On the tails of FI-modules

Abstract

We study the end-behavior of integer-valued FI-modules. Our first result describes the high degrees of an FI-module in terms of newly defined tail invariants. Our main result provides an equivalence of categories between FI-tails and finitely supported modules for a new category that we call FJ. Objects of FJ are natural numbers, and morphisms are infinite series with summands drawn from certain modules of Lie brackets.

Regularity and stable ranges of FI-modules

“…In this section we prove Theorem 3.3, which forms the additive structure part of Theorem D. We also observe that our stable range here is at least as good as that of Patzt-Wiltshire-Gordon [PWG19].…”