Bockstein homomorphisms for Hochschild cohomology of group algebras and of 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

We give an explicit approach for Bockstein homomorphisms of the Hochschild cohomology of a group algebra and of a block algebra of a finite group and we show some properties. To give explicit definitions for these maps we use an additive decomposition and a Product Formula for the Hochschild cohomology of group algebras given by Siegel and Witherspoon in 1999. For k an algebraically closed field of characteristic p and G a finite group we prove an additive decomposition and a Product Formula for the cohomology algebra of a defect group of a block ideal of kG with coefficients in the source algebra of this block, and we define similar Bockstein homomorphisms.

Keywords: finite, group, block, Hochschild, cohomology, Bockstein

1. Introduction

The Bockstein homomorphism in group cohomology is the connecting homomorphism in the long exact sequence associated to some short exact sequence of coefficients. It appears in the Bockstein spectral sequence, which is a tool for comparing integral and mod p cohomology (p is a prime), and has applications for Steenrod operations. We use results from [1], regarding Bockstein maps in group cohomology and with some minor adjustments we adopt the same notations. For the rest of the paper, if otherwise not specified, let G be a finite group, let 𝔽_{p} be the field with p elements and k be a field of characteristic p, with p dividing the order of G. Let r be a non-negative degree integer; since we work with more than one group we denote by β_{G} the Bockstein homomorphism

β_{G}: H^{r}(G,𝔽_{p}) → H^{r+1} (G,𝔽_{p})

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

Preprint submitted to Elsevier June 4, 2016

and by β̂_{G} the homomorphism

β̂_{G} : H^{r}(G,𝔽_{p}) → H^{r+1} (G,ℤ).

Another application of Bockstein homomorphism is that β_{G}gives a structure of (cochain) DG-algebra to the cohomology algebra H^{∗}(G,𝔽_{p}), that is β_{G} is a differential ([1, Lemma 4.3.2]) and satisfies the graded Leibniz rule, ([1, Lemma 4.3.3]). In this paper we will call a map which satisfies this rule a graded derivation. Bockstein maps appear also in a celebrated theorem of Serre [1, Theorem 4.7.3] on zero product of Bocksteins, given originally in a different form. This theorem is an important ingredient for proving a theorem of Carlson (or Quillen) which characterizes nilpotent elements of positive degree in Extalgebras over finitely generated kG-modules by nilpotence of restrictions to elementary abelian subgroups. Further, this theorem can be used for showing a remarkable theorem of Chouinard [2] which states that a finitely generated kG-module M is projective if and only if its restriction to every elementary abelian subgroup of G is projective.

In the case of group algebras it is well known that Hochschild cohomology HH^{∗}(kG) can be identified with the ordinary cohomology H^{∗}(G,kG), where kG is viewed as left kG-module by conjugation. This observation goes back to Eilenberg and MacLane [3]. However the isomorphism as graded k-algebras was given more recently by Siegel and Witherspoon [10, Proposition 3.1]. Here HH^{∗}(kG) is an associative graded algebra with the usual Hochschild cup product and H^{∗}(G,kG) is an associative graded algebra with the product called cup product with respect to the pairing μ, where μ is the multiplication map in kG; see [10, Sections 2, 3] for a general discussion regarding cup products in this context. For the rest of this paper, this identification allows us to say and have in mind Hochschild cohomology of group algebras HH^{∗}(kG), but we will work with H^{∗}(G,kG). In Section 2 we will define Bockstein homomorphisms for the Hochschild cohomology of group algebras HH^{∗}(kG). In the first part of this section we define the Bockstein homomorphism for H^{∗}(G,𝔽_{p}G). Using an additive decomposition and a Product Formula from [10], in the main result of this section, Theorem 2.4, we extend this definition to H^{∗}(G,kG) and we prove that H^{∗}(G,kG) is also a DG-algebra.

In Section 3 we deal with block algebras of kG, where from now k is an algebraically closed field of characteristic p. Cohomology of block algebras of finite groups was first introduced by Linckelmann in [5] and further studied in a series of papers [6], [7]. An important recent paper is [8] where the author shows that the varieties of the Hochschild cohomology of a block algebra and its block cohomology are isomorphic, hence answering some questions raised by Pakianathan and Witherspoon [9]. As in the case of group algebras we identify the Hochschild cohomology of a block algebra B of kG with the cohomology of G with coefficients in B, where G acts by conjugation on B, that is HH^{∗}(B) ≅ H^{∗}(G,B). With this identification we define Bockstein homomorphisms β_{G,B} for H^{∗}(G,B) in Definition 3.1. Using a trick we obtain in Proposition 3.2 an additive decomposition of H^{∗}(G,B) which allows us to give an alternative, explicit definition of the Bockstein homomorphisms β_{G,B}for H^{∗}(G,B). After we recall the definition of the block cohomology which we denote by H^{∗}(G,b,P_{δ}), where b is the primitive central idempotent corresponding to B and P_{δ} is a defect pointed group of b, it is easy to see that Bockstein homomorphisms of group cohomology can be restricted to Bockstein homomorphisms of Linckelmann block cohomology β_{G,b}. The canonical embedding [5, Theorem 5.6 (iii)] from block cohomology to HH^{∗}(B) is an important ingredient which appears in the main result of [6]. If β_{G,B}and β_{G,b}are somehow compatible through this canonical map remains a question for further investigations.

In Section 4 we study the graded k-algebra H^{∗}(ΔP,ikGi), where P_{δ} is a defect pointed group of b and i ∈ δ is a source idempotent. The k-algebra ikGi is called the source algebra and is a ΔP-module, see [11] for more details. Since the block algebra B is Morita equivalent to its source algebra ikGi an important ingredient for proving the main result of [6] was H^{∗}(ΔP,ikGi). So, to determine its properties could be useful for future applications. This is the goal of this section where we determine an additive decomposition and a Product Formula for H^{∗}(ΔP,ikGi) with respect to cohomology of some subgroups in ΔP with trivial coefficients, see Theorem 4.4. Theorem 4.6 is one of the main results of this article and can be viewed as a Product Formula of the cohomology algebra H^{∗}(ΔP,ikGi) through the additive decomposition of cohomology graded vector spaces of some finite subgroups in ΔP with trivial coefficients, from Theorem 4.4, and by using the cup products of restrictions in ordinary group cohomology with trivial coefficients.

In conclusion this article can be divided in two big parts. In the first part we prove similar results (Theorem 2.4, Remark 3.4, Theorem 5.2) for different cohomology algebras which can be summarized in the next theorem:

Theorem 1.1. With the above notations let H^{∗} be one of the cohomology algebras in the set {H^{∗}(G,kG),H^{∗}(G,b,P_{δ}),H^{∗}(ΔP,ikGi)}. Then there is a map β : H^{r} → H^{r+1} for any non-negative integer r such that β is a differential and satisfies the graded Leibniz rule. In particular H^{∗} is a DG-algebra with respect to β.

In the second part the main result is the Product Formula, Theorem 4.6, with its usefulness demonstrated in Section 5 where we define Bockstein homomorphisms for H^{∗}(ΔP,ikGi) and we show more properties.

For results, notations and definitions regarding block algebras we follow [11]. For results in homological algebra we follow [1] and [4]. For conjugation in a group G if x,a ∈ G then ^{x}a = xax^{−1} and a^{x} = x^{−1} ax; similarly for subgroups. If A is any set we denote by id_{A}the identity map and by μ the multiplication map in any k-algebra.

2. Hochschild cohomology of group algebras and Bockstein maps

The next lemma is a result which allows us to define Bockstein maps in Hochschild cohomology of group algebras. The proof is trivial and is left as an exercise.

Lemma 2.1. i) There exist a short exact sequence of ℤG-modules, where G acts by conjugation

\xymatrix{0 \ar[r] & 𝔽_{p}G \ar[r]^{i_{G}} & ℤ/p^{2}[G] \ar[r]^{π_{G}} & 𝔽_{p}G \ar[r] & 0},

and i_{G},π_{G} are given by i_{G}(âg) = \overline{pa}g, π_{G}(\overline{a}g) = âg for any â ∈ 𝔽_{p},\overline{a} ∈ ℤ/p^{2} and g ∈ G; here â = a + pℤ ∈ 𝔽_{p} and \overline{a} = a + p^{2}ℤ ∈ ℤ/p^{2}ℤ for any a ∈ ℤ.

ii) There exist a short exact sequence of ℤG-modules, where G acts by conjugation

\xymatrix{0 \ar[r] & ℤG \ar[r]^{i′_{G}} & ℤG \ar[r]^{π′_{G}} & 𝔽_{p}G \ar[r] & 0},

and i′_{G},π′_{G} are given by i′_{G}(ag) = pag, π′_{G}(ag) = âg for any a ∈ ℤ,g ∈ G.

