) ∩ N is a defect group of e_{P}. In particularly e_{P} is a nilpotent block of kC_{N}(P).

Proof From Lemma 4.2 we know that V (γ) is simple and projective as k\overline{C}-module, thus by [19, Theorem 39.1] V (γ) belongs to a block \overline{e} of k\overline{C} with defect 0. By [19, Proposition 39.2] \overline{e} lifts to a block e of kC_{N}(P) with defect group Z(P) ∩ N since Z(P) ∩ N is a central p-subgroup of C_{N}(P). Moreover there is a unique simple kC_{N}(P)e module up to isomorphism which is V (γ). Since e_{P} belongs to V (γ) it follows that e_{P} = e and the defect group is Z(P) ∩ N.

Since e_{P} has Z(P) ∩ N as defect group which is central in kC_{N}(P) by [19, Corollary 49.11] it follows that e_{P} is a nilpotent block.

If G = N we obtain the well known result that Z(P) is the defect group of e_{P} as a block of kC_{G}(P) and e_{P} is a nilpotent block.

Proposition 4.4 Let P_{γ} be a defect pointed group of G_{{c}} on A_{1} and (P,e_{P}) be the unique (c,G)-Brauer pair such that Br_{P}^{N}(i)e_{P} ≠ 0 for any i ∈ γ. Then P_{γ} is the unique pointed group on A_{1} with the above property and moreover we have that N_{G}(P,e_{P}) = N_{G}(P_{γ}). In this situation we have that N_{G}(P,e_{P})/PC_{N}(P) ≅ \overline{N}/\overline{C}.

Proof If there is another pointed group P_{γ ′}with Br_{P}^{N}(i′)e_{P} ≠ 0 for any i′∈ γ ′then there exists another simple kC_{N}(P)e_{P}-module denoted V ′= kC_{N}(P)Br_{P}^{N}(i ′)/J(kC_{N}(P))Br_{P}^{N}(i ′). But from [19, Corollary 21.1] V ′is in fact a simple k\overline{C}\overline{e_{P}}-module, where \overline{e_{P}} is the corresponding block of e_{P} in k\overline{C} of defect 0. [19, Theorem 39.1] says that there is a unique simple k\overline{C}\overline{e}-module thus γ = γ ′. Now, from the first conclusion of this proposition it is easy to check that N_{G}(P,e_{P}) = N_{G}(P_{γ}). Thus N_{G}(P,e_{P})/PC_{N}(P) ≅ N_{G}(P_{γ})/PC_{N}(P), which is isomorphic with \overline{N}/\overline{C}.

Proposition 4.5 Let N be a normal subgroup of G, c a G-stable block and P_{γ} a defect pointed group of G_{{c}} and let i ∈ γ. Consider ikGi as kP − kP-bimodule and [ζ] ∈ H^{∗}(G,N,c,P_{γ}). i) We have t_{ikGi}(δ_{P}([ζ])) = \frac{dim_{k}(ikGi)}{|P|}δ_{P}([ζ]); in particular π_{ikGi}= \frac{dim_{k}(ikGi)}{|P|}1_{kP}. ii) For any nonnegative integer n the following diagram is a homotopy commutative:

\xymatrix{P_{kP} \ar[r]^{ε_{ikGi}} \ar[d]^{δ_{P}(ζ_{n})} & (ikGi) ⊗_{kP}P_{kP}⊗_{kP}(ikGi) \ar[d]^{Id⊗δ_{P}(ζ_{n})⊗Id} \\ P_{kP}[n] \ar[r]^{ε_{ikGi}[n]} & (ikGi) ⊗_{kP}P_{kP}[n] ⊗_{kP}(ikGi)} ,

where ζ_{n} is the degree n component of ζ. Particularly δ_{P}([ζ]) is ikGi-stable.

Proof We denote by r the number of all indecomposable direct summands of ikGi as kP − kP-bimodule. By Remark 2.7 all indecomposable direct summands of ikGi as kP −kP-bimodule are kP ⊗_{kT_{m}}^{φm}(kP), where T_{m}is a subgroup of P and φ_{m}∈ Hom_{F} (T_{m},P) for any m ∈ {1,...,r}. Thus we have res_{T}^{P}([ζ]) = res_{φ_{m}}([ζ]) By [11, Proposition 2.11,i)] and the last statement in [11, Lemma 5.3] we have that

t_{ikGi}(δ_{P}([ζ])) = ∑_{m=1}^{r} t_{kP⊗_{kTm}_{φm} (kP)}(δ_{P}([ζ])) = ∑_{m=1}^{r}[P : T]δ_{P}([ζ]).

