On K0 of locally finite categories

Abstract

We calculate the Grothendieck group K0(A), where 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 SW of finite CW-complexes. We show that this group is a direct sum of a free group arising from localizations of the category 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 K0(SW).