Vanishing of Ext, cluster tilting modules and finite global dimension of endomorphism rings

TL;DR

This paper investigates conditions under which the endomorphism ring of a maximal Cohen-Macaulay module over a Cohen-Macaulay ring has finite global dimension, connecting Ext vanishing, cluster tilting modules, and non-commutative resolutions.

Contribution

It strengthens existing results linking Ext vanishing to finite global dimension and extends the theory of cluster tilting modules to positive characteristic.

Findings

Established new criteria for finite global dimension of endomorphism rings.

Extended cluster tilting theory to positive characteristic cases.

Connected higher-dimensional Auslander correspondence with non-commutative crepant resolutions.

Abstract

Let R be a Cohen-Macaulay ring and M a maximal Cohen-Macaulay R-module. Inspired by recent striking work by Iyama, Burban-Iyama-Keller-Reiten and Van den Bergh we study the question of when the endomorphism ring of M has finite global dimension via certain conditions about vanishing of $\Ext$ modules. We are able to strengthen certain results by Iyama on connections between a higher dimension version of Auslander correspondence and existence of non-commutative crepant resolutions. We also recover and extend to positive characteristics a recent Theorem by Burban-Iyama-Keller-Reiten on cluster-tilting objects in the category of maximal Cohen-Macaulay modules over reduced 1-dimensional hypersurfaces.