Regular Sequences On Squares Of Monomial Ideals

TL;DR

This paper investigates the depth of the square of monomial ideals in polynomial rings using initially regular sequences, providing conditions and criteria for depth properties.

Contribution

It introduces conditions for linear sums to form regular sequences on $R/I^2$ and establishes criteria for the depth of $R/I^2$ in monomial ideals.

Findings

Conditions for linear sums to be regular sequences

Criterion for $ ext{depth } R/I^2 > 1$

Lower bounds on the depth of $R/I^2$

Abstract

We use initially regular sequences that consist of linear sums to explore the depth of $R/I^2$, when $I$ is a monomial ideal in a polynomial ring $R$. We give conditions under which these linear sums form regular or initially regular sequences on $R/I^2$. We then obtain a criterion for when $\depth R/I^2>1$ and a lower bound on $\depth R/I^2$.