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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Gorenstein algebras, symmetric matrices,
self-linked ideals, and symbolic powers


Authors: Steven Kleiman and Bernd Ulrich
Journal: Trans. Amer. Math. Soc. 349 (1997), 4973-5000
MSC (1991): Primary 13C40, 13H10, 13A30, 14E05
MathSciNet review: 1422609
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Abstract: Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade $2$ as those with a Hilbert-Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade $1$ can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor, a symmetric syzygy matrix. In addition, we establish a correspondence, roughly speaking, between Gorenstein perfect algebras of grade $1$ that are birational onto their image, on the one hand, and self-linked perfect ideals of grade $2$ that have one of the self-linking elements contained in the second symbolic power, on the other hand. Finally, we provide another characterization of these ideals in terms of their symbolic Rees algebras, and we prove a criterion for these algebras to be normal.


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Additional Information

Steven Kleiman
Affiliation: Department of Mathematics, Room 2-278, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
Email: Kleiman@math.MIT.edu

Bernd Ulrich
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email: Ulrich@math.MSU.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01960-0
PII: S 0002-9947(97)01960-0
Received by editor(s): June 2, 1996
Additional Notes: The first author was supported in part by NSF grant 9400918-DMS. It is a pleasure for this author to thank the Mathematical Institute of the University of Copenhagen for its hospitality during the summer of 1995 when this work was completed
The second author was supported in part by NSF grant DMS-9305832
Dedicated: To David Eisenbud on his fiftieth birthday
Article copyright: © Copyright 1997 American Mathematical Society