Maximal families of Gorenstein algebras
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- by Jan O. Kleppe PDF
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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.References
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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
- Received by editor(s): August 13, 2004
- Published electronically: January 24, 2006
- © Copyright 2006 American Mathematical Society
- 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
- MathSciNet review: 2216262