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Endomorphism Rings of Finite Global Dimension

  Published:2007-04-01
 Printed: Apr 2007
  • Graham J. Leuschke
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Abstract

For a commutative local ring $R$, consider (noncommutative) $R$-algebras $\Lambda$ of the form $\Lambda = \operatorname{End}_R(M)$ 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 $\operatorname{Spec} R$. 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 $\Lambda=\operatorname{End}_R(M)$ 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.
Keywords: representation dimension, noncommutative crepant resolution, maximal Cohen--Macaulay modules representation dimension, noncommutative crepant resolution, maximal Cohen--Macaulay modules
MSC Classifications: 16G50, 16G60, 16E99 show english descriptions Cohen-Macaulay modules
Representation type (finite, tame, wild, etc.)
None of the above, but in this section
16G50 - Cohen-Macaulay modules
16G60 - Representation type (finite, tame, wild, etc.)
16E99 - None of the above, but in this section
 

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