Since for m ∈ {1,...,r} we have that

dim_{k}(kP ⊗_{kT_{m}}^{φm}(kP)) = [P : T] | P |,

it follows that \frac{dim_{k}(ikGi)}{|P|}≡| E | (mod p), where E contains all φ_{m} ∈ Aut_{F} (P) such that kP ⊗_{kP}^{φm}(kP) are pairwise non-isomorphic indecomposable direct summands. In this case since Aut_{F} (P) ≅ N_{G}(P,e_{P})/C_{N}(P) and the automorphisms induced by conjugation with elements of P gives isomorphic direct summands, we obtain that E ≅ N_{G}(P,e_{P})/PC_{N}(P)(see [19, Lemma 44.1 e)]). Thus, by Proposition 4.4 and Lemma 4.2 we have that \frac{dim_{k}(ikGi)}{|P|}≡| \overline{N}/\overline{C}| (mod p), which will be invertible in k. Then the last term of the above equality will be

∑_{m=1}^{r} \frac{dim_{k}(kP ⊗_{kT_{m}}^{φm}(kP))}{| P |}= \frac{dim_{k}(ikGi)}{| P |}δ_{P}([ζ]).

Statement ii) is a consequence of [11, Lemma 5.3,(i),(ii)] and the additivity of the adjunction maps [11, Proposition 2.7 (i)]

Remark 4.6 Let N be a normal subgroup of a finite group G, c a G-stable block of kN and P_{γ} a defect pointed group of G_{{c}} with i ∈ γ. There is an kP − kGc-bimodule isomorphism (kGi)^{∗} ≅ ikG.

We apply [11, 6.6] in the particular case of A = kGc,B = kP symmetric k-algebras with obvious symmetric forms s, respectively t and M = kGi and obtain a similar lemma with [11, Lemma 5.5] in our case.

Lemma 4.7 Let N be a normal subgroup of a finite group G, c a G-stable block of kN and P_{γ} a defect pointed group of G_{{c}} with i ∈ γ. With the identification through the isomorphism from Remark 4.6 and since by Proposition 2.5 the multiplication in kGc induces an isomorphism ikG ⊗_{kGc}kGi ≅ ikGi we have that the adjunction maps associated with kGi and his dual ikG are given as follows:

ϵ_{kGi}: kP −→ ikGi maps u ∈ P to ui;

η_{kGi}: kGi ⊗_{kP}ikG −→ kGc induced by multiplication in kGc;

ϵ_{ikG}: kGc −→ kGi ⊗_{kP}ikG maps a ∈ kGc to ∑_{x∈[G/P]}axi ⊗ ix^{−1} ;

η_{ikG}: ikGi −→ kP maps b ∈ ikGi to ∑_{u∈P}s(bu^{−1} )u.

Moreover, we have that π_{kGi}= T r_{P}^{G}(i) and π_{ikG}= s(i)1_{kP}= \frac{dim_{k}(ikG)}{|G|}1_{kP}.

Proof These maps are as stated follows, from explicit description of adjunction maps in [11, 6.6]. It follows that

π_{kGi}= (η_{kGi}∘ ϵ_{ikG})(c) = η_{kGi}( ∑_{x∈[G/P]}cxi ⊗_{kP}ix^{−1} ) = ∑_{x∈[G/P]}xix^{−1} = T r_{P}^{G}(i),

where T r_{P}^{G}: (kNc)^{P} −→ (kNc)^{G}, T r_{P}^{G}(a) = ∑_{x∈[G/P]}xax^{−1} ,a ∈ (kNc)^{P}.

Similar arguments to [11, Lemma 5.5] gives that π_{ikG}= \frac{dim_{k}(ikG)}{|G|}1_{kP}.

Proof of Theorem 1.2. (i) It follows from Lemma 4.7 that the projective elements are as stated in i). With the notations from the proof of Lemma 4.2 we proved that S(γ)\overline{N} ≅ k. Now, since k is algebraically closed field we may apply [18, Proposition 9.3], which shows that π_{kGi} is invertible. Since ker(Br_{P}^{N}) ⊂ ker(s), where s is the canonical symmetrizing form on kG we have that s(i) = s(Br_{P}^{N}(i)). Now Br_{P}^{N}(i) is a primitive idempotent

in kC_{N}(P), and applying the last statement of 4.7 to Br_{P}^{N}(i) instead of i shows that s(Br_{P}^{N}(i)) = \frac{dim_{k}Br_{P}^{N}(i)kC_{N}(P)}{|C_{N}(P)|}. e_{P} is a nilpotent block of kC_{N}(P)(see Proposition 4.3), then by [3] it follows that Br_{P}^{N}(i)kC_{N}(P) is the unique

projective indecomposable module of the nilpotent block e_{P}, which is split. Thus \frac{dim_{k}Br_{P}^{N}(i)kC_{N}(P)}{|C_{N}(P)|}is nonzero in k.

