Basic Morita equivalences and a conjecture of Sasaki for cohomology of block algebras of finite

groups

CONSTANTIN-COSMIN TODEA Technical University of Cluj-Napoca Department of Mathematics, Str. G. Baritiu 25 Cluj-Napoca 400027, Romania Constantin.Todea@math.utcluj.ro

June 13, 2015

Keywords: finite group; block; group cohomology; Hochschild cohomology; transfer map; basic Morita equivalence

2010 Mathematics Subject Classification: 20C20, 16EXX

Abstract

We show, using basic Morita equivalences between block algebras of finite groups, that the Conjecture of H. Sasaki from [9] is true for a new class of blocks called nilpotent covered blocks. When this Conjecture is true we define some transfer maps between cohomology of block algebras and we analyze them through transfer maps between Hochschild cohomology algebras.

1 Introduction

Let G be a finite group and let k be an algebraically closed field of prime characteristic p > 0 dividing the order of G. Let kGb be a block algebra where b is the primitive idempotent in the center of kG, b ∈ Z(kG), with defect group P. Let P_{γ} be a defect pointed group of G_{{b}}, (P, e_{P}) be a maximal b-Brauer pair associated to P_{γ} and let i ∈ γ be a source idempotent. For any 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 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) 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 equivalent to the Brauer category of b with respect to the choice of (P, e_{P}) and is a saturated fusion system. More than fifteen years ago M. Linckelmann defines in [4] the cohomology algebra of the block b, denoted H^{∗}(G, b, P_{γ}) which can also be viewed now as the cohomology algebra of a fusion system H^{∗}(F_{(P,e_{P})}(G, b)), see [1] for cohomology of fusion systems. The P -interior algebra ikGi is called the source algebra of kGb. As kP − kP -bimodule ikGi is a direct summand of kG, hence there is Y_{G,b,P_{γ}} ⊆ [P \G/P] such that

ikGi ≅ ⨁_{g∈Y_{G,b,Pγ}} k[P gP],

as kP − kP -bimodules, where [P \G/P] is a system of representatives of double cosets of P in G. The set Y_{G,b,P_{γ}} has the property that contains a system of representatives [N_{G}(P_{γ})/P C_{G}(P)] for any block b.

In [9] H. Sasaki defines a linear map as follows

t_{Y_{G,b,Pγ}} : H^{∗}(P, k) → H^{∗}(P, k); [ζ] ↦ ∑_{g∈Y_{G,b,Pγ}} tr_{P ∩^{g}P}^{P} res_{P ∩^{g}P}^{^{g}P}^{g}[ζ],

and he proposes the following conjecture.

Conjecture. With the above notations it follows that

H^{∗}(G, b, P_{γ}) = t_{Y_{G,b,Pγ}} (H^{∗}(P, k)).

For shortness we also use the notation

t_{Y_{G,b,Pγ}} = ∑_{g∈Y_{G,b,Pγ}} t_{g}

where for g ∈ Y_{G,b,P_{γ}} the graded linear map t_{g} is defined by

t_{g} : H^{∗}(P, k) → H^{∗}(P, k), [ζ] ↦ tr_{P ∩^{g}P}^{P} res_{P ∩^{g}P}^{^{g}P}^{g}[ζ].

Since we will we work with more than one block algebra we are forced to use a little cumbersome notation for t_{Y_{G,b,Pγ}} as opposed to the original one from [9]. Broto et al. [1] show that for any saturated fusion system on P there is a P − P -biset that induces a linear map analogous to t_{Y_{G,b,Pγ}} whose image is the cohomology of the saturated fusion system; such a biset is called characteristic biset. For the convenience of the reader we begin with a well-known remark giving a class of blocks for which the above Conjecture is true. The arguments for the proof of this remark are the same with the arguments from [9, Example 2]. For completeness we point out that we denote by F_{P} (N_{G}(P_{γ})) the fusion system on P in N_{G}(P_{γ}) (not necessarily saturated) with objects the subgroups of P and morphisms given by conjugation in N_{G}(P_{γ}); we use^{g}R = gRg^{−1} where R is a subgroup of P and g ∈ G.

Remark 1.1. With the above notations we assume that b is a block for which the inertial group N_{G}(P_{γ}) controls the fusion of b-Brauer pairs, F_{(P,e_{P})}(G, b) ⊆ F_{P} (N_{G}(P_{γ})). Then the above Conjecture is true. In particular, we have the following list of blocks for which the inertial group controls the fusion of b-Brauer pairs:

• P is normal in G; this is [9, Example 1].

• P is abelian; this is [9, Example 2].

• P is a p-group such that the hyperfocal subgroup of P_{γ} is a subgroup in the center of P (in particular this happens when P is abelian); this follows from the proof of [12, Theorem 1,(ii)]. Recall that by [6, 1.7] the subgroup ⟨[U, O^{p}(N_{G}(U, e_{U}))] | U ≤ P ⟩ is called the hyperfocal subgroup of P_{γ}, where O^{p}(N_{G}(U, e_{U})) is the unique smallest normal subgroup of N_{G}(U, e_{U}) of p-power index.

