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

Kleiman, Steven; Ulrich, Bernd

doi:10.1090/S0002-9947-97-01960-0

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.

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