HOCHSCHILD COHOMOLOGY AND LINCKELMANN COHOMOLOGY FOR BLOCKS OF FINITE GROUPS

JONATHAN PAKIANATHAN AND SARAH WITHERSPOON

Abstract. Let G be a finite group, 𝔽 an algebraically closed field of finite characteristic p, and let B be a block of 𝔽G.

We show that the Hochschild and Linckelmann cohomology rings of B are isomorphic, modulo their radicals, in the cases where (1) B is cyclic and

(2) B is arbitrary and G either a nilpotent group or a Frobenius group (p odd). (The second case is a consequence of a more general result).

We give some related results in the more general case that B has a Sylow p-subgroup P as a defect group, giving a precise local description of a quotient of the Hochschild cohomology ring. In case P is elementary abelian, this quotient is isomorphic to the Linckelmann cohomology ring of B, modulo radicals. 1991 Mathematics Subject Classification. Primary: 16E40, 20J06

1. Introduction

Let G be a finite group and 𝔽 an algebraically closed field of positive characteristic p dividing the order of G. Let B be a block of the group algebra 𝔽G, that is an indecomposable ideal direct summand of 𝔽G. In [14, 15], Linckelmann defines the cohomology ring LH^{∗}(B) (our notation) of the block B of 𝔽G to be a subring of certain stable elements in the group cohomology ring H^{∗}(P, 𝔽), where P is a defect group of B. (See Definition 2.1.) Linckelmann then defines an injective ring homomorphism γ from the block cohomology ring LH^{∗}(B) to the Hochschild cohomology ring HH^{∗}(B) of B [14].

We are interested in a better understanding of the map γ connecting these two cohomology rings. As HH^{0}(B) generally has dimension over 𝔽 larger than one, γ is not in general an isomorphism. However, if we take the quotient of each ring by its (Jacobson) radical, we still have an injective ring homomorphism, which is now an isomorphism in degree 0. One is now led to the following question:

Date: February 22, 2002.

Research of the second author supported in part by National Science Foundation Grant #DMS-9970119 and National Security Agency Grant #MDA904-01-1-0067.

When does Linckelmann’s injection

γ : LH^{∗}(B) → HH^{∗}(B)

induce an isomorphism \overline{γ} modulo radicals?

We point out that as these cohomology rings are finitely generated graded commutative rings, we need only check that such an isomorphism exists, and Linckelmann’s injection γ will then automatically induce an isomorphism \overline{γ}. It is known that LH^{∗}(B) and HH^{∗}(B) have the same Krull dimension, that of H^{∗}(P, 𝔽), the rank of P ([15, Corollary 4.3(ii)] or [11, Theorem 4.4]).

If B = B_{0} is the principal block, \overline{γ} is known to be an isomorphism in the cases where G is a p-group, G is abelian, and a few other specific cases [21, Sections 10 and 11], as well as the case where B_{0} is cyclic, that is its defect groups are cyclic [22, Theorem 3].

In §3, we extend these results to prove:

Theorem 3.1. Let G be a group with normal Sylow p-subgroup P such that for any k ∈ P C_{G}(P) − 1, C_{G}(k) ≤ P C_{G}(P). (For example, G a nilpotent group or G a Frobenius group where p divides the order of the Frobenius kernel.)

Then for any block B of 𝔽G, we have that LH^{∗}(B) and HH^{∗}(B) are isomorphic modulo their radicals.

As a consequence, we give an affirmative answer to the question for all cyclic blocks:

Corollary 3.5. Let G be any finite group, and B any block of 𝔽G having a cyclic defect group. Then the Linckelmann cohomology ring LH^{∗}(B) is isomorphic to the Hochschild cohomology ring HH^{∗}(B), modulo radicals.

We then give further examples: An analogous result is true for the principal blocks of A_{5} and SL_{2}(8). These examples use Theorem 3.1 and Broué’s abelian defect conjecture, which is known to hold for these groups by work of Rickard and Rouquier. They also suggest a strategy for handling a larger class of examples.

In §4, we give some related results in case P is a Sylow p-subgroup of G (now not necessarily normal), and B any block of G with defect group P (e.g. the principal block). We study the quotient ring HH_{P}^{∗} (B) of the Hochschild cohomology ring of B modulo the ideal of proper transfers (see Definition 4.1) after proving some general module-theoretic results about this cohomology quotient. We give the structure of this quotient HH_{P}^{∗} (B) in terms of local information:

Theorem 4.2. Let B be a block of 𝔽G with defect group the Sylow p-subgroup P of G. Let K = P C_{G}(P), and b a block of 𝔽K such that B is the unique block covering b. Then

HH_{P}^{∗} (B) ≅ (H_{P}^{∗} (P, 𝔽) ⊗ 𝔽Z(P))^{N_{G}(b)},

where 𝔽Z(P) is the group algebra of the center of P.

In particular, when P is elementary abelian, this quotient HH_{P}^{∗} (B) is isomorphic, modulo radicals, to Linckelmann cohomology (Corollary 4.10). We give further examples of blocks of symmetric groups of defect 2 (p odd). In this case, Hochschild cohomology and Linckelmann cohomology are again isomorphic, modulo their radicals. These examples use Theorem 3.1 and Chuang’s proof of Broué’s abelian defect conjecture for these blocks.

