# Asymptotic behaviors of representations of graded categories with inductive functors

## TL;DR

This paper develops an inductive framework to analyze the asymptotic properties of homology groups in representations of graded categories, establishing criteria for regularity and resolutions, with applications in representation stability.

## Contribution

It introduces inductive functors that generalize shift functors of $ ext{FI}$-modules, providing new tools to study regularity and resolutions in graded category representations.

## Key findings

- Finiteness of Castelnuovo-Mumford regularity is established for finitely generated representations.
- Inductive functors are shown to exist in key categories of interest in representation stability.
- Truncated representations have linear minimal resolutions, especially over fields of characteristic zero.

## Abstract

In this paper we describe an inductive machinery to investigate asymptotic behaviors of homology groups and related invariants of representations of certain graded combinatorial categories over a commutative Noetherian ring $k$, via introducing inductive functors which generalize important properties of shift functors of $\mathrm{FI}$-modules. In particular, a sufficient criterion for finiteness of Castelnuovo-Mumford regularity of finitely generated representations of these categories is obtained. As applications, we show that a few important infinite combinatorial categories appearing in representation stability theory are equipped with inductive functors, and hence the finiteness of Castelnuovo-Mumford regularity of their finitely generated representations is guaranteed. We also prove that truncated representations of these categories have linear minimal resolutions by relative projective modules, which are precisely linear minimal projective resolutions when $k$ is a field of characteristic 0.