An alternating sum for Hochschild cohomology of a block Burkhard Külshammer Fakultät Mathematik & Informatik Mathematisches Institut Universität Jena D07740 Jena Germany and ^{1} Geoffrey R. Robinson School of Mathematics and Statistics University of Birmingham Birmingham B15 2TT England

In this note, we point out that when p is a prime, the additive structure of the Hochschild cohomology ring of a p-block of a finite group G satisfies the now-familiar type of alternating sum formula using the p-subgroup complex of G. (Throughout, we work with the Tate version of Hochschild cohomology - which coincides with ordinary Hochschild cohomology in positive dimensions.) Such an expression ( in our situation) allows the Hochschild cohomology of the block to be computed from p-local information. The first such formula ( for the cohomology of a G-module) was obtained by P.J.Webb ( see, for example, [6]). We also give a formulation of our result in terms of subpairs. For illustrative purposes, we give an application in the situation when the block is controlled by the normalizer of a maximal subpair, and the outer automorphims of the defect group induced by the inertial quotient all act without non-trivial fixed-points. This application generalizes an earlier result of Kessar and Linckelmann [2], which dealt with such a situation in the case that the defect group is Abelian.

1This work was supported by the British Council-DAAD grants ARC 1166 and ARC 00/27756 respectively

Let F be an algebraically closed field of prime characteristic p ( we content ourselves with demonstrating the result with this ground ring, though the arguments readily adapt for other standard ground rings which arise when dealing with blocks. They can also be used to determine the p-primary components of the relevant Abelian groups when working over appropriate ground rings).

To prove the main result, we need two facts. The first, noted already in the paper of Siegel and Witherspoon [5], is that if B is a block of F X for a finite group X, then the F -vector space structure of ĤH^{∗}(B) is the same as the F -vector space structure of Ĥ^{∗}(X, B), where B is viewed as an F X-module under conjugation action. (Recall that we are using the Tate version of group and Hochschild cohomology.)

The second fact, proved in Knörr-Robinson [3] is that if B is a block of F G for a finite group G, then

∑_{σ∈S_{p}(G)/G}(−1)^{|σ|}Ind_{G_{σ}}^{G}(B_{σ})

is a virtual projective module in the Green ring for F G, where B_{σ} is viewed as a module for F G_{σ} under conjugation.

We recall that in the above notation, σ denotes a ( possibly empty) strictly increasing ( under usual inclusion) chain of non-trivial p-subgroups of G. The length of the chain is |σ|, which we consider here to be the number of non-trivial subgroups in the chain. The chain stabilizer is G_{σ}, which is just the intersection of the normalizers of the subgroups appearing in σ. If σ is empty, then we set G_{σ} = G. The sum of blocks B_{σ} is the sum of the Brauer correspondents of B for G_{σ}, which were shown to be defined ( with an appropriate notion of Brauer correspondence) in [3]. We point out, in particular, that B_{σ} = 0 unless the largest subgroup which appears in σ is contained in some defect group of B.

Using the virtual projectivity of the above alternating sum, taking (Tate) cohomology, and invoking the Eckmann-Shapiro lemma, shows that for each integer i, we have

∑_{σ∈S_{p}(G)/G}(−1)^{|σ|}dim_{F} (Ĥ^{i}(G_{σ}, B_{σ})) = 0.

Since we are viewing B_{σ} as an F G_{σ}- module under conjugation, we obtain : THEOREM 1: Let B be a block of F G, where G is a finite group. Then for each integer i, we have

∑_{σ∈S_{p}(G)/G}(−1)^{|σ|}dim_{F} (ĤH^{i}(B_{σ})) = 0.

It also follows from the results of [3] that we can replace the complex S_{p}(G) by the sub-complex N_{p}(G), which consists of chains of p-subgroups such that each subgroup on the chain is normal in the largest. If B has defect group D, since we are only concerned with chains up to conjugacy, we may consider normal chains σ such that the largest subgroup is contained in D.