A possible application of our work, particularly if it may be extended to include larger classes of groups and/or blocks, is to the study of varieties for blocks. In [15], Linckelmann develops such a theory, where the variety associated to a block B is the maximal ideal spectrum of the block cohomology ring LH^{∗}(B). Some unpublished work of Siegel [20] also gives a theory of varieties for blocks, where this time the variety associated to a block is the maximal ideal spectrum of its Hochschild cohomology ring HH^{∗}(B). In cases where Linckelmann’s block cohomology and the Hochschild cohomology of the block are isomorphic modulo their radicals (or more generally F-isomorphic), these two varieties associated to the block will be the same, and so both theories may potentially be exploited to obtain further information.

We would like to thank the referee for many useful comments and suggestions.

2. Preliminary remarks

We will use a number of results from [3] on subpairs and their partial order.

Let B be any block of 𝔽G. Let (P, B_{P}) be a Sylow B-subpair of G, unique up to conjugacy. In particular, P is a defect group of B and B_{P} is a block of 𝔽C_{G}(P). If R is any subgroup of P, then there exists a unique block B_{R} of 𝔽C_{G}(R) such that (R, B_{R}) ≤ (P, B_{P}). Let N_{G}(B_{R}) be the subgroup of N_{G}(R) fixing B_{R} setwise, under conjugation.

Definition 2.1 (Linckelmann). Let B be any block of 𝔽G with defect group P. The cohomology ring of the block B of G is the subring LH^{∗}(B) of H^{∗}(P, 𝔽) consisting of all [ζ] ∈ H^{∗}(P, 𝔽) satisfying

g res_{R}^{P}([ζ]) = res_{R}^{P}([ζ])

for any subgroup R of P, and any g ∈ N_{G}(B_{R}).

Our definition is equivalent to that of Linckelmann: By [5, Theorem 1.8], B_{R} is precisely the block of 𝔽C_{G}(R) with block idempotent e_{R} of [14, Definition 5.1]. (See also [15, p. 468].) We suppress Linckelmann’s pointed group P_{γ} in our notation, as we do not explicitly use pointed groups in our definition, and in any case the definition (up to isomorphism) does not depend on the choice of P_{γ}.

The next two remarks are due to Linckelmann [14, 15]. Remark 2.2. If B = B_{0} is the principal block, with defect group a Sylow p-subgroup P of G, it is easy to see that N_{G}(B_{R}) = N_{G}(R) for all R ≤ P. Thus by the Alperin Fusion Theorem [1] and the standard description of stable elements [9, Corollary 4.2.7], it follows that

LH^{∗}(B_{0}) = H^{∗}(G, 𝔽).

In this case, Linckelmann’s injection γ : LH^{∗}(B_{0}) → H^{∗}(B_{0}, B_{0}) is the composition of the canonical injection H^{∗}(G, 𝔽) ↪ H^{∗}(𝔽G, 𝔽G) with the canonical projection H^{∗}(𝔽G, 𝔽G) ↠ HH^{∗}(B_{0}). Remark 2.3. If the defect group P of B is abelian, then the inertial quotient E = N_{G}(B_{P})/C_{G}(P) controls fusion [3, Proposition 4.2], and is in general a p′-group [8, Theorem 61.15]. Therefore

LH^{∗}(B) ≅ H^{∗}(P, 𝔽)^{E} ≅ H^{∗}(P ⋊ E, 𝔽).

(Here the superscript E denotes fixed points.) If a block b of N_{G}(P) is the Brauer correspondent of B, it follows that LH^{∗}(B) ≅ LH^{∗}(b), that is their block cohomology rings are isomorphic. It is not known whether their Hochschild cohomology rings HH^{∗}(B) and HH^{∗}(b) are isomorphic in this case. This would be a consequence of Broué’s abelian defect conjecture, that B and b are derived equivalent. Broué’s conjecture is known to hold in case P is cyclic [13, 16], as well as in a number of other cases.

Let us look at an example so that the reader may see a sample of how Linckelmann’s injection induces an isomorphism modulo radicals. Example 2.4. Let G = S_{3}, the symmetric group on three letters, and p = 2. The principal block B_{0} of 𝔽S_{3} is isomorphic to 𝔽C_{2} (where C_{2} denotes a cyclic group of order 2). As C_{2} is abelian, [7, Theorem 2.1] or [21, Proposition 3.2] implies that

HH^{∗}(B_{0}) ≅ 𝔽C_{2} ⊗_{𝔽} H^{∗}(C_{2}, 𝔽).

On the other hand, by Remark 2.2, the Linckelmann cohomology of B_{0} is

LH^{∗}(B_{0}) ≅ H^{∗}(S_{3}, 𝔽) ≅ H^{∗}(C_{2}, 𝔽).

