Bounding the socles of powers of squarefree monomial ideals

Abstract

Let $S=K[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I\subset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u\in S$ belonging to the socle $\Soc(S/I^{k})$ of $S/I^k$, i.e., $u\notin I^k$ and $u{x}_i\in I^k$ for $1\le i\le n$. We prove that if a monomial $x_1^{a_1}\cdots x_n^{a_n}$ belongs to $\Soc(S/I^{k})$, then $a_i\le k-1$ for all $1\le i\le n$. We then discuss squarefree monomial ideals $I\subset S$ for which $x_{[n]}^{k-1} \in \Soc(S/I^{k})$, where $x_{[n]}=x_1{x}_2\cdots x_n$. Furthermore, we give a combinatorial characterization of finite graphs $G$ on $[n]=\{1,\dots ,n\}$ for which $\depth S/(I_{G})^{2}=0$, where $I_G$ is the edge ideal of $G$.

Cited by in corpus (5)

• Associated primes of powers of edge ideals and ear decompositions of graphs
• On the matroidal path ideals
• The socle module of a monomial ideal
• On the associated primes and the depth of the second power of squarefree monomial ideals
• Monomial ideals with primary components given by powers of monomial prime ideals