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Let be a Cohen-Macaulay ring and a maximal Cohen-Macaulay -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 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 -dimensional hypersurfaces.