Endomorphism rings of finite global dimension

Abstract

For a commutative local ring R, consider (noncommutative) R-algebras A of the form A = EndR(M) where M is a reflexive R-module with nonzero free direct summand. Such algebras A of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of Spec R. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal ℂ-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra A with finite global dimension and which is maximal Cohen-Macaulay over R (a "noncommutative crepant resolution of singularities"). We produce algebras A = EndR(M) having finite global dimension in two contexts: when R is a reduced one-dimensional complete local ring, or when R is a Cohen-Macaulay local ring of finite Cohen-Macaulay type. If in the latter case R is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh. © Canadian Mathematical Society 2007.