The long exact sequence theorem applied on the short exact sequences from Lemma 2.1 permit us to consider the next definition.

Definition 2.2. The Bockstein homomorphism in Hochschild cohomology of group algebras is the connecting homomorphism obtained from the short exact sequence of Lemma 2.1, i)

β_{G}: H^{r}(G,𝔽_{p}G) → H^{r+1} (G,𝔽_{p}G)

Similarly, for the short exact sequence from Lemma 2.1, ii) we have the connecting homomorphism

β̂_{G}: H^{r}(G,𝔽_{p}G) → H^{r+1} (G,ℤG).

Since we can identify H^{∗}(G,𝔽_{p}G) with Ext_{𝔽_{p}G}^{∗}(𝔽_{p},𝔽_{p}G) we have that β_{G} is also a homomorphism of 𝔽_{p}-vector spaces.

In order to prove the main result of this section we recall the additive decomposition and the product formula from [10, §4, §5] for H^{∗}(G,kG). We use the same notations and recall the results in our setting, when k is a field of characteristic p and G acts by conjugation on G; but notice that the results are true for any commutative ring and for any action by automorphisms of a finite group H on the group G. Let g_{1} = 1,g_{2},...,g_{s} be a system of representatives of the conjugacy classes of the action of G on G, where s is the number of the conjugacy classes. Denote by G_{i} := C_{G}(g_{i}), i ∈ {1,...,s} the centralizer subgroup of g_{i} in G. Next, fix g ∈ G and let i ∈ {1,...,s}. We consider the k(C_{G}(g))- module homomorphisms

θ_{g} : k → kG, θ_{g}(λ) = λg,

π_{g} : kG → k π_{g}( ∑_{a∈G}λ_{a}a) = λ_{g}.

If W is a subgroup of C_{G}(g) then θ_{g},π_{g} induce maps in cohomology

θ_{g}^{∗} : H^{∗}(W,k) → H^{∗}(W,kG),
π_{g}^{∗} : H^{∗}(W,kG) → H^{∗}(W,k).

Next, we define

γ_{i} : H^{∗}(G_{i},k) → H^{∗}(G,kG)

by γ_{i}:= tr_{G_{i}}^{G}∘θ_{g_{i}}^{∗}; for example γ_{1} : H^{∗}(G,k) → H^{∗}(G,kG) is γ_{1} = θ_{g_{1}}^{∗} = θ_{1}^{∗}, and is a graded k-algebra monomorphism. By [10, Lemma 4.2] there is an isomorphism of graded k-vector spaces, an additive decomposition

H^{∗}(G,kG) → ⨁_{i=1}^{s} H^{∗}(G_{i},k), [f] ↦ (π_{g_{i}}^{∗}(res_{G_{i}}^{G}[f]))_{i}  (2.1)

with its inverse sending [f_{i}] ∈ H^{∗}(G_{i},k) to γ_{i}([f_{i}]). For the Product Formula we fix i,j ∈ {1,...,s} and let D be a set of double coset representatives of G_{i}\G/G_{j}. For each x ∈ D there is a unique l = l(x) such that

g_{l} =^{y}g_{i}^{yx}g_{j}  (2.2)

for some y ∈ G. Let W :=^{yx}G_{j} ∩^{y}G_{i} and let [f_{i}] ∈ H^{∗}(G_{i},k), [f_{j}] ∈ H^{∗}(G_{j},k), then

γ_{i}([f_{i}]) ∪ γ_{j}([f_{j}]) = ∑_{x∈D}γ_{l}( tr_{W}^{G_{l}}(res_{W}^{^{y}G_{i}}(y^{∗}[f_{i}]) ∪ res_{W}^{^{yx}G_{j}}((yx)^{∗}[f_{j}])))  (2.3)

where y ∈ G,x ∈ D and l are like in equation (2.2) and y^{∗} : H^{∗}(G_{i},k) → H^{∗}(^{y}G_{i},k) is the conjugation map in group cohomology.

Lemma 2.3. With the above notations the following equality holds

β_{G}∘ γ_{i}^{r} = γ_{i}^{r+1} ∘ β_{G_{i}},

for any i ∈ {1,...,s}.

Proof. It is easy to verify the commutativity of the following diagram of short exact sequences of ℤG_{i}-modules

\xymatrix{0 \ar[r] & 𝔽_{p} \ar[r] \ar[d]^{θ_{g_{i}}} & ℤ/p^{2} \ar[r] \ar[d]^{θ′_{g_{i}}} & 𝔽_{p} \ar[d]^{θ_{g_{i}}} \ar[r] & 0 \\ 0 \ar[r] & 𝔽_{p}G \ar[r]^{i_{G}} & ℤ/p^{2}[G] \ar[r]^{π_{G}} & 𝔽_{p}G \ar[r] & 0},

where the first short exact sequence of trivial ℤG_{i}-modules induces the Bockstein map in group cohomology and θ′_{g_{i}} is the similar map to θ_{g_{i}} but with the coefficient ring ℤ/p^{2}. Since H^{∗}(G_{i},−) is a cohomolgical δ-functor the following diagram is commutative

\xymatrix{H^{r}(G_{i},𝔽_{p}) \ar[r]^{β_{G_{i}}} \ar[d]^{θ_{g_{i}}^{r}} & H^{r+1} (G_{i},𝔽_{p}) \ar[d]^{θ_{g_{i}}^{r+1}} \\ H^{r}(G_{i},𝔽_{p}G) \ar[r]^{β_{G_{i}}} & H^{r+1} (G_{i},𝔽_{p}G)},

where β_{G_{i}} is the connecting homomorphism of the down-side short exact sequence (of ℤG_{i}-modules) of the first diagram of this proof. Since tr_{G_{i}}^{G} is a morphism of δ-functors we also have the commutativity of the following diagram

\xymatrix{H^{r}(G_{i},𝔽_{p}G) \ar[r]^{β_{G_{i}}} \ar[d]^{_{tr{G \atop G_{i}}}} & H^{r+1} (G_{i},𝔽_{p}G) \ar[d]^{_{tr{G \atop G_{i}}}} \\ H^{r}(G,𝔽_{p}G) \ar[r]^{β_{G}} & H^{r+1} (G,𝔽_{p}G)}.

Gluing the above diagrams we obtain

β_{G}∘ tr_{G_{i}}^{G}∘ θ_{g_{i}}^{r} = tr_{G_{i}}^{G}∘ β_{G_{i}}∘ θ_{g_{i}}^{r} = tr_{G_{i}}^{G}∘ θ_{g_{i}}^{r+1} ∘ β_{G_{i}},

hence

β_{G}∘ γ_{i}^{r} = γ_{i}^{r+1} ∘ β_{G_{i}}.

Theorem 2.4. With the above notations the following statements are true:

i) β_{G}∘ β_{G} = 0;

ii) The Bockstein of a cup product (with respect to the pairing μ) is given by

β_{G}([a] ∪ [b]) = β_{G}([a]) ∪ [b] + (−1)^{deg([a])}[a] ∪ β_{G}([b]),

for any [a],[b] ∈ H^{∗}(G,𝔽_{p}G). In particular H^{∗}(G,𝔽_{p}G) is a DG-algebra.

iii) There is a k-linear map β_{G}: H^{r}(G,kG) → H^{r+1} (G,kG) such that H^{∗}(G,kG) is also a DG-algebra.

