Hochschild cohomology and support varieties for tame Hecke algebras

TL;DR

This paper computes the Hochschild cohomology ring of tame Hecke algebras, showing it is finitely generated with Krull dimension 2, and describes the support varieties of modules, revealing dimension differences based on algebra type.

Contribution

It provides a basis for the Hochschild cohomology ring of tame Hecke algebras and characterizes its structure and support varieties, highlighting differences between finite and tame types.

Findings

Hochschild cohomology ring modulo nilpotence is finitely generated with Krull dimension 2.

Support varieties of modules are described for these algebras.

Krull dimension of Hochschild cohomology ring is 1 for finite type, 2 for tame type.

Abstract

We give a basis for the Hochschild cohomology ring of tame Hecke algebras. We then show that the Hochschild cohomology ring modulo nilpotence is a finitely generated algebra of Krull dimension 2, and describe the support varieties of modules for these algebras. As a consequence we obtain the result that the Hochschild cohomology ring modulo nilpotence of a Hecke algebra has Krull dimension 1 if the algebra is of finite type and has Krull dimension 2 if the algebra is of tame type.