Relative Hilbert co-efficients

TL;DR

This paper investigates the properties of relative Hilbert coefficients in Cohen-Macaulay local rings, establishing constraints and implications for the depth of associated graded rings based on the vanishing of these coefficients.

Contribution

It introduces the concept of relative Hilbert coefficients and explores their constraints and impact on the depth of associated graded rings in Cohen-Macaulay local rings.

Findings

Constraints on relative Hilbert coefficients when $G_I(A)$ is Cohen-Macaulay.

Vanishing of certain $c_i(I,J)$ implies high depth of $G_{J^n}(A)$ for large $n$.

Provides new insights into the structure of Hilbert coefficients and associated graded rings.

Abstract

Let $(A,\m)$ be a \CM \ local ring of dimension $d$ and let $I \subseteq J$ be two $\m$-primary ideals with $I$ a reduction of $J$. For $i = 0,\ldots,d$ let $e_i^J(A)$ ($e_i^I(A)$) be the $i^{th}$ Hilbert coefficient of $J$ ($I$) respectively. We call the number $c_i(I,J) = e_i^J(A) - e_i^I(A)$ the $i^{th}$ relative Hilbert coefficient of $J$ \wrt \ $I$. If $G_I(A)$ is \CM \ then $c_i(I,J)$ satisfy various constraints. We also show that vanishing of some $c_i(I,J)$ has strong implications on $\depth G_{J^n}(A)$ for $n \gg 0$.