A theorem of Mislin for cohomology of fusion systems and applications to block algebras of finite groups

Constantin-Cosmin Todea^{∗}

Department of Mathematics, Technical University of Cluj-Napoca, Str. G. Baritiu 25, Cluj-Napoca 400027, Romania

Abstract

The aim of this short expository article is to give an algebraic proof for a theorem of Mislin in the case of cohomology of saturated fusion systems defined on p-groups when p is odd. Some applications of this theorem, which includes different proofs of known results regarding block algebras of finite groups, are also given.

Keywords: fusion system, cohomology, variety, block algebra 2010 MSC: 20C20, 16EXX

1. Introduction

A saturated fusion system F on a finite p-group P is a category whose objects are the subgroups of P and whose morphisms satisfy certain axioms mimicking the behavior of a finite group G having P as a Sylow subgroup. For the convenience of the reader we give, in Section 2, the definition of saturated fusion systems and some basic facts, following [2]. Let k be an algebraically closed field of characteristic p. We denote by H^{∗}(G, k) the cohomology algebra of a group G with trivial coefficients. We denote by H^{∗}(F) the subalgebra of F-stable elements in H^{∗}(P, k), i.e. the cohomology algebra of the saturated fusion system F, which is the subalgebra of H^{∗}(P, k) consisting of elements ζ ∈ H^{∗}(P, k) such that

res_{Q}^{P}(ζ) = res_{φ}(ζ),

for any φ ∈ Hom_{F}(Q, P) and any subgroup Q of P.

A celebrated theorem of Mislin in [15] regarding the control of fusion in group cohomology (stated for compact Lie groups) has now a new short algebraic

5 proof for p odd thanks to Benson, Grodal and Henke [4]. See [7], [19] for other algebraic proofs which uses Mackey functors and cohomology of trivial source modules; see also [21] for a different algebraic approach. Also, in [8, Remark

December 5, 2014

∗Corresponding author

Email address: constantin.todea@math.utcluj.ro (Constantin-Cosmin Todea)

5.8] Linckelmann suggests a topological proof for Mislin’s theorem in the case of block algebras of finite groups, more precisely for cohomology of fusion systems

10 associated to blocks. We prove this theorem of Mislin in the general context of cohomology of saturated fusion systems for p odd, by extending the proof of Benson, Grodal and Henke to saturated fusion systems.

The k-algebra H^{∗}(F) is a graded-commutative and finitely generated, hence we associate the spectrum of maximal ideals, i.e. the algebraic variety which we denote by V_{F}. Let G be a saturated fusion subsystem of F defined on the same finite p-group P. We have an inclusion map

i : H^{∗}(F) → H^{∗}(G),

which induces a map on varieties

i^{∗} : V_{G} → V_{F}.

The main result of this paper is the following theorem which contains Mislin’s theorem for saturated fusion systems as a special case, when p is odd.

15 Theorem 1.1. Let G be a saturated fusion subsystem of F defined on the same finite p-group P and p an odd prime. If for each ζ ∈ H^{∗}(G) we have ζ^{p^{r}} ∈ Im(i) for some r ≥ 0, then G = F. In particular we have H^{∗}(F) = H^{∗}(G) if and only if G = F.

The ingredients for the proof of the above theorem were already mentioned 20 by Benson, Grodal and Henke in [4, Remark 3.7]. We follow their suggestion and

we fill this gap in the literature. There are two ingredients for proving Theorem 1.1. To explain those ingredients, we first introduce the following terminology. Let G be a saturated fusion subsystem of F defined on the same finite p-group P. For shortness, we will say that G controls p-fusion in F on elementary abelian

25 p-subgroups if Hom_{G}(E_{1}, E_{2}) = Hom_{F}(E_{1}, E_{2}) for all E_{1}, E_{2} ≤ P, where E_{1}, E_{2} runs over the set of elementary abelian p-subgroups of P. Now one of the ingredients is a property of saturated fusion systems ([4, Theorem B]) which says that if p is odd and G controls p-fusion in F on elementary abelian psubgroups then G = F. Another ingredient is the following theorem which says

30 that control of p-fusion on elementary abelian subgroups happens if and only if i^{∗} is a bijective map. [1, Theorem 2] and [14, Proposition 10.9] are similar statements for group cohomology.

