For a commutative local ring R, consider (noncommutative) R-algebras
Λ of the form Λ=EndR(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 SpecR. 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=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.Comment: 13 pages, to appear in Canadian J. Mat