Restrictions on Hilbert coefficients give depths of graded domains

TL;DR

This paper establishes a criterion linking Hilbert coefficients and the depth of graded domains, providing new insights into the structure of prime ideals and their generic initial ideals in polynomial rings.

Contribution

It introduces a novel relationship between Hilbert coefficients and the depth of graded domains, offering restrictions on the generic initial ideal of prime ideals.

Findings

Adjoining general linear forms alters Hilbert coefficients predictably.

The depth of the quotient ring is directly related to changes in Hilbert coefficients.

Provides a new criterion for understanding the structure of prime ideals in polynomial rings.

Abstract

In this paper, we prove that if $P$ is a homogeneous prime ideal inside a standard graded polynomial ring $S$ with $\dim(S/P)=d$, and for $s \leq d$, adjoining $s$ general linear forms to the prime ideal changes the $(d-s)$-th Hilbert coefficient by 1, then $\text{depth}(S/P)=s-1$. This criterion also tells us about possible restrictions on the generic initial ideal of a prime ideal inside a polynomial ring.