G-groups of Cohen-Macaulay Rings with $n$-Cluster Tilting Objects

Abstract

Let $(R,\mathfrak{m},k)$ denote a local Cohen-Macaulay ring such that the category of maximal Cohen-Macaulay $R$-modules $\text{mcm}R$ contains an $n$-cluster tilting object $L$. In this paper, we compute $G_1(R):=K_1(\text{mod}R)$ explicitly as a direct sum of a free group and a specified quotient of ${\text{aut}}_R(L{)}_{\text{ab}}$ when $R$ is a $k$-algebra and $k$ is algebraically closed (and $\text{char}(k)\ne 2$). Moreover, we give some explicit computations of ${\text{aut}}_R(L{)}_{\text{ab}}$ and $G_1(R)$ for certain hypersurface singularities.

arXiv comment24 pages. Accepted to Algebras and Representation Theory

Cited by in corpus (1)

• K-theory and the singularity category of quotient singularities