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Research/Gorenstein Modules Finite Index and Finite Cohen-Macaulay Type

Abstract:

A Gorenstein module over a local ring R is a maximal Cohen-Macaulay module of finite injective dimension. We use existence of Gorenstein modules to extend a result due to S. Ding: A Cohen—Macaulay ring of finite index, with a Gorenstein module, is Gorenstein on the punctured spectrum. We use this to show that a Cohen-Macaulay local ring of finite Cohen-Macaulay type is Gorenstein on the punctured spectrum. Finally, we show that for a large class of rings (including all excellent rings), the Gorenstein locus of a finitely generated module is a Zariski-open set.

Notes:

My favorite memory about this paper is that I realized the key to the proof of the main theorem while driving from Indiana to Pennsylvania to visit my friend Nick. About halfway from Lafayette to Pittsburgh, somewhere in rural Ohio (which is very pretty in March), I suddenly had the proof in my mind, clear as day. I got off the highway at the next exit, which happened to be a park with a reservoir called Senecaville Lake, and wrote it down using the hood of my car as a desk. At my dissertation defense, I credited the theorem to “— and Senecaville Lake”, which seemed really witty at the time.

Bibtex code:

 @article {MR1894057,
    AUTHOR = {Leuschke, Graham J.},
     TITLE = {Gorenstein modules, finite index, and finite
              {C}ohen-{M}acaulay type},
   JOURNAL = {Comm. Algebra},
  FJOURNAL = {Communications in Algebra},
    VOLUME = {30},
      YEAR = {2002},
    NUMBER = {4},
     PAGES = {2023—2035},
      ISSN = {0092–7872},
     CODEN = {COALDM},
   MRCLASS = {13C14 (13D02 13D05 13H10)},
  MRNUMBER = {MR1894057 (2003a:13010)},
 MRREVIEWER = {Liam O’Carroll},
 }

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