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Algebras and Representation Theory

, Volume 18, Issue 3, pp 633–664 | Cite as

Noncommutative (Crepant) Desingularizations and the Global Spectrum of Commutative Rings

  • Hailong Dao
  • Eleonore Faber
  • Colin Ingalls
Article

Abstract

In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but generalizes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introduce and study the global spectrum of a ring R, that is, the set of all possible finite global dimensions of endomorphism rings of MCM R-modules. Finally, we use a variety of methods to compute global dimension for many endomorphism rings.

Keywords

Noncommutative (crepant) resolutions Rational singularities Global spectrum Endomorphism rings 

Mathematics Subject Classification (2010)

14B05 14A22 14E15 13C14 16E10 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA
  2. 2.Department of Computer and Mathematical SciencesUniversity of Toronto at ScarboroughTorontoCanada
  3. 3.Department of Mathematics and StatisticsUniversity of New BrunswickFrederictonCanada

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