(ii) Since ikGi ≅ ikG⊗_{kGc}kGi and π_{kGi}is invertible we can apply [11, Corollary 3.8] to A = C = kP, B = kGc,X = ikG,Y = kGi. By Proposition 4.5 ii) we have that δ_{P}(H^{∗}(G,N,c,P_{γ})) ⊆ HH_{ikGi}^{∗}(kP) ⊂ HH_{ikG}^{∗}(kP).

(iii) follows from (ii) together with [11, Theorem 3.6].

5 Transfer maps between Hochschild cohomology algebras and generalized restriction in block cohomology

We remind some properties of normalized transfer maps between Hochschild cohomology algebras of symmetric algebras from [11]. Let A,B,C be three symmetric R-algebras, X a complex of finite generated A − Bbimodules, projective as left A-modules and right B-modules and Y a complex of finite generated B − Cbimodules, projective as left B-modules and right C-modules. By X^{∗} we denote the dual complex of B − Abimodules. Let t_{X} : HH^{∗}(B) −→ HH^{∗}(A) be the transfer map associated with X defined in [11, Definition 2.9]. We denote abusively by t_{X}^{0} : Z(B) −→ Z(A) the degree 0 component of t_{X} since there are the natural isomorphisms Z(B) ≅ HH^{0}(B), Z(A) ≅ HH^{0}(A). The element π_{X} = t_{X}^{0}(1_{B}) is called the relatively X-projective element of Z(A). If π_{X} is invertible we know that there is the normalized transfer map associated with X is T_{X}. An element ζ ∈ HH^{∗}(A) is X-stable if there is τ ∈ HH^{∗}(B) satisfying the condition from [11, Definition 3.1]. We write this ζ ⊗_{A}1_{X} = 1_{X}⊗_{B}τ. We denote by HH_{X}^{∗}(A) the graded subalgebra of HH^{∗}(A) consisting of all X-stable elements. The following proposition is an easy consequence of [11, Proposition 2.11] and [11, Lemma 3.4, Proposition 3.5.(iv)].

Proposition 5.1 Under the above hypothesis we have:

i) π_{X⊗_{B}Y} = t_{X}^{0}(π_{Y}). ii) If X′ is a direct summand of X then HH_{X^{∗}}^{∗}(B) ⊂ HH_{X^{′∗}}^{∗}(B). Moreover if π_{X},π_{X′} are invertible then the

normalized transfer map T_{X′}coincides with T_{X} on HH_{X^{∗}}^{∗}(B).

The next proposition follows the ideas from [7].

Proposition 5.2 If π_{X},π_{Y},π_{X⊗_{B}Y} are invertible then T_{X}∘ T_{Y} coincides with T_{X⊗_{B}Y} on HH_{Y^{∗}⊗_{B}X^{∗}}^{∗}(C).

Proof Let ξ ∈ HH_{Y^{∗}⊗_{B}X^{∗}}^{∗}(C). By [11, Theorem 3.6] we have that T_{X⊗_{B}Y}(ξ) ∈ HH_{X⊗_{B}Y}^{∗}(A) ⊂ HH_{X}^{∗}(A). Where the last inclusion follows from [11, Corollary 3.8], since π_{Y} is invertible.

If we denote by ζ the element T_{X⊗_{B}Y}(ξ) we have that ζ ⊗ 1_{X⊗_{B}Y} = 1_{X⊗_{B}Y} ⊗ ξ. Since π_{Y} is invertible from [11, Corollary 3.8] we have that T_{Y}(HH_{Y^{∗}⊗_{B}X^{∗}}^{∗}(C)) ⊂ HH_{X^{∗}}^{∗}(B). From the argument in the proof of [11, Corollary 3.8] it follows that ζ ⊗1_{X} = 1_{X}⊗T_{Y}(ξ), which by the definition of T_{X} proves that T_{X}(T_{Y}(ξ)) = ζ = T_{X⊗_{B}Y}(ξ).

Let A = kNc, B = kGc, and X = ckGc = kGc as A − B-bimodule with X^{∗} = ckGc = ckG as B − A-bimodule. Let M = kP as kP − kQ-bimodule with M^{∗} = kP as kQ − kP-bimodule. By [11, Example 3.9] we know that π_{M} = [P : Q]1_{kP}∈ Z(kP) is not invertible and π_{M^{∗}} = 1_{kQ}∈ Z(kQ)^{×} .

