# Castelnuovo-Mumford regularity of representations of certain product categories

## TL;DR

This paper establishes that representations of certain product categories have finite Castelnuovo-Mumford regularity if and only if they are finitely presented, leading to the category being abelian, with applications to categories like FI^m.

## Contribution

It provides a characterization of finite Castelnuovo-Mumford regularity for representations of product categories under combinatorial conditions, extending known results to new categories.

## Key findings

- Representations have finite regularity iff finitely presented in finite degrees
- Category of such representations is abelian
- Results apply to categories like FI^m and FI_G^m

## Abstract

We show in this paper that representations of a finite product of categories satisfying certain combinatorial conditions have finite Castelnuovo-Mumford regularity if and only if they are presented in finite degrees, and hence the category consisting of them is abelian. These results apply to examples such as the categories $\mathrm{FI}^m$ and $\mathrm{FI}_G^m$.