Cohen-Macaulay representations of invariant subrings

TL;DR

This work classifies two-dimensional Cohen-Macaulay complete local rings of finite CM type over characteristic-zero fields by realizing them as invariant subrings $R\cong l[[x,y]]^G$ for finite Galois extensions $l/k$ and $G\subseteq GL_2(l)\rtimes\mathrm{Gal}(l/k)$. It extends the classical Auslander–Esnault results to non-algebraically closed residue fields, providing linearization techniques and ramification theory to control End refactorings and NCCRs. A central achievement is the explicit description of AR quivers via McKay quivers, including a method to compute irreducible skew-group representations and to obtain AR data as quotients by $\mathrm{Gal}(l/k)$; this yields a complete quiver classification for 2D Gorenstein finite CM type rings. The paper also develops higher-dimensional AR theory for NCCRs, establishes existence of $(d-1)$-almost split sequences, and demonstrates how NCCRs arise for invariant subrings, with concrete examples illustrating the theory and a detailed account of divisor class groups.

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 $l$.

Theorems & Definitions (149)

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• Theorem 10