Proof. i) From Lemma 2.1, Definition 2.2 and the long exact sequence theorem we obtain that β̂_{G} ∘ (π′_{G})^{∗} = 0. Next, we consider the commutative diagram of ℤGmodules

\xymatrix{0 \ar[r] & ℤG \ar[r]^{i′_{G}} \ar[d]^{π′_{G}} & ℤG \ar[r]^{π′_{G}} \ar[d]^{π′′_{G}} & 𝔽_{p}G \ar[d]^{_{id_{𝔽pG}}} \ar[r] & 0 \\ 0 \ar[r] & 𝔽_{p}G \ar[r]^{i_{G}} & ℤ/p^{2}[G] \ar[r]^{π_{G}} & 𝔽_{p}G \ar[r] & 0},

where π′′_{G}is the similar map to π′_{G} but with coefficients in ℤ/p^{2} for the codomain. Since H^{∗}(G,−) is a cohomological δ-functor we obtain the commutative diagram

\xymatrix{H^{∗}(G,𝔽_{p}G) \ar[r]^{β̂_{G}} \ar[d]^{id} & H^{∗+1} (G,ℤG) \ar[d]^{(π′_{G})^{∗+1}} \\ H^{∗}(G,𝔽_{p}G) \ar[r]^{β_{G}} & H^{∗+1} (G,𝔽_{p}G)}.

Now, β_{G} = (π′_{G})^{∗}∘ β̂_{G} hence

β_{G}∘ β_{G} = (π′_{G})^{∗+1} ∘β̂_{G}∘ (π′_{G})^{∗+1} ∘β̂_{G} = 0.

ii) The isomorphism (2.1) for k replaced with 𝔽_{p} and Lemma 2.3 give an explicit definition for β_{G}, since by (2.1) we will take

{γ_{i}([f_{i}]) | i ∈ {1,...,s}, [f_{i}] ∈ H^{∗}(G_{i},𝔽_{p})}

as a set of generators for H^{∗}(G,𝔽_{p}G) and then

β_{G}(γ_{i}([f_{i}])) = γ_{i}(β_{G_{i}}([f_{i}])),

for any i ∈ {1,...,s} and any [f_{i}] ∈ H^{∗}(G_{i},𝔽_{p}). Let [a] = γ_{i}([f_{i}]) and [b] = γ_{j}([f_{j}]) be two homogeneous elements in H^{∗}(G,𝔽_{p}G), where i,j ∈ {1,...,s} and [f_{i}] ∈ H^{∗}(G_{i},𝔽_{p}),[f_{j}] ∈ H^{∗}(G_{j},𝔽_{p}). For shortness, we denote by m the degree deg(res_{W}^{^{y}G_{i}}y^{∗}([f_{i}])) in the following equalities. From (2.3), using the same notations and Lemma 2.3, we have

β_{G}([a] ∪ [b])

= ∑_{x∈D}β_{G}[γ_{l}( tr_{W}^{G_{l}}(res_{W}^{^{y}G_{i}}y^{∗}([f_{i}]) ∪ res_{W}^{^{yx}G_{j}}(yx)^{∗}([f_{j}])))]

= ∑_{x∈D}γ_{l}[β_{G_{l}}( tr_{W}^{G_{l}}(res_{W}^{^{y}G_{i}}y^{∗}([f_{i}]) ∪ res_{W}^{^{yx}G_{j}}(yx)^{∗}([f_{j}])))]

= ∑_{x∈D}γ_{l}[tr_{W}^{G_{l}}( β_{W}(res_{W}^{^{y}G_{i}}y^{∗}([f_{i}]) ∪ res_{W}^{^{yx}G_{j}}(yx)^{∗}([f_{j}])))]

= ∑_{x∈D}γ_{l}[tr_{W}^{G_{l}}( β_{W}(res_{W}^{^{y}G_{i}}y^{∗}([f_{i}])) ∪ res_{W}^{^{yx}G_{j}}(yx)^{∗}([f_{j}])+

+(−1)^{m}res_{W}^{^{y}G_{i}}y^{∗}([f_{i}]) ∪ β_{W}(res_{W}^{^{yx}G_{j}}(yx)^{∗}([f_{j}]))) ]

= ∑_{x∈D}γ_{l}[tr_{W}^{G_{l}}( res_{W}^{^{y}G_{i}}(y^{∗}(β_{G_{i}}[f_{i}])) ∪ res_{W}^{^{yx}G_{j}}(yx)^{∗}([f_{j}])+

+(−1)^{m}res_{W}^{^{y}G_{i}}y^{∗}([f_{i}]) ∪ res_{W}^{^{yx}G_{j}}((yx)^{∗}(β_{G_{j}}[f_{j}]))) ]

= ∑_{x∈D}γ_{l}[tr_{W}^{G_{l}}( res_{W}^{^{y}G_{i}}(y^{∗}(β_{G_{i}}[f_{i}])) ∪ res_{W}^{^{yx}G_{j}}(yx)^{∗}([f_{j}]))]+

+ (−1)^{m} ∑_{x∈D}γ_{l}[tr_{W}^{G_{l}}( res_{W}^{^{y}G_{i}}y^{∗}([f_{i}]) ∪ res_{W}^{^{yx}G_{j}}((yx)^{∗}(β_{G_{j}}[f_{j}])))]

= γ_{i}(β_{G_{i}}[f_{i}]) ∪ γ_{j}([f_{j}]) + (−1)^{deg([f_{i}])}γ_{i}([f_{i}]) ∪ γ_{j}(β_{G_{j}}([f_{j}]))

= β_{G}([a]) ∪ [b] + (−1)^{deg([a])}[a] ∪ β_{G}([b]),

where the fourth equality is true since β_{W} is a graded derivation; some of the other equalities are true since restriction, transfer and conjugation map are homomorphisms of δ-functors.

iii) The field 𝔽_{p} is a subfield of k, hence k is an 𝔽_{p}-algebra. Then there is an isomorphism of graded k-algebras

H^{∗}(G,𝔽_{p}) ⊗_{𝔽_{p}} k ≅ H^{∗}(G,k),

see [4, p. 30] for an explicit description of this isomorphism. This isomorphism induces a k-linear map β_{G}: H^{r}(G,k) → H^{r+1} (G,k), for which by abuse of notation we use the same notation as in the case of H^{∗}(G,𝔽_{p}) and is obtained as the tensor product over 𝔽_{p} of the above Bockstein map with id_{k}. Obviously β_{G} is a differential and a graded derivation. The isomorphism (2.1) for k assure us that we may give the same explicit definition given in the proof of statement ii) for a Bockstein homomorphism in H^{∗}(G,kG)

β_{G}: H^{r}(G,kG) → H^{r+1} (G,kG)
β_{G}(γ_{i}([f_{i}])) = γ_{i}(β_{G_{i}}([f_{i}])),

for any i ∈ {1,...,s} and any [f_{i}] ∈ H^{∗}(G_{i},k). The same computations as in the proof of statement ii) assure us that β_{G} is a differential and a graded derivation.

Remark 2.5. We see that in the proof of statement iii) of the above theorem we have used the extension of the Bockstein map from H^{∗}(G,𝔽_{p}) to a k-linear map, which we will call still the Bockstein map of H^{∗}(G,k), via the isomorphism of graded k-algebras

H^{∗}(G,𝔽_{p}) ⊗_{𝔽_{p}} k ≅ H^{∗}(G,k),

as in [1, p.133] or in [4, p.30]. But in a revised version of [1] the author noticed that this extension is not a good thing to do since it caused problems for [1, Section 5.8]. Instead he uses a semi-linear extension via the Frobenius map of the Bockstein map from cohomology with coefficients in 𝔽_{p} to the cohomology with coefficients in k. It is possible that many of the above and next results to remain true by considering this map. However for our purposes the considered extension from 2.4, iii) is good and it will be used for the rest of this paper. Also notice that from Section 3 we will work, for simplicity, with an algebraically closed field k of characteristic p and in this case these two maps are quite similar, see [4, Exercise 3.4.2].

Remark 2.6. From Definition 2.2 we notice that if k = 𝔽_{p}, Bockstein homomorphisms for Hochschild cohomology of group algebras are defined as the connecting homomorphisms in some long exact sequences, however in general, for any field k of prime characteristic, from the proof of Theorem 2.4, iii) we see that we can define Bockstein homomorphism

β_{G}: H^{r}(G,kG) → H^{r+1} (G,kG)
β_{G}(γ_{i}([f_{i}])) = γ_{i}(β_{G_{i}}([f_{i}])),

for any i ∈ {1,...,s} and any [f_{i}] ∈ H^{∗}(G_{i},k), only using Siegel-Witherspoon additive decomposition. We proved its properties using Product Formula and properties of Bockstein homomorphisms for group cohomology with coefficients in k.