In the main theorem of this article we will show how basic Morita equivalent blocks give new information regarding the image of the above linear map in the Conjecture, if a condition is satisfied. This theorem provides a method for proving Sasaki’s Conjecture for the so-called nilpotent covered blocks. Let b be a block with the above notations and let H be a different finite group. Let c be a block of kH with defect pointed group Q_{δ}. The similar notations and definitions are considered for c: a source idempotent is j ∈ δ, the interior Qalgebra jkHj is the source algebra, the Brauer category is F_{(Q,f_{Q})}(H, c), the cohomology algebra is H^{∗}(H, c, Q_{δ}), etc. Let M be a kG − kH-bimodule such that bMc = M and M is indecomposable considered as k[G × H^{op}]-module. We take P′ ≤ G × H to be the vertex of M and N is a kP′-indecomposable source of M. We say that M defines a basic Morita equivalence between b and c if M gives a Morita equivalence such that the equivalent conditions of [5, Corollary 7.4] are satisfied. For example this happens if p does not divides the k-dimension of N or if S = End_{k}(N) is a Dade P′-algebra.

Theorem 1.2. Let b, c be two blocks with the above notations. We assume that b is basic Morita equivalent to c. If Y_{H,c,Q_{δ}} = [N_{H}(Q_{δ})/QC_{H}(Q)] then

Imt_{Y_{G,b,Pγ}} ⊆ H^{∗}(P, k)^{N_{G}(P_{γ})/P C_{G}(P)}.

A block c with normal defect group in H satisfies the condition

Y_{H,c,Q_{δ}} = [N_{H}(Q_{δ})/QC_{H}(Q)]

from Theorem 1.2. In particular the same arguments as for the proof of Remark 1.1 assure us that for these blocks the Conjecture of Sasaki is true. We say that b is a nilpotent covered block if there is G̃ a finite group such that G is a normal subgroup of G̃ and there is b̃ a nilpotent block of kG̃ which covers b. We know that e_{P} is lifted to a block ê_{P} of the inertial group with the same defect P and [7, Corrollary 4.3] assure us that a nilpotent covered block is basic Morita equivalent to ê_{P} as block of N_{G}(P_{γ}) which has the same defect pointed group P_{γ}. Since b is basic Morita equivalent to ê_{P} we know from [5] that F_{(P,e_{P})}(G, b) is isomorphic to F_{(P,e_{P})}(N_{G}(P_{γ}), ê_{P}) as finite categories hence by Theorem 1.2 we obtain

Imt_{Y_{G,b,Pγ}} ⊆ H^{∗}(P, k)^{N_{G}(P_{γ})/P C_{G}(P)} = H^{∗}(N_{G}(P_{γ}), ê_{P}, P_{γ}) ≅ H^{∗}(G, b, P_{γ}).

Since the last isomorphism is an isomorphism of graded vector spaces of finite type we obtain the next corollary.

Corollary 1.3. For any nilpotent covered block the Conjecture of Sasaki is true.

It might be possible to obtain a shorter proof for the above corollary since basic Morita equivalence preserves the Brauer categories and one of the blocks has normal defect group but we prefer the use of Theorem 1.2 to emphasize a new method. Section 2 is devoted for the proof of this theorem. The aim of Section 3 is to fill a gap in the literature, since for experts the fact that Hochschild cohomology of group algebras is a Mackey functor is known. So it is worth spending some time to analyze and to give an explicit approach for the maps defining the Mackey functor in Section 3, using bar resolution. The only new result is Proposition 3.9 which is used in the proof of Theorem 1.5, theorem which will be exposed below. We propose an approach for defining some transfer maps between cohomology of block algebras, maps which we define below. For this definition we need the existence of the map t_{Y_{G,b,Pγ}}. In Section 4 we will study these maps. Let b, c be blocks of kG, respectively kH with defect groups P respectively Q such that Q ≤ P and H ≤ G. We take P_{γ}, respectively Q_{δ} as defect pointed groups and we denote by (Q, f_{Q}) a maximal (H, c)-Brauer pair associated to Q_{δ}. Again j ∈ δ is a source idempotent for c, the fusion system associated to c is F_{(Q,f_{Q})}(H, c) and jkHj is the source algebra. So, if the Conjecture is true for b we can complete the following commutative diagram

[figure: H^{∗}(H, c, Q_{δ}); ^{tr{b \atop c}}; H^{∗}(G, b, P_{γ}); t_{Y_{G,b,Pγ}}; H^{∗}(Q, k); _{tr{P \atop Q}}; H^{∗}(P, k)] .

Definition 1.4. With the above notations we assume that the Conjecture is true for b. We define the graded k-linear map

tr_{c}^{b} : H^{∗}(H, c, Q_{δ}) → H^{∗}(G, b, P_{γ}); [ζ] ↦ tr_{c}^{b}([ζ]) = t_{Y_{G,b,Pγ}} (tr_{Q}^{P}([ζ])).

Explicitly tr_{c}^{b}([ζ]) = ∑_{g∈Y_{G,b,Pγ}} tr_{P ∩^{g}P}^{P} res_{P ∩^{g}P}^{^{g}P}^{g}(tr_{Q}^{P}([ζ])). We call tr_{c}^{b} a transfer map in cohomology of block algebras.

Some ideas for defining these transfer maps are suggested in [8]. For defining a restriction map between cohomology of block algebras, we need to assume that F_{(Q,f_{Q})}(H, c) is a subsystem of F_{(P,e_{P})}(G, b). In Section 4 and in the rest of this Introduction we work with blocks satisfying this condition. This inclusion and res_{Q}^{P} induce a well-defined restriction map between cohomology algebras of blocks, denoted

res_{c}^{b} : H^{∗}(G, b, P_{γ}) → H^{∗}(H, c, Q_{δ}); [ζ] ↦ res_{c}^{b}([ζ]) = res_{Q}^{P}([ζ]).