Theorem 1.2. Let G be a saturated fusion subsystem of F defined on the same finite p-group P. Then i^{∗} is surjective. Moreover we have that i^{∗} is an injective

35 map if and only if G controls p-fusion in F on elementary abelian p-subgroups.

To prove Theorem 1.2 we will use Quillen stratification for cohomology of saturated fusion systems given by Markus Linckelmann in [13], for which the proof is the same as for block algebras [11], with some minor adjustments. Since it has not appeared in this form in the literature we state it here for complete-

40 ness. For any subgroup Q of P denote by V_{Q} the maximal ideal spectrum of H^{∗}(Q, k), and set V_{Q}^{+} = V_{Q} \ ⋃_{R<Q}(res_{R}^{Q})^{∗}(V_{R}). Denote by V_{F,Q} and V_{F,Q}^{+} the images of V_{Q} and V_{Q}^{+} in V_{F} under the map r_{F,Q}^{∗} induced by the algebra homomorphism r_{F,Q} : H^{∗}(F) → H^{∗}(Q, k) given by composing the inclusion H^{∗}(F) ⊆ H^{∗}(P, k) with the restriction res_{Q}^{P} : H^{∗}(P, k) → H^{∗}(Q, k).

45 Theorem 1.3. ([13, Theorem 1]) With the notation above, the following hold.

(i) The variety V_{F} is the disjoint union of the locally closed subvarieties V_{F,E}^{+}, where E runs over a set of representatives of the F-isomorphism classes of elementary abelian subgroups of P.

(ii) Let E be an elementary abelian subgroup of P. The group W_{F}(E) = Aut_{F}(E) acts on V_{E}^{+} and the restriction map res_{E}^{P} induces an inseparable isogeny

V_{E}^{+}/W_{F}(E) → V_{F,E}^{+}.

We will not give the proof of Theorem 1.3 from [13], since the way to obtain 50 it is to copy the proof from [11] in the context of saturated fusion systems.

This proof follows in turn very closely Bensons presentation in [3] of parts of Quillens original work in [14], with only additional ingredient the fusion stable bisets whose existence was proved by Broto, Levi, and Oliver in [5].

Remark 1.1. From [16] we know that any saturated fusion system F can be 55 identified with F_{P} (G) for some finite group G containing P as a subgroup, where

in the last category the morphisms are given by conjugation with elements from G. It follows that for any elementary abelian p-subgroup E of P we can identify Aut_{F}(E) with N_{G}(E)/C_{G}(E). Now, arguments similar to [6, Theorem 9.1.7] assure us that Aut_{F}(E) acts freely on V_{E}^{+}.

60 In Section 3 we will prove Theorem 1.1 and 1.2. In Section 4 we give applications of Mislin’s theorem to the case of block algebras of finite groups. It is known that if two blocks are basic Morita equivalent then there is an isomorphism between the defect groups which induces an equivalence between their local categories [17, 7.6.6]. We will prove that in some cases Mislin’s theorem and an inclusion

65 between some subalgebras of stable elements in Hochschild cohomology algebras of group algebras over the defect groups (with respect to the source algebras of the blocks; see [8]) can give us an alternative method to obtain an equality between the associated fusion systems (Brauer categories). Mislin’s Theorem and cohomological techniques have implications for other results involving block

70 algebras of finite groups. For example, under some assumptions (which are required to apply Mislin’s theorem as above) we will show that two block algebras, which are splendid stable equivalent in the sense of Linckelmann [10], have the same fusion systems (Theorem 4.2). One of the main features of this article appears in Section 4 where we show how homological methods have applications

75 in proving old/new results for block algebras of finite groups, although some of these results are already known and proved by specific methods.

