Generalized Hilbert Functions

Claudia Polini, Yu Xie

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Let M be a finite module, and let I be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of I on M using the zeroth local cohomology functor. We show that our definition reconciliates with that of Ciupercǎ. By generalizing Singh's formula (which holds in the case of λ(M/IM) < ∞), we prove that the generalized Hilbert coefficients j{fraktur}0,., j{fraktur}d-2 are preserved under a general hyperplane section, where d = dim M. We also keep track of the behavior of j{fraktur}d-1. Then we apply these results to study the generalized Hilbert function for ideals that have minimal j-multiplicity or almost minimal j-multiplicity. We provide counterexamples to show that the generalized Hilbert series of ideals having minimal or almost minimal j-multiplicity does not have the 'expected' shape described in the case where λ(M/IM) < ∞. Finally, we give a sufficient condition such that the generalized Hilbert series has the desired shape.

Original languageEnglish (US)
Pages (from-to)2411-2427
Number of pages17
JournalCommunications in Algebra
Volume42
Issue number6
DOIs
StatePublished - Jun 1 2014

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Hilbert Function
Generalized Functions
Hilbert Series
Multiplicity
Local Cohomology
Zeroth
Noetherian Ring
Local Ring
Hyperplane
Functor
Hilbert
Counterexample
Module
Sufficient Conditions
Arbitrary
Coefficient

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

Polini, Claudia ; Xie, Yu. / Generalized Hilbert Functions. In: Communications in Algebra. 2014 ; Vol. 42, No. 6. pp. 2411-2427.
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Generalized Hilbert Functions. / Polini, Claudia; Xie, Yu.

In: Communications in Algebra, Vol. 42, No. 6, 01.06.2014, p. 2411-2427.

Research output: Contribution to journalArticle

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AU - Xie, Yu

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