In order that the definition of the transfer map tr_{c}^{b} to be reasonable we need that good properties such as Frobenius reciprocity and transitivity law

tr_{c}^{b} ∘ t_{Y_{H,c,P_{δ}}} = t_{Y_{G,b,Pγ}}

to hold. We prove Frobenius reciprocity in Proposition 4.1 but the law of transitivity is proved in Proposition 4.2 for b and c blocks such that

Ker t_{Y_{H,c,P_{δ}}} ⊆ Ker t_{Y_{G,b,Pγ}} .

In the next theorem, which will be proved in Section 4, we try to understand transfer maps in block cohomology via transfer maps between Hochschild cohomology of block algebras, which are in Brauer correspondence; therefore from now we take Q = P. For the rest of Section 4 and of this Introduction let H be a subgroup of G containing P C_{G}(P) where P is the common defect group of b and c, where c is the Brauer correspondent of b. We choose i ∈ γ and j ∈ δ such that the source modules A = kGi, B = kHj are in Green correspondence with respect to (G×P^{op}, ΔP, H×P^{op}). We denote by L = L(kGb, kHc) the Green correspondent of kHc, viewed as an indecomposable kH − kH-bimodule with respect to (G × H^{op}, ΔP, H × H^{op}). In [4] Linckelmann defines the subalgebra of stable elements in the Hochschild cohomology algebra of a symmetric algebra and transfer maps between Hochschild cohomology of symmetric algebras. For example, since ikGi is a kP − kP -bimodule, projective as left and right module, there is a transfer map associated to ikGi denoted t_{ikGi} : HH^{∗}(kP) → HH^{∗}(kP). We denote by T the transfer t_{ikGi} restricted to HH_{B^{∗}}^{∗}(kP) ∩ δ_{P} (H^{∗}(P, k)) (which is isomorphic to H^{∗}(H, c, P_{δ}) by [9, Theorem 1]). Recall that HH_{B^{∗}}^{∗}(kP) is the subalgebra of B^{∗}-stable elements; that is [ζ] ∈ HH_{B^{∗}}^{∗}(kP) if there is [θ] ∈ HH^{∗}(kHc) such that [ζ]⊗_{kP} [Id_{B^{∗}}] = [Id_{B^{∗}}]⊗_{kHc}[θ] in Ext_{kP ⊗_{k}(kHc)^{op}}(B^{∗}, B^{∗}), see [9, Definition 2].

Theorem 1.5. With the above conditions we assume that F_{(P,f_{P})}(H, c) ⊆ F_{(P,e_{P})}(G, b). If we assume that the Conjecture is true for b then the following diagram of graded k-linear maps is commutative

[figure: H^{∗}(H, c, P_{δ}); R_{B}∘T ∘δ_{P}; _{tr{b \atop c}}; HH_{L^{∗}}^{∗}(kHc); R_{L}; H^{∗}(G, b, P_{γ}); R_{A}∘δ_{P}; HH_{L}^{∗}(kHc)] .

The homomorphisms of graded k-algebras R_{A}, R_{B} and R_{L} are obtained by restricting the normalized transfer maps T_{A}, T_{B} and T_{L}, respectively, to some subalgebras of stable elements, see [4, Theorem 3.6]. For basic facts in block theory we refer to [11] and for properties and notations regarding transfer maps in Hochschild cohomology we refer to [4] and [9].

2 Proof of Theorem 1.2

We denote by C | D the fact that C is isomorphic to a direct summand of D,

where C, D are some kG-modules.

Proof. Since b is basic Morita equivalent to c, by [5, Corollary 7.4] we can identify P = Q, hence P_{γ} ≤ G_{{b}} is a defect pointed group with i ∈ γ and P_{δ} ≤ H_{{c}} is a defect pointed group with j ∈ δ. By [5, Theorem 6.9] we know that we can construct two embeddings of P -interior algebras

ikGi → S ⊗ jkHj

and

jkHj → S^{op} ⊗ ikGi.

In particular we obtain

ikGi | S ⊗ jkHj  (1)

as k[P ×P^{op}]-module. Let x ∈ Y_{G,b,P_{γ}} then k[P xP] is an indecomposable direct summand of ikGi as k[P × P^{op}]-module, hence by (1) we get that

k[P xP] | ⨁_{y∈Y_{H,c,P_{δ}}}(S ⊗ k[P yP])

as k[P × P^{op}]-module. It follows that there is y ∈ Y_{H,c,P_{δ}} such that

k[P xP] | S ⊗ k[P yP]

as k[P × P^{op}]-module. It is known that k[P yP] is a trivial source module of vertex ^{(y,1)}Δ(^{y^{−1}}P ∩ P) which satisfies

k[P yP] | Ind_{^{(y,1)}Δ(^{y^{−1}} P ∩P)}^{P ×P^{op}}k;

the similar statements are true for k[P xP]. It follows that

k[P xP] | S ⊗ Ind_{^{(y,1)}Δ(^{y^{−1}} P ∩P)}^{P ×P^{op}}k

as k[P × P^{op}] and since we have the k[P × P^{op}]-modules isomorphism

S ⊗ Ind_{^{(y,1)}Δ(^{y^{−1}} P ∩P)}^{P ×P^{op}}k ≅ Ind_{^{(y,1)}Δ(^{y^{−1}} P ∩P)}^{P ×P^{op}}(Res_{^{(y,1)}Δ(^{y^{−1}} P ∩P)}^{P ×P^{op}}(S))

we get that k[P xP] is ^{(y,1)}Δ(^{y^{−1}}P ∩ P)-projective. But its vertex is

(x,1)Δ(^{x^{−1}}P ∩ P)

so we obtain that there is (a, b) ∈ P × P^{op} such that

(x,1)Δ(^{x^{−1}}P ∩ P) ≤ ^{(a,b)(y,1)}Δ(^{y^{−1}}P ∩ P).

