On the Hochschild Cohomology of Tame Hecke Algebras

In this paper we are interested in Hochschild cohomology of finite-dimensional algebras; the main motivation is to generalize group cohomology to larger classes of algebras. If suitable finite generation holds, one can define support varieties of modules as introduced by [SS]. Furthermore, when the algebra is self-injective, many of the properties of group representations generalize to this setting as was shown in [EHSST]. Although Hecke algebras do not have a Hopf algebra structure, one may expect that being deformations of group algebras, they should have good homological properties. Furthermore, the study of Hochschild cohomology for blocks of group algebras with cyclic or dihedral defect groups has made use of the fact that the basic algebras in this case are special biserial [Ho1, Ho2]. These results suggest that Hochschild cohomology of special biserial algebras might be accessible more generally. This paper is concerned with the self-injective special biserial algebra A which occurs as the basic algebra of the Hecke algebra Hq(S4) when q = −1. For this algebra we will explicitly give a minimal projective bimodule resolution and use it to calculate the dimensions of the Hochschild cohomology. By [EN] any tame block of some Hecke algebra Hq(Sn) is derived equivalent to A. Since Hochschild cohomology is invariant under derived equivalence, our result gives information for arbitrary tame blocks Hecke algebras of type A. In order for the Hochschild cohomology ring to be finitely generated, the dimensions of HH(A) must have at most polynomial growth. Therefore our results give evidence towards the finite generation of the Hochschild cohomology. Further evidence towards this comes from [ES], which shows by a different approach, via derived equivalence, that the finite generation hypothesis in [EHSST] holds for the special biserial algebra under consideration in this paper.