ON BLOCKS WITH FROBENIUS INERTIAL QUOTIENT

Radha Kessar, Markus Linckelmann

January 2001

1. Introduction

Let p be a prime number, k a field of characteristic p, G a finite group and b a block of kG, that is, a primitive idempotent of the center of kG. In [18] Okuyama and Tsushima showed that the center Z(kGb) of the block algebra kGb is a symmetric algebra if and only if b has abelian defect groups and trivial inertial quotient, in other words if and only if b is a nilpotent block with abelian defect groups.

In this paper we study the connection between symmetry properties of the stable center and the p−local structure of group algebras and their blocks. The stable center of a finite dimensional k-algebra A is defined as follows. Denoting by A^{0} the opposite algebra of A, the algebra A has a natural structure as an A ⊗_{k} A^{0}−module via left and right multiplication. It is easy to see that the map (z → (a → az)) gives an isomorphsim between the center Z(A) of A and the ring End_{A⊗_{k}A^{o}}(A) of A ⊗_{k} A^{o}-module endomorphisms of A. We denote by Z^{pr}(A) the ideal in Z(A) consisting of those elements of Z(A) whose image under the above isomorphism is an endomorphism which factors through a projective A ⊗_{k} A^{o}- module. The stable center Z̄(A) is then defined to be the quotient Z(A)/Z^{pr}(A).

It turns out that the property of Z̄(kGb) being symmetric puts a strong restriction on the p−local structure of b. We assume that k is large enough for all the algebras appearing in the statements of this section to be split. In other words we assume that the semi-simple quotients of these algebras are direct products of matrix algebras over k.

Theorem 1.1. Let G be a finite group having a non-trivial Sylow-p-subgroup P and let b be the principal block of kG. Then Z̄(kGb) is a symmetric algebra if and only if P is abelian and N_{G}(P)/C_{G}(P) acts regularly on the group P.

One implication in the preceding theorem can be formulated for arbitrary blocks of kG, using a result of Puig in [22], which gives a stable equivalence between the block algebra of a block b with abelian defect groups and Frobenius intertial quotient, and the block algebra of the Brauer correspondent of b.

Theorem 1.2. Let b be a block of kG and let (P, e) be a maximal b-Brauer pair. If P is non-trivial abelian and N_{G}(P, e)/C_{G}(P) acts regularly on the group P, then Z̄(kGb) is a symmetric algebra.

The question of whether the converse of Theorem 1.2 holds for arbitrary blocks is open. We will give some necessary and sufficient conditions for a block to have

Typeset by AMS-TEX

symmetric stable center in Theorem 3.1 below. Combining 1.1 and 1.2 yields the following statement.

Corollary 1.3. Let G be a finite group having a non-trivial Sylow p-subgroup P. Then Z̄(kG) is a symmetric algebra if and only if P is abelian and N_{G}(P)/C_{G}(P) acts regularly on the group P.

Proof. If Z̄(kG) is symmetric, so is Z̄(kGb), where b is the principal block of kG. Thus P is abelian and N_{G}(P)/C_{G}(P) acts regularly on P by 1.1. Conversely, if P is abelian and N_{G}(P)/C_{G}(P) acts regularly on P, then for any non-trivial subgroup Q, N_{G}(Q)/C_{G}(Q) acts regularly on Q. Thus, for any block b of kG with a non-trivial defect group, Z̄(kGb) is symmetric by 1.2. If b is a block of kG with the trivial defect group 1, then kGb is a separable algebra, and hence Z(kGb) = Z^{pr}(kGb), or equivalently, Z̄(kGb) = {0}. Consequently, Z̄(kG) is symmetric. □

The stable center of a symmetric algebra A is the degree zero component of the Tate analogue of the Hochschild cohomology (whose definition is given in §5 below). The Hochschild cohomology of Brauer tree algebras has been computed by Holm [12] and Erdmann-Holm [6]; in the particular case of blocks with cyclic defect groups, an alternative approach appears in the work of Siegel and Witherspoon [25]. We extend their methods to computing ĤH^{∗}(kGb) in terms of the Tate cohomology of P for blocks fulfilling the hypotheses of 1.2.

Theorem 1.4. Let G be a finite group, b a block of kG and (P, e) a maximal b−Brauer pair. Suppose that P is non-trivial abelian and that E = N_{G}(P, e)/C_{G}(P) acts regularly on P. There is an isomorphism of graded k−algebras

ĤH^{∗}(kGb) ≅ (kP ⊗_{k} Ĥ^{∗}(P, k))^{E} ,

where E acts diagonally on the tensor product via its natural action on P.

Remark 1.5 Theorem 1.4 holds more generally for a complete local Noetherian ring with k as residue field instead of k, as follows easily from the proof. Also, we should point out that Z̄(kGb) is symmetric if and only if its socle has dimension one. More generally, a finite-dimensional split local commutative k−algebra is symmetric if and only if its socle has dimension one.

This paper is divided into five sections. In section 2, we collect some general results on symmetric algebras. In particular, we give necessary and sufficient conditions for such an algebra to have a symmetric stable center. We then interpret these results in the specific context of block algebras in section 3. In section 4, we prove Theorems 1.1 and 1.2. In section 5, we describe the Tate analogue of the Hochschild cohomology of a symmetric algebra and prove 1.4. We will introduce relevant notation and terminology in each section as the need arises.

2. Some results on symmetric algebras

Let k be a field of prime characteristic p. For a finite-dimensional k-algebra A, we say that k is large enough for A if A is split. We denote by l(A) the number of isomorphism classes of simple A−modules. Recall that a finite-dimensional k-algebra A is called symmetric if there exists a k-linear form : A → k such that s(ab) = s(ba) for every pair of elements a, b of A, and such that no non zero left or right ideal of A is contained in the kernel of s. Any such form s is called a symmetrising form of A. Note that the commutator subspace [A, A] of a symmetric algebra A is contained in the kernel of any symmetrising form of A.

Any group algebra kG of a finite group G is symmetric, with the canonical symmetrising form mapping 1_{G} to 1_{k} and any non-trivial element of G to zero. Furthermore, if E is a p′−subgroup acting on G, then the algebra (kG)^{E}, of E−fixed points in kG is still symmetric, since the restriction to (kG)^{E} of the canonical symmetrising form on kG remains a symmetrising form.

Given a symmetrising form s of A, for any k-subspace U of A, we denote by U^{⊥} the k-subspace {a ∈ A : s(au) = 0 for all u ∈ U} (this is a slight abuse of notation, since U^{⊥} depends in general on the choice of the symmetrising form). It is easy to check that (U^{⊥})^{⊥} = U and that dim_{k}(U) + dim_{k}(U^{⊥}) = dim_{k}(A). In the following proposition we gather a few well-known standard facts about symmetric algebras; we refer to Külshammer [14] for proofs as well as further properties of symmetric algebras.