2. Saturated fusion systems

The axioms of saturated fusion systems were invented by Puig in early 1990’s. He called them ”Frobenius categories” but his approach was taken up and ex-

80 tended by others and in the process an alternate terminology evolved which is now commonly used. In particular, we call today a saturated fusion system what Puig refers to a Frobenius category. The next definitions which we recall are modified versions of [18], but equivalent to them.

Definition 2.1. ([18], [5]) A fusion system over a p-group S is a category F, 85 where Ob(F) is the set of all subgroups of S and which satisfies the following

two properties for all P, Q ≤ S:

• Hom_{S}(P, Q) ≤ Mor_{F}(P, Q) ≤ Inj(P, Q) and

• each φ ∈ Mor_{F}(P, Q) is the composite of an F-isomorphism followed by an inclusion.

Here Inj(P, Q) is the set of all injective group homomorphisms from Q to P and Hom_{S}(P, Q) is the set of group homomorphisms

{φ ∈ Hom(P, Q)|φ = c_{s} for some s ∈ S such that ^{g}P ≤ Q}

where c_{s} : P → Q is defined by c_{s}(x) = ^{x}s = xsx^{−1} for any x ∈ P. Composition in a fusion system F is always given by the composition of homomorphisms. We usually write Hom_{F}(P, Q) = Mor_{F}(P, Q) to emphasize that morphisms in F are actually group homomorphisms and we set

Aut_{F}(P) = Hom_{F}(P, P).

90 A saturated fusion system is a fusion system satisfying certain axioms which had origins in some properties of fusion in finite groups. The following version of these axioms is due to Roberts and Shpectorov and is taken from [2].

Definition 2.2. ([2, Definition 2.2]) Let F be a fusion system over a p-group S.

95 • Two subgroups P, Q ≤ S are F-conjugate if they are isomorphic as objects of F.

• A subgroup P ≤ S is fully automized in F if Aut_{S}(P) is a Sylow psubgroup of Aut_{F}(P).

• A subgroup P ≤ S is receptive in F if it has the following property: for each Q ≤ S and each φ ∈ Mor_{F}(Q, P) an isomorphism if we set

N_{φ} = {g ∈ N_{S}(Q)|^{φ}c_{g} ∈ Aut_{S}(P)},

then there is \overline{φ} ∈ Hom_{F}(N_{φ}, S) such that \overline{φ}|_{Q} = φ.

100 • A fusion system F over a p-group S is saturated if each subgroup of S is F-conjugate to a subgroup which is fully automized and receptive.

The fusion category F_{S}(G) of a finite group G with morphisms given by conjugation in G and objects the subgroups of a Sylow p-subgroup S in G is the first example of saturated fusion systems. A different, important example which

105 will be given in detail in Section 4 is the saturated fusion system associated to a block algebra b of a finite group G, with morphisms given by conjugation between b-Brauer pairs.

The theory of fusion systems is an emerging area of mathematics and we copy a paragraph from the Introduction of [2] to emphasize this aspect: ”Puig

110 created his theory of Frobenius categories largely as a tool in modular representation theory, motivated in part by work of Alperin and Broué. Later, homotopy theorists used this theory to provide a formal setting for, and prove results about, the p-completed classifying spaces of finite groups. As part of this process, objects called p-local finite groups associated to abstract fusion

115 systems were introduced by Broto, Levi and Oliver in [5]; these also possess interesting p-completed classifying spaces. Finally, local finite group theorists became interested in fusion systems, in part because methods from local group theory proved to be effective in the study of fusion systems, but also because certain results in finite group theory seem to be easier to prove in the category

120 of saturated fusion systems.”

3. Proofs of theorems from Section 1

Proof of Theorem 1.2. Clearly Ker(i) is a nilpotent ideal and H^{∗}(G) is finitely generated as H^{∗}(F)-module. This yields the surjectivity of i^{∗}.

