On the embedded associated primes of monomial ideals

TL;DR

This paper investigates the embedded associated primes of monomial ideals, establishing bounds on the powers for which the maximal ideal appears, and explores properties like torsion-freeness and corner-elements in relation to these ideals.

Contribution

It proves a lower bound on the power of the maximal ideal in associated primes of monomial ideals and examines torsion-freeness and corner-elements for specific classes of monomial ideals.

Findings

If the maximal ideal is associated to a power of a monomial ideal, then the power is at least one more than the maximum size of an independent set.

Under certain conditions, unmixed K"onig ideals are normally torsion-free and have the strong persistence property.

Square-free transversal polymatroidal ideals are shown to be normally torsion-free.

Abstract

Let $I$ be a square-free monomial ideal in a polynomial ring $R=K[x_1,\ldots, x_n]$ over a field $K$, $\mathfrak{m}=(x_1, \ldots, x_n)$ be the graded maximal ideal of $R$, and $\{u_1, \ldots, u_{\beta_1(I)}\}$ be a maximal independent set of minimal generators of $I$ such that $\mathfrak{m}\setminus x_i \notin \mathrm{Ass}(R/(I\setminus x_i)^t)$ for all $x_i\mid \prod_{i=1}^{\beta_1(I)}u_i$ and some positive integer $t$, where $I\setminus x_i$ denotes the deletion of $I$ at $x_i$ and $\beta_1(I)$ denotes the maximum cardinality of an independent set in $I$. In this paper, we prove that if $\mathfrak{m}\in \mathrm{Ass}(R/I^t)$, then $t\geq \beta_1(I)+1$. As an application, we verify that under certain conditions, every unmixed K\"onig ideal is normally torsion-free, and so has the strong persistence property. In addition, we show that every square-free transversal polymatroidal ideal is normally torsion-free. Next, we state some results on the corner-elements of monomial ideals. In particular, we prove that if $I$ is a monomial ideal in a polynomial ring $R=K[x_1, \ldots, x_n]$ over a field $K$ and $z$ is an $I^t$-corner-element for some positive integer $t$ such that $\mathfrak{m}\setminus x_i \notin \mathrm{Ass}(I\setminus x_i)^t$ for some $1\leq i \leq n$, then $x_i$ divides $z$.