On $G(A)_\mathbb{Q}$ of rings of finite representation type

TL;DR

This paper estimates the rational Grothendieck group of certain finite representation type rings when the AR-quiver is unknown, providing explicit computations and applications to specific classes of rings.

Contribution

It offers new estimates for the rational Grothendieck group of rings of finite representation type without requiring the AR-quiver, extending previous explicit computations.

Findings

Explicit estimates for G(A)_Q when AR-quiver is unknown.

G(A)_Q is isomorphic to Q for certain Gorenstein rings of positive even dimension.

Provides conditions under which G(A)_Q simplifies to a rational vector space.

Abstract

Let $(A,\mathfrak{m})$ be an excellent Henselian Cohen-Macaulay local ring of finite representation type. If the AR-quiver of $A$ is known then by a result of Auslander and Reiten one can explicity compute $G(A)$ the Grothendieck group of finitely generated $A$-modules. If the AR-quiver is not known then in this paper we give estimates of $G(A)_\mathbb{Q} = G(A)\otimes_\mathbb{Z} \mathbb{Q}$ when $k = A/\mathfrak{m}$ is perfect. As an application we prove that if $A$ is an excellent equi-characteristic Henselian Gornstein local ring of positive even dimension with $\text{char} \ A/\mathfrak{m} \neq 2,3,5$ (and $A/\mathfrak{m}$ perfect) then $G(A)_\mathbb{Q} \cong \mathbb{Q}$.