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 $\textbf{mcm}\ R$ 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.