Linckelmann’s injection γ : LH^{∗}(B_{0}) → HH^{∗}(B_{0}) sends H^{∗}(C_{2}, 𝔽) to 1 ⊗ H^{∗}(C_{2}, 𝔽) ⊂ 𝔽C_{2} ⊗ H^{∗}(C_{2}, 𝔽) (see [15, Theorem 4.2(ii)]). As 𝔽C_{2} is a local ring, γ is indeed an isomorphism, modulo radicals.

3. Frobenius groups

In this section, we will assume the following: G is a group with normal Sylow p-subgroup P, such that for any k ∈ P C_{G}(P) − 1, the centralizer C_{G}(k) is contained in P C_{G}(P). In particular, this is true of any group G that is equal to P C_{G}(P) for a Sylow p-subgroup P, for example any nilpotent group. It is also true of any Frobenius group G = K ⋊ H with Frobenius kernel K and complement H, in which p divides |K|, by [10, Theorem 2.7.6(ii),(iv) and Theorem 10.3.1(iii)]. In this case, K = P C_{G}(P).

The main aim of this section is to show that for any block B of such a group G, the Linckelmann and Hochschild cohomology rings of the block are isomorphic modulo radicals. Theorem 3.1. Let G be a group with normal Sylow p-subgroup P such that for any k ∈ P C_{G}(P) − 1, C_{G}(k) ≤ P C_{G}(P). Let B be a block of 𝔽G. Then LH^{∗}(B) and HH^{∗}(B) are isomorphic modulo their radicals.

Let K = P C_{G}(P), and H = G/K. First we will prove the following result on the structure of the Hochschild cohomology ring HH^{∗}(𝔽G). Lemma 3.2. There is an additive decomposition

HH^{∗}(𝔽G) ≅ HH^{∗}(𝔽K)^{G} ⊕ (𝔽(G − K))^{G}.

The first summand is a subalgebra, and the second is an ideal consisting of nilpotent elements.

Proof. Let ΔG = {(g, g) | g ∈ G}. Consider 𝔽K as a module for the subgroup (K × K)ΔG of G × G, where the element (x, y) acts as left multiplication by x and right multiplication by y^{−1}. There is an isomorphism of 𝔽(G × G)-modules:

𝔽G ≅ 𝔽K ↑_{(K×K)ΔG}^{G×G},

where the arrow denotes induction from the subgroup (K × K)ΔG. Thus by the Eckmann-Shapiro Lemma,

(1)  HH^{∗}(𝔽G) ≅ Ext_{(K×K)ΔG}^{∗}(𝔽K, 𝔽G).

As K × K is normal in (K × K)ΔG of index prime to p, the latter is isomorphic to (Ext_{K×K}^{∗}(𝔽K, 𝔽G))^{G} (see e.g. [4, Proposition 3.8.2]). The 𝔽(K × K)-module 𝔽G is the direct sum 𝔽K ⊕ 𝔽(G − K), each summand of which is invariant under the G-action, so we obtain the additive decomposition

(2) HH^{∗}(𝔽G) ≅ (Ext_{K×K}^{∗}(𝔽K, 𝔽K))^{G} ⊕ (Ext_{K×K}^{∗}(𝔽K, 𝔽(G − K))^{G}.

The first term is (HH^{∗}(𝔽K))^{G}. As an 𝔽(K×K)-module, 𝔽K ≅ 𝔽 ↑_{ΔK}^{K×K}, so we may apply the Eckmann-Shapiro Lemma to the second term to obtain (Ext_{K}^{∗}(𝔽, 𝔽(G−K))^{G}. The hypotheses imply that 𝔽(G−K) is a free 𝔽K-module, so in fact the second term is (Ext_{K}^{0}(𝔽, 𝔽(G − K))^{G} ≅ (𝔽(G − K))^{G}. We have therefore proven the first statement of the lemma.

Next we will consider the ring structure of Hochschild cohomology, which is induced by the ring structure of 𝔽G in (1). If g ∈ G − K, note that the sum of the elements in its conjugacy class may be written as c_{g}·κ for a sum c_{g} of group elements, where κ = ∑_{k∈K} k. If g, h ∈ G−K, then

(c_{g}κ)(c_{h}κ) = c_{g}c_{h}κ^{2} = |K|c_{g}c_{h}κ = 0,

as K is normal in G and p divides |K|. This shows in particular that (𝔽(G − K))^{G} consists of nilpotent elements.

Finally, the image of the multiplication map on 𝔽K × 𝔽(G − K) is 𝔽(G − K), so the second term in (2) is an ideal. Clearly the first is a subalgebra. □

Next we look at the structure of the cohomology ring LH^{∗}(B). Lemma 3.3. Let G be a group with normal Sylow p-subgroup P and set K = P C_{G}(P). Suppose that for any k ∈ K − 1, C_{G}(k) ≤ K. Let B be a block of 𝔽G and let b be a block of 𝔽K that is covered by B. Then LH^{∗}(B) ≅ H^{∗}(P, 𝔽)^{N_{G}(b)}.

Proof. Since K = P × Q where Q = O_{p′}(K), we may write C_{G}(P) = Z(P) × Q. The blocks of 𝔽C_{G}(P) correspond bijectively with blocks of 𝔽Q, and may be written 𝔽Z(P) ⊗ b′, where b′ is a block of 𝔽Q.