3. Hochschild cohomology of block algebras and Bockstein homomorphisms

In this section let k be an algebraically closed field of characteristic p and B = {b|b ∈ Z(kG)} be the finite set of block idempotents of the group algebra kG. Each block idempotent determines a block algebra kGb, which is an indecomposable factor of kG as k-algebra. We have a well-known decomposition

kG = ⨁_{b∈B} kGb

as k[G×G]-modules, where the action of G×G on kG is given by (g,h)a = gah^{−1} for any g,h ∈ G,a ∈ kG. It follows that

kG = ⨁_{b∈B} kGb

as kG-modules, where G acts by conjugation, hence

H^{∗}(G,kG) ≅ ⨁_{b∈B} H^{∗}(G,kGb)

as graded k-algebras. In particular, for the fixed non-negative integer r we have

H^{r}(G,kG) ≅ ⨁_{b∈B} H^{r}(G,kGb)

as k-vector spaces. Next, fix a block algebra from {kGb|b ∈ B}, which we denote by B = kGb, where b is its fixed block idempotent. Again B is a kG-module by conjugation, since we identify kG ≅ kΔG, where ΔG ≤ G × G is the diagonal subgroup. We define the following kG-module homomorphisms

𝕚_{G,B}: B ↪ kG
𝕡_{G,B}: kG → B,

where 𝕚_{G,B} is the canonical inclusion map and 𝕡_{G,B} is the canonical projection, obtained by multiplication with b. Clearly 𝕡_{G,B}∘ 𝕚_{G,B}= id_{B} and these maps induce the following maps in cohomology:

𝕚_{G,B}^{∗}: H^{∗}(G,B) → H^{∗}(G,kG)
𝕡_{G,B}^{∗}: H^{∗}(G,kG) → H^{∗}(G,B)

such that 𝕡_{G,B}^{∗}∘ 𝕚_{G,B}^{∗} = id_{H^{∗}(G,B)}. Moreover, since 𝕚_{G,B},𝕡_{G,B} are homomorphisms of kalgebras as well, we have that 𝕚_{G,B}^{∗},𝕡_{G,B}^{∗} are homomorphisms of graded commutative kalgebras, where H^{∗}(G,B) is again an associative graded algebra with cup product with respect to the pairing μ; here μ is the multiplication map in B.

Definition 3.1. The Bockstein homomorphism in Hochschild cohomology of a block algebra B is the map

β_{G,B}: H^{r}(G,B) → H^{r+1} (G,B)

given by

β_{G,B}([f]) = 𝕡_{G,B}^{r+1}(β_{G}(𝕚_{G,B}^{r}([f]))

for any [f] ∈ H^{r}(G,B).

With the notations from Section 2 for each i ∈ {1,...,s} we define the homomorphism of graded k-vector spaces

γ_{i,G,B}:= 𝕡_{G,B}^{∗}∘ γ_{i}
γ_{i,G,B}: H^{∗}(G_{i},k) → H^{∗}(G,B)

and

H_{B}^{∗}(G_{i},k) := Im(π_{g_{i}}^{∗} ∘ res_{G_{i}}^{G}∘ 𝕚_{G,B}^{∗})

as a graded k-subspace of H^{∗}(G_{i},k). Since π_{g_{i}} is only additive in general, H_{B}^{∗}(G_{i},k) is not necessarily a subring of H^{∗}(G_{i},k).

Proposition 3.2. With the above notations the map

Φ : H^{∗}(G,B) → ⨁_{i=1}^{s} H_{B}^{∗}(G_{i},k)

Φ([f]) = π_{g_{i}}^{∗}(res_{G_{i}}^{G}(𝕚_{G,B}^{∗}[f])), [f] ∈ H^{∗}(G,B)

is an isomorphism of graded k-vectors spaces with its inverse sending an element [f_{i}] ∈ H_{B}^{∗}(G_{i},k) to γ_{i,G,B}([f_{i}]).

Proof. Clearly Φ is well-defined and surjective. Since

γ_{i,G,B}(Φ([f])) = γ_{i,G,B}[π_{g_{i}}^{∗}(res_{G_{i}}^{G}(𝕚_{G,B}^{∗}([f])))]
= 𝕡_{G,B}^{∗}( (γ_{i}∘ π_{g_{i}}^{∗} ∘ res_{G_{i}}^{G})(𝕚_{G,B}^{∗}([f])))
= (𝕡_{G,B}^{∗}∘ 𝕚_{G,B}^{∗})([f]) = [f],

for any [f] ∈ H^{∗}(G,B) we obtain the conclusion.

From Proposition 3.2 we obtain that the set

{γ_{i,G,B}([f_{i}])|i ∈ {1,...,s},[f_{i}] ∈ H_{B}^{r}(G_{i},k)}

is a k-generating set for H^{r}(G,B). This allows us to state the next remark.

Remark 3.3. It is obvious that by Proposition 3.2 an equivalent formulation for β_{G,B}is

β_{G,B}(γ_{i,G,B}[f_{i}]) = γ_{i,G,B}(β_{G_{i}}([f_{i}]))

for any i ∈ {1,...,s} and any [f_{i}] ∈ H_{B}^{r}(G_{i},k). Equivalently

β_{G,B}(γ_{i,G,B}[f_{i}]) = 𝕡_{G,B}^{r+1}(β_{G}(γ_{i}([f_{i}])))

for any i ∈ {1,...,s} and [f_{i}] ∈ H_{B}^{r}(G_{i},k). Theorem 2.4 and the fact that 𝕡_{G,B}^{∗},𝕚_{G,B}^{∗} are homomorphisms of graded k-algebras assure us that β_{G,B}is also a graded derivation.

We end this section with a remark regarding Bockstein homomorphisms for Linckelmann cohomology of block algebras. Let P_{δ} be a defect pointed group in G_{{b}} and (P,e_{P}) be an associated maximal b-Brauer pair. It is well known that for any Q ≤ P there is a unique block e_{Q} in kC_{G}(Q) such that (Q,e_{Q}) is a subpair of (P,e_{P}) and the set of b-Brauer pairs is a G-set. Block cohomology was introduced by Linckelmann in [5] and is denoted by H^{∗}(G,b,P_{δ}). Recall that H^{∗}(G,b,P_{δ}) is formed by all elements [ζ] ∈ H^{∗}(P,k) such that

res_{Q}^{P}([ζ]) = (g^{∗}∘ res_{Q}^{P})([ζ]),

for any g ∈ N_{G}(Q,e_{Q}). Moreover as a graded k-algebra, we have the equality

H^{∗}(G,b,P_{δ}) = ⨁_{r≥0} H^{r}(G,b,P_{δ}).

Remark 3.4. Restriction and conjugation are homomorphisms of cohomological δ-functors, hence for k = 𝔽_{p} we know that res_{Q}^{P} and g^{∗} commutes with β_{P}, respectively β_{Q} for any Q ≤ P and g ∈ N_{G}(Q). But also for any field k of characteristic p it is easy to prove that these properties hold, since res_{Q}^{P}⊗_{𝔽_{p}} id_{k} and g^{∗} ⊗_{𝔽_{p}} id_{k} can be identified with res_{Q}^{P} respectively g^{∗}. From the proof of Theorem 2.4, iii) recall that β_{P}: H^{r}(P,k) → H^{r+1} (P,k) is the tensor product over 𝔽_{p} of the Bockstein map for 𝔽_{p} with id_{k}, it follows that there is a well-defined homomorphism

β_{G,b}: H^{r}(G,b,P_{δ}) → H^{r+1} (G,b,P_{δ}),

which is the restriction of β_{P} to H^{∗}(G,b,P_{δ}). Since β_{P} is a differential and a graded derivation we have that β_{G,b} is also a differential and a graded derivation.

4. A Product Formula for cohomology of defect groups with coefficients in source algebras

Before obtaining the main results of this section let us introduce some notations and collect some properties in the first lemma of this section, without proof, for some subgroups in ΔP. For any subgroup P ≤ G let x ∈ G,(u,v) ∈ P × P. We denote the subgroups

Q_{x}:= {(a,a^{x})|a ∈ P ∩^{x}P}

and Q_{u,v,x}:= ΔP ∩^{(u,v)}Q_{x}. If w ∈ ΔP, by abuse of notation, we sometimes denote w := (w,w).

