leuschke.org

Research/Endomorphism Rings of Finite Global Dimension

Download (PDF)

Abstract:

For a commutative local ring R, consider (noncommutative) R-algebras Λ of the form Λ = EndR(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 = 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.

Notes:

I started working on this paper at MSRI in Spring 2003, after a series of conversations with Ragnar-Olaf Buchweitz. We spent hours talking about Auslander’s work on representation dimension, the McKay correspondence and particularly the work of Ito-Nakamura and Bridgeland-King-Reid, Van den Bergh’s recent preprints on noncommutative crepant resolutions of singularities, Lipman’s work on Arf rings, and a whole circle of ideas I hadn’t encountered before. In the end it turned out that Osamu Iyama had beaten me to the punch on one of the main results of the paper (the case of dimension one), though my result gives information that his doesn’t (and I understand mine) The other main result probably raises more questions than it answers, but it is a nice bridge from the topic that I’ve been working on for the last several years to several new ones. This is probably the paper that I’ve learned the most from writing.

Erratum: There is (at least) one error in the paper; thanks to Hailong Dao and Igor Burban for pointing it out.

In the discussion in section 3, I falsely assert that “Theorem 6 implies that if R is Gorenstein of finite CM type and dimension 2 or 3, then R has a non-commutative crepant resolution”. In dimension 2, this is correct, but fails in dimension 3. A counterexample is already in Van den Bergh’s paper “Three-dimensional flops and noncommutative rings”: the 3-dimensional A1 singularity has no non-commutative crepant resolution. It is apparently well-known to geometers that, among the 3-dimensional simple singularities, only A2n+1 and D2n have non-commutative crepant resolutions.

Bibtex code: 

 @article {MR2310620,
     AUTHOR = {Leuschke, Graham J.},
      TITLE = {Endomorphism rings of finite global dimension},
    JOURNAL = {Canad. J. Math.},
   FJOURNAL = {Canadian Journal of Mathematics. 
               Journal Canadien de Math\’ematiques},
     VOLUME = {59},
       YEAR = {2007},
     NUMBER = {2},
      PAGES = {332—342},
       ISSN = {0008–414X},
      CODEN = {CJMAAB},
    MRCLASS = {16G60},
   MRNUMBER = {MR2310620},
 }

Up: Papers index
Previous: Appendix: Some examples in tight closure
Next: Factoring the adjoint and maximal Cohen-Macaulay modules over the generic determinant