On cohomology of saturated fusion systems and support varieties

TL;DR

This paper explores the cohomology algebra of saturated fusion systems by leveraging finite group realizations and Hochschild cohomology, extending classical results to fusion systems related to block algebras.

Contribution

It introduces new connections between fusion system cohomology and Hochschild cohomology, extending Alperin's theorem to fusion systems associated with block algebras.

Findings

Established a relationship between fusion system cohomology and Hochschild cohomology.

Proved a theorem similar to Alperin's for cohomology varieties of fusion systems.

Extended classical cohomological results to the context of fusion systems and block algebras.

Abstract

In this short note we study the cohomology algebra of saturated fusion systems using finite groups which realize saturated fusion systems and Hochschild cohomology of group algebras. A similar result to a theorem of Alperin is proved for varieties of cohomology algebras of fusions systems associated to block algebras of finite groups.