On the cohomology rings of small categories

Let CC be a small category and RR a commutative ring with identity. The cohomology ring of CC with coefficients in RR 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.