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Endomorphism Rings of Finite Global Dimension

Published online by Cambridge University Press:  20 November 2018

Graham J. Leuschke*
Affiliation:
Mathematics Department, Syracuse University, Syracuse, NY 13244, U.S.A. email: gjleusch@math.syr.edu
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2007

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