Proposition 2.1. Suppose that A is a symmetric k-algebra with symmetrising form s : A → k. Then the following hold.

(i) Z(A)^{⊥} = [A, A].

(ii) J(A)^{⊥} = soc(A).

(iii) Z(A) ∩ soc(A) ⊆ soc(Z(A)).

(iv) Z^{pr}(A) ⊆ Z(A) ∩ soc(A).

Moreover, if A is split then

(v) dim_{k}(Z(A) ∩ soc(A)) = dim_{k}(A/(J(A) + [A, A])) = l(A).

Lemma 2.2. Let T be a finite-dimensional local commutative k-algebra and suppose that J is an ideal of T such that T/J is a symmetric algebra. Then, for every ideal I of T such that J ⊆ I, either soc(T) ⊆ I or I = J.

Proof. Let T and J be as in the proposition. Since T/J is local and symmetric, soc(T/J) is simple. Thus, if I is an ideal of T properly containing J then soc(T/J) ⊆ I/J. In particular, (soc(T) + J)/J ⊆ I/J, hence soc(T) ⊆ I. □

The following proposition is key.

Proposition 2.3. Let A be a symmetric k−algebra such that Z(A) is split local. Assume that A has a simple module of dimension prime to p. Then Z(A) ∩ soc(A) ⊈ Z(A) ∩ [A, A]. In particular, soc(Z(A)) ⊈ Z(A) ∩ [A, A].

Proof. We first show that we may assume that A is split. Let k′ be an extension field of k such that the k′−algebra k′ ⊗_{k} A is split. Since k′ ⊗_{k} J(A) ⊆ J(k′ ⊗_{k} A), we have

soc(k′ ⊗_{k} A) ⊆ k′ ⊗_{k} soc(A). Clearly [k′ ⊗_{k} A, k′ ⊗_{k} A] = k′ ⊗_{k} [A, A], and Z(k′ ⊗_{k} A) =

k′ ⊗_{k} Z(A) is split local. Moreover, if S is a simple A−module of k−dimension prime

to p, then the k′ ⊗_{k} A−module k′ ⊗_{k} S has a composition factor of k′-dimension prime

to p. Hence, we may indeed assume that A is split. The commutator subspace of a matrix algebra M_{n}(k), where n is a positive integer, consists precisely of the matricies whose trace is zero. Thus the identity of M_{n}(k) is in the commutator subspace if and only if p divides n. Since A has a simple module of dimension prime to p, it follows that 1_{A/J(A)} ∉ [A/J(A), A/J(A)]. In other words, 1_{A} ∉ J(A) + [A, A], and since Z(A) is a local algebra and (J(A) + [A, A]) ∩ Z(A)is an ideal in Z(A), this means that (J(A) + [A, A]) ∩ Z(A) ⊆ J(Z(A)) ⊆ J(A). Applying 2.1 implies that soc(A) ⊆ (Z(A) ∩ soc(A)) + [A, A].

Now, if Z(A) ∩ soc(A) ⊆ Z(A) ∩ [A, A], then soc(A) ⊆ [A, A]. But this is clearly impossible as [A, A] ⊆ ker(s) and soc(A) ⊈ ker(s), for any symmetrising form s on A. This proves the first statement of the proposition; the second follows from part (iv) of Proposition 2.1. □

Corollary 2.4. Let A be a symmetric k−algebra such that Z(A) is split local. Assume that A has a simple module of dimension prime to p. Let J be an ideal of Z(A) contained in Z(A) ∩ [A, A]. Then Z(A)/J is symmetric if and only if J = Z(A) ∩ [A, A].

Proof. By 2.3, we have soc(Z(A)) ⊈ Z(A) ∩ [A, A]. Thus, if Z(A)/J is symmetric, then J = Z(A) ∩ [A, A] by 2.2. Conversely, since Z(A)^{⊥} = [A, A] by 2.1(i), any symmetrising form on A induces a symmetrising form on Z(A)/(Z(A) ∩ [A, A]). □

Corollary 2.4 is going to be applied below in the case where J = Z^{pr}(A). In an arbitrary symmetric algebra A, the projective ideal Z^{pr}(A) need not be contained in [A, A], but the following easy (and well-known) observation will imply that whenever A is a block algebra or block source algebra with non trivial defect groups, then indeed Z^{pr}(A) ⊆ [A, A].

Let H, L be subgroups of a finite group G and let A be a finite-dimensional k−algebra on which G acts. We denote as usual by A^{H} the subalgebra of elements a ∈ A which are fixed under the action of H on A. If L ⊂ H, we denote by

Tr_{L}^{H} : (kG)^{L} → (kG)^{H}

the relative trace map (cf. [10]) which sends a ∈ A^{L} to ∑_{x∈[H/L]} ^{x}a ∈ A^{H} and we set

A_{L}^{H} = Im(Tr_{L}^{H}); this is easily seen to be an ideal in A^{H}. If P is a p−subgroup of G, we set A(P) = A^{P} /∑_{Q} A_{Q}^{P}, where Q runs over the set of proper subgroups of P, and we

denote by Br_{P}^{A} : A^{P} → A(P) the canonical surjective algebra homomorphism (called Brauer homomorphism; see [26, §11]). If A = kG, there is a canonical isomorphism A(P) ≅ kC_{G}(P). Recall from [20] (see also [26]), that an interior G−algebra is an algebra A endowed with a group homomorphism G → A^{×}. In particular, G acts on A by conjugation with the images in A^{×} of the elements of G.

Lemma 2.5. Let G be a finite group and let A be an interior G−algebra of finite dimension over k. Then for any two subgroups, H and L of G such that L ⊆ H and such that p divides the index of L in H, we have A_{L}^{H} ⊆ [A, A]. In particular, for every non-trivial p-subgroup P of G, we have A_{1}^{P} ⊆ ker(Br_{P}^{A}) ⊆ [A, A].