By [3, (2.9)(3)], the block idempotent E of B is just the trace of the block idempotent e of 𝔽Z(P) ⊗ b′, from N_{G}(e) to G, where B covers 𝔽Z(P)⊗b′. Therefore we have (1, B) ≤ (P, 𝔽Z(P)⊗b′) by [3, Definition 3.2 and Theorem 3.4]. That is, we have shown that (P, 𝔽Z(P) ⊗ b′) is a Sylow B-subpair. We will write B_{P} = 𝔽Z(P) ⊗ b′.

As in fact the block idempotent of 𝔽Z(P) ⊗ b′ lies in 𝔽Q, and the same is true for the block idempotent of 𝔽Z(R) ⊗ b′ for any R ≤ P, R ≠ 1, we now have B_{R} = 𝔽Z(R) ⊗ b′, with

(R, B_{R}) ≤ (P, B_{P}).

Next we show that N_{G}(B_{R}) ≤ N_{G}(B_{P}) = N_{G}(b), where b = 𝔽P ⊗ b′. To see this, note that as P is normal in G, we also have that Z(P) is normal in G. Thus the normalizer in G of B_{P} = 𝔽Z(P) ⊗ b′ is just the normalizer of b′, which is the same as N_{G}(b). Further, any element of N_{G}(B_{R}) must normalize b′.

Applying the definition of LH^{∗}(B), we now have that

LH^{∗}(B) ≅ H^{∗}(P, 𝔽)^{N_{G}(b)}.
□

We are now ready to finish the proof of Theorem 3.1.

Proof of Theorem 3.1. Let B be a block of G, and b a block of K that is covered by B. By Lemma 3.3,

LH^{∗}(B) ≅ H^{∗}(P, 𝔽)^{N_{G}(b)} ≅ LH^{∗}(b).

It remains to show that the Hochschild cohomology ring HH^{∗}(B) is isomorphic to H^{∗}(P, 𝔽)^{N_{G}(b)}, modulo radicals.

Let E and e be the primitive central idempotents of 𝔽G and 𝔽K corresponding to B and b, respectively. We have HH^{∗}(B) ≅ HH^{∗}(𝔽G)E, where we identify E with an element of HH^{0}(𝔽G) ≅ Z(𝔽G). Therefore by Lemma 3.2,

HH^{∗}(B) ≅ (HH^{∗}(𝔽K)E)^{G} + (𝔽(G − K)E)^{G}.

Here we have used the fact that the G-invariant elements are the image of the trace map tr_{K}^{G} as |G : K| is prime to p, and E is itself G-invariant. By [3, (2.9)(3)], we also have E = tr_{N_{G}(b)}^{G}(e), and so modulo radicals, HH^{∗}(B) ≅ HH^{∗}(b)^{N_{G}(b)}. Now K = P × Q, and the idempotent e may be considered to be an element of 𝔽Q, as a block idempotent involves only p′-elements. Therefore b = 𝔽Ke ≅ 𝔽P ⊗_{𝔽}𝔽Qe. As Q is a p′-group, 𝔽Qe is a matrix algebra, and so HH^{∗}(b) ≅ HH^{∗}(𝔽P). We thus have

HH^{∗}(B) ≅ HH^{∗}(𝔽P)^{N_{G}(b)}.

By [21, Theorem 10.1], HH^{∗}(𝔽P) is isomorphic to H^{∗}(P, 𝔽), modulo radicals. As N_{G}(b)/K is a p′-group, N_{G}(b)/K-invariant elements of the quotient by an ideal are the same as the quotient of the N_{G}(b)/Kinvariant elements, and so HH^{∗}(𝔽P)^{N_{G}(b)} is isomorphic to H^{∗}(P, 𝔽)^{N_{G}(b)}, modulo their radicals. □

As stated at the beginning of the section, any nilpotent group, or Frobenius group with p dividing the order of the Frobenius kernel, satisfies the hypotheses of Theorem 3.1. Thus we have the following corollary. Corollary 3.4. Let G be a nilpotent group, or a Frobenius group in which p divides the order of the Frobenius kernel. Let B be a block of 𝔽G. Then LH^{∗}(B) and HH^{∗}(B) are isomorphic, modulo their radicals.

□

We further obtain the following general result about cyclic blocks. Corollary 3.5. Let G be any finite group, and B any block of 𝔽G having a cyclic defect group. Then the Linckelmann cohomology ring LH^{∗}(B) is isomorphic to the Hochschild cohomology ring HH^{∗}(B), modulo radicals.

Proof. Suppose P is a defect group of B, and P is cyclic. Let E denote the inertial quotient, e = |E|, and m = (|P | − 1)/e. By Remark 2.3,

LH^{∗}(B) ≅ H^{∗}(P ⋊ E, 𝔽).

Note that by [8, Lemma 60.9], 𝔽(P ⋊E) has only one block. This block has cyclic defect group P, and so is a Brauer tree algebra with inertial index e and multiplicity m (see [2, p. 123]).

