The Hochschild cohomology ring of a class of special biserial 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.