We may work instead with the complex N (B), consisting of (normal) chains of B-subpairs, that is, chains of B-subpairs of the form (σ, b_{n}) : (Q_{1}, b_{1}) < (Q_{2}, b_{2}) < … < (Q_{n}, b_{n}), where each (Q_{i}, b_{i}) ⊲ (Q_{n}, b_{n}) ( and is a B-subpair). The stabilizer of this chain of subpairs is just I_{G_{σ}}(b_{n}), where G_{σ} is the stabilizer of the chain of subgroups Q_{1} < Q_{2} < … < Q_{n}. There is a unique block of F I_{G_{σ}}(b_{n}) covering the block b_{n} of F C_{G}(Q_{n}), and we denote this block of F I_{G_{σ}}(b_{n}) by β_{(σ,b_{n})}. The block β_{(σ,b_{n})} induces to a Morita equivalent block of F G_{σ}, which is one of the summands of B_{σ} in the earlier definition. Furthermore, all summands of B_{σ} arise in this fashion, and the uniqueness of maximal subpairs up to conjugacy ensures that there is no unwanted repetition. Since Hochschild cohomology is invariant under Morita equivalence, we may recast Theorem 1 in terms of B-subpairs as: THEOREM 2: Let B, G be as before. Then for each integer i, we have

∑_{(σ,b)∈N (B)/G}(−1)^{|σ|}dim_{F} (ĤH^{i}(β_{(σ,b)})) = 0.

The application promised in the introduction is:

THEOREM 3 : Let B be a block of a finite group G with maximal B-subpair (P, b_{P}). Suppose that N_{G}(P, b_{P}) controls the fusion of B-subpairs (Q, b_{Q}) ≤ (P, b_{P}) and that every p′-element in N_{G}(P, b_{P}) \ C_{G}(P) acts fixed-point-freely on P^{♯}. Then dim_{F} (ĤH^{i}(B)) = dim_{F} (ĤH^{i}(β_{(P,b_{P})})) for all i.

Proof. We fix an integer i. For a nonempty chain (σ, b_{n}) : (Q_{0}, b_{0}) < … < (Q_{n}, b_{n}) in N (B), Q_{n} is a normal p-subgroup of the stabilizer G_{(σ,b_{n})} = I_{G_{σ}}(b_{n}) of (σ, b_{n}) in G, and b_{n} may be viewed as a G_{(σ,b_{n})}-stable block of the normal subgroup Q_{n}C_{G}(Q_{n}) of G_{(σ,b_{n})}. The block β_{(σ,b_{n})} := b_{n}^{I_{Gσ} (b_{n})} is defined, and β_{(σ,b_{n})} is the only block of I_{G_{σ}}(b_{n}) covering b_{n}. Since C_{G}(Q_{n}) = C_{G_{σ}}(Q_{n}), (Q_{n}, b_{n}) is also a β_{(σ,b_{n})}-subpair. For (σ, B) the empty chain ∅, we set G_{(σ,B)} := G and β_{(σ,B)} := B. We denote by N_{1}(B) the set of non-empty chains (σ, b_{n}) : (Q_{0}, b_{0}) < … < (Q_{n}, b_{n}) in N (B) such that (Q_{n}, b_{n}) is a maximal β_{(σ,b_{n})}-subpair, and we denote by N_{0}(B) the set of remaining chains in N (B). Both N_{0}(B) and N_{1}(B) are stable under conjugation by elements of G. For a chain (σ, b_{n}) : (Q_{0}, b_{0}) < … < (Q_{n}, b_{n}) in N_{1}(B), we denote by (σ′, b_{n−1}) the chain (Q_{0}, b_{0}) < … < (Q_{n−1}, b_{n−1}) in N (B). We claim that (σ′, b_{n−1}) ∈ N_{0}(B).

