TY - JOUR
T1 - Endomorphism rings of finite global dimension
AU - Leuschke, Graham J.
N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2007/4
Y1 - 2007/4
N2 - For a commutative local ring R, consider (noncommutative) R-algebras A of the form A = EndR(M) where M is a reflexive R-module with nonzero free direct summand. Such algebras A 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 A with finite global dimension and which is maximal Cohen-Macaulay over R (a "noncommutative crepant resolution of singularities"). We produce algebras A = EndR(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.
AB - For a commutative local ring R, consider (noncommutative) R-algebras A of the form A = EndR(M) where M is a reflexive R-module with nonzero free direct summand. Such algebras A 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 A with finite global dimension and which is maximal Cohen-Macaulay over R (a "noncommutative crepant resolution of singularities"). We produce algebras A = EndR(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.
KW - Maximal Cohen-Macaulay modules
KW - Noncommutative crepant resolution
KW - Representation dimension
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U2 - 10.4153/CJM-2007-014-1
DO - 10.4153/CJM-2007-014-1
M3 - Article
AN - SCOPUS:34248672540
VL - 59
SP - 332
EP - 342
JO - Canadian Journal of Mathematics
JF - Canadian Journal of Mathematics
SN - 0008-414X
IS - 2
ER -