K-groups for rings of finite Cohen-Macaulay type

TL;DR

This paper computes the first Quillen K-group for local Cohen-Macaulay rings of finite CM-type, extending previous work on Grothendieck groups and providing explicit examples.

Contribution

It introduces a novel computation of the first Quillen K-group for such rings, complementing existing Grothendieck group results.

Findings

Computed the first Quillen K-group for rings of finite CM-type

Described the induced group homomorphism from projective modules to all modules

Provided concrete examples illustrating the theoretical results

Abstract

For a local Cohen-Macaulay ring R of finite CM-type, Yoshino has applied methods of Auslander and Reiten to compute the Grothendieck group of the category mod(R) of finitely generated R-modules. For the same type of rings we compute in this paper the first Quillen K-group of mod(R). We also describe the group homomorphism induced by the inclusion of proj(R) into mod(R) and illustrate our results with concrete examples.