For assume the contrary. Then (Q_{n−1}, b_{n−1}) is a maximal β_{(σ′,b_{n−1})}-subpair. On the other hand, Q_{n} is contained in I_{G_{σ′}} (b_{n−1}), b_{n} is a block of Q_{n}C_{G}(Q_{n}) = Q_{n}C_{G_{σ′}} (Q_{n}), and (Q_{n−1}, b_{n−1}) < (Q_{n}, b_{n}) as β_{(σ′,b_{n−1})}-subpairs. This contradiction shows that (σ′, b_{n−1}) ∈ N_{0}(B). This procedure yields a G-equivariant map N_{1}(B) → N_{0}(B), (σ, b_{n}) ↦ (σ′, b_{n−1}). This induces a well-defined map N_{1}(B)/G → N_{0}(B)/G. We will show next that this last induced map is a bijection. Suppose first that (σ, b_{n}) : (Q_{0}, b_{0}) < … < (Q_{n}, b_{n}) and (τ, c_{n}) : (R_{0}, c_{0}) < … < (R_{n}, c_{n}) are chains in N_{1}(B) such that (σ′, b_{n−1}) and (τ′, c_{n−1}) are conjugate in G. We need to prove that (σ, b_{n}) and (τ, c_{n}) are conjugate in G. After conjugation, if necessary, we may assume that (σ′, b_{n−1}) = (τ′, c_{n−1}). As above, (Q_{n}, b_{n}) is also a β_{(σ′,b_{n−1})}-subpair. Suppose that (Q_{n}, b_{n}) is not a maximal β_{(σ′,b_{n−1})}-subpair. Then there exists a β_{(σ′,b_{n−1})}-subpair (Q_{n+1}, b_{n+1}) such that (Q_{n}, b_{n}) < (Q_{n+1}, b_{n+1}). Then b_{n+1} is a block of Q_{n+1}C_{G_{σ′}} (Q_{n+1}) = Q_{n+1}C_{G_{σ}}(Q_{n+1}), and we have (Q_{n}, b_{n}) < (Q_{n+1}, b_{n+1}) as β_{(σ,b_{n})}-subpairs. This contradiction shows that (Q_{n}, b_{n}) is in fact a maximal β_{(σ′,b_{n−1})}-subpair. Similarly, (R_{n}, c_{n}) is a maximal β_{(σ′,b_{n−1})}-subpair. Hence (Q_{n}, b_{n}) and (R_{n}, c_{n}) are conjugate in I_{G_{σ′}} (b_{n−1}). Thus (σ, b_{n}) and (τ, c_{n}) are conjugate under I_{G_{σ′}} (b_{n−1}), and we have proved that the map N_{1}(B)/G → N_{0}(B)/G induced by (σ, b_{n}) ↦ (σ′, b_{n−1}) is one-to-one. We show next that the map N_{1}(B)/G → N_{0}(B)/G is onto. Let us first consider the empty chain (σ, B) = ∅ in N_{0}(B). Then the singleton chain (σ̃, b_{P}) : (P, b_{P}) belongs to N_{1}(B) and satisfies (σ̃′, b_{P}) = (σ, B). Suppose now that σ : (Q_{0}, b_{0}) < … < (Q_{n}, b_{n}) is a nonempty chain in N_{0}(B). Then (Q_{n}, b_{n}) is not a maximal β_{(σ,b_{n})}-subpair. We denote by (Q_{n+1}, b_{n+1}) a maximal β_{(σ,b_{n})}-subpair such that (Q_{n}, b_{n}) < (Q_{n+1}, b_{n+1}). Then b_{n+1} is a block of Q_{n+1}C_{G_{σ}}(Q_{n+1}) = Q_{n+1}C_{G}(Q_{n+1}), and (Q_{n}, b_{n}) < (Q_{n+1}, b_{n+1}) as B-subpairs. Since Q_{n+1} ≤ I_{G_{σ}}(b_{n}), the chain (σ̃, b_{n+1}) : (Q_{0}, b_{0}) < … < (Q_{n}, b_{n}) < (Q_{n+1}, b_{n+1}) is in N (B). Since (Q_{n+1}, B_{n+1}) is a β_{(σ̃,b_{n+1})}-subpair, Q_{n+1} is contained in a defect group of β_{(σ̃,b_{n+1})}. On the other hand, Q_{n+1} is a defect group of β_{(σ,b_{n})} =

b_{n}^{I_{Gσ} (b_{n})} = (b_{n}^{I_{Gσ̃} (b_{n+1})})^{I_{Gσ} (b_{n})} = bI_{Gσ} (b_{n}) ; σ̃. Thus Q_{n+1} is a defect group of β_{(σ̃,b_{n+1})}, so (Q_{n+1}, b_{n+1})

is a maximal β_{(σ̃,b_{n+1})}-subpair. Hence (σ̃, b_{n+1}) ∈ N_{1}(B), and certainly (σ̃′, b_{n+1}) = (σ, b_{n}). Thus we have shown that the map (σ, b_{n}) ↦ (σ′, b_{n−1}) induces a bijection N_{1}(B)/G → N_{0}(B)/G. By the previous theorem, we have

