Bounding regularity of $FI^m$-modules

TL;DR

This paper establishes an upper bound on the regularity of $FI^m$-modules based on their generation and relation degrees, extending understanding of their algebraic properties in a multi-parameter setting.

Contribution

It provides a new bound on the regularity of $FI^m$-modules, linking it explicitly to the module's generation and relation degrees across multiple parameters.

Findings

Regularity of $FI^m$-modules is bounded by a function of $m$, $d$, and $r.

The bound applies to modules generated in degree ≤ d and related in degree ≤ r.

This advances the theoretical understanding of the algebraic structure of $FI^m$-modules.

Abstract

Let $FI$ be a skeleton of the category of finite sets and injective maps, and $FI^m$ the product of $m$ copies of $FI$. We prove that if an $FI^m$-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 $r$.