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

TL;DR

This paper establishes bounds on the regularity of finitely generated $ ext{VI}^m$-modules over noetherian rings, using a shift theorem, which advances understanding of their structural properties.

Contribution

It introduces a bound on the regularity of $ ext{VI}^m$-modules based on generation and relation degrees, with a new shift theorem for these modules.

Findings

Regularity is bounded by a function of $m$, $d$, and $r$.

A shift theorem for finitely generated $ ext{VI}^m$-modules is proved.

Provides tools for analyzing the structure of $ ext{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 $\leqslant d$ and related in degree $\leqslant r$, 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.