Arkiv för Matematik

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if $I = I_\Delta$ is the Stanley–Reisner ideal of a simplicial complex $\Delta$, then $\operatorname{reg}(I^{(n)}) \leqslant \delta (n-1)+b$ for all $n \geqslant 1$, where $\delta = \operatorname{lim}_{n \to \infty} \operatorname{reg}(I^{(n)}) / n$, $b=\max \lbrace \operatorname{reg}(I_\Gamma ) \vert \Gamma$ is a subcomplex of $\Delta$ with $\mathcal{F}(\Gamma) \subseteq F(\Delta ) \rbrace$, and $F(\Gamma)$ and $F(\Delta)$ are the set of facets of $\Gamma$ and $\Delta$, respectively. This bound is sharp for any $n$. When $I=I(G)$ is the edge ideal of a simple graph $G$, we obtain a general linear upper bound $\operatorname{reg}(I^{(n)}) \leqslant 2n+ \operatorname{ord-match} (G) - 1$, where $\operatorname{ord-match} (G)$ is the ordered matching number of $G$.