Since c ∈ Z(kN) is an idempotent such that c is G-stable then c is an idempotent in Z(kG). We apply [7, Proposition 3.2] and obtain that:

t_{X}^{0} : Z(kGc) −→ Z(kNc), α ↦ cη_{N}^{G}(α) α ∈ Z(kGc),
t_{X^{∗}}^{0} : Z(kNc) −→ Z(kGc), β ↦ cT r_{N}^{G}(β) β ∈ Z(kNc).

Where for H a subgroup of G and R a commutative ring, in general we have:

η_{H}^{G}: RG −→ RH, η_{H}^{G}( ∑_{x∈G}α_{x}x) = ∑_{x∈H} α_{x}x,

Tr_{H}^{G} : (RG)^{H} −→ Z(RG), Tr_{H}^{G}(β) = ∑_{x∈[G/H]}xβx^{−1} ,

for β ∈ (RG)^{H}.

Proposition 5.3 We have the following:

i) π_{X} = c, π_{X^{∗}} = [G: N]c. ii) π_{X} ∈ Z(kNc) is invertible. π_{X^{∗}} is invertible in Z(kGc) if and only if p does not divide [G: N].

Proof We have that π_{X} = t_{X}^{0}(c) = cη_{N}^{G}(c) = cc = c. For π_{X^{∗}} we have

π_{X^{∗}} = t_{X^{∗}}(c) = cT r_{N}^{G}(c) = c ∑_{g∈[G/N]}gcg^{−1} = [G: N]c,

since c is G-stable.

For b) since p does not divide [G : N] it follows that [G : N]1_{k} is invertible, thus there is u a nonnegative integer such that u1_{k} is the inverse. We have that uc · [G: N]c = c.

Under the situation (∗) since Q_{δ} ≤ P_{γ} we choose i ∈ γ and j ∈ δ such that j = ij = j i. In the next lines set Y = kGi as B − kP bimodule and Z = kNj as A − kQ-bimodule. Then we have the following descriptions:

ikGj = ikG ⊗_{B}X^{∗}⊗_{A}kNj = Y^{∗}⊗_{B}X^{∗}⊗_{A}Z;
jkGi = Z^{∗}⊗_{A}X ⊗_{B}Y.

Lemma 5.4 The following hold:

a) HH_{ikGj}^{∗}(kP) ⊂ HH_{Y^{∗}⊗_{B}X^{∗}}^{∗}(kP).

b) T_{Y}(HH_{Y^{∗}⊗_{B}X^{∗}}^{∗}(kP)) ⊂ HH_{X^{∗}}^{∗}(kGc).

Proof Since π_{Z} invertible by [11, Theorem 5.6] and π_{Y} is invertible by Theorem 1.2, we apply now [11, Corollary 3.8] for the inclusions.

Lemma 5.5 With the above notations the following hold:

a) The kQ − kQ-bimodule jkNj is a direct summand of ikGj.

b) The A − kP-bimodule X ⊗_{B}Y is a direct summand of Z ⊗_{kQ}jkGi.

c) We have that M is isomorphic with a direct summand of ikGj as kP − kQ-bimodules.

Proof In the first case we work with kQ−kQ-bimodules and we have the decomposition ikGj = ikNj ⊕ik[G\ N]j, where k[G \ N] is the k-algebra which as k-vector space has the basis G \ N. Since i ∈ (kN)^{P} ⊂ (kN)^{Q} and j appears in a primitive decomposition of i in (kN)^{Q}, it follows that jkNj is a direct summand of ikGj. This proves a).

For b) we know from a general property of source idempotents that the canonical map Z ⊗_{kQ}Z^{∗} −→ A splits as a homomorphism of A − A-bimodules, since Q is a defect group of the block c. Then we have that A is a direct summand of Z ⊗_{kQ}Z^{∗} as A − A-bimodules. We obtain that A ⊗_{A}X ⊗_{B}Y is a direct summand of Z ⊗_{kQ}Z^{∗}⊗_{A}X ⊗_{B}Y as A − kP-bimodules.

For c) we consider M and ikGj as k[P × Q]-left modules. From [17, Lemma 10.22] we know that any indecomposable direct summand of ikGj as k[P × Q]-module is isomorphic with Ind_{Δ_{x}(P^{x}∩Q)}^{P×Q}k, where x ∈ [P \

G/Q] and Δ_{x}(P^{x}∩ Q) is the subgroup of P × Q which contains all the pairs (u,u^{x}) with u ∈ P ∩ Q^{x^{−1}} . By [19, Lemma 27.1] we obtain that Δ_{x}(P^{x}∩ Q) is the vertex of Ind_{Δ_{x}(P^{x}∩Q)}^{P×Q}k.