Suppose that G controls p-fusion in F on elementary abelian p-subgroups. Let m_{1}, m_{2} ∈ V_{G} such that i^{∗}(m_{1}) = i^{∗}(m_{2}). By Quillen stratification for V_{G} (Theorem 1.3), there are two elementary abelian p-subgroups E_{1}, E_{2} ≤ P unique up to G-isomorphism and γ_{1} ∈ V_{E_{1}}^{+}, γ_{2} ∈ V_{E_{2}}^{+} such that

m_{1} = r_{G,E_{1}}^{∗}(γ_{1}), m_{2} = r_{G,E_{2}}^{∗}(γ_{2})

hence

(i^{∗} ∘ r_{G,E_{1}}^{∗})(γ_{1}) = (i^{∗} ∘ r_{G,E_{2}}^{∗})(γ_{2}),

and since r_{G,E_{1}} ∘ i = r_{F,E_{1}}, r_{G,E_{2}} ∘ i = r_{F,E_{2}} we obtain that r_{F,E_{1}}^{∗}(γ_{1}) = r_{F,E_{2}}^{∗}(γ_{2}). Quillen stratification for V_{F} (Theorem 1.3) gives us that E_{1} is Fisomorphic to E_{2} hence E_{1} is G-isomorphic to E_{2}. This allow us to choose E_{1} = E_{2} = E such that r_{F,E}^{∗}(γ_{1}) = r_{F,E}^{∗}(γ_{2}). The inseparable isogeny from Theorem 1.3, (ii) is given by r_{F,E}^{∗} so γ_{1}, γ_{2} are in the same orbit of the action of W_{F}(E) on V_{E}^{+}. The control of fusion on elementary abelian p-subgroups assure us that γ_{1}, γ_{2} are in the same orbit of the action of W_{G}(E) on V_{E}^{+}. In conclusion

m_{1} = r_{G,E}^{∗}(γ_{1}) = r_{G,E}^{∗}(γ_{2}) = m_{2}.

Conversely suppose that i^{∗} is injective. First we prove that if E_{1}, E_{2} are F-isomorphic elementary abelian p-subgroups then they are also G-isomorphic. So let E_{1}, E_{2} be F-isomorphic. Then r_{F,E_{1}}^{∗}(V_{E_{1}}^{+}) = r_{F,E_{2}}^{∗}(V_{E_{2}}^{+}), hence

i^{∗}(r_{G,E_{1}}^{∗}(V_{E_{1}}^{+})) = i^{∗}(r_{G,E_{2}}^{∗}(V_{E_{2}}^{+})).

Since i^{∗} is injective we obtain that V_{G,E_{1}}^{+} = V_{G,E_{2}}^{+}, hence E_{1}, E_{2} are G-isomorphic. 125 Secondly we prove that Aut_{F}(E) = Aut_{G}(E) for any elementary abelian

p-subgroup E of P. Since V_{F,E}^{+} = i^{∗}(V_{G,E}^{+}) we obtain that i^{∗} induces a bijection between V_{F,E}^{+} and V_{G,E}^{+}. This bijection, the inclusion Aut_{G}(E) ⊆ Aut_{F}(E), Remark 1.1 and the similar statements for V_{G,E}^{+} give the desired equality; the definitions of the bijection from V_{F,E}^{+} to V_{E}^{+}/Aut_{F}(E) (Theorem 1.3) and of i^{∗}

130 are important for showing this equality.

Finally, let φ ∈ Hom_{F}(E_{1}, E_{2}), which gives us the decomposition

\xymatrix{E_{1} \ar[r]^{φ_{1}} & φ(E_{1}) \ar@{^{(}->}[r] & E_{2}} ,

where φ_{1} : E_{1} → φ(E_{1}) is an isomorphism in F hence, from the above, we have that E_{1} is G-isomorphic to φ(E_{1}). It follows that there is α : E_{1} → φ(E_{1}) an isomorphism in G such that α^{−1} ∘ φ_{1} ∈ Aut_{F}(E_{1}), that is α^{−1} ∘ φ_{1} ∈ Aut_{G}(E_{1}). It is easy to see now that φ ∈ Hom_{G}(E_{1}, E_{2}).