Since B is also a Brauer tree algebra with e edges and multiplicity m (see [2]), B and 𝔽(P ⋊ E) are derived equivalent by [16, Theorem 4.2]. By [18, Proposition 2.5], derived equivalent algebras have isomorphic Hochschild cohomology rings, and so HH^{∗}(B) ≅ HH^{∗}(𝔽(P ⋊ E)).

Finally, as P ⋊ E satisfies the hypotheses of Theorem 3.1, we have

HH^{∗}(𝔽(P ⋊ E)) ≅ LH^{∗}(𝔽(P ⋊ E)),

modulo radicals. Since LH^{∗}(𝔽(P ⋊ E)) ≅ H^{∗}(P ⋊ E, 𝔽), this shows HH^{∗}(B) ≅ LH^{∗}(B), modulo radicals, as desired.

□

The case in which G is a Frobenius group and p divides the order of the Frobenius complement H is covered by Corollary 3.5 in all cases except when p = 2 and a Sylow 2-subgroup of H is generalized quaternion (otherwise a Sylow p-subgroup of G is cyclic, see [10, Theorem 10.3.1(iv)]). We do not know whether LH^{∗}(B) and HH^{∗}(B) are isomorphic, modulo their radicals, in this case.

Finally, as an application of Corollary 3.4, we give two further examples in which Linckelmann’s injection γ : LH^{∗}(B) → HH^{∗}(B) is an isomorphism modulo radicals. These groups do not satisfy the hypotheses of this section; instead the corollary is applied to the normalizers of Sylow p-subgroups.

Example 3.6. Let G = A_{5}, the alternating group on five letters, and p = 2. Let P = ⟨(12)(34), (13)(24)⟩, a Sylow 2-subgroup, and H = N_{G}(P) ≅ P ⋊ ⟨(123)⟩ ≅ A_{4}. Let b_{0} be the principal block of H. As H is a Frobenius group, Corollary 3.4 implies that LH^{∗}(b_{0}) and HH^{∗}(b_{0}) are isomorphic modulo their radicals. By Remark 2.2 or 2.3, LH^{∗}(b_{0}) ≅ H^{∗}(H, 𝔽) ≅ H^{∗}(P, 𝔽)^{H}.

On the other hand, the principal block B_{0} of 𝔽G is Rickard equivalent to b_{0} by [17] or [19, Example 1]. Therefore HH^{∗}(B_{0}) ≅ HH^{∗}(b_{0}). By Remark 2.2 and [9, Theorem 4.2.8],

LH^{∗}(B_{0}) ≅ H^{∗}(G, 𝔽) ≅ H^{∗}(P, 𝔽)^{H}.

Considering the previous statements about the cohomology rings of b_{0}, we now have LH^{∗}(B_{0}) ≅ HH^{∗}(B_{0}), modulo radicals. Example 3.7. Let G = SL_{2}(8) and

P = \{( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) | b ∈ 𝔽_{8}\}

the Sylow 2-subgroup. Then

H = N_{G}(P) = \{( \begin{matrix}a & b \\ 0 & a^{−1} \end{matrix} ) | a, b ∈ 𝔽_{8}, a ≠ 0\}

is a semidirect product of P with a cyclic group of order 7. Let b_{0} be the principal block of H, and note that H is a Frobenius group with Frobenius kernel P. Therefore LH^{∗}(b_{0}) and HH^{∗}(b_{0}) are isomorphic modulo their radicals. Further, LH^{∗}(b_{0}) ≅ H^{∗}(H, 𝔽) ≅ H^{∗}(P, 𝔽)^{H}.

Rouquier gives a Rickard equivalence between the principal block B_{0} of 𝔽G and b_{0} [19, Example 2], and so HH^{∗}(B_{0}) ≅ HH^{∗}(b_{0}). As before,

LH^{∗}(B_{0}) ≅ H^{∗}(G, 𝔽) ≅ H^{∗}(P, 𝔽)^{H},

and we now have LH^{∗}(B_{0}) ≅ HH^{∗}(B_{0}), modulo radicals.

4. Sylow defect blocks

Let P be a Sylow p-subgroup of G, and B a block of G with defect group P (e.g. the principal block). To obtain some results in this general case, we will consider quotients of cohomology rings by the ideals generated by the images of transfers from all p-subgroups properly contained in a Sylow p-subgroup. We will write Q <_{G} P to indicate that Q is properly contained in a subgroup conjugate to P by an element of G.

We take a module-theoretic view of blocks, as in [2]. Thus we will first make some general statements about modules, and later specialize to blocks. We refer the reader to [2] or [4] for the necessary facts about relatively projective modules, Green correspondence, and Brauer correspondence.

Definition 4.1. If M is an 𝔽G-module, we let

H_{P}^{∗} (G, M) := H^{∗}(G, M)/ ∑_{Q<_{G}P} cor_{Q}^{G}(H^{∗}(Q, M)).

If B is a block of G, we let

HH_{P}^{∗} (B) := H_{P}^{∗} (G, B), and similarly HH^{∗}(𝔽G) := H_{P}^{∗} (G, 𝔽G),

where the G-action on 𝔽G and B is by conjugation.

We will sometimes use the notation

X_{G} = {Q ≤ G | Q <_{G} P },

so that the sum in the definition is over all elements of X_{G}.