Next we prove that ikGj has an indecomposable direct summand of vertex ΔQ by working with the Brauer construction associated to k[P × Q]-modules (see [2]). There is the Brauer homomorphism

Br_{ΔQ} : (ikGj)^{ΔQ} −→ (ikGj)(ΔQ).

We have that (ikGj)(ΔQ) = Br_{ΔQ}(i(kG)^{ΔQ}j) = Br_{ΔQ}(i)kC_{G}(Q)Br_{ΔQ}(j), which is not equal to 0 (since j ∈ (ikGj)^{ΔQ} and Br_{ΔQ}(j) ≠ 0). Thus there is an indecomposable direct summand N of ikGj such that N(ΔQ) ≠ 0. Since for N there is x ∈ [P \ G/Q] such that Δ_{x}(P^{x}∩ Q) is the vertex of N it follows, by [2, Theorem 3.2(1)], that there is an element (r,q) ∈ P × Q such that

ΔQ ⊆ Δ_{x}(P^{x}∩ Q)^{(r,q)}.

Then we have that Q ⊆ P ∩ Q^{x^{−1} r} and u^{r} = q^{xq} for all u ∈ Q. We obtain that P ∩ Q^{x^{−1} r} = Q, which concludes that Δ_{x}(P^{x}∩ Q)^{(r,q)} = ΔQ.

By the statement c) of the above lemma we will identify M with a direct summand of ikGj. Since π_{M^{∗}} = 1_{kQ} the normalized transfer T_{M^{∗}} is t_{M^{∗}}. Suppose that we are in situation (∗), then by [11, Proposition 4.7] the following diagram of graded k-algebras homomorphisms is commutative:

\xymatrix{H^{∗}(G,N,c,P_{γ}) \ar[r]^{δ_{P}} \ar[d]^{res_{N,c}^{G,N,c}} & HH^{∗}(kP) \ar[d]^{T_{M^{∗}}} \\ H^{∗}(N,c,Q_{δ}) \ar[r]^{δ_{Q}} & HH^{∗}(kQ)}.

By [11, Proposition 4.8] we have that δ_{P}(H^{∗}(P,k)) ⊂ HH_{M}^{∗}(kP). Thus we obtain a slightly different diagram in the next remark where, we abusively denote with T_{M^{∗}} the surjective homomorphism R_{M^{∗}}.

Remark 5.6 Under the above assumptions the following diagram of graded k-algebras homomorphisms is commutative:

\xymatrix{H^{∗}(G,N,c,P_{γ}) \ar[r]^{δ_{P}} \ar[d]^{res_{N,c}^{G,N,c}} & HH_{M}^{∗}(kP) \ar[d]^{T_{M^{∗}}} \\ H^{∗}(N,c,Q_{δ}) \ar[r]^{δ_{Q}} & HH_{M^{∗}}^{∗}(kQ)}.

Proposition 5.7 Suppose that δ_{P}(H^{∗}(G,N,c,P_{γ})) ⊆ HH_{ikGj}^{∗}(kP), where ikGj is the kP − kQ-bimodule. Then the following diagram of graded k-algebras homomorphisms is commutative:

\xymatrix{H^{∗}(G,N,c,P_{γ}) \ar[r]^{δ_{P}} \ar[d]^{_{res{G,N,c \atop N,c}}} & HH_{ikGj}^{∗}(kP) ∩ HH_{M}^{∗}(kP) \ar[d]^{T_{M^{∗}}} \\ H^{∗}(N,c,Q_{δ}) \ar[r]^{δ_{Q}} & HH_{jkNj}^{∗}(kQ) ∩ HH_{M^{∗}}^{∗}(kQ)}.

Proof Since δ_{Q}(H^{∗}(N,c,Q_{δ})) ⊂ HH_{jNj}^{∗}(kQ) (see [11, Proposition 5.4]) by Remark 5.6 it is enough to prove that T_{M^{∗}}(HH_{ikGj}^{∗}(kP) ∩ HH_{M}^{∗}(kP)) ⊆ HH_{jkNj}^{∗}(kQ). We apply [7, Lemma 2.1] and obtain that

T_{M^{∗}}(HH_{ikGj}^{∗}(kP) ∩ HH_{M}^{∗}(kP)) ⊂ π_{M^{∗}}HH_{kP⊗_{kP}ikGj}^{∗}(kQ) = HH_{ikGj}^{∗}(kQ),

where ikGj is considered as kQ − kQ-bimodule. From Lemma 5.5 a) and Proposition 5.1 it follows that HH_{ikGj}^{∗}(kQ) ⊆ HH_{jkNj}^{∗}(kQ).