135 Proof of Theorem 1.1. From the hypothesis we obtain that i^{∗} is an F - isomorphism. In particular i^{∗} is bijective. By Theorem 1.2 we get that G controls p-fusion in F on elementary abelian p-subgroups. Hence [4, Theorem B] assure us that G = F.

4. Applications of Mislin’s theorem to block algebras of finite groups

140 Let H, G be two finite groups with a common p-subgroup P. Let b be a block of kG and let c be a block of kH with the same defect group P. Let P_{γ}, P_{δ} be defect pointed groups of G_{{b}}, respectively H_{{c}} and i ∈ γ, j ∈ δ some source idempotents. Let (P, e_{P}) be a maximal (G, b)-Brauer pair associated to P_{γ} and (P, f_{P}) be a maximal (H, c)-Brauer pair associated to P_{δ}. For any

145 subgroup R of P, there is a unique block e_{R} of C_{G}(R) such that Br_{R}(i)e_{R} ≠ 0. Then (R, e_{R}) is a (G, b)-Brauer pair and e_{R} is also the unique block of C_{G}(R) such that (R, e_{R}) ≤ (P, e_{P}). We define F_{(P,e_{P})}(G, b) as the category which has as objects the set of subgroups of P; for any two subgroups R, S of P the set of morphisms from R to S in F_{(P,e_{P})}(G, b) is the set of (necessarily injective)

150 group homomorphisms φ : R → S for which there is an element x ∈ G satisfying φ(u) = xux^{−1} for all u ∈ R and satisfying ^{x}(R, e_{R}) ≤ (S, e_{S}). The category F_{(P,e_{P})}(G, b) is sometimes called the Brauer category of b with respect to the choice of (P, e_{P}) and is a saturated fusion system. The analogous definitions give F_{(P,f_{P})}(H, c). For the rest of this section we assume that F_{(P,f_{P})}(H, c) is

155 a subsystem of F_{(P,e_{P})}(G, b) and p is an odd prime.

We refer the reader to [8] for results regarding transfer maps and stable elements in Hochschild cohomology algebras. Recall that if A, B are two symmetric k-algebras and X is a bounded complex of A − B-bimodules whose components are projective as left and right modules, there is a graded k-linear map t_{X} : HH^{∗}(B) → HH^{∗}(A) called the transfer map associated to X [8, Definition 2.9]. If π_{X} = t_{X}^{0}(1_{B}) ∈ Z(A) (the relatively X-projective element) is invertible then T_{X} = π_{X}^{−1}t_{X} is called the normalized transfer map associated to X. If both π_{X}, π_{X^{∗}} are invertible then T_{X} induces a graded k-algebra isomorphism which by abuse of notation we denote still by T_{X} : HH_{X^{∗}}^{∗}(B) → HH_{X}^{∗}(A), with its inverse T_{X^{∗}}. Here HH_{X}^{∗}(A) is the graded subalgebra of X-stable elements in HH^{∗}(A); more precisely ζ ∈ HH_{X}^{∗}(A) if there is θ ∈ HH^{∗}(B) such that ζ ⊗ Id_{X} = Id_{X} ⊗ θ in Ext_{A⊗B^{op}}(X, X), see [20, Definition 2]. For example, we denote by HH_{ikGi}^{∗}(kP) (respectively HH_{jkHj}^{∗}(kP)) the subalgebra of ikGi-stable (jkHj-stable) elements in the Hochschild cohomology algebra of kP. The kalgebra ikGi (respectively jkHj) is called the source algebra of b (respectively c) and we obtain decompositions as kP − kP -bimodules

ikGi ≅ ⨁_{g∈Y_{G,b}} k[P gP], jkHj ≅ ⨁_{h∈Y_{H,c}} k[P hP]

where Y_{G,b} ⊆ [P \ G/P] and Y_{H,c} ⊆ [P \ H/P], see [22, Theorem 44.3].

