Let $S=K[x_1,\ldots,x_n]$ 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., $u \not\in I^{ k}$ and $ux_{ i} \in I^{ k}$ for $1 \leq i \leq n$. We prove that if a monomial $x_1^{ a_1}\cdots x_n^{ a_n}$ belongs to $\Soc(S/I^{ k})$, then $a_i\leq k-1$ for all $1 \leq i \leq n$. 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 $[n] = \{ 1, \ldots, n\}$ for which $\depth S/(I_{ G})^{ 2}=0$, where $I_{ G}$ is the edge ideal of $G$.
Categories: math.AC