Hilbert Coefficients of Parameter Ideals

TL;DR

This paper investigates the non-positivity of Hilbert coefficients associated with parameter ideals in commutative Noetherian local rings, establishing conditions under which these coefficients are non-positive or zero.

Contribution

It provides new results on the non-positivity of higher Hilbert coefficients for parameter ideals with certain depth conditions, extending previous understanding.

Findings

Second Hilbert coefficient is non-positive when depth ≥ d-1.

Conditions identified for the vanishing of the second Hilbert coefficient.

All Hilbert coefficients up to d are non-positive when the associated graded ring has depth at least d.

Abstract

We consider the non-positivity of the Hilbert coefficients for a parameter ideal of a commutative Noetherian local ring. In particular, we show that the second Hilbert coefficient of a parameter ideal of depth at least d-1 is always non-positive and give a condition for the coefficient to be zero. With the added condition that the depth of the associated graded ring is also at least d- we show $e_{i}(q) \leq 0$ for $i=1, ..., d$.