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Endomorphism Rings of Finite Global Dimension
Published online by Cambridge University Press: 20 November 2018
Abstract
For a commutative local ring $R$, consider (noncommutative)
$R$-algebras
$\Lambda$ of the form
$\Lambda \,=\,\text{En}{{\text{d}}_{R}}\left( M \right)$ 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
$\mathbb{C}$-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 \,=\,\text{En}{{\text{d}}_{R}}\left( M \right) $ 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.
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- Copyright © Canadian Mathematical Society 2007
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