For a commutative local ring R, consider (noncommutative) R-algebras lamda of the form lamda = End R(M) where M is a reflexive R-module with nonzero free direct summand. Such algebras lamda 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 lamda with finite global dimension and which is maximal Cohen–Macaulay over R (a “noncommutative crepant resolution of singularities”). We produce algebras lamda = 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.
Leuschke, Graham J., "Endomorphism Rings of Finite Global Dimension" (2005). Mathematics Faculty Scholarship. 34.
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