It follows that for any s ∈ ^{x^{−1}}P ∩ P there is a unique elements t ∈ ^{y^{−1}}P ∩ P such that

xs = ^{ay}t, s^{−1} = (b · t · b^{−1})^{−1}

where b · t · b^{−1} = b^{−1}tb in P^{op}, thus

xs = ^{ay}t, s^{−1} = ^{b^{−1}}t^{−1}.

We obtain ^{x}s = ^{ayb}s for any s ∈ ^{x^{−1}}P ∩ P, hence there is c ∈ C_{G}(P ∩^{x}P) such that

ayb = cx.  (2)

Let t_{x}([ζ]) ∈ Imt_{x} for some [ζ] ∈ H^{∗}(P, k). Then we have

t_{x}([ζ]) = tr_{P ∩^{x}P}^{P} res_{P ∩^{x}P}^{^{x}P}^{x}[ζ]
= tr_{P ∩^{x}P}^{P}^{c}(res_{P ∩^{x}P}^{^{x}P}^{x}[ζ]) (∵ c ∈ C_{G}(P ∩^{x}P))
= tr_{P ∩^{x}P}^{P} res_{P ∩^{x}P}^{^{cx}P}^{cx}[ζ])
= tr_{P ∩^{x}P}^{P} res_{P ∩^{x}P}^{^{ayb}P}^{ayb}[ζ]) (∵ (2))
= tr_{P ∩^{x}P}^{P}^{a}(res_{^{a^{−}1}(P ∩^{x}P)}^{^{y}P}^{y}[ζ]) (∵ b ∈ P)
= tr_{^{a^{−1}} (P ∩^{x}P)}^{P}(res_{^{a^{−}1}(P ∩^{x}P)}^{^{y}P}^{y}[ζ]) (∵ a ∈ P)
= |P : ^{a^{−1}}(P ∩^{x}P)|t_{y}([ζ]) (∵ y ∈ N_{H}(P_{δ}))

Since P = ^{a^{−1}}(P ∩^{x}P) if and only if x ∈ N_{G}(P_{γ}) we obtain that

t_{x}([ζ]) = t_{y}([ζ])

for some y which depends on x if x ∈ N_{G}(P_{γ}) and t_{x}([ζ]) is 0 if x ∉ N_{G}(P_{γ}). It follows that for any [η] ∈ Imt_{Y_{G,b,Pγ}} there is [ζ] ∈ H^{∗}(P, k) such that

[η] = ∑_{x∈Y_{G,b,Pγ}} |P : ^{a^{−1}}(P ∩^{x}P)|t_{y}([ζ]) = ∑_{x∈[N_{G}(P_{γ})/P C_{G}(P)]} t_{x}([ζ])

= tr_{P C_{G}(P)}^{N_{G}(P_{γ})}[ζ],

which is what we need.

3 Hochschild cohomology of group algebras as Mackey functor

In this section we use transfer maps between Hochschild cohomology algebras of symmetric algebras, defined originally in [4], to give a structure of Mackey functor for Hochschild cohomology of group algebras. We will work in this section under the general assumption that k is a commutative ring. Let H be a subgroup of G and let g ∈ G. For shortness we denote by _{H}kG_{G} the kH − kGbimodule structure on kG given by multiplication in G, with its k-dual _{G}kG_{H}. Also we consider _{^{g}H}k[gH]_{H} to be the (k[^{g}H] − kH)-bimodule given by

gh_{1}g^{−1}(gh)h_{2} = gh_{1}hh_{2},

for any h_{1}, h_{2}, h ∈ H. The following notations are used

r_{H}^{G} = t_{_{H}kG_{G}}, t_{H}^{G} = t_{_{G}kG_{H}} , c_{g,H} = t_{_{^{g} H}k[gH]_{H}} ,

hence we have the following graded k-linear maps

r_{H}^{G} : HH^{∗}(kG) → HH^{∗}(kH);
t_{H}^{G} : HH^{∗}(kH) → HH^{∗}(kG);
c_{g,H} : HH^{∗}(kH) → HH^{∗}(k[^{g}H]);

which can be viewed as: restriction, transfer and conjugation maps. The next well-known lemma is straightforward.

Lemma 3.1. Let K, H ≤ G and g ∈ G. Then

kK ⊗_{k[K∩^{g}H]} k[gH] ≅ k[KgH]

as (kK − kH)-bimodules.

In the next proposition we verify that the next quadruple is a Mackey functor, see [11, §53]:

(HH^{∗}(kH), r_{H}^{G}, t_{H}^{G}, c_{g,H})_{H≤G,g∈G}.

