On the embedded associated primes of monomial ideals

Abstract

Let $I$ be a square-free monomial ideal in a polynomial ring $R=K[x_1,\dots ,x_n]$ over a field $K$, $\mathfrak{𝔪}=(x_1,\dots ,x_n)$ be the graded maximal ideal of $R$, and $\{u_1,\dots ,u_{{\beta }_1(I)}\}$ be a maximal independent set of minimal generators of $I$ such that $\mathfrak{𝔪}\setminus x_i\notin \text{Ass}(R∕{(I\setminus x_i)}^t)$ 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 $I$. We prove that if $\mathfrak{𝔪}\in \text{Ass}(R∕I^t)$, then $t\ge {\beta }_1(I)+1$. As an application, we verify that under certain conditions, every unmixed Kőnig 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 $R=K[x_1,\dots ,x_n]$ over a field $K$ and $z$ is an $I^t$ corner element for some positive integer $t$ such that $\mathfrak{𝔪}\setminus x_i\notin \text{Ass}{(I\setminus x_i)}^t$ for some $1\le i\le n$, then $x_i$ divides $z$.