Bounds on Multigraded Regularity

Multigraded Castelnuovo--Mumford regularity of a module $M$ over the total coordinate ring $S$ of a smooth projective toric variety $X$ is a region $\operatorname{reg} M \subset \operatorname{Pic} X$ invariant under translation by the nef cone $\operatorname{Nef} X$. We prove that the multigraded regularity of a finitely generated faithful module is contained in a translate of $\operatorname{Nef} X$ determined by the degrees of the generators of $M$, and thus contains only finitely many minimal elements. We show that this condition can fail even for cyclic modules if $M$ has torsion and the rank of the Picard group is at least two. As an application, we exhibit asymptotic bounds for the multigraded regularity of powers of ideals. For $I$ an ideal in $S$, we bound $\operatorname{reg}(I^n)$ by proving that it contains a translate of $\operatorname{reg} S$ and is contained in a translate of $\operatorname{Nef} X$, where each bound translates by a fixed vector as $n$ increases.