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

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 $G(A)_\mathbb{Q} \cong \mathbb{Q}$.