Proposition 3.2. Let K ≤ H ≤ G and g, h ∈ G. The following statements are true.

i) r_{K}^{H} ∘ r_{H}^{G} = r_{K}^{G}, t_{H}^{G} ∘ t_{K}^{H} = t_{K}^{G};

ii) r_{H}^{H} = t_{H}^{H} = Id_{HH^{∗}(kH)};

iii) c_{gh,H} = c_{g,^{h}H} ∘ c_{h,H};

iv) c_{h,H} = Id_{HH^{∗}(kH)} if h ∈ H;

v) c_{g,K} ∘ r_{K}^{H} = r_{^{g}K}^{^{g}H} ∘ c_{g,H} and c_{g,H} ∘ t_{K}^{H} = t_{^{g}K}^{^{g}H} ∘ c_{g,K};

vi) r_{K}^{G} ∘t_{H}^{G} = ∑_{g∈[K\G/H]} t_{K∩^{g}H}^{K} ∘r_{K∩^{g}H}^{^{g}H} ∘c_{g,H,} where [K\G/H] is a system of representatives of double cosets KgH with g ∈ G.

Proof. Statements ii) and iv) are an easy exercise if we use [3, Proposition 2.7 (4)]. The second part of statements (i) and (v) are analogous to the first part and are left as an exercise. The rest of the proof is a consequence of [3, Proposition 2.7 (1)] and some bimodule isomorphisms. For statement(i) we have

r_{K}^{H} ∘ r_{H}^{G} = t_{_{K}kH_{H}} ∘ t_{_{H}kG_{G}} = t_{_{K}kH⊗_{kH}kG_{G}} = r_{K}^{G}.

For statement iii) we have

c_{gh,H} = t_{_{^{gh}H}k[ghH]_{H}};
c_{g,^{h}H} ∘ c_{h,H} = t_{_{^{gh}H}k[g(^{h}H)]_{^{h}H}} ∘ t_{_{^{h}H}k[hH]_{H}} = t_{_{^{gh}H}k[g(^{h}H)]⊗_{k[^{h}H]}k[hH]_{H}};

Since

k[g(^{h}H)] ⊗_{k[^{h}H]} k[hH] ≅ k[ghH], ghxh^{−1} ⊗ hy ↦ ghxy,

for any x, y ∈ H, is an isomorphism of (k[^{gh}H] − kH)-bimodules, we obtain statement iii). Statement v) is similar to statement iii) and is left for the reader. For statement vi) we have

r_{K}^{G} ∘ t_{H}^{G} = t_{_{K}kG_{G}} ∘ t_{_{G}kG_{H}} = t_{_{K}kG_{H}};

∑_{g∈[K\G/H]} t_{K∩^{g}H}^{K} ∘ r_{K∩^{g}H}^{^{g}H} ∘ c_{g,H} = ∑_{g∈[K\G/H]} t_{_{K}kK⊗_{k[K∩^{g} H]}k[^{g}H]⊗_{k[^{g} H]}k[gH]_{H}}

= ∑_{g∈[K\G/H]} t_{_{K}k[KgH]_{H}}

= t_{_{K}kG_{H}} ,

where the second equality holds by Lemma 3.1 and the last equality is given by [4, Proposition 2.11].

Remark 3.3. The above proposition is generalized in [2] from group algebras to some fully group-graded algebras which are symmetric k-algebras and satisfies a condition which states that all fully group-graded subalgebras with respect to subgroups must be parabolic subalgebras.

In [3, Proposition 2.5] the authors give explicit definitions for transfer maps between Hochschild cohomology algebras of symmetric algebras in terms of chain maps using the bar resolution. It is worth spending some time in giving these technical definitions for the above maps. For each of the following map we find a suitable set of generators and some maps (denoted φ_{i} in [3]) as in [3, Proposition 2.5], but we omit these details which are left for the reader. Let n ≥ 0 be an integer and denote by Bar_{∗}(kG) the bar resolution of kG as (kG)^{e}-module. Then (C^{∗}(kG), d) is the Hochschild cochain complex where

C^{n}(kG) = Hom_{(kG)^{e}}(Bar_{n}(kG), kG) ≅ Hom_{k}((kG)^{⊗n}, kG),

and d are differentials induced by the bar resolution. It is well known that

HH^{n}(kG) = H^{n}(C^{∗}(kG), d).

Remember that kG, kH, k[^{g}H] are symmetric k-algebras and we denote by s_{G}, s_{H}, s_{^{g}H} their symmetrizing forms.

3.4. Restriction map:

r_{H}^{G} : HH^{∗}(kG) → HH^{∗}(kH), [f] ↦ r_{H}^{G}([f]) = [r_{H}^{G}(f)],

where

r_{H}^{G}(f)(h_{1} ⊗ … ⊗ h_{n}) = ∑_{h∈H} s_{G}(h^{−1}f(h_{1} ⊗ … ⊗ h_{n}))h,

for h_{1}, …, h_{n} ∈ H and f ∈ C^{n}(kG).

3.5. Conjugation map:

c_{g,H} : HH^{∗}(kH) → HH^{∗}(k[^{g}H]), [f] ↦ c_{g,H}([f]) = [c_{g,H}(f)],

where

c_{g,H}(f)(^{g}h_{1} ⊗ … ⊗^{g} h_{n}) = ∑_{h∈H} s_{H}(h^{−1}f(h_{1} ⊗ … ⊗ h_{n}))ghg^{−1}

= gf(h_{1} ⊗ … ⊗ h_{n})g^{−1},

for h_{1}, …, h_{n} ∈ H and f ∈ C^{n}(kH).

