On K_0 of locally finte categories

Abstract

We calculate the Grothendieck group $K_0(\cal A)$, where $\cal A$ is an additive category, locally finite over a Dedekind ring and satisfying some additional conditions. The main examples are categories of modules over finite algebras and the stable homotopy category $\mathsf{SW}$ of finite CW-complexes. We show that this group is a direct sum of a free group arising from localizations of the category $\cal A$ and a group analogous to the groups of ideal classes of maximal orders. As a corollary, we obtain a new simple proof of the Freyd's theorem describing the group $K_0(\mathsf{SW})$.