Lemma 4.1. Let x ∈ G,w ∈ ΔP and (u,v) ∈ P × P. The following statements hold.

i) Q_{u,v,x}= ΔP ∩^{(ux,v)}Δ(P^{x}∩ P) = ΔP ∩^{(u,vx^{−1} )}Δ(P ∩^{x}P);

ii) Let (a,a^{x}) ∈ Q_{x} where a ∈ P ∩^{x}P. Then Q_{wu,wv,x}=^{w}Q_{u,v,x}and Q_{ua,va^{x},x}= Q_{u,v,x};

iii) If w ∈ Q_{u,v,x}then w ∈ C_{P}(uxv^{−1} ).

From now, for the rest of this section, we consider as in the previous section a block algebra B = kGb with defect pointed group P_{δ}. Let i ∈ δ be a source idempotent, then ikGi is a P-interior algebra, called the source algebra of the block b. In particular ikGi is also a kP − kP-bimodule, hence a left k[P × P]-module by (u,v)(imi) = iumv^{−1} i, for any u,v ∈ P,m ∈ kG. By [11, Theorem 44.3] we have a decomposition

ikGi ≅ ⨁_{x∈Y_{G,b,P_{δ}}}k[PxP]

as indecomposable k[P × P]-modules, where Y_{G,b,P_{δ}}is a subset of a system of representatives of double cosets of P in G. We denote this isomorphism by

s: ikGi → ⨁_{x∈Y_{G,b,P_{δ}}}k[PxP],

and its inverse by s^{−1} . For x ∈ Y_{G,b,P_{δ}}, any element (u,v) ∈ P × P and W ≤ Q_{u,v,x}, by Lemma 4.1, iii) we define the following homomorphisms of kW -modules

\begin{aligned}& θ_{u,v,x}: k → ikGi \\ θ_{u,v,x}(α) & = αs^{−1} (uxv^{−1} ), α ∈ k \end{aligned}  (4.1)

\begin{aligned}& π_{u,v,x}: ikGi → k \\ π_{u,v,x}(m) = δ_{u,v,x} & (π_{x}(s(m))), m ∈ ikGi \end{aligned}  (4.2)

where δ_{u,v,x}: k[PxP] → k is given by

δ_{u,v,x}(𝔲x𝔳) = \{ \begin{matrix}1, if (𝔲,𝔳^{−1} ) ∈ Q_{u,v,x}(u,v)Q_{x} \\ 0, if (𝔲,𝔳^{−1} ) ∉ Q_{u,v,x}(u,v)Q_{x} \end{matrix}

and π_{x}: ⨁ _{x∈Y_{G,b,P_{δ}}}k[PxP] → k[PxP] is the k[P × P]-module canonical projection.

Lemma 4.2. Let x ∈ Y_{G,b,P_{δ}},(u,v) ∈ P × P,w ∈ Q_{u,v,x} and (a,a^{x}) ∈ Q_{x}. The following statements hold.

i) θ_{wua,wva^{x},x}= θ_{u,v,x};

ii) π_{wua,wva^{x},x}= π_{u,v,x}.

Proof. Statement ii) is left for the reader. For statement i) let us take an element α ∈ k. It follows

θ_{wua,wva^{x},x}(α) = αs^{−1} (wuax(a^{−1} )^{x}v^{−1} w^{−1} ) = αs^{−1} (wuaxx^{−1} a^{−1} xv^{−1} w^{−1} )
= αs^{−1} (uxv^{−1} ),

where the last equality is true by Lemma 4.1, iii).

We emphasize that in this section, in contrast to Sections 2 and 3, we prefer to use the more suggestive notation for cohomology of a group with coefficients in the group algebra or in the source algebra H^{∗}(ΔP,ikGi), opposed to H^{∗}(P,ikGi) where P acts by conjugation on ikGi. With the above notations π_{u,v,x},θ_{u,v,x}induce maps in cohomology

\begin{matrix}θ_{u,v,x}^{∗}: H^{∗}(W,k) → H^{∗}(W,ikGi) \\ π_{u,v,x}^{∗}: H^{∗}(W,ikGi) → H^{∗}(W,k) \end{matrix}  (4.3)

Moreover we define the maps in cohomology

\begin{matrix}γ_{u,v,x} & : H^{∗}(Q_{u,v,x},k) → H^{∗}(ΔP,ikGi) \\ γ_{u,v,x}:= tr_{Q_{u,v,x}}^{ΔP}∘ θ_{u,v,x}^{∗} \end{matrix}  (4.4)

here θ_{u,v,x}^{∗}is obtained for W = Q_{u,v,x}. From Lemma 4.2 and Lemma 4.1 we immediately collect some properties of the above maps (4.3), (4.4) in the next proposition.

Proposition 4.3. Let x ∈ Y_{G,b,P_{δ}},(u,v) ∈ P ×P,W ≤ Q_{u,v,x},w ∈ Q_{u,v,x}and (a,a^{x}) ∈ Q_{x}. The following statements hold.

i) θ_{wua,wva^{x},x}^{∗}= θ_{u,v,x}^{∗}and γ_{wua,wva^{x},x}^{∗}= γ_{u,v,x}^{∗};

ii) π_{wua,wva^{x},x}^{∗}= π_{u,v,x}^{∗}.

For x ∈ Y_{G,b,P_{δ}}by [11, Lemma 44.1, c)] there is an isomorphism of k[P × P]-modules

k[PxP] ≅ Ind_{Q_{x}}^{P×P}k,

hence it follows the isomorphism

ikGi ≅ ⨁_{x∈Y_{G,b,P_{δ}}}Res_{ΔP}^{P×P}Ind_{Q_{x}}^{P×P}k,

as kΔP-modules. For each x ∈ Y_{G,b,P_{δ}}we choose once and for all a system of representatives, which we denote by [Q_{x}], of double cosets ΔP\P ×P/Q_{x}. By Mackey decomposition we get

Res_{ΔP}^{P×P}Ind_{Q_{x}}^{P×P}k ≅ ⨁_{(u_{x},v_{x})∈[Q_{x}]}Ind_{ΔP∩^{(ux,vx)}Q_{x}}^{ΔP}Res_{ΔP∩^{(ux,vx)}Q_{x}}^{^{(ux,vx)}Q_{x}}^{(u_{x},v_{x})}k.

For shortness, we will identify from now ^{(u_{x},v_{x})}k with k. We can do this, since k is the trivial k[ΔP ∩^{(u_{x},v_{x})}Q_{x}]-module; moreover we will use the notations

Q_{u_{x},v_{x}}= Q_{u_{x},v_{x},x},θ_{u_{x},v_{x}}= θ_{u_{x},v_{x},x},π_{u_{x},v_{x}}= π_{u_{x},v_{x},x}, etc.

Thus we have the decomposition

ikGi ≅ ⨁_{\begin{matrix}x∈Y_{G,b,P_{δ}} \\ (u_{x},v_{x})∈[Q_{x}] \end{matrix}}Ind_{Q_{ux,vx}}^{ΔP}k  (4.5)

as kΔP-modules. In the first main result of this section we obtain an additive decomposition for the graded k-vector space H^{∗}(ΔP,ikGi).

Theorem 4.4. With the above notations the map

H^{∗}(ΔP,ikGi) → ⨁_{\begin{matrix}x∈Y_{G,b,P_{δ}} \\ (u_{x},v_{x})∈[Q_{x}] \end{matrix}}H^{∗}(Q_{u_{x},v_{x}},k)

[ζ] ↦ ∑_{\begin{matrix}x∈Y_{G,b,P_{δ}} \\ (u_{x},v_{x})∈[Q_{x}] \end{matrix}}(π_{u_{x},v_{x}}^{∗}∘ res_{Q_{ux,vx}}^{ΔP})([ζ])

is an isomorphism of graded k-vector spaces with is inverse sending an element [f] ∈ H^{∗}(Q_{u_{x},v_{x}},k) to γ_{u_{x},v_{x}}([f]).

