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Research/Two Theorems About Maximal Cohen-Macaulay Modules

Joint work with Craig Huneke.

Abstract:

This paper contains two theorems concerning the theory of maximal Cohen-Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen-Macaulay modules M and N must have finite length, provided only finitely many isomorphism classes of maximal Cohen—Macaulay modules exist having ranks up to the sum of the ranks of M and N. This has several corollaries. In particular it proves that a Cohen-Macaulay local ring of finite Cohen-Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen-Macaulay local ring of finite Cohen-Macaulay type is again of finite Cohen-Macaulay type. The second theorem proves that a complete local Gorenstein domain of positive characteristic p and dimension d is F-rational if and only if the number of copies of R splitting out of R^{1/p^e} divided by p^de has a positive limit. This result generalizes work of Smith and Van den Bergh. We call this limit the F-signature of the ring and give some of its properties.

Notes:

This was the first paper that Craig Huneke and I wrote together, shortly after I got to Kansas. It’s got two completely separate sections, one about Ext groups and one about rings of prime characteristic. I thought of the first section as being essentially about rings of finite Cohen-Macaulay type, but Craig saw that it was really a theorem about how Exts behave. Like all of his best theorems (and like our result in On a Conjecture of Auslander and Reiten), it’s an effective result, in that it gives more precise numerical data than is a priori needed. In retrospect, I almost wish we’d published this first part separately — I think it’s important enough to stand on its own, and even to get placed in a very good journal. The second section was my first introduction to thinking about tight closure, and I’m still very pleased to have been able to bring something to that theory.

Bibtex code:

 @article {MR1933863,
    AUTHOR = {Huneke, Craig and Leuschke, Graham J.},
     TITLE = {Two theorems about maximal {C}ohen-{M}acaulay modules},
   JOURNAL = {Math. Ann.},
  FJOURNAL = {Mathematische Annalen},
    VOLUME = {324},
      YEAR = {2002},
    NUMBER = {2},
     PAGES = {391—404},
      ISSN = {0025–5831},
     CODEN = {MAANA},
   MRCLASS = {13C14 (13A35 13D07 13H10)},
  MRNUMBER = {MR1933863 (2003j:13011)},
 MRREVIEWER = {{\=I}. {\=I}. Burban},
 }

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