Homology of FI-modules

TL;DR

This paper establishes explicit bounds on the regularity of FI-modules and applies these results to improve stability theorems for homology of congruence subgroups, including small characteristics and integral cases.

Contribution

It provides a sharp upper bound for the Castelnuovo-Mumford regularity of FI-modules and refines stability results for homology of congruence subgroups, extending previous work.

Findings

Explicit upper bounds for FI-module regularity.

Refined stability results for homology in small characteristics.

Extended stability theorems to integral homology.

Abstract

We prove an explicit and sharp upper bound for the Castelnuovo-Mumford regularity of an FI-module V in terms of the degrees of its generators and relations. We use this to refine a result of Putman on the stability of homology of congruence subgroups, extending his theorem to previously excluded small characteristics and to integral homology while maintaining explicit bounds for the stable range.