Endomorphism rings of finite global dimension

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dc.creatorLeuschke, Graham J.
dc.date2005-05-16
dc.date.accessioned2026-07-07T05:19:55Z
dc.date.available2026-07-07T05:19:55Z
dc.descriptionFor a commutative local ring $R$, consider (noncommutative) $R$-algebras $Λ$ of the form $Λ= End_R(M)$ 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 $Spec R$. 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 $Λ$ with finite global dimension and which is maximal Cohen--Macaulay over $R$ (a ``noncommutative crepant resolution of singularities''). We produce algebras $Λ=End_R(M)$ 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.
dc.description13 pages, to appear in Canadian J. Math
dc.identifierhttps://arxiv.org/abs/math/0505323
dc.identifierhttp://arxiv.org/abs/math/0505323
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/75204
dc.subjectCommutative Algebra
dc.subjectRings and Algebras
dc.subject16G50, 16G60
dc.titleEndomorphism rings of finite global dimension
dc.typetext

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