Vanishing of the Hochschild cohomology for some self-injective special biserial algebra of rank four

TL;DR

This paper calculates the Hochschild cohomology dimensions for certain self-injective special biserial algebras of rank four, providing a negative answer to a longstanding question in the field.

Contribution

It offers the first explicit computation of Hochschild cohomology for these specific algebras, addressing a question posed by Happel.

Findings

Hochschild cohomology dimensions determined for the algebras

Negative answer to Happel's question

Advances understanding of self-injective special biserial algebras

Abstract

In this paper, we determine the dimensions of the Hochschild cohomology groups of some self-injective special biserial algebra whose Grothendieck group is of rank $4$. This result provides us with a negative answer to Happel's question in [H].