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

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Lawrence, KS, 66045-7523, USA

    Hailong Dao

  2. Department of Computer and Mathematical Sciences, University of Toronto at Scarborough, Toronto, Ontario, M1A 1C4, Canada

    Eleonore Faber

  3. Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

    Colin Ingalls

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  1. Hailong Dao
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  2. Eleonore Faber
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  3. Colin Ingalls
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Correspondence to Eleonore Faber.

Additional information

This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000, while the authors were in residence at the Mathematical Science Research Institute (MSRI) in Berkeley, California, during the spring semester of 2013. H.D. was partially supported by NSF grant DMS 1104017. E.F. was supported by the Austrian Science Fund (FWF) in frame of project J3326. C.I. was supported by an NSERC Discovery grant.

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Dao, H., Faber, E. & Ingalls, C. Noncommutative (Crepant) Desingularizations and the Global Spectrum of Commutative Rings. Algebr Represent Theor 18, 633–664 (2015). https://doi.org/10.1007/s10468-014-9510-y

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