Graph and depth of a monomial squarefree ideal

TL;DR

This paper introduces a graph-based method to determine the depth of certain monomial squarefree ideals in polynomial rings, showing that the depth is independent of the field's characteristic and that these ideals satisfy Stanley's Conjecture.

Contribution

It provides a novel graph-theoretic approach to compute the depth of monomial squarefree ideals and proves these ideals satisfy Stanley's Conjecture.

Findings

Depth can be read from an associated graph.

Depth is independent of the characteristic of the field.

The ideals satisfy Stanley's Conjecture.

Abstract

Let $I$ be a monomial squarefree ideal of a polynomial ring $S$ over a field $K$ such that the sum of every three different of its minimal prime ideals is the maximal ideal of $S$, or more general a constant ideal. We associate to $I$ a graph on $[s]$, $s=|\Min S/I|$ on which we may {\em read} the depth of $I$. In particular, $\depth_S\ I$ does not depend of char $K$. Also we show that $I$ satisfies the Stanley's Conjecture.