On central stability

TL;DR

This paper generalizes the concept of central stability from FI-modules to broader categories, establishing key equivalences and introducing the notion of d-step central stability linked to relation degrees.

Contribution

It extends central stability to a wide class of categories and characterizes modules presented in finite degrees as centrally stable, also defining d-step stability based on relation degrees.

Findings

Modules presented in finite degrees are equivalent to being centrally stable.

If relations are generated in degrees ≤ d, modules are d-step centrally stable.

The notion of central stability is generalized beyond FI-modules.

Abstract

The notion of central stability was first formulated for sequences of representations of the symmetric groups by Putman. A categorical reformulation was subsequently given by Church, Ellenberg, Farb, and Nagpal using the notion of FI-modules, where FI is the category of finite sets and injective maps. We extend the notion of central stability from FI to a wide class of categories, and prove that a module is presented in finite degrees if and only if it is centrally stable. We also introduce the notion of $d$-step central stability, and prove that if the ideal of relations of a category is generated in degrees at most $d$, then every module presented in finite degrees is $d$-step centrally stable.