Hochschild and ordinary cohomology rings of small categories

TL;DR

This paper investigates the relationship between Hochschild cohomology of category algebras and the cohomology of classifying spaces, proving a split surjective homomorphism and providing a counterexample to a conjecture about finite generation.

Contribution

It establishes a split surjective homomorphism between Hochschild cohomology and space cohomology for small categories and constructs a counterexample to a finite generation conjecture.

Findings

Proves the homomorphism HH*(kC) --> H*(|C|,k) is split surjective.

Constructs a category algebra with non-finitely generated Hochschild cohomology.

Generalizes known results from groups and posets to small categories.

Abstract

Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C|=BC. We study the ring homomorphism HH*(kC) --> H*(|C|,k) and prove it is split surjective. This generalizes the well-known results for groups and posets. Based on this result, we construct a seven-dimensional category algebra whose Hochschild cohomology ring modulo nilpotents is not finitely generated, against a conjecture of Snashall and Solberg.