0 = ∑_{(σ,b_{n})∈N (B)/G}(−1)^{|σ|} dim_{F} (ĤH^{i}(β_{(σ,b_{n})}))
= ∑_{(σ,b_{n})∈N_{1}(B)/G}(−1)^{|σ|}[dim_{F} (ĤH^{i}(β_{(σ,b_{n})})) − dim_{F} (HH^{i}(β_{(σ′,b_{n−1})}))].

We have to show that the summand corresponding to the singleton chain σ : (P, b_{P}) in N_{1}(B) vanishes. It suffices to prove that the remaining summands in this sum vanish. In the following, let (σ, b_{n}) : (Q_{0}, b_{0}) < … < (Q_{n}, b_{n}) be a chain in N_{1}(B) such that σ′ ≠ ∅. We will show that β_{(σ,b_{n})} and β_{(σ′,b_{n−1})} are Morita equivalent; this in turn implies that dim_{F} (ĤH^{i}(β_{(σ,b_{n})})) = dim_{F} (ĤH^{i}(β_{(σ′,b_{n−1})})). Note first that Q_{n−1} is a normal p-subgroup of I_{G_{σ′}} (b_{n−1}), so Q_{n−1}C_{G}(Q_{n−1}) is a normal subgroup of I_{G_{σ′}} (b_{n−1}). Moreover, b_{n−1} is an I_{G_{σ′}} (b_{n−1})-stable block of Q_{n−1}C_{G}(Q_{n−1}) such that b_{n−1}^{I_{G_{σ′}} (b_{n−1})} = β_{(σ′,b_{n−1})}. Thus β_{(σ′,b_{n−1})} is the only block of I_{G_{σ′}} (b_{n−1}) covering b_{n−1}. We claim that the block b_{n−1} of Q_{n−1}C_{G}(Q_{n−1}) is nilpotent. In fact, let (V, b_{V}) be a b_{n−1}-subpair. We need to show that N_{Q_{n−1}C_{G}(Q_{n−1})}(V, b_{V})/C_{Q_{n−1}C_{G}(Q_{n−1})}(V) is a p-group. Since Q_{n−1} is a normal p-subgroup of Q_{n−1}C_{G}(Q_{n−1}) we may assume, by the Alperin-Goldschmidt fusion theorem for subpairs, that Q_{n−1} ≤ V. Then b_{V} is a block of V C_{Q_{n−1}C_{G}(Q_{n−1})}(V) = V C_{G}(V) such that b_{V}^{Q_{n−1}C_{G}(Q_{n−1})} = b_{n−1}. Hence b_{V}^{V C_{G}(Q_{n−1})} = b_{n−1}^{V C_{G}(Q_{n−1})} which implies that (Q_{n−1}, b_{n−1}) ≤ (V, b_{V}) as B-subpairs. We may, and do, assume that (V, b_{V}) ≤ (P, b_{P}). Since N_{G}(P, b_{P}) controls the fusion of B-subpairs (Q, b_{Q}) ≤ (P, b_{P}), we see that

N_{G}(V, b_{V}) = [N_{G}(V, b_{V}) ∩ N_{G}(P, b_{P})]C_{G}(V).

Since C_{G}(V) ≤ C_{G}(Q_{n−1}), Dedekind’s law yields

N_{Q_{n−1}C_{G}(Q_{n−1})}(V, b_{V}) = [N_{G}(V, b_{V}) ∩ N_{G}(P, b_{P}) ∩ Q_{n−1}C_{G}(Q_{n−1})]C_{G}(V)
= [N_{G}(V, b_{V}) ∩ N_{G}(P, b_{P}) ∩ C_{G}(Q_{n−1})]Q_{n−1}C_{G}(V).

