Auslander's Theorem and n-Isolated Singularities

TL;DR

This paper generalizes Auslander's Theorem to non-commutative rings and modules that are high syzygies of MCM modules, focusing on complete Gorenstein local domains and path algebras.

Contribution

It introduces a new generalization of Auslander's Theorem by restricting to high syzygy modules over non-commutative rings, expanding the theorem's applicability.

Findings

Generalization of Auslander's Theorem to non-commutative rings.

Analysis of modules that are high syzygies of MCM modules.

Application to path algebras over complete Gorenstein local domains.

Abstract

One of the most stunning results in the representation theory of Cohen-Macaulay rings is Auslander's well known theorem which states a CM local ring of finite CM type can have at most an isolated singularity. There have been some generalizations of this in the direction of countable CM type by Huneke and Leuschke. In this paper, we focus on a different generalization by restricting the class of modules. Here we consider modules which are high syzygies of MCM modules over non-commutative rings, exploiting the fact that non-commutative rings allow for finer homological behavior. We then generalize Auslander's Theorem in the setting of complete Gorenstein local domains by examining path algebras, which preserve finiteness of global dimension.