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

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- by Steven Kleiman and Bernd Ulrich PDF
- Trans. Amer. Math. Soc.
**349**(1997), 4973-5000 Request permission

## 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.## References

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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
- MR Author ID: 175910
- Email: Ulrich@math.MSU.edu
- 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 - © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**349**(1997), 4973-5000 - MSC (1991): Primary 13C40, 13H10, 13A30, 14E05
- DOI: https://doi.org/10.1090/S0002-9947-97-01960-0
- MathSciNet review: 1422609

Dedicated: To David Eisenbud on his fiftieth birthday