Depth in a pathological case

TL;DR

This paper investigates the depth of certain squarefree monomial ideals and establishes conditions under which Stanley's Conjecture holds, providing insights into the algebraic structure of these ideals.

Contribution

It introduces specific conditions on monomial ideals that guarantee an upper bound on depth and verifies Stanley's Conjecture in these cases.

Findings

Depth of I/J is at most d+1 under given conditions.

Stanley's Conjecture holds for I/J when conditions are satisfied.

Provides a criterion for the depth and Stanley's Conjecture in pathological cases.

Abstract

Let $I$ be a squarefree monomial ideal of a polynomial algebra over a field minimally generated by $f_1,...,f_r$ of degree $ d\geq 1$, and a set $E$ of monomials of degree $\geq d+1$. Let $J\subsetneq I$ be a squarefree monomial ideal generated in degree $\geq d+1$. Suppose that all squarefree monomials of $I\setminus (J\cup E)$ of degree $d+1$ are some least common multiples of $f_i$. If $J$ contains all least common multiples of two of $(f_i)$ of degree $d+2$ then $\depth_SI/J\leq d+1$ and Stanley's Conjecture holds for $I/J$.