Proof. Let a ∈ A^{L}. Since p divides the index of H in L, identifying the elements of G to their images in A yields Tr_{L}^{H}(a) = ∑_{x∈H/L} xax^{−1} = ∑_{x∈H/L} xax^{−1} − [H : L]a = ∑_{x∈H/L}(xax^{−1} − a) = ∑_{x∈H/L}[xa, x^{−1}] ∈ [A, A]. The second part of the Lemma is an immediate consequence of the first since ker(Br_{P}^{A}) = ∑_{R⊊P} A_{R}^{P}. □

3 A criterion for blocks with symmetric stable center

Let G be a finite group and let b be a block of kG; that is, b is a primitive idempotent in Z(kG). For any p−subgroup P of G, we denote by Br_{P} : (kG)^{P} → kC_{G}(P) the Brauer homomorphism obtained from composing Br_{P}^{kG} with the canonical isomorphism (kG)(P) ≅ kC_{G}(P). A b−Brauer pair is a pair (P, e) consisting of a p−subgroup P of G and a block e of C_{G}(P) satisfying Br_{P} (b)e ≠ 0. Similarly, a b−Brauer element is a pair (u, f) consisting of a p−element in G and a block f of C_{G}(u) satisfying Br_{<u>}(b)f ≠ 0. The set of b−Brauer pairs is partially ordered, and the group G acts transitively by conjugation on the set of maximal b−Brauer pairs (cf. [1]). We say that the b−Brauer element (u, f)belongs to the b−Brauer pair (P, e) if (< u >, f) ≤ (P, e). A p−subgroup P of G is a defect group of b if it is minimal such that b ∈ (kG)_{P}^{G}, or equivalently, if it is maximal such that Br_{P} (b) ≠ 0. In particular, the maximal b−Brauer pairs are precisely the b−Brauer pairs (P, e) in which P is a defect group of b. Since Br_{P} (b) ≠ 0 when P is a defect group of b, there is a primitive idempotent i in (kGb)^{P} such that Br_{P} (i) ≠ 0. The interior P −algebra ikGi, with structural homomorphism mapping u ∈ P to ui ∈ (ikGi)^{×}, is a source algebra of b. This concept is due to L. Puig [20]. We refer to [26] for a detailed exposition of the material from block theory that we use here.

The above definitions make sense for arbitrary fields of characteristic p, but for the remainder of the paper, unless stated otherwise, we assume that k is large enough for all the block algebras appearing below.

The stable Grothendieck group of a block b of G is the quotient of the Grothendieck group of finitely generated kGb−modules by the subgroup generated by the images of the finitely generated projective kGb−modules. This is an abelian p−group whose order is the determinant of the Cartan matrix of kGb, and whose p−rank is the number of non-trivial elementary divisors (with multiplicities) of the Cartan matrix.

Theorem 3.1. Let G be a finite group, b a block of kG and let (P, e) be a maximal b−Brauer pair. Suppose that P is non-trivial. The following are equivalent.

(i) Z̄(kGb) is a symmetric algebra.

(ii) The stable Grothendieck group of kGb is cyclic, for every non-trivial b−Brauer element (u, f), the algebra kC_{G}(u)f has a unique isomorphism class of simple modules, and there exists x ∈ G such that ^{x}(u, f) ∈ (P, e) and ^{x}u ∈ Z(P).

We break up the proof in a series of Lemmas. The first one is a collection of elementary (and well-known) observations which hold without the assumption on k being large enough; we include a proof for the convenience of the reader.

Lemma 3.2. Let G be a finite group, let b be a block of kG and let P be a defect group of b.

(i) Z(kGb) ∩ (kGb)_{Q}^{P} ⊆ (kGb)_{Q}^{G} for any subgroup Q of P.

(ii) Z(kGb)∩ker(Br_{P}) = ∑_{Q} (kGb)_{Q}^{G}, where Q runs over the set of proper subgroups

of P.

(iii) Z(kGb) ∩ ker(Br_{Q}) ⊆ Z(kGb) ∩ ker(Br_{P}) for any subgroup Q of P.

(iv) Z(kGb) ∩ (kGb)_{1}^{P} = (kGb)_{1}^{G} = Z^{pr}(kGb).

Proof. There is c ∈ (kGb)^{P} such that b = Tr_{P}^{G}(c). Let z ∈ Z(kGb) ∩ ker(Br_{P}); that is, z = ∑_{Q} Tr_{Q}^{P}(d_{Q}), where Q runs over the set of proper subgroups of P and where

d_{Q} ∈ (kGb)^{Q} for any such Q. Then z = bz = Tr_{P}^{G}(c)z = Tr_{P}^{G}(cz) = ∑_{Q} Tr_{Q}^{G}(cd_{Q}). This

shows the inclusion Z(kGb) ∩ ker(Br_{P}) ⊆ ∑_{Q} (kGb)_{Q}^{G}, with Q running over the set of proper subgroups of P. By applying the same argument to the case where all but one of the d_{Q} are zero one gets the inclusion in (i). The Mackey formula for relative traces implies that (kGb)_{Q}^{G} ⊆ ker(Br_{P}) for any proper subgroup Q of P, from which we get the equality in (ii). If Q ⊆ P then C_{G}(P) ⊆ C_{G}(Q). Thus, if z ∈ Z(kGb) such that Br_{P} (z) ≠ 0 then Br_{Q}(z) ≠ 0, which shows (iii). We have Z(kGb) ∩ (kGb)_{1}^{P} ⊆ (kGb)_{1}^{G} by (i) applied to Q = 1, and clearly (kGb)_{1}^{G} ⊆ (kGb)_{1}^{P}, which shows the first equality in (iv). The second equality in (iv) is a consequence of Higman’s criterion [26, 17.3] applied to the group G × G^{0} and the k(G × G^{0})−module kGb. □

The first statement of the next Lemma is due to L. Puig [20, 3.5].

Lemma 3.3. Let G be a finite group, b a block of G, P a defect group of b and let i be a primitive idempotent in (kGb)^{P} such that Br_{P} (i) ≠ 0. Multiplication by i induces an isomorphism of the centers Z(kGb) ≅ Z(ikGi). This isomorphism maps Z^{pr}(kGb) onto Z^{pr}(ikGi) and Z(kGb) ∩ ker(Br_{P}) onto Z(ikGi) ∩ ker(Br_{P}).

Proof. By [20, 3.5], multiplication by i induces a Morita equivalence between kGb and ikGi. Any Morita equivalence preserves centers and projective ideals. Since i commutes with P, clearly multiplication by i maps Z(kGb)∩ker(Br_{P}) into Z(ikGi)∩ ker(Br_{P}). If z ∈ Z(kGb) such that Br_{P} (iz) = 0, then Br_{P} (Tr_{P}^{G}(i)z) = Br_{P} (Tr_{P}^{G}(iz)) = Tr_{P}^{N_{G}(P)}(Br_{P} (iz)) = 0, where we use the formula [26, 11.9]. By [19, Prop. 1] (see also [26, 9.3]), Tr_{P}^{G}(i) is invertible (this is where we use that k is large enough), and since Br_{P} is an algebra homomorphism, it follows that Br_{P} (z) = 0. □

Lemma 3.4. Let G be a finite group, let b be a block of kG, let P be a defect group of b and let i be a primitive idempotent in (kGb)^{P} such that Br_{P} (i) ≠ 0. Then

Z(ikGi) ∩ ker(Br_{P}) = Z(ikGi) ∩ [ikGi, ikGi] .

