Endomorphism rings of finite global dimension

Abstract

For a commutative local ring $R$, consider (noncommutative) $R$-algebras $\Lambda$ of the form $\Lambda = End_R(M)$ where $M$ is a reflexive $R$-module with nonzero free direct summand. Such algebras $\Lambda$ 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 $\Lambda$ with finite global dimension and which is maximal Cohen--Macaulay over $R$ (a ``noncommutative crepant resolution of singularities''). We produce algebras $\Lambda=End_R(M)$ 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.