Proof of Theorem 1.3. By Proposition 5.7 and Lemma 5.4 it is enough to prove that the following diagram is commutative:

\xymatrix{HH_{ikGj}^{∗}(kP) ∩ HH_{M}^{∗}(kP) \ar@{^{(}->}[r] \ar[d]^{T_{M^{∗}}} & HH_{Y^{∗}⊗_{B}X^{∗}}^{∗}(kP) ∩ HH_{M}^{∗}(kP) \ar[r]^{T_{Y}} & HH_{X^{∗}}^{∗}(kGc) \ar[d]^{T_{X}} \\ HH_{jkNj}^{∗}(kQ) ∩ HH_{M^{∗}}^{∗}(kQ) \ar@{^{(}->}[r] & HH_{Z^{∗}}^{∗}(kQ) ∩ HH_{M^{∗}}^{∗}(kQ) \ar[r]^{T_{Z}} & HH^{∗}(kNc)}.

From Proposition 5.1 i) we have that π_{X⊗_{B}Y} = t_{X}^{0}(π_{Y}) = t_{X}^{0}(T r_{P}^{G}(i)) = cη_{N}^{G}(T r_{P}^{G}(i)) = cπ_{Y} ∈ Z(kNc)^{×} , where the last equality is true since π_{Y}∈ (kNc)^{G}. Now π_{jkGi} = π_{Z^{∗}⊗_{A}X⊗_{B}Y} = t_{Z^{∗}}^{0}(π_{X⊗_{B}Y}), then by [7, Proposition 3.5] we obtain that π_{jkGi} is invertible.

By Lemma 5.5 c) and Proposition 5.1 ii) we have that HH_{ikGj}^{∗}(kP) ⊂ HH_{M}^{∗}(kP) . It follows that we only have to prove that the following diagram is commutative:

\xymatrix{HH_{ikGj}^{∗}(kP) \ar@{^{(}->}[r] \ar[d]^{T_{jkGi}} & HH_{Y^{∗}⊗_{B}X^{∗}}^{∗}(kP) ∩ HH_{M}^{∗}(kP) \ar[r]^{T_{Y}} & HH_{X^{∗}}^{∗}(kGc) \ar[d]^{T_{X}} \\ HH_{jkNj}^{∗}(kQ) ∩ HH_{M^{∗}}^{∗}(kQ) \ar@{^{(}->}[r] & HH_{Z^{∗}}^{∗}(kQ) ∩ HH_{M^{∗}}^{∗}(kQ) \ar[r]^{T_{Z}} & HH^{∗}(kNc)}.

Since π_{X}, π_{Y}, π_{X⊗_{B}Y} are invertible it follows that T_{X}∘ T_{Y} coincides with T_{X⊗_{B}Y} on HH_{Y^{∗}⊗_{B}X^{∗}}^{∗}(kP) (see Proposition 5.2).

We prove next that the element π_{Z⊗_{kQ}jkGi} = t_{Z}^{0}(π_{jkGi})) is invertible in Z(kNc). Notice that t_{Z}^{0}(τ) = Tr_{Q}^{N}(jτ), for τ ∈ Z(kQ)(see [11, Lemma 5.5]), where we have that Tr_{Q}^{N}: (kNc)^{Q} −→ (kNc)^{N}. It follows that π_{Z⊗_{kQ}jkGi} = Tr_{Q}^{N}(jπ_{jkGi}) = Tr_{Q}^{N}(jπ_{jkGi}cj). Since π_{jkGi} ∈ Z(kQ) and Z(kQ)c is a subalgebra of (kNc)^{Q} it follows that π_{jkGi}c ∈ (kNc)^{Q}. Now, we apply [18, Proposition 9.3] and obtain that Tr_{Q}^{N}(jπ_{jkGi}cj) ∉ J(Z(kNc)). Then π_{Z⊗_{kQ}jkGi} is an invertible element in Z(kNc), since Z(kNc) = (kNc)^{N} is a commutative local ring.

Since π_{Z}, π_{jkGi} andπ_{Z⊗_{kQ}jkGi} are invertible it follows, by Proposition 5.2, that T_{Z} ∘ T_{jkGi} coincides with T_{Z⊗_{kQ}jkGi} on HH_{ikGj⊗_{kQ}Z^{∗}}^{∗} (kP). By Proposition 5.1 ii) and Lemma 5.5 b) it follows that T_{Z⊗_{kQ}jkGi} coincides with T_{X⊗_{B}Y} on HH_{ikGj}^{∗}(kP) ⊆ HH_{Y^{∗}⊗_{B}X^{∗}}^{∗}(kP) ∩ HH_{M}^{∗}(kP).