Proof. If P is trivial, both sides in 3.4 are zero. If P ≠ 1, the left side is contained in the right side by 2.5. By 3.3, multiplication by i maps Z(kGb) ∩ ker(Br_{P}) onto Z(ikGi) ∩ ker(Br_{P}). It follows from an observation of Watanabe (which appears in the proof of [24, Theorem 3]), that the quotient Z(ikGi)/(Z(ikGi) ∩ ker(Br_{P})) is isomorphic to the symmetric algebra (kZ(P))^{E}. Since ikGi has a simple module of dimension prime to p (cf. [21, 14.6]), the equality in 3.4 follows from 2.4. □

Lemma 3.5. Let G be a finite group and let b be a block of kG having a non-trivial defect group P. Denote by r_{b} the p−rank of the stable Grothendieck group of b. Then

l(b) = dim_{k}(Z^{pr}(kGb)) + r_{b} ≤ dim_{k}(Z(kGb) ∩ ker(Br_{P})) + 1 .

In particular, the stable Grothendieck group of b is cyclic if and only if dim_{k}(Z^{pr}(kGb)) = l(b) − 1.

Proof. The proof is based on Brauer’s work on lower defect groups, as exposed in [17]. For any conjugacy class C of G, let δ(C) denote the set of G-conjugates of the Sylow p-subgroups of the centralizer in G of an element x of C, and let \underline{C} denote the sum in kG of all elements of C.

Let Cl(G) denote the set of conjugacy classes of G and let Bl(G) denote the set of blocks of kG. Then, by Theorem 11.3, Chapter 5 of [17], we have a partition

Cl(G) = ∪_{e∈Bl(G)}Ω_{e}, (Ω_{e} ∩ Ω_{e′} = ϕ, if e ≠ e′)

such that \{\underline{C}e| C ∈ Ω_{e}\} is a k-basis of Z(kGe) for every e in Bl(G). Such a partition is called a block partition of Cl(G).

Let G_{p′} denote the set of p-regular elements of G, and Cl(G_{p′}), the set of conjugacy classes of p-regular elements of G. We have Z^{pr}(kGb) = (kGb)_{1}^{G} (cf. 2.3(iii)). For any conjugacy class C in G and any x ∈ C, we have Tr_{1}^{G}(x) = |C_{G}(x)|\underline{C}; this is non zero if and only if |C_{G}(x)| is prime to p. Thus the set

\{\underline{C}b | C ∈ Ω_{b} ∩ Cl(G_{p′}); δ(C) = 1\}  (∗)

is a k-basis for Z^{pr}(kGb).

On the other hand, for any C ∈ Ω_{b} such that δ(C) consists of conjugates of proper subgroups of P, we have \underline{C}b ∈ ker(Br_{P}^{kGb}) ∩ Z(kGb).

Now by Theorem 11.5 of Chapter 5 of [17],

|Ω_{b} ∩ Cl(G_{p′})| = l(b) ,  (∗∗)

there exists a unique C in Ω_{b}∩Cl(G_{p′}) such that δ(C) consists of the G−conjugates of P, and for all other C′ in Ω_{b} ∩ Cl(G_{p′}), the elements of δ(C′) are conjugate to proper subgroups of P. This implies the inequality l(b) − 1 ≤ dim_{k}(Z(kGb) ∩ ker(Br_{P})).

Finally, the orders of the groups in δ(C), as C varies over the set of p′−conjugacy classes in Ω_{b}, are exactly the elementary divisors of the Cartan matrix of kGb, and thus the p−rank r_{b} of the stable Grothendieck group of kGb is precisely the number of p′−conjugacy classes C in Ω_{b} such that δ(C) ≠ {1}. Combining (*) and (**) yields the first equality. In particular, the stable Grothendieck group of b is cyclic if and only if δ(C) = {1} for all but one p′−conjugacy class C in Ω_{b}, from which the last statement follows. □

Proposition 3.6. Let G be a finite group, let b be a block of kG and let (P, e) be a maximal b−Brauer pair. Denote by Z a set of representatives of the G−conjugacy classes of b−Brauer elements (u, f) for which there is x ∈ G such that ^{x}(u, f) ∈ (P, e) and ^{x}u ∈ Z(P); denote by Z′ a set of representatives of the G−conjugacy classes of all other b−Brauer elements. Then

dim_{k}(Z(kGb) ∩ ker(Br_{P})) = ∑_{(u,f)∈Z} (l(f) − 1) + ∑_{(u,f)∈Z′} l(f) .

Proof. We use a counting argument similar to what appears in the proof of [24, Theorem 3]. Set E = N_{G}(P, e)/P C_{G}(P). By considering Watanabe’s isomorphism Z(kGb)/(ker(Br_{P}) ∩ Z(kGb)) ≃ (kZ(P))^{E}, we get that

dim_{k}(Z(kGb)) = dim_{k}(Z(kGb)) ∩ ker(Br_{P})) + dim_{k}((kZ(P))^{E}) .

By [17, Ch. 5, Theorem 4.13], we have that

dim_{k}(Z(kGb)) = ∑_{(u,f)∈Z∪Z′} l(f) .

Now, let (u, f), (v, g) be two b−Brauer elements contained in (P, e) such that u, v ∈ Z(P), and suppose that (u, f) = ^{x}(v, g) for some element x ∈ G. Then, in fact (u, f) = ^{n}(v, g) for some element n ∈ N_{G}(P, e) (cf. [17, Ch. 5, Lemma 9.9]). Thus

dim_{k}((kZ(P))^{E}) = |Z| .

The Proposition follows from combining these three equalities. □

As a consequence of 3.6 we get a criterium for when the inequality in 3.5 is an equality.

Corollary 3.7. Let G be a finite group, b a block of G and let (P, e) be a maximal b−Brauer pair. The following are equivalent:

(i) dim_{k}(Z(kGb) ∩ ker(Br_{P})) = l(b) − 1.

(ii) For every non-trivial b−Brauer element (u, f) we have l(f) = 1, and there is x ∈ G such that ^{x}(u, f) ∈ (P, e) and ^{x}u ∈ Z(P).

Proof. We use the notation of 3.6. The set Z contains the trivial Brauer pair (1, b), which contributes the value l(b) − 1 to the sum in 3.6. Thus (i) holds if and only if all other summands are zero. This is clearly equivalent to l(f) = 1 for every non-trivial Brauer element (u, f) ∈ Z and Z′ = ∅, thus to statement (ii). □

Lemma 3.8. Let G be a finite group, b a block of kG, let (P, e) be a maximal b−Brauer pair and set E = N_{G}(P, e)/P C_{G}(P). Suppose that P is non-trivial. Let i be a primitive idempotent in (kGb)^{P} such that Br_{P} (i) ≠ 0. The following are equivalent.