3.6. Transfer map:

Let {x_{i}}_{1≤i≤l} be a system of representatives of left cosets of H in G. Let φ_{i} : kG → kH be the right kH-module homomorphisms defined by φ_{i}(g) = h if g = x_{i}h ∈ x_{i}H, and φ_{i}(g) = 0 if g ∉ x_{i}H, where x_{i} such that 1 ≤ i ≤ l.

t_{H}^{G} : HH^{∗}(kH) → HH^{∗}(kG), [f] ↦ t_{H}^{G}([f]) = [t_{H}^{G}(f)],

where

t_{H}^{G}(f)(g_{1} ⊗ … ⊗ g_{n}) = ∑_{1≤i_{0},i_{1},...,i_{n}≤l} x_{i_{1}}f(φ_{i_{1}}(g_{1}x_{i_{0}}) ⊗ … ⊗ φ_{i_{n}}(g_{n}x_{i_{0}}))x_{i_{0}}^{−1},

for g_{1}, …, g_{n} ∈ G and f ∈ C^{n}(kH).

In [4] Linckelmann gives the diagonal induction δ_{G} : H^{∗}(G, k) → HH^{∗}(kG) as an injective graded k-algebra homomorphisms in language of chain homotopy maps. We prefer to use the bar resolution for group cohomology. Recall that H^{n}(G, k) = H^{n}(C^{∗}(G, k), ∂^{∗}) where

C^{n}(G, k) = {f | f : G^{n} → k is a set map},

and ∂^{n} : C^{n}(G, k) → C^{n+1}(G, k) is defined for any g_{1}, …, g_{n+1} ∈ G by

∂^{n}(f)(g_{1}, …, g_{n+1}) =

g_{1}f(g_{2}, …, g_{n+1}) + ∑_{i=1}^{n}(−1)^{i}f(g_{1}, …, g_{i}g_{i+1}, …, g_{n+1}) + (−1)^{n+1}f(g_{1}, …, g_{n}).

3.7. Diagonal induction:

δ_{G} : H^{∗}(G, k) → HH^{∗}(kG), [f] ↦ δ_{G}([f]) = [δ_{G}(f)],

where

δ_{G}(f) ∈ C^{n}(kG), δ_{G}(f)(g_{1} ⊗ … ⊗ g_{n}) = f(g_{1}, …, g_{n})g_{1} … g_{n},

for f ∈ C^{n}(G, k) and g_{1}, …, g_{n} ∈ G. Notice that δ_{G} can be obtained in these terms as δ_{G} = α^{∗} ∘ θ_{1_{G}}^{∗}, where θ_{1_{G}}^{∗} : H^{∗}(G, k) → H^{∗}(G, kG) is defined in [10, p.138] and α^{∗} : H^{∗}(G, kG) → HH^{∗}(kG) is the isomorphism obtained from

α^{∗} : C^{∗}(G, kG) → C^{∗}(kG, kG)

(by abuse of notation) given by

f ↦ α^{n}(f), α^{n}(f)(g_{1} ⊗ … ⊗ g_{n}) = f(g_{1}, …, g_{n})g_{1} … g_{n},

for f ∈ C^{n}(G, kG) and α^{n}(f) ∈ C^{n}(kG, kG).

By [13, Proposition 25.1] the conjugation map in cohomology of groups can be defined with bar resolution explicitly.

3.8. Conjugation map for group cohomology:

g· : H^{∗}(H, k) → H^{∗}(^{g}H, k), [f] ↦^{g} [f] = [^{g}f],

where

gf ∈ C^{n}(^{g}H, k), ^{g}f(^{g}h_{1}, …,^{g} h_{n}) = gf(h_{1}, …, h_{n}) = f(h_{1}, …, h_{n}),

for f ∈ C^{n}(H, k) and h_{1}, …, h_{n} ∈ H.

Proposition 3.9. With the above notations the following diagram is commutative

\xymatrix{H^{∗}(H, k) \ar[r]^{^{g}·} \ar[d]^{δ_{H}} & H^{∗}(^{g}H, k) \ar[d]^{δ_{^{g} H}} \\ HH^{∗}(kH) \ar[r]^{c_{g,H}} & HH^{∗}(k[^{g}H])} .

Proof. Let f ∈ C^{n}(H, k),^{g} h_{1}, …,^{g} h_{n} ∈ ^{g}H. The following hold

c_{g,H}(δ_{H}(f))(^{g}h_{1} ⊗ … ⊗^{g}h_{n}) = gδ_{H}(f)(h_{1} ⊗ … ⊗ h_{n})g^{−1}  (∵ 3.5)

= gf(h_{1}, …, h_{n})h_{1} … h_{n}g^{−1} (∵ 3.7)

= f(h_{1}, …, h_{n}) ^{g}(h_{1} … h_{n});
δ_{^{g}H}(^{g}f)(^{g}h_{1} ⊗ … ⊗^{g}h_{n}) = (^{g}f)(^{g}h_{1}, …,^{g} h_{n}) ^{g}h_{1} …^{g} h_{n}  (∵ 3.7)

= (gf(h_{1}, …, h_{n})) ^{g}(h_{1} … h_{n}) (∵ 3.8)

= f(h_{1}, …, h_{n}) ^{g}(h_{1} … h_{n}).

