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Restrictions on Hilbert coefficients give depths of graded domains

Cheng Meng

TL;DR

The paper develops a framework linking Hilbert coefficients to the depth of graded quotients by homogeneous prime ideals via hyperplane sections and projections. It proves a sharp depth criterion: if, after projecting by $s$ general linear forms, the $(d-s)$-th Hilbert coefficient satisfies $e_{d-s}(S/P)=e_{d-s}(S(r)/P_s)-(-1)^{d-s}$ with $r=n-s$, then $\depth(S/P)=s-1$. It further derives structural restrictions on the generic initial ideal of primes, and provides a height-one prime characterization together with an Artinian hyperplane-section result, showing how depth drops under sections and how certain monomial ideals cannot occur as $\mathrm{gin}(P)$. These results tie Hilbert coefficient behavior to depth and offer concrete consequences for the possible $\mathrm{gin}(P)$ of primes 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.

Restrictions on Hilbert coefficients give depths of graded domains

TL;DR

The paper develops a framework linking Hilbert coefficients to the depth of graded quotients by homogeneous prime ideals via hyperplane sections and projections. It proves a sharp depth criterion: if, after projecting by general linear forms, the -th Hilbert coefficient satisfies with , then . It further derives structural restrictions on the generic initial ideal of primes, and provides a height-one prime characterization together with an Artinian hyperplane-section result, showing how depth drops under sections and how certain monomial ideals cannot occur as . These results tie Hilbert coefficient behavior to depth and offer concrete consequences for the possible of primes in polynomial rings.

Abstract

In this paper, we prove that if is a homogeneous prime ideal inside a standard graded polynomial ring with , and for , adjoining general linear forms to the prime ideal changes the -th Hilbert coefficient by 1, then . This criterion also tells us about possible restrictions on the generic initial ideal of a prime ideal inside a polynomial ring.
Paper Structure (6 sections, 18 theorems, 8 equations)

This paper contains 6 sections, 18 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Generic initial ideal under projection and saturation
  4. Initial and generic initial ideal of prime ideals
  5. Difference in the Hilbert coefficients
  6. Height 1 prime ideal and Artinian hyperplane section

Key Result

Theorem 1

Let $P$ be a homogeneous prime ideal in $S$, $\dim(S/P)=d$. Take $1 \leq s \leq d$ and let $r=n-s$. Choose $s$ general linear forms $l_1,\ldots,l_s$. Set $P_s=\pi_{l_1}\ldots\pi_{l_s}P \subset S(r)$. If $e_{d-s}(S/P)=e_{d-s}(S(r)/P_s)-(-1)^{d-s}$, then $\textup{depth}(S/P)=n-r-1$.

Theorems & Definitions (37)

  • Conjecture 1.1: eisenbud1984linear
  • Theorem : See \ref{['5.4']}
  • Proposition 2.1: ginchar02 and gincharp
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 3.1
  • Proposition 3.2: elias1998six, Proposition 2.14 and bayer1987criterion, Lemma 2.2
  • Remark 3.3
  • Proposition 3.4
  • ...and 27 more