  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.