Proof. We have the isomorphisms of kΔP-modules

ikGi ≅ ⨁_{x∈Y_{G,b,P_{δ}}}k[PxP] ≅ ⨁_{x∈Y_{G,b,P_{δ}}}Res_{ΔP}^{P×P}Ind_{Q_{x}}^{P×P}k,

where the first isomorphism is s and the second isomorphism is given by

𝔲x𝔳 ↦ (𝔲,𝔳^{−1} ) ⊗_{kQ_{x}}1

with its inverse

(𝔲,𝔳) ⊗_{kQ_{x}}1 ↦ 𝔲x𝔳^{−1}

for any 𝔲,𝔳 ∈ P and x ∈ Y_{G,b,P_{δ}}. We obtain an isomorphism in cohomology

H^{∗}(ΔP,ikGi) ≅ ⨁_{x∈Y_{G,b,P_{δ}}}H^{∗}(ΔP,Res_{ΔP}^{P×P}Ind_{Q_{x}}^{P×P}k).

Now for each x ∈ Y_{G,b,P_{δ}}as we have seen in (4.5) that Mackey decomposition tells us

Res_{ΔP}^{P×P}Ind_{Q_{x}}^{P×P}k ≅ ⨁_{(u_{x},v_{x})∈[Q_{x}]}Ind_{Q_{ux,vx}}^{ΔP}k.

Explicitly this isomorphism is obtained from the decomposition of P ×P in double cosets, that is for any (𝔲,𝔳^{−1} ) ∈ P × P there is a unique representative (u_{x},v_{x}) ∈ [Q_{x}] such that

(𝔲,𝔳^{−1} ) = (𝔲′,𝔲′)(u_{x},v_{x})(a,a^{x})

for some 𝔲′∈ ΔP and (a,a^{x}) ∈ Q_{x}. The isomorphism is given by

(𝔲,𝔳^{−1} ) ⊗_{kQ_{x}}1 ↦ (𝔲′,𝔲′) ⊗_{k[Q_{ux,vx}]}1.

This induces an isomorphism in cohomology

H^{∗}(ΔP,Res_{ΔP}^{P×P}Ind_{Q_{x}}^{P×P}k) ≅ ⨁_{(u_{x},v_{x})∈[Q_{x}]}H^{∗}(ΔP,Ind_{Q_{ux,vx}}^{ΔP}k).

Composing it with the isomorphism of [10, Lemma 4.1] one obtains the required isomorphism. Recall that in our case the isomorphism from [10, Lemma 4.1] is obtained by composing res_{Q_{ux,vx}}^{ΔP} with the map in cohomology induced by

Ind_{Q_{ux,vx}}^{ΔP}k → k,

which, with our notations (for 𝔲′∈ P), is given by

(𝔲′,𝔲′) ⊗_{k[Q_{ux,vx}]}1 ↦ \{ \begin{matrix}1, if (𝔲′,𝔲′) ∈ Q_{u_{x},v_{x}} \\ 0, if (𝔲′,𝔲′) ∉ Q_{u_{x},v_{x}} \end{matrix} .

Remark 4.5. From [11, Lemma 44.6] we know that ikGi always has a direct summand isomorphic to kP as k[P × P]-module. It follows that we can choose Y_{G,b,P_{δ}}such that there is x ∈ PC_{G}(P) in Y_{G,b,P_{δ}}(we take x = 1), and in this case Q_{1} = ΔP with

kP ≅ Ind_{ΔP}^{P×P}k.

Moreover we obtain

Res_{ΔP}^{P×P}Ind_{ΔP}^{P×P}k ≅ ⨁_{(u_{1},v_{1})∈[Q_{1}]}Ind_{Q_{u_{1},v_{1}}}^{ΔP}k.

In the set [Q_{1}] there is a representative of the double coset ΔP, for example we take (u_{1},v_{1}) = (1,1) ∈ N_{P×P}(ΔP). It follows that Q_{1,1} = ΔP and the map

θ_{1,1}^{∗} : H^{∗}(ΔP,k) → H^{∗}(ΔP,ikGi)

is actually the homomorphism i_{P}^{ikGi}constructed and studied for any interior P-algebra by Linckelmann in [8, Sections 2,3] and applied for the source algebra in [8, Section 5].

Next we will give a way to describe the cup product in H^{∗}(ΔP,ikGi) in terms of direct sums decomposition described in Theorem 4.4. We accomplish this as follows. Fix x,y ∈ Y_{G,b,P_{δ}}and (u_{x},v_{x}) ∈ [Q_{x}],(u_{y},v_{y}) ∈ [Q_{y}] as in the statements after Proposition 4.3, where an element from [Q_{x}] was denoted (u_{x},v_{x}) and similarly for [Q_{y}]. For each w ∈ ΔP there is a unique subset

Z = {z|z = z(x,y,u_{x},v_{x},wu_{y},wv_{y})}

in Y_{G,b,P_{δ}}, a set A_{z} = {(u′_{z},v′_{z})|(u′_{z},v′_{z}) ∈ P × P} and α_{z} ∈ k,z ∈ Z such that

s^{−1} (u_{x}xv_{x}^{−1} )s^{−1} (wu_{y}yv_{y}^{−1} w^{−1} ) = ∑_{\begin{matrix}z∈Z \\ (u′_{z},v′_{z})∈A_{z} \end{matrix}}α_{z}s^{−1} (u′_{z}z(v′_{z})^{−1} )  (4.6)

For two different z’s, for example z ≠ z′,z,z′∈ Z, we can have the same pairs (u′_{z},v′_{z}) = (u′_{z′},v′_{z′}) ∈ A_{z}∩ A_{z′}, or not. If the elements w are from a chosen set of representatives of Q_{u_{x},v_{x}}\ΔP/Q_{u_{y},v_{y}}then any element z ∈ Z is independent of the choice of representatives of double cosets Q_{u_{x},v_{x}}\ΔP/Q_{u_{y},v_{y}}; this is a consequence of Lemma 4.1, iii).

For each w ∈ [Q_{u_{x},v_{x}}\ΔP/Q_{u_{y},v_{y}}] with the corresponding attached unique set Z, and for an element (u′_{z},v′_{z}) ∈ A_{z} (where z ∈ Z) such that equation (4.6) holds, there is a unique element (u_{z},v_{z}) ∈ [Q_{z}] such that

(ru′_{z},rv′_{z}) = (u_{z}a,v_{z}a^{z})  (4.7)

for some r ∈ ΔP and (a,a^{z}) ∈ Q_{z}. The set of all r ∈ ΔP which satisfies (4.7) is also a double coset, that is Q_{u_{z},v_{z}}\ΔP/W′; for shortness, we denoted (and we will denote) the subgroup Q_{u_{x},v_{x}}∩ Q_{wu_{y},wv_{y}}by W′and the set [Q_{u_{x},v_{x}}\ΔP/Q_{u_{y},v_{y}}], of representatives of double cosets, by D .

Theorem 4.6. With the above notations let [f] ∈ H^{∗}(Q_{u_{x},v_{x}},k) and [g] ∈ H^{∗}(Q_{u_{y},v_{y}},k). Then

γ_{u_{x},v_{x}}([f]) ∪ γ_{u_{y},v_{y}}([g])

= ∑_{\begin{matrix}w∈D \\ z∈Z \\ (u_{z},v_{z}) \end{matrix}}α_{z}γ_{u_{z},v_{z}}( tr_{Q_{uz,vz}∩^{r}W′}^{Q_{uz,vz}}(res_{Q_{uz,vz}∩^{r}W′}^{^{r}Q_{ux,vx}}r^{∗}([f]) ∪ res_{Q_{uz,vz}∩^{r}W′}^{^{rw}Q_{uy,vy}}(rw)^{∗}([g])))

where D is a set of representatives of double cosets Q_{u_{x},v_{x}}\ΔP/Q_{u_{y},v_{y}}, the elements z ∈ Z, (u_{z},v_{z}) = (u_{z}(w),v_{z}(w)) and r = r(w),r ∈ ΔP are chosen to satisfy equations (4.6), (4.7) and W′:= Q_{u_{x},v_{x}}∩^{w}Q_{u_{y},v_{y}}.

To prove the theorem, we first deduce some lemmas where we collect more properties for the maps θ_{u,v,x}^{∗}and π_{u,v,x}^{∗}.

