On the cohomology rings of small categories

Fei Xu

Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, Scotland, U.K.

\begin{abstract}
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.
\end{abstract}

To appear in Journal of Pure and Applied Algebra, May 2008.