6 Varieties

In this section we follow the notations and assumptions of Theorem 1.3. First we recall the definition of varieties applied to generalized block cohomology. The generalized block cohomology algebra H^{∗}(G,N,c,P_{γ}) is a finitely generated, graded commutative algebra. We denote its maximal ideal spectrum by V_{G,N,c}. Let U be a finitely generated kGc-module and let I_{G,N,c,P_{γ}}^{∗}(U) be the kernel of the composite of the graded k-algebras homomorphisms

\xymatrix{H^{∗}(G,N,c,P_{γ}) \ar[r]^{T_{kGi}∘δ_{P}} & HH^{∗}(kGc) \ar[r]^{α_{U}} & Ext_{kGc}^{∗}(U,U)},

where α_{U} is induced by the functor − ⊗_{kGc}U. The variety V_{G,N,c}(U) is defined to be the subvariety of V_{G,N,c}, which consists of all maximal ideals containing I_{G,N,c,P_{γ}}^{∗}(U). We denote still by U the structure of U as kNcmodule and with I_{N,c,Q_{δ}}^{∗} the kernel of the composite

\xymatrix{H^{∗}(N,c,Q_{δ}) \ar[r]^{T_{kNj}∘δ_{Q}} & HH^{∗}(kNc) \ar[r]^{α_{U}} & Ext_{kNc}^{∗}(U,U)}.

The variety V_{N,c}(U) is the subvariety of V_{N,c} which consists of all maximal ideals containing I_{N,c,Q_{δ}}^{∗}. The cohomology variety V_{G}(U) of U, introduced by Carlson [5], is defined to be the subvariety of the maximal ideal spectrum of H^{∗}(G,k) (denoted V_{G}) determined by I_{G}^{∗}(U). Here I_{G}^{∗}(U) is the kernel of the graded k-algebra homomorphism H^{∗}(G,k) −→ Ext_{kGc}^{∗}(U,U) induced by the functor − ⊗_{k}U. For more details, see [1],[5].

The following proposition is an analogue result to [14, Theorem 2.1].

Proposition 6.1 Let G be a finite group, c a block of kN, P_{γ} a defect pointed group of G_{{c}} and i ∈ γ a source idempotent. The inclusion i_{P} : H^{∗}(G,N,c,P_{γ}) −→ H^{∗}(P,k) induces a finite surjective map i_{P}^{∗} : V_{P} −→ V_{G,N,c}, and for any finitely generated kGc-module U we have V_{G,N,c}(U) = i_{P}^{∗}(V_{P}(iU)), where iU is considered as kP-module.

Proof The following inclusions hold:

res_{P}^{G}(H^{∗}(G,k)) ⊆ H^{∗}(G,N,c,P_{γ}) ⊆ H^{∗}(P,k),

and H^{∗}(P,k) is Noetherian as module over res_{P}^{G}(H^{∗}(G,k)), whence H^{∗}(P,k) is finitely generated as module over H^{∗}(G,N,c,P_{γ}). Therefore the inclusion i induces a finite surjective map i_{P}^{∗} : V_{P} −→ V_{G,N,c}, see [1, II, Section 5.4]. For the last equality, Theorem 1.2 and Lemma 2.4 allow us to remark that the proof is analogue to the proof of [14, Theorem 2.1].

For any T ≤ P, the graded algebra homomorphism r_{T} (which is the inclusion i_{P}: H^{∗}(G,N,c,P_{γ}) ↪ H^{∗}(P,k) followed by the restriction res_{T}^{P}: H^{∗}(P,k) −→ H^{∗}(T,k) ) induces a map on varieties r_{P,T}^{∗}: V_{T} −→ V_{G,N,c}.

Lemma 6.2 With the notations in Proposition 6.1 we have that V_{G,N,c}(U) = ⋃_{E}r_{P,E}^{∗}(V_{E}(iU)), where E runs over the set of elementary abelian subgroups of P.

Proof By Proposition 6.1 we have that V_{G,N,c}(U) = i_{P}^{∗}(V_{P}(iU)). We apply [6, Theorem 9.4.3] with P and iU instead of G and M respectively, and obtain that:

i_{P}^{∗}(V_{P}(iU)) = i_{P}^{∗}(⋃_{E} (res_{E}^{P})^{∗}(V_{E}(iU))) = ⋃_{E} r_{P,E}^{∗}(V_{E}(iU)).

Proof of Theorem 1.5.

