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Maximal families of Gorenstein algebras


Author: Jan O. Kleppe
Journal: Trans. Amer. Math. Soc. 358 (2006), 3133-3167
MSC (2000): Primary 14C05, 13D10, 13D03, 13D07, 13C40
DOI: https://doi.org/10.1090/S0002-9947-06-03845-1
Published electronically: January 24, 2006
MathSciNet review: 2216262
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Abstract: The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring $ R$. Let $ \operatorname{GradAlg}^H(R)$ be the scheme parametrizing graded quotients of $ R$ with Hilbert function $ H$. We prove there is a close relationship between the irreducible components of $ \operatorname{GradAlg}^H(R)$, whose general member is a Gorenstein codimension $ (c+1)$ quotient, and the irreducible components of $ \operatorname{GradAlg}^{H'}(R)$, whose general member $ B$ is a codimension $ c$ Cohen-Macaulay algebra of Hilbert function $ H'$ related to $ H$. If the Castelnuovo-Mumford regularity of the Gorenstein quotient is large compared to the Castelnuovo-Mumford regularity of $ B$, this relationship actually determines a well-defined injective mapping from such ``Cohen-Macaulay'' components of $ \operatorname{GradAlg}^{H'}(R)$ to ``Gorenstein'' components of $ \operatorname{GradAlg}^{H}(R)$, in which generically smooth components correspond. Moreover the dimension of the ``Gorenstein'' components is computed in terms of the dimension of the corresponding ``Cohen-Macaulay'' component and a sum of two invariants of $ B$. Using linkage by a complete intersection we show how to compute these invariants. Linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem.


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Additional Information

Jan O. Kleppe
Affiliation: Faculty of Engineering, Oslo University College, Postboks 4, St. Olavs plass, N-0130 Oslo, Norway
Email: JanOddvar.Kleppe@iu.hio.no

DOI: https://doi.org/10.1090/S0002-9947-06-03845-1
Keywords: Parametrization, Gorenstein algebra, Artinian algebra, liaison, licci, Cohen-Macaulay, canonical module, normal module, Hilbert scheme.
Received by editor(s): August 13, 2004
Published electronically: January 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society

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