On the Hilbert series of Hochschild cohomology of block algebras

We show that the degrees and relations of the Hochschild cohomology of a p-block algebra of a finite group over an algebraically closed field of prime characteristic p are bounded in terms of the defect groups of the block and that for a fixed defect d, there are only finitely many Hilbert series of Hochschild cohomology algebras of blocks of defect d. The main ingredients are Symondsʼ proof of Bensonʼs regularity conjecture and the fact that the Hochschild cohomology of a block is finitely generated as a module over block cohomology, which is an invariant of the fusion system of the block on a defect group.