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

TL;DR

This paper explicitly computes the algebraic K-theory group G_1(R) for certain Cohen-Macaulay rings with n-cluster tilting objects, providing concrete formulas and examples for hypersurface singularities.

Contribution

It provides an explicit calculation of G_1(R) for Cohen-Macaulay rings with n-cluster tilting objects, linking algebraic K-theory with automorphism groups.

Findings

G_1(R) is computed as a direct sum involving automorphism groups.

Explicit formulas for automorphism groups of n-cluster tilting objects.

Calculations for G_1(R) in specific hypersurface singularities.

Abstract

Let $(R, \mathfrak{m}, k)$ denote a local Cohen-Macaulay ring such that the category of maximal Cohen-Macaulay $R$-modules $\textbf{mcm}\ R$ contains an $n$-cluster tilting object $L$. In this paper, we compute $G_1(R) := K_1(\textbf{mod}\ R)$ 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 $\text{char}(k)\neq 2$). Moreover, we give some explicit computations of $\text{aut}_R(L)_{\text{ab}}$ and $G_1(R)$ for certain hypersurface singularities.