Hochschild cohomology and support varieties for tame Hecke algebras

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.

💡 Research Summary

The paper investigates the Hochschild cohomology and support varieties of tame Hecke algebras, providing a complete description of the cohomology ring, its nilpotent quotient, and the geometry of module varieties. The authors begin by fixing a concrete presentation of a tame Hecke algebra (A) as a bound quiver algebra. The quiver has two vertices and several arrows, and the relations are the usual Hecke relations depending on a parameter (q). This presentation guarantees that (A) is of tame representation type, i.e., its indecomposable modules occur in one‑parameter families.

Using this presentation, the authors construct an explicit minimal projective bimodule resolution (P_{\bullet}\to A) over the enveloping algebra (A^{e}=A\otimes_k A^{op}). By applying (\operatorname{Hom}_{A^{e}}(-,A)) they obtain the Hochschild cochain complex (C^{*}(A,A)). The low‑degree cohomology groups are computed directly: (HH^{0}(A)) is the centre of (A); (HH^{1}(A)) is generated by derivations coming from the arrows; and (HH^{2}(A)) is generated by classes corresponding to the defining relations. The authors exhibit a basis for each (HH^{n}(A)) in terms of explicit cocycles built from the arrows and relations. For higher degrees they show a periodic pattern: every class in degree (n\ge 3) can be expressed as a product of degree‑2 classes, reflecting the Koszul‑like nature of the algebra.

The next step is to isolate the ideal of nilpotent elements (\mathcal N\subset HH^{*}(A)). By analysing the multiplication table of the basis elements, they prove that (\mathcal N) is generated by sufficiently high powers of the degree‑1 and degree‑2 generators. After factoring out (\mathcal N) the resulting ring \