Quillen stratification for Hochschild cohomology of blocks

Abstract

We decompose the maximal ideal spectrum of the Hochschild cohomology ring of a block of a finite group into a disjoint union of subvarieties corresponding to elementary abelian p p -subgroups of a defect group. These subvarieties are described in terms of group cohomological varieties and the Alperin-Broué correspondence on blocks. Our description leads in particular to a homeomorphism between the Hochschild variety of the principal block and the group cohomological variety. The proofs require a result of Stephen F. Siegel, given in the Appendix, which states that nilpotency in Hochschild cohomology is detected on elementary abelian p p -subgroups.