The Hochschild cohomology ring of a class of special biserial algebras

TL;DR

This paper computes the Hochschild cohomology ring of certain self-injective special biserial algebras, showing it is finitely generated and confirming a conjecture about its structure modulo nilpotence.

Contribution

It demonstrates that the Hochschild cohomology ring of these algebras is finitely generated and verifies the Snashall-Solberg conjecture for this class.

Findings

Hochschild cohomology ring is finitely generated.

Modulo nilpotence, the ring is a 2-dimensional commutative algebra.

The Snashall-Solberg conjecture holds for these algebras.

Abstract

We consider a class of self-injective special biserial algebras $\Lambda_N$ over a field $K$ and show that the Hochschild cohomology ring of $\Lambda_N$ is a finitely generated $K$-algebra. Moreover the Hochschild cohomology ring of $\Lambda_N$ modulo nilpotence is a finitely generated commutative $K$-algebra of Krull dimension two. As a consequence the conjecture of Snashall-Solberg \cite{SS}, concerning the Hochschild cohomology ring modulo nilpotence, holds for this class of algebras.