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Hilbert polynomials and powers of ideals

Published online by Cambridge University Press:  01 November 2008

JÜRGEN HERZOG ,
TONY J. PUTHENPURAKAL  and
JUGAL K. VERMA
JÜRGEN HERZOG
Affiliation:
Fachbereich Mathematik und Informatik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany e-mail: [email protected]
TONY J. PUTHENPURAKAL
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai 400076, India e-mail: [email protected], [email protected]
JUGAL K. VERMA
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai 400076, India e-mail: [email protected], [email protected]
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Abstract

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal I in the polynomial ring S = K[x1, . . ., xn] and a finitely generated graded S-module M, the Hilbert coefficients ei(M/IkM) are polynomial functions. Given two families of graded ideals (Ik)k≥0 and (Jk)k≥0 with Jk ⊂ Ik for all k with the property that JkJℓ ⊂ Jk+ℓ and IkIℓ ⊂ Ik+ℓ for all k and ℓ, and such that the algebras and are finitely generated, we show the function k ↦ e0(Ik/Jk) is of quasi-polynomial type, say given by the polynomials P0,. . ., Pg−1. If Jk = Jk for all k, for a graded ideal J, then we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that , if I is a monomial ideal. We also study analogous statements in the local case.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 3 , November 2008 , pp. 623 - 642
Copyright
Copyright © Cambridge Philosophical Society 2008

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