Remark 4.2. Let Ω denote the Heller operator, that is ΩV is the kernel of the projective cover of a module V. Then

(i) H_{P}^{n} (G, M) = Hom_{𝔽G}(Ω^{n}𝔽, M)/ ∑_{Q<_{G}P} tr_{Q}^{G}(Hom_{𝔽Q}(Ω^{n}𝔽, M)),

where tr_{Q}^{G}(f)(v) = ∑_{g∈G/Q} gf(g^{−1}v).

(ii) In particular, if M is a relatively X_{G}-projective 𝔽G-module, then H_{P}^{∗} (G, M) = 0 by Higman’s criterion.

The goal of this section is to prove the following theorem and derive consequences for Linckelmann cohomology in the case P is elementary abelian. Theorem 4.3. Let B be a block of 𝔽G with defect group the Sylow p-subgroup P of G. Let K = P C_{G}(P), and b a block of 𝔽K such that B is the unique block covering b. Then

HH_{P}^{∗} (B) ≅ (H_{P}^{∗} (P, 𝔽) ⊗ 𝔽Z(P))^{N_{G}(b)}.

Before we can prove this theorem, we must develop some preliminary properties of the functor H_{P}^{∗} (G, −). The proof of the following lemma is a straightforward consequence of the proof of the Eckmann-Shapiro Lemma. Lemma 4.4. Let H be a subgroup of G containing P. Let M be an 𝔽H-module. Then

H_{P}^{∗} (G, M ↑_{H}^{G}) ≅ H_{P}^{∗} (H, M). □

We next observe that H_{P}^{∗} is determined locally:

Lemma 4.5. Let L be any subgroup of G such that L ≥ N_{G}(P), and M an 𝔽G-module. Then res_{L}^{G} induces an isomorphism

H_{P}^{∗} (G, M) ≅ H_{P}^{∗} (L, M_{L}).

If M is indecomposable, Q is a vertex of M, and V its Green correspondent, then

H_{P}^{∗} (G, M) ≅ \{ \begin{matrix}H_{P}^{∗} (L, V), & if Q =_{G} P \\ 0, & if Q <_{G} P. \end{matrix}

Proof. This follows from Remark 4.2(i) and [4, Theorem 3.12.2(v)]. (For relevant facts about Ω, see [2, Theorem 20.5 and 20.7].) Alternatively, the lemma follows directly from Remark 4.2(ii) and Lemma 4.4. □

The next lemma allows us further to express the H_{P}^{∗} -cohomology groups as fixed points of cohomology groups of 𝔽P -modules. Lemma 4.6. Let K, L be any subgroups of G such that P ≤ K ≤ L ≤ N_{G}(P), K is normal in L, and M an 𝔽L-module. Then res_{K}^{L} induces an isomorphism

H_{P}^{∗} (L, M) ≅ H_{P}^{∗} (K, M_{K})^{L}.

Proof. As P is the Sylow p-subgroup of L, K is normal in L, and K contains P, standard arguments show that res_{K}^{L} induces an isomorphism on cohomology, H^{∗}(L, M) ≅ H^{∗}(K, M_{K})^{L}. It may be checked that this induces the isomorphism in the lemma. □

We will next turn our attention to blocks of group algebras, first deriving a result for the group algebras themselves. Let K = P C_{G}(P). As K ≤ N_{G}(P) and P is the Sylow p-subgroup of N_{G}(P), the arguments of Section 3 show that K ≅ P × Q for a p′-group Q ≅ C_{G}(P)/Z(P). Lemma 4.7. (i) HH_{P}^{∗} (𝔽G) ≅ (H_{P}^{∗} (P, 𝔽) ⊗ 𝔽C_{G}(P))^{N_{G}(P)}.

(ii) HH_{P}^{∗} (𝔽G) is isomorphic to (H_{P}^{∗} (P, 𝔽) ⊗ 𝔽Q)^{N_{G}(P)}, modulo radicals.

Proof. By Lemmas 4.5 and 4.6, we have

H_{P}^{∗} (G, 𝔽G) ≅ H_{P}^{∗} (P, 𝔽G)^{N_{G}(P)}.

Now write 𝔽G ≅ 𝔽K ⊕ 𝔽(G − K) as 𝔽P -modules. If g ∈ G − K, then C_{P} (g) ≠ P, so 𝔽(G − K) is relatively X_{P} -projective as an 𝔽P -module. By Remark 4.2(ii),

H_{P}^{∗} (P, 𝔽G) ≅ H_{P}^{∗} (P, 𝔽K).

As K ≅ P × Q, and P acts trivially on Q by conjugation,

H^{∗}(P, 𝔽K) ≅ H^{∗}(P, 𝔽P) ⊗ 𝔽Q

(see [9, p. 18]). By analyzing this isomorphism, we find that it factors to yield

H_{P}^{∗} (P, 𝔽K) ≅ H_{P}^{∗} (P, 𝔽P) ⊗ 𝔽Q.

Taking fixed points, we now have

H_{P}^{∗} (G, 𝔽G) ≅ (H_{P}^{∗} (P, 𝔽P) ⊗ 𝔽Q)^{N_{G}(P)},

which by [21, Theorem 10.2] yields statement (i) of the theorem.

