Presentations of rings with non-trivial semidualizing modules

Joint work with David A. Jorgensen and Sean Sather-Wagstaff.

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A result of Foxby, Reiten and Sharp says that a commutative noetherian local ring R admits a dualizing module if and only if R is Cohen–Macaulay and a homomorphic image of a local Gorenstein ring Q. We establish an analogous result by showing that such a ring R having a dualizing module admits a non-trivial finitely generated self-orthogonal module C satisfying Hom_R(C,C) \cong R if and only if R is the homomorphic image of a Gorenstein ring in which the defining ideal decomposes in a non-trivial way, forcing significant structural requirements on the ring R.


We started working this out at the conference for Mel Hochster in Ann Arbor in 2008. I think the paper contains a very satisfying answer to a natural question. It’s a bit technical, but still quite pretty in my opinion.

BibTeX code:

@article {MR2909823,
    AUTHOR = {Jorgensen, David A. and Leuschke, Graham J. and
              Sather-Wagstaff, Sean},
     TITLE = {Presentations of rings with non-trivial semidualizing modules},
   JOURNAL = {Collect. Math.},
  FJOURNAL = {Collectanea Mathematica},
    VOLUME = {63},
      YEAR = {2012},
    NUMBER = {2},
     PAGES = {165--180},
      ISSN = {0010-0757},
   MRCLASS = {13D07 (13H10)},
  MRNUMBER = {2909823},
MRREVIEWER = {Lars Winther Christensen},
       DOI = {10.1007/s13348-010-0024-6},
       URL = {},