Endomorphism Rings of Finite Global Dimension

Abstract

Abstract For a commutative local ring R , consider (noncommutative) R -algebras Λ of the form Λ = End R ( M ) where M is a reflexive R -module with nonzero free direct summand. Such algebras Λ 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 Λ with finite global dimension and which is maximal Cohen–Macaulay over R (a “noncommutative crepant resolution of singularities”). We produce algebras Λ = End R ( 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.