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

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.