Bounds for Hochschild cohomology of block algebras

TL;DR

This paper establishes bounds on the dimensions of Hochschild cohomology groups for block algebras of finite groups, depending only on the cohomology degree and the defect of the algebra.

Contribution

It provides a new bound for Hochschild cohomology dimensions of block algebras based solely on defect and degree, extending previous results for n=0.

Findings

Dimension of HH^n(B) is bounded by a function of n and the defect of B.

The proof leverages a theorem of Brauer and Feit for the case n=0.

The result applies to all block algebras over algebraically closed fields of prime characteristic.

Abstract

We show that for any block algebra B of a finite group over an algebraically closed field of prime characteristic the dimension of HH^n(B) is bounded by a function depending only on the nonnegative integer n and the defect of B. The proof uses in particular a theorem of Brauer and Feit which implies the result for n=0.