Hochschild and ordinary cohomology rings of small categories

Abstract

Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra k C and the classifying space | C | = B C . We study the ring homomorphism HH ∗ ( k C ) → H ∗ ( | C | , k ) and prove it is split surjective, using the factorization category of Quillen [D. Quillen, Higher algebraic K-theory I, in: Lecture Notes in Math., vol. 341, Springer-Verlag, Berlin, 1973, pp. 85–147] and certain techniques from functor cohomology theory. This generalizes the well-known theorems 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, disproving a conjecture of Snashall and Solberg [N. Snashall, Ø. Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc. 88 (3) (2004) 705–732].