Remark 3.10. The above proposition is an analogue of [4, Propositions 4.6, 4.7] for conjugation map in group cohomology. Alternative proofs for [4, Propositions 4.6, 4.7] can be given using the explicit descriptions from 3.4, 3.6 and 3.7. Although the elegant way to define the above transfer maps are given in [4] it is our strong belief that the explicit approach given in this section could prove to be useful in further applications.

4 Transfer maps between cohomology algebras of block algebras of finite groups

First we give the reciprocity laws in the following proposition.

Proposition 4.1. Let b, c be blocks as above such that the Conjecture is true for b. Let [ζ] ∈ H^{∗}(G, b, P_{γ}), [τ] ∈ H^{∗}(H, c, Q_{δ}). The following statements hold.

(i) tr_{c}^{b}(res_{c}^{b}([ζ])[τ]) = [ζ]tr_{c}^{b}([τ]);

(ii) tr_{c}^{b}([τ]res_{c}^{b}([ζ])) = tr_{c}^{b}([τ])[ζ].

Proof. Since (ii) is analogue to (i) we only prove (i).

tr_{c}^{b}(res_{c}^{b}([ζ])[τ]) = ∑_{g∈Y_{G,b}} tr_{P ∩^{g}P}^{P} (res_{P ∩^{g}P}^{^{g}P}^{g}(tr_{Q}^{P}(res_{Q}^{P}([ζ])[τ])))

= ∑_{g∈Y_{G,b}} tr_{P ∩^{g}P}^{P} (res_{P ∩^{g}P}^{^{g}P}^{g}([ζ]tr_{Q}^{P}([τ]))

= ∑_{g∈Y_{G,b}} tr_{P ∩^{g}P}^{P} (res_{P ∩^{g}P}^{^{g}P} (^{g}[ζ])res_{P ∩^{g}P}^{^{g}P} (^{g}tr_{Q}^{P}([τ])))

= ∑_{g∈Y_{G,b}} tr_{P ∩^{g}P}^{P} (res_{P ∩^{g}P}^{P} ([ζ])res_{P ∩^{g}P}^{^{g}P} (^{g}tr_{Q}^{P}([τ])))

= ∑_{g∈Y_{G,b}}[ζ]tr_{P ∩^{g}P}^{P} (res_{P ∩^{g}P}^{^{g}P} (^{g}tr_{Q}^{P}([τ])))

= [ζ]tr_{c}^{b}([τ]),

where the fourth equality is true since [ζ] ∈ H^{∗}(G, b, P_{γ}).

From Proposition 4.1 we obtain the transitivity laws when a supplementary condition is satisfied.

Proposition 4.2. We keep the above assumptions.

(a) If Q < P then tr_{c}^{b} ∘ res_{c}^{b} = 0;

(b) If Q = P then res_{c}^{b} is the inclusion map and tr_{c}^{b}(res_{c}^{b}([ζ])) = \frac{dim_{k}(ikGi)}{|P |}[ζ] for any [ζ] ∈ H^{∗}(G, b, P_{γ}). Moreover in this case if the Conjecture is also true for c and Ker t_{Y_{H,c,P_{δ}}} ⊆ Ker t_{Y_{G,b,Pγ}} the following diagram is commutative

\xymatrix{H^{∗}(H, c, P_{δ}) \ar@{->>}[rr]^{^{tr{b \atop c}}} & & H^{∗}(G, b, P_{γ}) \\ & H^{∗}(P, k) \ar@{->>}[ul]^{t_{Y_{H,c,P_{δ}}}} \ar@{->>}[ur]_{t_{Y_{G,b,Pγ}}}} .

Proof. Let [1] be the unity of H^{∗}(G, b, P_{γ}) and H^{∗}(H, c, Q_{δ}). It is clear from Definition 1.4 that

tr_{c}^{b}([1]) = ∑_{g∈Y_{G,b,Pγ}} tr_{P ∩^{g}P}^{P} res_{P ∩^{g}P}^{^{g}P}^{g}(tr_{Q}^{P}([1]))

is equal to 0 if Q < P or it is equal to

∑_{g∈Y_{G,b,Pγ}} |P : P ∩^{g}P |[1] = \frac{dim_{k} ikGi}{|P |}[1],

if Q = P. Then, by Proposition 4.1, (ii) we have

tr_{c}^{b}(res_{c}^{b}([ζ])) = tr_{c}^{b}([1])[ζ]

which is equal to 0 if Q < P or it is equal to \frac{dim_{k} ikGi}{|P |}[ζ] if Q = P.