Statement (ii) follows as P acts trivially on H_{P}^{∗} (P, 𝔽) ⊗ 𝔽C_{G}(P), N_{G}(P)/P is a p′-group, and 𝔽Z(P) is a local ring. □

Now we will show how the H_{P}^{∗} -cohomology of B is determined locally. Lemma 4.8. Let B be a block of 𝔽G. If B has a defect group properly contained in P, then HH_{P}^{∗} (B) = 0. If P is a defect group of B and b is the Brauer correspondent of B in 𝔽N_{G}(P), then

HH_{P}^{∗} (B) ≅ HH_{P}^{∗} (b).

Proof. By Lemma 4.5, H_{P}^{∗} (G, B) = 0 if B has a defect group Q properly contained in P, since the vertices of its components as an 𝔽Δ(G)- module will all be contained in Q. Otherwise, Lemma 4.5 implies that H_{P}^{∗} (G, B) ≅ H_{P}^{∗} (N_{G}(P), B).

As b is the Brauer correspondent of B, we have B ≅ b ⊕ T as 𝔽(N_{G}(P) × N_{G}(P))-modules, where T is a module that is relatively projective for the set

{s(Δ(P))s^{−1} ∩ (N_{G}(P) × N_{G}(P)) | s ∈ G × G, s ∉ N_{G}(P) × N_{G}(P)}.

Restricting to Δ(N_{G}(P)), we have

B_{Δ(N_{G}(P))} ≅ b_{Δ(N_{G}(P))} ⊕ T_{Δ(N_{G}(P))},

where by the Mackey formula, T_{Δ(N_{G}(P))} is relatively projective for the set

{s(Δ(P))s^{−1} ∩ Δ(N_{G}(P)) | s ∈ G × G, s ∉ N_{G}(P) × N_{G}(P)}.

The elements in this set are all properly contained in Δ(P). Therefore by Remark 4.2(ii), H_{P}^{∗} (N_{G}(P), B) ≅ H_{P}^{∗} (N_{G}(P), b). □

We are now ready to prove Theorem 4.3.

Proof of Theorem 4.3. By Lemma 4.8, we may assume that P is normal in G. As 𝔽(K × K)-modules, B divides

𝔽G ≅ 𝔽K ⊕ 𝔽(G − K),

and by [2, Lemma 13.7(3)], no indecomposable summand of 𝔽(G − K) has a vertex containing Δ(P). Therefore all summands of B_{K×K} with vertex containing Δ(P) are in fact blocks of K. By [2, Theorem 15.1], these form a single G-conjugacy class of blocks of K. Restricting to Δ(K) and applying Mackey’s formula and Remark 4.2(ii), we obtain

H_{P}^{∗} (K, B) ≅ ⨁_{g∈G/N_{G}(b)} H_{P}^{∗} (K,^{g}b).

By Lemma 4.6 with L = G, restricting from G to K yields

H_{P}^{∗} (G, B) ≅ H_{P}^{∗} (K, B)^{G}
≅ ⨁_{g∈G/N_{G}(b)}(H_{P}^{∗} (K,^{g}b))^{G}
≅ H_{P}^{∗} (K, b)^{N_{G}(b)}
≅ H_{P}^{∗} (P, b)^{N_{G}(b)},

by Lemma 4.6 again (with K taking the place of L, and P taking the place of K). As in §3, the block b of 𝔽K must have the form b = 𝔽P ⊗b′, where b′ is a block of the semisimple algebra 𝔽Q. Therefore

H^{∗}(P, b) ≅ H^{∗}(P, 𝔽P) ⊗ b′.

By analyzing this isomorphism as in Lemma 4.7, we find that it factors to yield an isomorphism

H_{P}^{∗} (P, b) ≅ H_{P}^{∗} (P, 𝔽P) ⊗ b′.

By [21, Theorem 10.2], H_{P}^{∗} (P, 𝔽P) ≅ H_{P}^{∗} (P, 𝔽) ⊗ 𝔽Z(P), and so we have

H_{P}^{∗} (G, B) ≅ (H_{P}^{∗} (P, 𝔽) ⊗ 𝔽Z(P) ⊗ b′)^{N_{G}(b)}.

Now Q ≤ N_{G}(b) acts trivially on H_{P}^{∗} (P, 𝔽) ⊗ 𝔽Z(P), and (b′)^{Q} ≅ 𝔽 as (b′)^{Q} is the center of the matrix algebra b′. □

As P acts trivially on H_{P}^{∗} (P, 𝔽) ⊗ 𝔽Z(P), and 𝔽Z(P) is a local ring, we have the following corollary. Corollary 4.9. HH_{P}^{∗} (B) is isomorphic to H_{P}^{∗} (P, 𝔽)^{N_{G}(b)}, modulo radicals.

In case P is elementary abelian, this result has the following consequence for the cohomology ring of the block B. Corollary 4.10. Let G be a finite group with elementary abelian Sylow p-subgroup P, and B a block of G with defect group P. Then LH_{P}^{∗} (B) is isomorphic to HH^{∗}(B), modulo radicals.