Lemma 4.7. Let x,y ∈ Y_{G,b,P_{δ}},(u,v),(u′,v′) ∈ P × P, the subgroups W_{1} ≤ W ≤ Q_{u,v,x}and w ∈ ΔP. The following statements hold.

i) w^{∗}∘ θ_{u,v,x}^{∗}= θ_{wu,wv,x}^{∗}∘ w^{∗};

ii) The maps θ_{u,v,x}^{∗}and π_{u,v,x}^{∗}commute with res_{W_{1}}^{W}respectively tr_{W_{1}}^{W}.

iii) We have that π_{u,v,x}^{∗}∘ θ_{u′,v′,y}^{∗}is the identity map if x = y and (u,v) ∈ Q_{u,v,x}(u′,v′)Q_{x}, and is the zero map otherwise. In particular if (u,v) = (u_{x},v_{x}) ∈ [Q_{x}] and (u′,v′) = (u′_{y},v′_{y}) ∈ [Q_{y}] then π_{u_{x},v_{x}}^{∗}∘ θ_{u′_{y},v′_{y}}^{∗} is the identity map if x = y and (u_{x},v_{x}) = (u′_{x},v′_{x}), and is the zero map otherwise.

Proof. i) Recall that by Lemma 4.1, ii) we know that ^{w}Q_{u,v,x}= Q_{wu,wv,x}and that w^{∗} : H^{∗}(W,k) → H^{∗}(^{w}W,k) is the conjugation map in cohomology. Let [f] ∈ H^{∗}(W,k) and 𝔞 ∈ P_{∗} where P_{∗} → k is a kΔP-projective resolution of k as trivial kΔP-module. We have that (w^{∗}∘ θ_{u,v,x}^{∗})[f] = [w^{∗}(θ_{u,v,x}∘ f)] with

w^{∗}(θ_{u,v,x}∘ f)(𝔞) = w(θ_{u,v,x}∘ f)(w^{−1} 𝔞) = w(f(w^{−1} 𝔞)s^{−1} (uxv^{−1} ))
= f(w^{−1} 𝔞)s^{−1} (wuxv^{−1} w^{−1} ).

Also we have that (θ_{wu,wv,x}^{∗}∘ w^{∗})[f] = [θ_{wu,wv,x}∘ w^{∗}(f)] with

θ_{wu,wv,x}(w^{∗}(f)(𝔞)) = w^{∗}(f)(𝔞)s^{−1} (wuxv^{−1} w^{−1} ) = (wf(w^{−1} 𝔞)s^{−1} (wuxv^{−1} w^{−1} )) = f(w^{−1} 𝔞)s^{−1} (wuxv^{−1} w^{−1} ).

Notice that this statement is true for any w ∈ ΔP. ii) We verify only that tr_{W_{1}}^{W}∘ θ_{u,v,x}^{∗}= θ_{u,v,x}^{∗}∘ tr_{W_{1}}^{W}, the other statements are similar and

left for the reader. Let [f] ∈ H^{∗}(W,k). We obtain

tr_{W_{1}}^{W}(θ_{u,v,x}(f))(𝔞) = ∑_{w∈W /W_{1}}wθ_{u,v,x}(f(w^{−1} 𝔞))

= ∑_{w∈W /W_{1}}f(w^{−1} 𝔞)s^{−1} (wuxv^{−1} w^{−1} ) = ∑_{w∈W /W_{1}}f(w^{−1} 𝔞)s^{−1} (uxv^{−1} ),

where the last equality holds since w ∈ Q_{u,v,x}by Lemma 4.1, iii).

θ_{u,v,x}(tr_{W_{1}}^{W}(f))(𝔞) = θ_{u,v,x}( ∑_{W /W_{1}}wf(w^{−1} 𝔞)) = ∑_{W /W_{1}}f(w^{−1} 𝔞)s^{−1} (uxv^{−1} ).

iii) Let α ∈ k. It follows that

(π_{u,v,x}∘ θ_{u′,v′,y})(α) = π_{u,v,x}(αs^{−1} (u′y(v′)^{−1} ))

= αδ_{u,v,x}[π_{x}(s(s^{−1} (u′y(v′)^{−1} )))]

= \{ \begin{matrix}αδ_{u,v,x}(u′y(v′)^{−1} ), if x = y \\ 0, if x ≠ y \end{matrix}

= \{ \begin{matrix}α, if x = y and (u′,v′) ∈ Q_{u,v,x}(u,v)Q_{x} \\ 0, otherwise \end{matrix}

The particular case is an immediate consequence.

Lemma 4.8. With the notations of (4.6) and(4.7) the following statement holds

θ_{u_{x},v_{x}}^{∗}([f]) ∪ θ_{wu_{y},wv_{y}}^{∗}([g]) = ∑_{\begin{matrix}z∈Z \\ (u′_{z},v′_{z})∈A_{z} \end{matrix}}θ_{u′_{z},v′_{z}}^{∗}([f] ∪ [g])

for any [f] ∈ H^{∗}(W′,k) and [g] ∈ H^{∗}(W′,k).

Proof. Let D: P_{∗} → P_{∗}⊗ P_{∗} be the diagonal approximation, where P_{∗} → k is a kΔPprojective resolution of the trivial kΔP-module k. By the definition of the cup product we have

θ_{u_{x},v_{x}}^{∗}([f]) ∪ θ_{wu_{y},wv_{y}}^{∗}([g]) = [μ ∘ (θ_{u_{x},v_{x}}⊗ θ_{wu_{y},wv_{y}}) ∘ (f ⊗ g) ∘ D],

where μ is the multiplication map in the source algebra. But since, for any α ∈ k

(μ ∘ (θ_{u_{x},v_{x}}⊗ θ_{wu_{y},wv_{y}}))(α) = αs^{−1} (u_{x}xv_{x}^{−1} )s^{−1} (wu_{y}yv_{y}^{−1} w^{−1} )

= ∑_{\begin{matrix}z∈Z \\ (u′_{z},v′_{z}) \end{matrix}}αα_{z}s^{−1} (u′_{z}z(v′_{z})^{−1} ) = ∑_{\begin{matrix}z∈Z \\ (u′_{z},v′_{z})∈A_{z} \end{matrix}}α_{z}θ_{u′_{z},v′_{z}}(α),

we are done.

Proof of Theorem 4.6. We apply [4, Propostion 4.2.4, Theorem 4.2.6] to obtain the following equalities

γ_{u_{x},v_{x}}([f]) ∪ γ_{u_{y},v_{y}}([g])

= tr_{Q_{ux,vx}}^{ΔP}(θ_{u_{x},v_{x}}^{∗}([f])) ∪ tr_{Q_{uy,vy}}^{ΔP}(θ_{u_{y},v_{y}}^{∗}([g]))

= tr_{Q_{ux,vx}}^{ΔP}[θ_{u_{x},v_{x}}^{∗}([f]) ∪ (res_{Q_{ux,vx}}^{ΔP}∘ tr_{Q_{uy,vy}}^{ΔP})(θ_{u_{y},v_{y}}^{∗}([g]))]

= ∑_{w∈D}tr_{Q_{ux,vx}}^{ΔP}[θ_{u_{x},v_{x}}^{∗}([f]) ∪ (tr_{^{w}Q_{uy,vy}∩Q_{ux,vx}}^{Q_{ux,vx}}∘

∘res_{^{w}Q_{uy,vy}∩Q_{ux,vx}}^{^{w}Q_{uy,vy}})(w^{∗}(θ_{u_{y},v_{y}}^{∗}([g])))]

By Lemma 4.7, Lemma 4.8 and again [4, Proposition 4.2.4] the above sum is

∑_{w∈D}tr_{Q_{ux,vx}}^{ΔP}[θ_{u_{x},v_{x}}^{∗}([f]) ∪ (tr_{W′}^{Q_{ux,vx}}∘ res_{W′}^{Q_{wuy,wvy}})(θ_{wu_{y},wv_{y}}^{∗}(w^{∗}([g])))]

= ∑_{w∈D}tr_{W′}^{ΔP}[res_{W′}^{Q_{ux,vx}}(θ_{u_{x},v_{x}}^{∗}([f])) ∪ res_{W′}^{Q_{wuy,wvy}}(θ_{wu_{y},wv_{y}}^{∗}(w^{∗}([g])))]

= ∑_{w∈D}tr_{W′}^{ΔP}[res_{W′}^{Q_{ux,vx}}(θ_{u_{x},v_{x}}^{∗}([f])) ∪ res_{W′}^{Q_{wuy,wvy}}(θ_{wu_{y},wv_{y}}^{∗}(w^{∗}([g])))]