(i) Z̄(kGb) is a symmetric algebra.

(ii) Z̄(ikGi) is a symmetric algebra.

(iii) Z̄(kGb) ≅ Z̄(ikGi) ≅ (kZ(P))^{E}.

(iv) Z^{pr}(ikGi) = Z(ikGi) ∩ [ikGi, ikGi].

(v) Z^{pr}(kGb) = Z(kGb) ∩ ker(Br_{P}).

(vi) Z^{pr}(kGb) = Z(kGb) ∩ ker(Br_{Q}) for every non-trivial subgroup Q of P.

(vii) Z^{pr}(kGb) = (kGb)_{Q}^{G} for any proper subgroup Q of P.

(viii) dim_{k}(Z^{pr}(kGb)) = l(b) − 1 = dim_{k}(ker(Br_{P}) ∩ Z(kGb)).

Proof. Multiplication by i induces an isomorphism Z(kGb) ≅ Z(ikGi) mapping Z^{pr}(kGb) onto Z^{pr}(ikGi) by 3.3, whence (i) and (ii) are equivalent. The equivalence of (ii) and (iv) is just a particular case of 2.4, because the source algebra ikGi has a simple module of dimension prime to p (cf. [21, 14.6]). The equivalence of (iv) and (v) follows from combining 3.3 and 3.4. As we have inclusions Z^{pr}(kGb) ⊆ Z(kGb) ∩ ker(Br_{Q}) ⊆ Z(kGb) ∩ ker(Br_{P}) for any non-trivial subgroup Q of P by 3.2(iii) and 3.2(iv), we get the equivalence of (v) and (vi). The equivalence of (v) and (vii) follows from the inclusions Z^{pr}(kGb) ⊆ (kGb)_{Q}^{G} for any subgroup Q of G and 3.2(ii). The equivalence of (v) and (viii) is an immediate consequence of 3.5. Finally, the equivalence of (v) and (iii) follows from Watanabe’s isomorphism Z(kGb)/(Z(kGb) ∩ ker(Br_{P}) ≅ (kZ(P))^{E}, which is a symmetric algebra. □

The above Lemmas contain all the required information to prove 3.1.

Proof of Theorem 3.1. Statement (i) is equivalent to the statement in 3.8 (viii). This, in turn, is equivalent to (ii) by 3.5 and 3.7. □

As a consequence of 3.8 we get the following necessary condition for a block to have symmetric stable center:

Proposition 3.9. Let G be a finite group, let b be a block of kG, let P be a defect group of b and let (Q, f) be a b−Brauer pair such that Q is conjugate to a non-trivial proper subgroup of P. Suppose that Z̄(kGb) is symmetric. We have

(kC_{G}(Q)f)_{Q}^{N_{G}(Q,f)} = {0} .

Proof. Let Q be a non-trivial proper subgroup of P. Then Br_{Q}(Z^{pr}(kGb)) = {0}. From the equivalence of 3.8 (i) and (vii) follows that Br_{Q}((kGb)_{Q}^{G}) = {0}. By the formula in [26, 11.9], this is equivalent to (kC_{G}(Q)Br_{Q}(b))_{Q}^{N_{G}(Q)} = {0}. Since different blocks of kC_{G}(Q) appearing in a decomposition of Br_{Q}(b) are orthogonal, this is equivalent to the statement in the Proposition. □

Corollary 3.10. With the notation and hypotheses of 3.8, set C̄ = C_{G}(Q)/Z(Q) ≅ QC_{G}(Q)/Q and N̄ = N_{G}(Q, f)/Q. Denote by f̄ the image of f in kC̄. We have

TrN̄ ; C̄(Z^{pr}(kC̄f̄)) = {0} .

Proof. We have (kC_{G}(Q)f)_{Q}^{N_{G}(Q,f)} = Tr_{QC_{G}(Q)}^{N_{G}(Q,f)}((kC_{G}(Q)f)_{Q}^{QC_{G}(Q)}). The image of

(kC_{G}(Q)f)_{Q}^{QC_{G}(Q)} in kC̄f̄ is precisely Z^{pr}(kC̄f̄). Thus 3.10 follows from 3.9. □

4. Proofs of the Theorems 1.1 and 1.2

Proof of Theorem 1.2. Let us use the notation of the Theorem. Further, let c be the Brauer correspondent of b in N_{G}(P). If P is abelian and if E = N_{G}(P, e)/C_{G}(P) acts regularly on P, then by [22, 6.8] there is a stable equivalence between kGb and kN_{G}(P)c. It follows from the construction in [22, 6.2], that this is a stable equivalence of Morita type in the sense of Broué [3]. In particular, the two algebras have isomorphic stable centers (see [3]). Also, kN_{G}(P)c is Morita equivalent to a twisted semidirect product k_{∗}(P ⋊ Ê) where Ê is a central extension of E by k^{∗} (see [21, 14.6]). Thus Theorem 1.2 follows from Proposition 4.2 below. □

Remark 4.1 With the notation of the proof of 1.2, since E acts regularly on P, the Sylow-q-subgroups of E are cyclic for q odd and cyclic or generalised quaternion for q = 2 (cf. [9, Ch. 5, Theorem 4.11]). Thus E has a trivial Schur multiplier (cf. [13, Kap. V, Satz 25.3]). Therefore, since is k is assumed to be large enough here, the central extension 1 → k^{×} → Ê → E → 1 splits; in other words, k_{∗}(P ⋊Ê) ≅ k(P ⋊E).

The following Proposition holds without the assumption on k being large enough.

Propsition 4.2. Let P be a finite non-trivial abelian p-group, E a p′-subgroup of Aut(P) and let Ê be a central k^{×}-extension of E. Suppose that E acts regularly on P. Then

Z̄(k_{∗}(P ⋊ Ê)) ≅ (kP)^{E} ;

in particular, the stable center of k_{∗}(P ⋊ Ê) is symmetric.

Proof. The version [26, 17.7] of Higman’s criterion for twisted group algebras implies that

Z^{pr}(k_{∗}(P ⋊ Ê)) = Tr_{1}^{P ⋊E}(k_{∗}(P ⋊ Ê)) .

Let u ∈ P and e ∈ E. Denote by ê any inverse image of e in Ê. We have Tr_{1}^{P ⋊E}(u) = 0, while for e ≠ 1, the free action of e on P − {1} implies that Tr_{1}^{P ⋊E}(uê) = Tr_{1}^{E}(ê)( ∑_{u∈P} u). Since E is a p′group, we have Z(k_{∗}(P ⋊ Ê)) =

Tr_{P}^{P ⋊E}((k_{∗}(P ⋊ Ê))^{P}). The free action of E on P implies that

(k_{∗}(P ⋊ Ê))^{P} = kP ⊕ ( ⊕_{e∈E−{1}} k(∑_{u∈P} u)ê) .