Since the Conjecture holds for both b and c the next two short exact sequences split as sequences of H^{∗}(G, b, P_{γ})-modules, respectively of H^{∗}(H, c, P_{δ})- modules; see [9, Remark 1]

\xymatrix{0 \ar[r] & Kert_{Y_{G,b,Pγ}} \ar@{^{(}->}[r] & H^{∗}(P, k) \ar@{->>}[r]^{t_{Y_{G,b,Pγ}}} & H^{∗}(G, b, P_{γ}) \ar[r] & 0},

\xymatrix{0 \ar[r] & Kert_{Y_{H,c,P_{δ}}} \ar@{^{(}->}[r] & H^{∗}(P, k) \ar@{->>}[r]^{t_{Y_{H,c,P_{δ}}}} & H^{∗}(H, c, P_{δ}) \ar[r] & 0},

hence

H^{∗}(P, k) = Ker t_{Y_{H,c,P_{δ}}} ⨁ H^{∗}(H, c, P_{δ}) = Ker t_{Y_{G,b,Pγ}} ⨁ H^{∗}(G, b, P_{γ})

as graded k-vector spaces. Now we take

[ζ] = [ζ_{1}] + [ζ_{2}] ∈ Ker t_{Y_{H,c,P_{δ}}} ⨁ H^{∗}(H, c, P_{δ})

and we obtain

tr_{c}^{b}(t_{Y_{H,c,P_{δ}}} ([ζ_{1}] + [ζ_{2}])) = tr_{c}^{b}([ζ_{2}]) = t_{Y_{G,b,Pγ}} ([ζ_{2}]).

Since Ker t_{Y_{H,c,P_{δ}}} ⊆ Ker t_{Y_{G,b,Pγ}} we get

t_{Y_{G,b,Pγ}} ([ζ_{1}] + [ζ_{2}]) = t_{Y_{G,b,Pγ}} ([ζ_{2}]).

Remark 4.3. From Proposition 4.2, (ii) we can notice the fact that the name ”transfer map” in Definition 1.4 is more suitable for blocks b, c for which the Conjecture of Sasaki is true and for which we have Kert_{Y_{H,c,P_{δ}}} ⊆ Kert_{Y_{G,b,Pγ}}.

We end with the proof of Theorem 1.5.

Proof of Theorem 1.5 First we prove the commutativity of the next diagram

\xymatrix{H^{∗}(H, c, P_{δ}) \ar[r]^{δ_{P}} \ar[d]^{_{tr{b \atop c}}} & HH^{∗}(kP) \ar[d]^{t_{ikGi}} \\ H^{∗}(G, b, P_{γ}) \ar[r]^{δ_{P}} & HH^{∗}(kP).} .

Let [ζ] ∈ H^{∗}(H, c, P_{δ}). The following equalities hold

(δ_{P} ∘ tr_{c}^{b})([ζ]) = ∑_{g∈Y_{G,b,Pγ}} (δ_{P} ∘ tr_{P ∩^{g}P}^{P})(res_{P ∩^{g}P}^{^{g}P} (^{g}[ζ])

= ∑_{g∈Y_{G,b,Pγ}} (t_{_{P} kP_{P ∩^{g} P}} ∘ δ_{P} ∘ res_{P ∩^{g}P}^{^{g}P})(^{g}[ζ])

= ∑_{g∈Y_{G,b,Pγ}} (t_{_{P} kP_{P ∩^{g} P}} ∘ t_{_{P ∩^{g} P} k[^{g}P]_{^{g} P}} ∘ δ_{P})(^{g}[ζ])

= ∑_{g∈Y_{G,b,Pγ}} t_{_{P} kP ⊗_{k[P ∩^{g} P]}k[^{g}P]_{^{g} P}} (c_{g,P} (δ_{P} ([ζ])))

= ∑_{g∈Y_{G,b,Pγ}} t_{_{P} kP ⊗_{k[P ∩^{g} P]}k[^{g}P]⊗_{k[^{g} P]}k[gP]_{P}} (δ_{P} ([ζ]))

= ∑_{g∈Y_{G,b,Pγ}} t_{_{P} k[P gP]_{P}} (δ_{P} ([ζ]))

= (t_{ikGi} ∘ δ_{P})([ζ]),

where the fourth and sixth equality follow from Proposition 3.9 and Lemma 3.1; the other equalities are straightforward applications of properties of transfer maps from [4]. Since by [9, Theorem 1]

δ_{P} : H^{∗}(H, c, P_{δ}) → HH_{B^{∗}}^{∗}(kP) ∩ δ_{P} (H^{∗}(P, k))

is an isomorphism and

F_{(P,f_{P})}(H, c) ⊆ F_{(P,e_{P})}(G, b),

the above diagram and [9, Theorem 2] assure us that

Im(T) ⊆ HH_{A^{∗}⊗_{kGb}L⊗_{kHc}B}^{∗}(kP).

Consequently, the above diagram gives us

[figure: H^{∗}(H, c, P_{δ}); δ_{P}; _{tr{b \atop c}}; HH_{B^{∗}}^{∗}(kP) ∩ δ_{P} (H^{∗}(P, k)); T; HH_{B^{∗}⊗_{kHc}L^{∗}⊗_{kGb}A}^{∗}(kP); H^{∗}(G, b, P_{γ}); δ_{P}; HH_{A^{∗}⊗_{kGb}L⊗_{kHc}B}^{∗}(kP)] .

Furthermore, we glue this diagram with the commutative diagrams from the center of [9, Theorem 4, (6)]

[figure: HH_{B^{∗}⊗_{kHc}L^{∗}⊗_{kGb}A}^{∗}(kP); R_{B}; HH_{L^{∗}}^{∗}(kHc) ∩ HH_{B}^{∗}(kHc); R_{L}; HH_{L^{∗}}^{∗}(kHc); R_{L}; HH_{A^{∗}⊗_{kGb}L⊗_{kHc}B}^{∗}(kP); R_{A}; HH_{L⊗_{kHc}B}^{∗}(kGb); HH_{L}^{∗}(kGb)]

to obtain the conclusion.

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