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Remarks on the Stanley depth and Hilbert depth of monomial ideals with linear quotients

Andreea I. Bordianu, Mircea Cimpoeas

Abstract

We prove that if $I$ is a monomial ideal with linear quotients in a ring of polynomials $S$ in $n$ indeterminates and $\operatorname{depth}(S/I)=n-2$, then $\operatorname{sdepth}(S/I)=n-2$ and, if $I$ is squarefree, $\operatorname{hdepth}(S/I)=n-2$. Also, we prove that $\operatorname{sdepth}(S/I)\geq \operatorname{depth}(S/I)$ for a monomial ideal $I$ with linear quotients which satisfies certain technical conditions.

Remarks on the Stanley depth and Hilbert depth of monomial ideals with linear quotients

Abstract

We prove that if is a monomial ideal with linear quotients in a ring of polynomials in indeterminates and , then and, if is squarefree, . Also, we prove that for a monomial ideal with linear quotients which satisfies certain technical conditions.

Paper Structure

This paper contains 3 sections, 12 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Remarks on the Hilbert depth

Key Result

Proposition 2.1

Let $I\subset S$ be a monomial ideal and $u\in S\setminus I$ a monomial. Then:

Theorems & Definitions (25)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 15 more