G-Groups of Cohen-Macaulay Rings with n-Cluster Tilting Objects

Abstract

Let (R, m, k) denote a local Cohen-Macaulay ring such that the category of maximal Cohen-Macaulay R-modules mcm R contains an n-cluster tilting object L. In this paper, we compute the Quillen K-group G 1 (R) := K 1 (mod R) explicitly as a direct sum of a finitely generated free abelian group and an explicit quotient of Aut R (L) ab when R is a k-algebra and k is algebraically closed with characteristic not two. Moreover, we compute Aut R (L) ab and G 1 (R) for certain hypersurface singularities.

K-theory and the singularity category of quotient singularities

“…Many of these properties follow directly from the work of Orlov [56,57] once one makes sure that the relevant triangulated functors are induced from dg enhancements. For a similar perspective on studying homological invariants of the singularity category see [69,70,35], and for an algebraic approach to K 0 and K 1 of the singularity category via MCM modules see [41,52,28].…”