Together we get Z(k_{∗}(P ⋊Ê)) = (kP)^{E} ⊕Z^{pr}(k_{∗}(P ⋊Ê)), from which the Proposition follows. □

Proof of Theorem 1.1. If P is abelian and N_{G}(P)/C_{G}(P) acts regularly on P, then Z̄(kHb) is symmetric by 1.2. Suppose conversely that Z̄(kGb) is symmetric. From Theorem 3.1 follows that for every non-trivial element u of P, the principal block of kC_{G}(u) has one simple module; hence C_{G}(u) is a p-nilpotent group. Also, a conjugate of u lies in the center of P. Thus we are done by Proposition 4.3 below. □

Proposition 4.3. Let G be a finite group and let P be a Sylow−p−subgroup of G. Suppose that every non-trivial p-element u of G is conjugate to an element in Z(P) and that C_{G}(u) is p-nilpotent. Then P is abelian and N_{G}(P)/C_{G}(P) acts regularly on P.

Proof. We first consider the case that p is odd. We follow the first part of the proof of [2, 9.2]. Since the centralisers of non-identity p elements are p-nilpotent, (Z_{p} × Z_{p}) ⋊ Sl(2, p) is not involved in G. Let P be a Sylow p-subgroup of G. By a theorem of Glauberman in [7], we know that N_{G}(Z(J(P))), where J(P) denotes the Thompson subgroup of P, controls p-fusion in G. Let x be an element of P . If there exists g ∈ G such that y = ^{g}x ∈ Z(P), then y = ^{t}x for some t ∈ N_{G}(Z(J(P))); hence x = ^{t^{−}1}y ∈ Z(J(P)). If every p-element of G is conjugate to an element of Z(P),then P ⊂ Z(J(P)), and P is abelian.

Now , we consider the case that p = 2. Here we follow a strategy suggested by R. Solomon. Let G be a minimum counter-example to the proposition. Then it is clear that O_{2′} (G) = 1.

We first show that G is a simple group. Let N be maximal normal subgroup of G and suppose, if possible, that 1 ≠ N ≠ G. Let P be a Sylow 2-subgroup of G and Q = P ∩ N, a Sylow 2-subgroup of N. Since O_{2′} (G) = 1, we have Q ≠ 1. On the other hand, N satisfies the hypothesis of the proposition , namely the centraliser of every non-trivial 2-element of N is 2-nilpotent and contains a Sylow 2-subgroup of N. Hence Q is abelian and in particular a proper subgroup of P.

By the Frattini argument, G = N.N_{G}(Q), and hence N_{G}(Q)/N_{N}(Q) ≃ G/N is a simple group. Hence, either C_{G}(Q) = N_{G}(Q) or C_{G}(Q) ⊂ N_{G}(Q) ⊂ N. Let x ∈ P − Q. By the hypothesis, some conjugate of x centralises P,and in particular, Q; hence C_{G}(Q) is not contained in N.

Thus, C_{G}(Q) = N_{G}(Q). Since C_{G}(Q)/C_{N}(Q) = N_{G}(Q)/N_{N}(Q) is a simple group, O_{2}′(C_{G}(Q)) = O_{2}′(C_{N}(Q)). On the other hand, C_{G}(Q) is 2-nilpotent, thus C_{G}(Q)/C_{N}(Q) is a simple 2-group, hence of order 2. Thus N has index 2 in G. Let x ∈ P − Q, and suppose that gxg^{−1} ∈ Z(P). Then, P =< gxg^{−1}, Q > is abelian, which contradicts the choice of G. Thus G is simple.

In [8], Gorenstein has classified all simple groups, the centralisers of whose involutions are 2-nilpotent. Explicitly, G must be isomorphic to one of the groups P SL(2, 2^{n}) (with n ≥ 4), P SL(2, q) (with q > 3, q odd), Sz(2^{n}) ( with n ≥ 3), A_{7} or P SL(3, 4).

The Sylow 2-subgroups of P SL(2, 2^{n}) are abelian, while for the other groups, it can be checked that either the Sylow 2-subgroups are abelian or the exponent of a Sylow 2-subgroup is strictly greater than the exponent of the center of a Sylow 2-subgroup. In the latter case, it is impossible for every element of a Sylow 2 group to be conjugate to an element of the center of the Sylow 2-subgroup.Hence, P is abelian.

Let e ∈ E and u ∈ P − {1} such that e ∈ C_{G}(u). Since P ⊆ C_{G}(u) and C_{G}(u) is p−nilpotent, we have e = 1. Thus E acts regularly on P, which concludes the proof. □

We do not know at this stage whether the converse of 4.2 holds in general, but we have the following partial result (where again k need not be large enough):

Proposition 4.4. Let P be a non-trivial finite p−group, let E be a p′−subgroup of Aut(P) and let Ê be a central k^{×}−extension of E such that k_{∗}(P ⋊Ê) has a symmetric stable center. Then P is abelian, and if Ê is the split extension of E by k^{×} or if E is abelian, then E acts regularly on P.

Proof. The algebra k_{∗}(P ⋊ Ê) has a split local center and its semi-simple quotient is isomorphic to the separable algebra k_{∗}Ê, and therefore has a simple module of dimensiom prime to p. Thus, by 2.4,

Z^{pr}(k_{∗}(P ⋊ Ê)) = Z(k_{∗}(P ⋊ Ê)) ∩ [k_{∗}(P ⋊ Ê), k_{∗}(P ⋊ Ê)] .

Let u ∈ P and e ∈ E. Denote by ê any inverse image of e in Ê. We have Tr_{1}^{P ⋊E}(u) = 0 and for a non-identity element e of E, Tr_{1}^{P ⋊E}(uê) is a linear combination of elements of the form vf̂, where v ∈ P and f ∈ E −{1}. In particular, kP ∩Z^{pr}(k_{∗}(P ⋊Ê)) = 0.

We show now that P is abelian. If not, let u ∈ P − Z(P), and set Q = C_{P} (u). Then 1 ⊂ Q ⊂ P. Consider the element z = Tr_{Q}^{P ⋊E}(u). Since P is normal in the group P ⋊ Ê, and since E is a p′− group, it follows that z is a non zero element of Z(k_{∗}(P ⋊Ê)). Also, z ∈ [k_{∗}(P ⋊Ê), k_{∗}(P ⋊Ê)] since Q is proper in P (cf. 2.5). Thus z ∈ Z^{pr}(k_{∗}(P ⋊ Ê)), which is impossible as z ∈ kP. This shows that P is abelian.

