Generalized Hilbert Functions

TL;DR

This paper introduces a new generalized Hilbert function for modules over Noetherian local rings, explores its properties, and examines its behavior under hyperplane sections, especially for ideals with minimal or almost minimal j-multiplicity.

Contribution

It defines a generalized Hilbert function using local cohomology, generalizes Singh's formula, and studies the behavior of Hilbert coefficients under hyperplane sections for broader classes of ideals.

Findings

Generalized Hilbert coefficients are preserved under hyperplane sections.

Counterexamples show the generalized Hilbert series shape differs in certain cases.

A sufficient condition is provided for the Hilbert series to have the expected shape.

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 0th local cohomology functor. We show that our definition re-conciliates with that of Ciuperc$\breve{{\rm a}}$. By generalizing Singh's formula (which holds in the case of $\lambda(M/IM)<\infty$), we prove that the generalized Hilbert coefficients $j_0,..., j_{d-2}$ are preserved under a general hyperplane section, where $d={\rm dim}\,M$. We also keep track of the behavior of $j_{d-1}$. 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 $\lambda(M/IM)<\infty$. Finally we give a sufficient condition such that the generalized Hilbert series has the desired shape.