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Hilbert Functions, Residual Intersections, and Residually S2 Ideals

Published online by Cambridge University Press:  04 December 2007

Marc Chardin ,
David Eisenbud  and
Bernd Ulrich
Marc Chardin
Affiliation:
Institut de Mathématiques, CNRS & Université Paris 6, 4, place Jussieu, F–75252 Paris Cedex 05, France. E-mail: chardin@math.jussieu.fr
David Eisenbud
Affiliation:
Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720, U.S.A. E-mail: de@msri.org
Bernd Ulrich
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A. E-mail: ulrich@math.msu.edu
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Abstract

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Let R be a homogeneous ring over an infinite field, I⊂R a homogeneous ideal, and $\frak a$⊂I an ideal generated by s forms of degrees d1,…,ds so that codim($\frak a$:I)[ges ]s. We give broad conditions for when the Hilbert function of R/$\frak a$ or of R/($\frak a$:I) is determined by I and the degrees d1,…,ds. These conditions are expressed in terms of residual intersections of I, culminating in the notion of residually S2 ideals. We prove that the residually S2 property is implied by the vanishing of certain Ext modules and deduce that generic projections tend to produce ideals with this property.

Keywords

Hilbert functionHilbert polynomialresidual intersectionresidually S2parsimoniousthrifty
Type
Research Article
Information
Compositio Mathematica , Volume 125 , Issue 2 , January 2001 , pp. 193 - 219
Copyright
© 2001 Kluwer Academic Publishers