On K0 of locally finite categories

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 algebras stable homotopy category SW CW-complexes. show that this direct sum free arising from localizations analogous to groups ideal classes maximal orders. As corollary, we obtain new simple proof Freyd's theorem describing K0(SW).