Title: Hochschild and ordinary cohomology rings of small categories

Author: Fei Xu

Address: UMR 6629 CNRS/UN, Laboratoire de Mathematiques Jean Leray, Universite de Nantes, 2 Rue de la
Houssiniere, 44322 Nantes, France.


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, using
the factorization category of Quillen 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.

Status: Advances in Mathematics, to appear, final version 10/07/2008.