On the cohomology rings of small categories

Abstract

Communicated by M. Broué MSC: 16E30 18A25 18A40 20J06 20J99a b s t r a c t Let C be a small category and R a commutative ring with identity. The cohomology ring of C with coefficients in R is defined as the cohomology ring of the topological realization of its nerve. First we give an example showing that this ring modulo nilpotents is not finitely generated in general, even when the category is finite EI. Then we study the relationship between the cohomology ring of a category and those of its subcategories and extensions. The main results generalize certain theorems in group cohomology theory.

Hochschild and ordinary cohomology rings of small categories

“…The ordinary cohomology ring of C with coefficients in k can be defined as Ext * kC (k, k), which is isomorphic to H * (|C|, k) [22,23] and hence is graded commutative. Such an ordinary cohomology ring modulo nilpotents is not finitely generated in general, see for example [24].…”

“…3.1.The category E 0 . In[24] we presented an example, by Aurélien Djament, Laurent Piriou and the author, of the mod-2 ordinary cohomology ring of the followingcategory E 0 , in which [1 x ] ∼ = [h] ∼ = [g] ∼ = [gh] and [α] ∼ = [β]. For the purpose of computation, we use the skeleton F ′ (E 0 ) of F (E 0 ) (which is equivalent to F (E 0 ) hence the two category algebras and their module categories are Morita equivalent)…”

Resolutions, relation modules and Schur multipliers for categories

“…We now quote a result of Fei Xu [20] which generalizes the Lyndon-Hochschild-Serre spectral sequences in the cohomology of groups. Our notation is slightly different to Xu's, in that he defines homology groups H q (E, B) when B is a left ZE-module, whereas in this paper we define these homology groups when B is a right ZE-module.…”

Representations of categories and their applications

“…However, the finite generation is not true in general for finite categories, see Xu [40]. Direct computation of cohomology groups is in general very difficult, and so people have been searching for reduction formulas.…”