Hochschild and ordinary cohomology rings of small categories

Fei Xu (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

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.

Mots clés: Hochschild cohomology ring; ordinary cohomology ring; category algebra; category of factorizations; left Kan extension; finite EI-categories; finite generation; nilpotent element

Domaines: Topologie algébrique [math.AT]

Fei Xu. Hochschild and ordinary cohomology rings of small categories. Advances in Mathematics, 2008, 219 (6), pp.1872-1893. ⟨hal-00300800⟩