Cohen-Macaulay representations of invariant subrings

Abstract

We classify two-dimensional complete local rings $(R,\mathfrak{m},k)$ of finite Cohen-Macaulay type where $k$ is an arbitrary field of characteristic zero, generalizing works of Auslander and Esnault for algebraically closed case. Our main result shows that they are precisely of the form $R=l[[x_1,x_2]]^G$ where $l/k$ is a finite Galois extension and $G$ is a finite group acting on $l[[x_1,x_2]]$ as a $k$-algebra. In fact, $G$ can be linearized to become a subgroup of $GL_2(l)\rtimes{\rm Gal}(l/k)$. Moreover, we establish algebraic McKay correspondence in this general setting and completely describe its McKay quiver, which is often non-simply laced, as a quotient of another certain McKay quiver. Combining these results, we classify the quivers that may arise as the Auslander-Reiten quivers of two-dimensional Gorenstein rings of finite Cohen-Macaulay type of equicharacteristic zero. These are shown to be either doubles of (not necessarily simply-laced!) extended Dynkin diagrams or of type $\widetilde{A}_0$ or $\widetilde{CL}_n$ having loops. More generally, we consider higher dimensional $R=l[[x_1,\cdots,x_d]]^G\ (G\subseteq GL_d(l)\rtimes{\rm Gal}(l/k))$ and show they have non-commutative crepant resolutions (NCCRs). Furthermore, we explicitely determine the quivers of the NCCRs as quotients of another certain quivers. To accomplish these, we establish two results which are of independent interest. First, we prove the existence of $(d-1)$-almost split sequences for arbitrary $d$-dimensional Cohen-Macaulay rings having NCCR, even when their singularities are not isolated. Second, we give an explicit recipe to determine irreducible representations of skew group algebras $l*G$ in terms of those over the group algebras $lH$ where $H$ is the kernel of the action of $G$ on $l$.