  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.