By our hypothesis, every p-regular element in N_{G}(P, b_{P}) \ C_{G}(P) acts fixed-point-freely on P^{♯}. Hence N_{G}(P, b_{P})∩C_{G}(Q_{n−1})/C_{G}(P) is a p-group. Thus N_{Q_{n−1}C_{G}(Q_{n−1})}(V, b_{V})/C_{Q_{n−1}C_{G}(Q_{n−1})}(V) is indeed a p-group. We have thus shown that the block B_{n−1} of Q_{n−1}C_{G}(Q_{n−1}) is nilpotent. In the following, we will assume (as we may) that (Q_{n}, b_{n}) ≤ (P, b_{P}). Then

N_{G}(Q_{n−1}, b_{n−1}) = [N_{G}(Q_{n−1}, b_{n−1}) ∩ N_{G}(P, b_{P})]C_{G}(Q_{n−1}).

Since C_{G}(Q_{n−1}) ≤ I_{G_{σ′}} (b_{n−1}) ≤ N_{G}(Q_{n−1}, b_{n−1}), Dedekind’s law implies that

I_{G_{σ′}} (b_{n−1}) = [N_{G}(Q_{n−1}, b_{n−1}) ∩ N_{G}(P, b_{P}) ∩ I_{G_{σ′}} (b_{n−1})]C_{G}(Q_{n−1})
= N_{I_{G_{σ′}} (b_{n−1})}(P, b_{P})C_{G}(Q_{n−1}).

We claim that P ∩ I_{G_{σ′}} (b_{n−1}) = Q_{n}. In fact, we certainly have Q_{n} ≤ P ∩ I_{G_{σ′}} (b_{n−1}) =: R. On the other hand, there is a B-subpair (R, b_{R}) such that (Q_{n}, b_{n}) ≤ (R, b_{R}) ≤ (P, b_{P}). Then b_{R} is a block of RC_{G}(R) = RC_{I_{Gσ} (b_{n})}(R), and we have (Q_{n}, b_{n}) ≤ (R, b_{R}) as β_{(σ,b_{n})}-subpairs. Since (Q_{n}, b_{n}) is a maximal β_{(σ,b_{n})}-subpair, this implies that Q_{n} = R = P ∩ I_{G_{σ′}} (b_{n−1}). Hence I_{G_{σ′}} (b_{n−1}) = N_{G_{σ′}} (Q_{n}, b_{n})C_{G}(Q_{n−1}), so that Q_{n}C_{G}(Q_{n−1}) is a normal subgroup of I_{G_{σ′}} (b_{n−1}). By a well-known theorem of Cabanes [1], the block b_{n−1}^{Q_{n}C_{G}(Q_{n−1})} = b_{n}^{Q_{n}C_{G}(Q_{n−1})} covering b_{n−1} is also nilpotent. Moreover, (Q_{n}, b_{n}) is a b_{n}^{Q_{n}C_{G}(Q_{n−1})}-subpair. It is easy to see that (Q_{n}, b_{n}) is a maximal b_{n}^{Q_{n}C_{G}(Q_{n−1})}-subpair. By the results of of Külshammer-Puig [4], the block β_{(σ′,b_{n−1})} of I_{G_{σ′}} (b_{n−1}) is Morita equivalent to its Harris-Knörr correspondent in N_{G_{σ′}} (Q_{n}, b_{n}) = I_{G_{σ}}(b_{n}). However, this is just the block β_{(σ,b_{n})} of I_{G_{σ}}(b_{n}), and we are done. Acknowledgements. The authors would like to thank the referee for some useful suggestions.

REFERENCES

[1] Cabanes, M., Extensions of p-groups and construction of characters, Comm. Algebra 15 (1987), 1297-1322

[2] Kessar, R., Linckelmann, M., On blocks with Frobenius inertial quotient, preprint

[3] Knörr, R., Robinson, G.R, Some remarks on a conjecture of Alperin, J. London Math. Soc., (2),39,1,(1989), 48-60.

[4] Külshammer, B., Puig, L., Extensions of nilpotent blocks, Invent. math. 102 (1990), 17-71 [5] Siegel, S., Witherspoon,S., The Hochschild cohomology ring of a cyclic block, Proc. Amer. Mat. Soc., 128,5,(2000), 1263-1268.

[6] Webb, P.J., A split exact sequence of Mackey functors, Comment. Math. Helv., 66,1,(1991), 34-69.