Next proposition may be regarded as a tool for checking when two blocks of two finite groups have the same local structure and is a first application of Mislin’s theorem (Theorem 1.1) to block algebras. Recall that δ_{P} : H^{∗}(P, k) →

160 HH^{∗}(kP) is the embedding defined in [8, Proposition 4.5].

Proposition 4.1. With the above assumptions and notations if

HH_{jkHj}^{∗}(kP) ∩ δ_{P} (H^{∗}(P, k)) ⊆ HH_{ikGi}^{∗}(kP)

then F_{(P,f_{P})}(H, c) = F_{(P,e_{P})}(G, b). In particular if ikGi is isomorphic with a direct summand of jkHj as kP −kP -bimodules then F_{(P,f_{P})}(H, c) = F_{(P,e_{P})}(G, b).

Proof. The second statement follows from the first if we use [8, Proposition 3.5, (iv)]. For the first statement, we know from [8, Proposition 5.4] that

δ_{P} (H^{∗}(F_{(P,e_{P})}(G, b))) ⊆ HH_{ikGi}^{∗}(kP).

Moreover [20, Theorem 1] is a nice result which tells us more precisely that

δ_{P} (H^{∗}(F_{(P,e_{P})}(G, b))) = HH_{ikGi}^{∗}(kP) ∩ δ_{P} (H^{∗}(P, k)),
δ_{P} (H^{∗}(F_{(P,f_{P})}(H, c))) = HH_{jkHj}^{∗}(kP) ∩ δ_{P} (H^{∗}(P, k)).

It follows that δ_{P} (H^{∗}(F_{(P,e_{P})}(G, b))) = δ_{P} (F_{(P,f_{P})}(H, c)), hence

H^{∗}(F_{(P,e_{P})}(G, b)) = H^{∗}(F_{(P,f_{P})}(H, c)).

Now, Theorem 1.1 gives us our desired conclusion.

Remark 4.1. If kGb and kHc are two block algebras for which there is an 165 indecomposable direct summand M of the kGb − kHc-bimodule kGi ⊗_{kP} jkH

which induces a Morita equivalence (i.e. there is a Morita equivalence between the blocks induced by a p-permutation bimodule); see [10, Theorem 4.1] or [17, 7.5.1], then we actually have an isomorphism of kP − kP -bimodules between the source algebras. It follows that the above proposition gives an alternative

170 proof (when p is odd) of the fact that a splendid Morita equivalence preserves block fusion systems.

It is well known that if kHc and kGb are splendid stable equivalent then their cohomology algebras are isomorphic H^{∗}(F_{(P,e_{P})}(G, b)) ≅ H^{∗}(F_{(P,f_{P})}(H, c)). With our assumptions Mislin’s theorem applied to block algebras gives a new

175 method to prove that if two block algebras are splendid stable equivalent then the associated fusion systems are the same.

Theorem 4.2. Let b, c be two blocks with the above assumptions. Let X be a bounded complex of kHc − kGb-bimodules whose components are isomorphic to direct sums of direct summands of kHj ⊗_{kQ} ikG, where Q runs over the set of

180 subgroups of P. If X induces a splendid stable equivalence (i.e. X ⊗_{kGb} X^{∗} ≅ kHc ⊕ U_{c}, X^{∗} ⊗_{kHc} X ≅ kGb ⊕ U_{b}, where U_{c} is a bounded complex of projective kHc − kHc-bimodules and U_{b} is a bounded complex of projective kGb − kGbbimodules) then F_{(P,f_{P})}(H, c) = F_{(P,e_{P})}(G, b).