Proof. As cor_{Q}^{P} : H^{∗}(Q, 𝔽) → H^{∗}(P, 𝔽) is 0 in case Q < P, Corollary 4.9 implies that HH_{P}^{∗} (B) is isomorphic to H^{∗}(P, 𝔽)^{N_{G}(b)}, modulo radicals. By Remark 2.3, this is precisely LH^{∗}(B). □

We conclude with an example suggesting that Corollary 4.10 may be strengthened in some cases to yield an isomorphism, modulo radicals, between HH^{∗}(B) and LH^{∗}(B). Example 4.11. Let p > 2, t < p, and G = S_{t+2p}, the symmetric group on t+2p letters. See [12] for the block theory of G. A Sylow p-subgroup P of G is elementary abelian of rank 2. The normalizer of P is

N = N_{G}(P) ≅ ((C_{p} ⋊ C_{p−1}) ≀ C_{2}) × S_{t},

where C_{i} denotes a cyclic group of order i. Let B be a block of G with defect group P. By [6, Corollary 3.2], B is derived equivalent to its Brauer correspondent b in N_{G}(P). Further, Remark 2.3 applies as P is abelian, and so

HH^{∗}(B) ≅ HH^{∗}(b) and LH^{∗}(B) ≅ LH^{∗}(b).

Now b is Morita equivalent to the principal block b′ of

N′ = (C_{p} ⋊ C_{p−1}) ≀ C_{2}

(see the introduction of [6]). Therefore HH^{∗}(b) ≅ HH^{∗}(b′). Another application of Remark 2.3 shows that

LH^{∗}(b) ≅ H^{∗}(N, 𝔽) ≅ H^{∗}(N′, 𝔽) ≅ LH^{∗}(b′),

as C_{N}(P) = P = C_{N′}(P) implies b_{p} = 𝔽P = b′_{P} and N = N′ × S_{t} with S_{t} a p′-group.

Finally, N′ satisfies the hypotheses of Theorem 3.1, so that HH^{∗}(b′) and LH^{∗}(b′) are isomorphic modulo their radicals. Combining all of the above isomorphisms, we now have that LH^{∗}(B) is isomorphic to HH^{∗}(B), modulo their radicals.

References

[1] J. L. Alperin, Sylow intersections and fusion, J. Algebra 6 (1967), 222–241.

[2] J. L. Alperin, Local Representation Theory, Cambridge University Press, 1986.

[3] J. L. Alperin and M. Broué, Local methods in block theory, Ann. Math. 110 (1979), 143–157.

[4] D. J. Benson, Representations and Cohomology I: Basic representation theory of finite groups and associative algebras, Cambridge University Press, 1991.

[5] M. Broué and L. Puig, Characters and local structure in G-algebras, J. Algebra 63 (1980), 306–317.

[6] J. Chuang, The derived categories of some blocks of symmetric groups and a conjecture of Broué, J. Algebra 217 (1999), 114–155.

[7] C. Cibils and A. Solotar, Hochschild cohomology algebra of abelian groups, Arch. Math. 68 (1997), 17–21.

[8] C. Curtis and I. Reiner, Methods of Representation Theory, with Applications to Finite Groups and Orders, volume II, Wiley, 1987.

[9] L. Evens, Cohomology of Groups, Oxford University Press, 1991.

[10] D. Gorenstein, Finite Groups, Chelsea, 1980.

[11] T. Holm, The Hochschild Cohomology Ring of a Modular Group Algebra: The Commutative Case, Comm. in Alg., 24(6), (1996), 1957–1969.

[12] G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and Its Applications, Vol. 16, Addison-Wesley, 1981.

[13] M. Linckelmann, Derived equivalences for cyclic blocks over a p-adic ring, Math. Z. 207 (1991), 293–304.

[14] M. Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups, Algebras and Representation Theory 2 (1999), 107–135.

[15] M. Linckelmann, Varieties in block theory, J. Algebra 215 (1999), 460–480.

[16] J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Alg. 61 (1989), 303–317.

[17] J. Rickard, Derived equivalences for principal blocks of A_{4} and A_{5}, preprint 1990.

[18] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37–48.

[19] R. Rouquier, From stable equivalences to Rickard equivalences for blocks with cyclic defect, Groups ’93 Galway/St. Andrews, Vol. 2, 512–523, London Math. Soc. Lecture Note Ser. 212, Cambridge University Press, 1995.

[20] S. F. Siegel, Varieties for Hochschild cohomology of group algebras and blocks, unpublished notes, 1997.

[21] S. F. Siegel and S. J. Witherspoon, The Hochschild cohomology ring of a group algebra, Proc. London Math. Soc. 79 (1999), 131–157.

[22] S. F. Siegel and S. J. Witherspoon, The Hochschild cohomology ring of a cyclic block, Proc. Amer. Math. Soc. 128 (2000), 1263–1268.

Department of Mathematics, University of Rochester, Rochester, NY 14627

  E-mail address: jonpak@math.rochester.edu

Department of Mathematics and Computer Science, Amherst College, Amherst, MA 01002

  E-mail address: sjwitherspoon@amherst.edu