Note that for x ∈ P and f̂ ∈ Ê, Tr_{1}^{P} (xf̂) = ∑_{y∈P} x[f, y]f̂ where [f, y] denotes the commutator of f and y. Thus Tr_{1}^{P} (xf̂) and consequently Tr_{1}^{P ⋊E}(xf̂) is zero unless C_{P} (f) = 1. In other words, the elements of Tr_{1}^{P ⋊E}(k_{∗}(P ⋊ Ê)) consist of linear combinations of elements of the type xf̂, where x ∈ P and f ∈ E such that C_{P} (f) = 1.

Let 1 ≠ e ∈ E and suppose that Q = C_{P} (e) is non-trivial. Let 1 ≠ v ∈ Q and let ê be any lift of e in Ê. Consider the element a = Tr_{Q}^{P ⋊E}(vê). Since Q is a proper subgroup of E, by 2.5 a is in the commutator subspace of k_{∗}(P ⋊ Ê), and hence, by Lemma 2.4, a ∈ Tr_{1}^{P ⋊E}(k_{∗}(P ⋊ Ê)). So a = 0, and hence, Tr_{Q}^{P ⋊C_{E}(e)}(vê) = 0.

Now, Tr_{Q}^{P}(vê) = ∑_{x∈[P/Q]} [x, e]vê. The element u = ∑_{x∈[P/Q]} [x, e] is C_{E}(e)- invariant. and thus we have

0 = Tr_{Q}^{P ⋊C_{E}(e)}(vê) = Tr_{1}^{C_{E}(e)}(uvê) = uTr_{1}^{C_{E}(e)}(vê).

Since Q is C_{E}(e) invariant, and P = Q × [P, < e >], the above equations imply that Tr_{1}^{C_{E}(e)}(vê) = 0. Consequently, Tr_{1}^{C_{E}(e)∩C_{E}(v)}(ê) = 0 for every pair of commuting non-identity elements e ∈ E and v ∈ P.

If Ê is the split extension of E by k^{×}, that is, k_{∗}(P ⋊ Ê) ≅ k(P ⋊ E), the above condition implies immediately that E acts regularly on P, since for all e ∈ E, and v ∈ P, Tr_{1}^{C_{E}(e)∩C_{E}(v)}(e) = |C_{E}(e) ∩ C_{E}(v)|e which is clearly non-zero.

Suppose finally that E is abelian and suppose, if possible, that E does not act regularly on P. We are going to show that there exists a non-identity element v of P such that C_{E}(v) is cyclic. Write

Ω_{1}(P) = Q_{1} × ⋯ × Q_{r}

such that for 1 ≤ i ≤ r, Q_{i} is E-invariant and E acts indecomposably on Q_{i}. Since E is abelian, E/C_{E}(Q_{i}) acts regularly on Q_{i} for 1 ≤ i ≤ r; in particular, by [9, Theorem 5.3.2], E/C_{E}(Q_{i}) is cyclic.

Let s, 1 ≤ s ≤ r, be the least integer such that ∩_{i=1}^{i=s}C_{E}(Q_{i}) = 1. Such an s exists since by [9, Theorem 5.2.4], ∩_{i=1}^{i=r}C_{E}(Q_{i}) = 1. If s = 1, then E is cyclic. If s > 1, then ∩_{i=1}^{i=s−1}C_{E}(Q_{i}) is a non-trivial cyclic subgroup of E. This is because, by choice of s, the canonical surjection of E onto E/C_{E}(Q_{s}) restricts to an injection on ∩_{i=1}^{i=s−1}C_{E}(Q_{i}). For 1 ≤ i ≤ s − 1 choose a non-identity element v_{i} of Q_{i}, and let v = v_{1} ⋯ v_{s−1}. Since C_{E}(v_{i}) = C_{E}(Q_{i}) and each Q_{i} is E-invariant, we have that C_{E}(v) = ∩_{i=1}^{i=s−1}C_{E}(Q_{i}) is non-trivial and cyclic. Let e be a generator of C_{E}(v). Then C_{E}(e) ∩ C_{E}(v) = C_{E}(v) and since ê commutes to all of its powers, it is fixed under the action of C_{E}(v); hence Tr_{1}^{C_{E}(v)}(ê) = |C_{E}(v)|ê is non zero. This contradiction completes the proof of the Proposition. □

5 On the Hochschild cohomology of blocks with abelian defect

Let A be a finite-dimensional symmetric k−algebra. We define the Tate analogue of the Hochschild cohomology of A by setting

ĤH(A) = \overline{Hom}_{A⊗_{k}A^{0}}(Ω_{A⊗_{k}A^{0}}^{n}(A), A)

for any integer n. Here \overline{Hom}_{A⊗_{k}A^{0}} denotes the homomorphism space in the stable

category of A ⊗_{k} A^{0}−modules, and Ω_{A⊗_{k}A^{0}} is the Heller operator, mapping an A ⊗_{k} A^{0}-

module to the kernel of a projective cover of that module. This makes sense, because A and thus A ⊗_{k} A^{0} are symmetric, and therefore the Heller operator Ω_{A⊗_{k}A^{0}} induces

an equivalence on the stable module category of A ⊗_{k} A^{0}.

It is well-known, that for positive n, we have canonical isomorphisms ĤH^{n}(A) ≅ HH^{n}(A) = Ext_{A⊗_{k}A^{0}}^{n}(A, A). In degree zero, we have canonical isomorphisms

HH^{0}(A) ≅ Z(A) and ĤH^{0}(A) ≅ Z̄(A). While the Hochschild cohomology is zero in negative degrees, its Tate analogue fulfills the analogue of the Tate duality; that is, we have isomorphisms ĤH^{−n}(A) ≅ (ĤH^{n−1}(A))^{∗} for any integer n.

For the proof of 1.4 we need the following (both 5.1 and 5.2 below hold over an arbitrary field k of characteristic p).

Lemma 5.1. Let P be a non-trivial finite p−group, and let α be an automorphism on P having no non-trivial fixed point. Then Êxt_{k(P ×P^{0})}^{∗}(kP, (kP)_{α}) = {0}.

