On K_0 of locally finte categories

TL;DR

This paper computes the Grothendieck group K_0 for certain locally finite additive categories, revealing its structure as a sum of a free group and an ideal class group, with applications to module categories and stable homotopy.

Contribution

It provides a new explicit description of K_0 for locally finite categories over Dedekind rings, including a simplified proof of Freyd's theorem for the stable homotopy category.

Findings

K_0 is a direct sum of a free group and an ideal class group.

The result applies to module categories over finite algebras.

A new proof of Freyd's theorem is established.

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 $K_0(\mathsf{SW})$.