Bounding regularity of ${\mathrm{VI}}^m$-modules

Abstract

Fix a finite field $\mathbb{F}$. Let $\mathrm{VI}$ be a skeleton of the category of finite dimensional $\mathbb{F}$-vector spaces and injective $\mathbb{F}$-linear maps. We study ${\mathrm{VI}}^m$-modules over a noetherian commutative ring in the nondescribing characteristic case. We prove that if a finitely generated ${\mathrm{VI}}^m$-module is generated in degree $⩽d$ and related in degree $⩽r$, then its regularity is bounded above by a function of $m$, $d$, and $r$. A key ingredient of the proof is a shift theorem for finitely generated ${\mathrm{VI}}^m$-modules.