Admissible matchings and the Castelnuovo-Mumford regularity of square-free powers

Abstract

Let $I$ be any square-free monomial ideal, and ${\mathcal{H}}_I$ denote the hypergraph associated with $I$. Refining the concept of $k$-admissible matching of a graph defined by Erey and Hibi, we introduce the notion of generalized $k$-admissible matching for any hypergraph. Using this, we give a sharp lower bound on the (Castelnuovo-Mumford) regularity of $I^{[k]}$, where $I^{[k]}$ denotes the $k^{\text{th}}$ square-free power of $I$. In the special case when $I$ is equigenerated in degree $d$, this lower bound can be described using a combinatorial invariant $\mathrm{aim}({\mathcal{H}}_I,k)$, called the $k$-admissible matching number of ${\mathcal{H}}_I$. Specifically, we prove that $\mathrm{reg}(I^{[k]})\ge (d-1)\mathrm{aim}({\mathcal{H}}_I,k)+k$, whenever $I^{[k]}$ is non-zero. Even for the edge ideal $I(G)$ of a graph $G$, it turns out that $\mathrm{aim}(G,k)+k$ is the first general lower bound for the regularity of $I(G{)}^{[k]}$. In fact, when $G$ is a forest, $\mathrm{aim}(G,k)$ coincides with the $k$-admissible matching number introduced by Erey and Hibi. Next, we show that if $G$ is a block graph, then $\mathrm{reg}(I(G{)}^{[k]})=\mathrm{aim}(G,k)+k$, and this result can be seen as a generalization of the corresponding regularity formula for forests. Additionally, for a Cohen-Macaulay chordal graph $G$, we prove that $\mathrm{reg}(I(G{)}^{[2]})=\mathrm{aim}(G,2)+2$. Finally, we propose a conjecture on the regularity of square-free powers of edge ideals of chordal graphs.

33 pages, 3 figures, comments are welcome!