Proof. We mimic the proof of [9, Theorem 5.5., (ii)] to obtain the following commutative diagram

\xymatrix{H^{∗}(F_{(P,e_{P})}(G, b)) \ar[r]^{T_{kGi}∘δ_{P}} \ar@{^{(}->}[d]^{i} & HH_{X^{∗}}^{∗}(kGb) \ar[d]^{T_{X}} \\ H^{∗}(F_{(P,f_{P})}(H, c)) \ar[r]^{T_{kHj}∘δ_{P}} & HH_{X}^{∗}(kHc)} .

where T_{X}, T_{kGi}, T_{kHj} are the normalized transfer maps. Since the projective elements π_{X}, π_{X^{∗}} are invertible ([9, Theorem 5.5, (i)]) the map T_{X} is an isomorphism. For completeness we repeat some of the arguments of that proof. By [8, 3.2.3] we may choose symmetrizing forms such that π_{kGi} = 1_{kGb}, π_{X} = 1_{kHc}, or equivalently T_{kGi} = t_{kGi}, T_{X} = t_{X}. The relatively projective elements π_{kHj}, π_{jkH} are still invertible, [8, Proposition 5.4, (iv)]. In order to show the commutativity of the above diagram we need to show that T_{jkH} ∘T_{X} ∘T_{kGi}∘δ_{P} = δ_{P} ∘ i. This is equivalent to showing that

π_{jkH}^{−1} ∘ t_{jkH} ∘ t_{X} ∘ t_{kGi} ∘ δ_{P} = δ_{P} ∘ i,

hence equivalent to

π_{jkH}^{−1} ∘ t_{jXi} ∘ δ_{P} = δ_{P} ∘ i.

With our notations by [8, Proposition 5.7, (iv)] we know that the map t_{jXi} acts as multiplication by π_{jXi} on δ_{P} (H^{∗}(F_{(P,e_{P})}(G, b))), so all we need to show is that π_{jXi} = π_{jkH}. But this is true since by [8, 3.2] we have

π_{jXi} = t_{jXi}^{0}(1_{kP}) = (t_{jkH}^{0}∘t_{X}^{0}∘t_{kGi}^{0})(1_{kP}) = t_{jkH}^{0}(t_{X}^{0}(1_{kGb})) = t_{jkH}^{0}(1_{kHc}) = π_{jkH}.

We denote by τ_{b} the injective graded k-algebra homomorphism T_{kGi} ∘ δ_{P}. Similarly τ_{c} is T_{kHj} ∘ δ_{P}. By [12, Theorem 1.1] τ_{b}, τ_{c} induce isomorphisms of varieties

τ_{b}^{∗} : X_{kGb} → V_{F_{(P,e_{P} )}(G,b)}
τ_{c}^{∗} : X_{kHc} → V_{F_{(P,f_{P} )}(H,c)}

hence τ_{b}^{∗}, τ_{c}^{∗} are bijective maps; where X_{kGb}, X_{kHc} are the varieties of the Hochschild cohomology algebras HH^{∗}(kGb), HH^{∗}(kHc). The first diagram yields

T_{X} ∘ τ_{b} = τ_{c} ∘ i

hence

τ_{b}^{∗} ∘ T_{X}^{∗} = i^{∗} ∘ τ_{c}^{∗}.

Since T_{X} is an isomorphism it follows that i^{∗} is injective. Now Theorem 1.1 185 give us the conclusion.

In the last remark of this paper we give a new consequence of Mislin’s theorem. We obtain a generalization of a theorem of Watanabe [23, Theorem 2] to the class of all finite groups, not just for p-solvable groups.

Remark 4.2. Let R be a normal subgroup of P such that N_{G}(P) ≤ N_{G}(R). 190 We denote by d the Brauer correspondent of b in N_{G}(R), that is d is the unique

block of N_{G}(R) such that Br_{P} (d) = Br_{P} (b), where Br_{P} is the Brauer homomorphism from (kG)^{P} to kC_{G}(P). Set N = N_{G}(R). Then b = d^{G} and (P, e_{P}) is a maximal (N, d)-Brauer pair. Moreover F_{(P,e_{P})}(N, d) is a subsystem of F_{(P,e_{P})}(G, b). Now from Theorem 1.1 we have that H^{∗}(F_{(P,e_{P})}(N, d)) =

195 H^{∗}(F_{(P,e_{P})}(G, b)) if and only if F_{(P,e_{P})}(N, d) = F_{(P,e_{P})}(G, b).

Acknowledgments. We are grateful to Professor Markus Linckelmann for some email discussions and for the reference [13].

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