Tensor structure on kC-mod and cohomology

Abstract

AbstractLet $\mathcal{C}$ be a finite category and let k be a field. We consider the category algebra $k\mathcal{C}$ and show that $k\mathcal{C}$-mod is closed symmetric monoidal. Through comparing $k\mathcal{C}$ with a co-commutative bialgebra, we exhibit the similarities and differences between them in terms of homological properties. In particular, we give a module-theoretic approach to the multiplicative structure of the cohomology rings of small categories. As an application, we prove that the Hochschild cohomology rings of a certain type of finite category algebras are finitely generated.