Proof. Set ΔP = {(u, u^{−1})}_{u∈P} ⊂ P × P^{0}. Then kP ≅ Ind_{ΔP}^{P ×P^{0}}(k) as k(P × P^{0})− module. By the Eckmann-Shapiro lemma, we have therefore Êxt_{k(P ×P^{0})}^{∗}(kP, (kP)_{α}) ≅ Êxt_{kΔP}^{∗} (k, (kP)_{α}) = Ĥ^{∗}(ΔP, (kP)_{α}). Since α has no nontrivial fixed point, if u runs over all elements of P then so does uα(u^{−1}). Thus ΔP acts regularly on the basis P of (kP)_{α}. This shows that (kP)_{α} is projective as kΔP −module, and hence its Tate cohomology vanishes, which implies the result. □

Proposition 5.2. Let P be a non-trivial finite abelian p−group, let E be a p′−subgroup of Aut(P) and let Ê be a central k^{×}− extension of E. Assume that E acts regularly on P. Then

ĤH^{∗}(k_{∗}(P ⋊ Ê)) ≅ (kP ⊗_{k} Ĥ^{∗}(P, k))^{E} .

Proof. Set ΔE = {(e, e^{−1})}_{e∈E}. The algebra k((P × P^{0}) ⋊ ΔE)) can be identified to a subalgebra of k_{∗}(P ⋊ Ê) ⊗_{k} (k_{∗}(P ⋊ Ê))^{0} by identifying P × P^{0} to its obvious image

and by sending (e, e^{−1}) to ê ⊗ ê^{−1}, where e ∈ E and ê ∈ Ê is any element which lifts e. There is a canonical isomorphism of k_{∗}(P ⋊ Ê) − k_{∗}(P ⋊ Ê)−bimodules

k_{∗}(P ⋊ Ê) ≅ k_{∗}(P ⋊ Ê) ⊗_{k} (k_{∗}(P ⋊ Ê))^{0} ⊗_{k((P ×P^{0})⋊ΔE)} kP ,

and thus, by Eckmann-Shapiro, again we get

\begin{aligned}ĤH^{∗}(k_{∗}(P ⋊ Ê)) & ≅ Êxt_{k((P ×P^{0})⋊ΔE)}^{∗}(kP, k_{∗}(P ⋊ Ê)) \\ & ≅ (Êxt_{k(P ×P^{0})}^{∗}(kP, k_{∗}(P ⋊ Ê)))^{E}, \end{aligned}

where the last isomorphism comes from the fact that E is a p′−group. Now as a kP − kP −bimodule, we have k_{∗}(P ⋊ Ê) ≅ k(P ⋊ E) = ⊕_{e∈E} kP e. Since non trivial

elements of E act without fixed points on P − {1}, it follows from 5.1 that

ĤH^{∗}(k_{∗}(P ⋊ Ê)) ≅ (Êxt_{k(P ×P^{0})}^{∗}(kP, kP))^{E} = (ĤH^{∗}(kP))^{E} .

By a result of T. Holm [11], we have an isomorphism HH^{∗}(kP) ≅ kP ⊗_{k} H^{∗}(P, k). It

is easy to see that this isomorphism extends to an isomorphism ĤH^{∗}(kP) ≅ kP ⊗_{k}

Ĥ^{∗}(P, k), which is E−invariant (see e.g. [16] for an explicit way of describing this isomorphism). □

Proof of Theorem 1.4. By [22], there is a stable equivalence of Morita type between the block algebra kGb and a twisted group algebra k_{∗}(P ⋊ Ê) for some central k^{×}−extension Ê of E. Then ĤH^{∗}(kGb) ≅ ĤH^{∗}(k_{∗}(P ⋊Ê)) by the arguments in [15, 2.13], and now 1.4 follows from 5.2. □

6 Further remarks

Block algebras are symmetric algebras, but not every symmetric algebra is Morita equivalent to a block algebra:

Proposition 6.1. Let A be a split symmetric k-algebra which is Morita equivalent to a block algebra with non-trivial defect groups of some finite group. Then the ideal Z^{pr}(A) is strictly smaller than Z(A) ∩ soc(A).

Proof. By 3.5 and 2.1(v) we have dim_{k}(Z^{pr}(A)) ≤ l(A) − 1 < dim_{k}(Z(A) ∩ soc(A)) = l(A). Alternatively, we may assume that A is a block source algebra, in which case we have Z^{pr}(A) ⊆ [A, A] by 2.5, but Z(A) ∩ soc(A) ⊈ [A, A] by 2.4. □

Corollary 6.2. Let G be a finite group and let b be a block of kG having non-trivial defect groups. Assume that kGb has exactly one isomorphism class of simple modules. Then Z^{pr}(kGb) = {0}.

The previous statement is not true for arbitrary symmetric algebras with one isomorphism class of simple modules: if A is a split local symmetric algebra of dimension prime to p, then Z^{pr}(A) = Z(A) ∩ soc(A) ≅ k.

Theorem 3.1 can be viewed as a generalisation of the following Theorem, due to L. Puig and A. Watanabe:

Theorem 6.3. ([24, Theorem 3]) Let G be a finite group and let b be a block with an abelian defect group P. If l(f) = 1 for any b−Brauer element (u, f), then the block b is nilpotent.

Proof. We may assume that P is non-trivial. If l(f) = 1 for any b−Brauer element then in particular l(b) = 1. Then the stable Grothendieck group of the block algebra kGb is cyclic and Z^{pr}(kGb) = {0} by 3.5. Thus, by 3.1, Z̄(kGb) ≅ Z(kGb) is symmetric, and now it follows from the results in [18], that b is a nilpotent block. □

By a Theorem of Puig [23, Theorem 8.2], if there is a stable equivalence of Morita type between two block algebras over a complete discrete valuation ring of characteristic zero, and if one of the two blocks is nilpotent, so is the other. Under the assumption that the nilpotent block has abelian defect groups, we show that this result holds more generally over the field k.

Theorem 6.4. Let G, H be finite groups, let b be a block of kG, let c be a block of kH, and let P, Q be defect groups of b, c, respectively. Suppose that there is a stable equivalence of Morita type between the algebras kGb and kHc. If P is abelian and b is nilpotent, then Q ≅ P and c is nilpotent; in particular, kGb and kHc are Morita equivalent to kP.

Proof. We may assume that P and Q are non-trivial. By [4], if P is abelian and b is nilpotent, then kGb is Morita equivalent to kP. In particular, Z̄(kGb) ≅ Z(kGb) ≅ kP is symmetric. Thus Z̄(kHc) ≅ kP is symmetric. Therefore, by 3.8, kP ≅ (kZ(Q))^{E}, where E is the inertial quotient of c. Since kGb and kHc are stably equivalent, their Cartan matrices have the same elementary divisors, and hence |P | = |Q|. Thus dim_{k}(kZ(Q)^{E}) = |Q|, which forces Q = Z(Q) and E = 1. Thus c is nilpotent with the abelian defect group Q such that kP ≅ kQ. For abelian p−groups such an isomorphism implies an isomorphism of the groups P ≅ Q by [5]. □

Acknowledgements. The authors would like to thank Ron Solomon for his help regarding the proof of Proposition 4.3 and Christin Borge for pointing out the reference [5].

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