A family of Koszul self-injective algebras with finite Hochschild cohomology

TL;DR

This paper introduces an infinite family of Koszul self-injective algebras with finite-dimensional Hochschild cohomology rings, including examples with arbitrarily large finite dimensions, extending previous known cases.

Contribution

It constructs a new family of Koszul self-injective algebras with finite Hochschild cohomology, generalizing earlier examples and providing explicit dimension examples for each N ≥ 5.

Findings

Family includes and generalizes known 4-dimensional algebras.

Provides examples with Hochschild cohomology dimension N for each N ≥ 5.

Shows these algebras have finite Hochschild cohomology despite infinite global dimension.

Abstract

This paper presents an infinite family of Koszul self-injective algebras whose Hochschild cohomology ring is finite-dimensional. Moreover, for each $N \geq 5$ we give an example where the Hochschild cohomology ring has dimension $N$. This family of algebras includes and generalizes the 4-dimensional Koszul self-injective local algebras of Buchweitz, Green, Madsen and Solberg, which were used to give a negative answer to Happel's question, in that they have infinite global dimension but finite-dimensional Hochschild cohomology.