(a) The following inclusions hold:

res_{Q}^{G}(H^{∗}(G,k)) = res_{Q}^{P}(res_{P}^{G}(H^{∗}(G,k))) ⊆ res_{Q}^{P}(H^{∗}(G,N,c,P_{γ})) ⊆ H^{∗}(N,c,Q_{δ}) ⊆ H^{∗}(Q,k).

Since H^{∗}(Q,k) is Noetherian as a module over res_{Q}^{G}(H^{∗}(G,k)) it follows that H^{∗}(N,c,Q_{δ}) is finitely generated as module over res_{N,c}^{G,N,c}(H^{∗}(G,N,c,P_{γ})). Thus res_{N,c}^{G,N,c} induces a finite map on varieties, which we denote r_{G,N,c}^{∗}: V_{N,c}−→ V_{G,N,c}.

(b) Now, [12, Theorem 5.1] says that the following diagram commutes:

\xymatrix{HH_{X^{∗}}^{∗}(kGc) \ar[r]^{α_{U}} \ar[d]^{T_{X}} & Ext_{kGc}^{∗}(U,U) \ar[d]^{β_{X,U}} \\ HH^{∗}(kNc) \ar[r]^{α_{X⊗_{kGc}U}} & Ext_{kNc}^{∗}(U,U)}

where β_{X,U}is induced by the functor X ⊗_{kGc}−, more precise is the restriction res_{kNc}^{kGc}. Composing the above diagram with the commutative diagram from Theorem 1.3 we obtain the following commutative diagram:

\xymatrix{H^{∗}(G,N,c,P_{γ}) \ar[r]^{α_{U}∘T_{kGi}∘δ_{P}} \ar[d]^{_{res{G,N,c \atop N,c}}} & Ext_{kGc}^{∗}(U,U) \ar[d]^{β_{X,U}} \\ H^{∗}(N,c,Q_{δ}) \ar[r]^{α_{X⊗_{kGc}U}∘T_{kNj}∘δ_{Q}} & Ext_{kNc}^{∗}(U,U)}.

It follows that res_{N,c}^{G,N,c}(I_{G,N,c,P_{γ}}^{∗}(U)) ⊆ I_{N,c,Q_{δ}}^{∗}(U) (since X ⊗_{kGc}U ≅ U as kNc-modules), thus

V_{N,c}(U) ⊆ (r_{G,N,c}^{∗})^{−1} (V_{G,N,c}(U)).

Suppose now that α ∈ V_{N,c} and β ∈ V_{G,N,c}(U). Then by the obvious analogue for block cohomology of Quillen stratification obtained by combining [14, Theorem 4.2] and [14, Proposition 4.3] we obtain that there is an elementary abelian subgroup E ≤ Q and a maximal ideal α′∈ V_{E} such that r_{Q,E}^{∗}(α′) = α. From Lemma 6.2 it follows that there is an elementary abelian subgroup E′≤ P and a maximal ideal β′∈ V_{E′}(iU) such that r_{P,E′}^{∗}(β′) = β. Now, the following equalities holds:

r_{G,N,c}^{∗}(α) = r_{G,N,c}^{∗}(r_{Q,E}^{∗}(α′)) = ((res_{Q}^{P}∘ i_{P})^{∗}∘ (res_{E}^{Q}∘ i_{Q})^{∗})(α′)
= (res_{E}^{Q}∘ i_{Q}∘ res_{Q}^{P}∘ i_{P})^{∗}(α′) = (res_{E}^{P})^{∗}(α′),

where i_{Q} is the inclusion i_{Q}: H^{∗}(N,c,Q_{δ}) −→ H^{∗}(Q,k). We also have

r_{G,N,c}^{∗}(α) = β = r_{P,E′}^{∗}(β′) = (res_{E′}^{P})^{∗}(β′),

where the last equality holds since we can consider r_{P,E′}^{∗}as (res_{E′}^{P})^{∗} restricted to V_{G,N,c}. We apply now [6, Proposition 9.6.1] and obtain that α′∈ V_{E}(iU) since β′∈ V_{E′}(iU). Since iU is a direct summand of jU as kQ-module then iU is a direct summand of jU as kE-module and then, by [6, Proposition 9.7.2] it follows that V_{E}(iU) ⊂ V_{E}(jU). We conclude that α′∈ V_{E}(jU) and r_{Q,E}^{∗}(α′) = α ∈ V_{N,c}(U) (see [14, Lemma 4.4]).

Acknowledgements The author would like to thank Professor Andrei Mărcuşfor the continuous help given in writing this article and also to Professor Markus Linckelmann and Hiroaki Kawai for their helpful answers to some questions.

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