= ∑_{w∈D}tr_{W′}^{ΔP}[θ_{u_{x},v_{x}}^{∗}(res_{W′}^{Q_{ux,vx}}([f])) ∪ θ_{wu_{y},wv_{y}}^{∗}(res_{W′}^{Q_{wuy,wvy}}(w^{∗}([g])))]

= ∑_{w∈D} ∑_{\begin{matrix}z∈Z \\ (u′_{z},v′_{z})∈A_{z} \end{matrix}}α_{z}( tr_{W′}^{ΔP}∘ θ_{u′_{z},v′_{z}}^{∗}) ( res_{W′}^{Q_{ux,vx}}([f]) ∪ res_{W′}^{^{w}Q_{uy,vy}}(w^{∗}([g])))

Next we apply the isomorphism from Theorem 4.4, Mackey decomposition [4, Theorem 4.2.6] and Lemma 4.7 i), ii), hence the above sum becomes

∑_{\begin{matrix}w∈D \\ z∈Z \\ (u′_{z},v′_{z})∈A_{z} \end{matrix}} ∑_{\begin{matrix}t∈Y_{G,b,P_{δ}} \\ (u_{t},v_{t})∈[Q_{t}] \end{matrix}}α_{z}(γ_{u_{t},v_{t}}∘ π_{u_{t},v_{t}}^{∗}∘ res_{Q_{ut,vt}}^{ΔP}∘ tr_{W′}^{ΔP}∘ θ_{u′_{z},v′_{z}}^{∗})(res_{W′}^{Q_{ux,vx}}([f])∪

∪ res_{W′}^{Q_{wuy,wvy}}(w^{∗}([g]))

= ∑_{w∈D} ∑_{\begin{matrix}z∈Z \\ (u′_{z},v′_{z})∈A_{z} \end{matrix}}α_{z} ∑_{\begin{matrix}t∈Y_{G,b,P_{δ}} \\ (u_{t},v_{t})∈[Q_{t}] \end{matrix}} ∑_{r∈[Q_{ut,vt}\ΔP/W′]}(γ_{u_{t},v_{t}}∘ π_{u_{t},v_{t}}^{∗}∘ tr_{Q_{ut,vt}∩^{r}W′}^{Q_{ut,vt}}∘

∘ res_{Q_{ut,vt}∩^{r}W′}^{^{r}W′}∘ θ_{ru′_{z},rv′_{z}}^{∗})(res_{^{r}W′}^{^{r}Q_{ux,vx}}(r^{∗}[f]) ∪ res_{^{r}W′}^{^{r}Q_{wuy,wvy}}((rw)^{∗}([g]))

= ∑_{w∈D} ∑_{\begin{matrix}z∈Z \\ (u′_{z},v′_{z})∈A_{z} \end{matrix}}α_{z} ∑_{\begin{matrix}t∈Y_{G,b,P_{δ}} \\ (u_{t},v_{t})∈[Q_{t}] \end{matrix}} ∑_{r∈[Q_{ut,vt}\ΔP/W′]}(γ_{u_{t},v_{t}}∘ π_{u_{t},v_{t}}^{∗}∘ θ_{ru′_{z},rv′_{z}}^{∗}∘

∘ tr_{Q_{ut,vt}∩^{r}W′}^{Q_{ut,vt}})(res_{Q_{ut,vt}∩^{r}W′}^{^{r}Q_{ux,vx}}(r^{∗}[f]) ∪ res_{Q_{ut,vt}∩^{r}W′}^{^{r}Q_{wuy,wvy}}((rw)^{∗}([g])).

By Lemma 4.7, iii) each t must be equal to some z to obtain non-zero terms, and in these cases by (4.7) we know that

(ru′_{z},rv′_{z}) = (u_{z}a,v_{z}a^{z})

for some a ∈ Q_{z}. We obtain the shorter sum

∑_{\begin{matrix}w∈D \\ z∈Z \\ (u_{z},v_{z}) \end{matrix}}α_{z}(γ_{u_{z},v_{z}}∘ π_{u_{z},v_{z}}^{∗}∘ θ_{u_{z}a,v_{z}a^{z}}^{∗}∘ tr_{Q_{uz,vz}∩^{r}W′}^{Q_{uz,vz}})(res_{Q_{uz,vz}∩^{r}W′}^{^{r}Q_{ux,vx}}(r^{∗}[f])∪

∪res_{Q_{uz,vz}∩^{r}W′}^{^{r}Q_{wuy,wvy}}((rw)^{∗}([g]))

= ∑_{\begin{matrix}w∈D \\ z∈Z \\ (u_{z},v_{z}) \end{matrix}}α_{z}γ_{u_{z},v_{z}}(tr_{Q_{uz,vz}∩^{r}W′}^{Q_{uz,vz}}(res_{Q_{uz,vz}∩^{r}W′}^{^{r}Q_{ux,vx}}(r^{∗}[f]) ∪ res_{Q_{uz,vz}∩^{r}W′}^{^{rw}Q_{uy,vy}}((rw)^{∗}([g]))).

The last equality is true since we apply Proposition 4.3, i) and Lemma 4.7, iii). We are done.

5. Bockstein homomorphisms for cohomology of defect groups with coefficients in source algebras and further properties

We continue with the notations of Section 4. The additive decomposition from Theorem 4.4 allows us to obtain a similar definition as in Remark 2.6 for Bockstein homomorphisms of H^{∗}(ΔP,ikGi).

Definition 5.1. The Bockstein homomorphism for the cohomology of a defect group P with coefficients in the source algebra ikGi is the map

β_{P,ikGi}: H^{r}(ΔP,ikGi) → H^{r+1} (ΔP,ikGi)
β_{P,ikGi}(γ_{u_{x},v_{x}}([f])) := γ_{u_{x},v_{x}}(β_{Q_{ux,vx}}([f])),

for any x ∈ Y_{G,b,P_{δ}},(u_{x},v_{x}) ∈ [Q_{x}] and [f] ∈ H^{r}(Q_{u_{x},v_{x}},k).

Some consequences of Theorem 4.6 are included in the following theorem where we collect some properties of β_{P,ikGi}.

Theorem 5.2. With the above notations the following statements hold:

i) β_{P,ikGi}is a differential;

ii) The Bockstein map β_{P,ikGi} is a graded derivation, hence H^{∗}(ΔP,ikGi) is a DGalgebra;

iii) The Bockstein map β_{P,ikGi}is compatible to β_{G,b}, i.e. the next diagram is commutative

\xymatrix{H^{r}(G,b,P_{δ}) \ar[r]^{β_{G,b}} \ar[d]^{γ_{u_{1},v_{1}}} & H^{r+1} (G,b,P_{δ}) \ar[d]^{γ_{u_{1},v_{1}}} \\ H^{r}(ΔP,ikGi) \ar[r]^{β_{P,ikGi}} & H^{r+1} (ΔP,ikGi)}.

Proof. In order to prove statements i) and ii) we apply similar methods as in the proof of Theorem 2.4; for showing these statements Theorem 4.6 is a crucial ingredient. By Definition 5.1 we know that

β_{P,ikGi}(γ_{u_{1},v_{1}}([f])) := γ_{u_{1},v_{1}}(β_{Q_{u_{1},v_{1}}}([f])),

for any [f] ∈ H^{r}(Q_{u_{1},v_{1}},k). But by Remark 4.5 we choose (u_{1},v_{1}) = (1,1) ∈ ΔP such that Q_{1,1} = ΔP and γ_{u_{1},v_{1}}= θ_{1,1}^{r} hence

γ_{u_{1},v_{1}}: H^{r}(ΔP,k) → H^{r}(ΔP,ikGi).

Next, Remark 3.4 implies that if [f] is in H^{r}(G,b,P_{δ}) then β_{G,b}([f]) = β_{P}([f]) is in H^{r+1} (G,b,P_{δ}).

Theorem 4.6 and Bockstein map defined in Definition 5.1 can be applied in order to compute the graded algebra structure of the cohomology algebra H^{∗}(ΔP,ikGi) for some particular cases, for example when P is cyclic of order p. This can be done similarly to [4, Example p.28], but we do not pursue this here.

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