$FI$-modules over preadditive categories and torsion theories

TL;DR

This paper explores torsion theories in $FI$-modules over preadditive categories, focusing on finitely generated modules and their inductive descriptions, advancing the understanding of their algebraic structure.

Contribution

It introduces new torsion theories for $FI$-modules over preadditive categories and studies their properties, especially for finitely generated and shift finitely generated modules.

Findings

Developed torsion theories for $FI$-modules over preadditive categories

Characterized finitely generated and shift finitely generated $FI$-modules

Provided inductive descriptions of $FI$-modules over $\\mathcal R$

Abstract

We work with $FI$-modules over a small preadditive category $\mathcal R$, viewed as a ring with several objects. Our aim is to study torsion theories for $FI$-modules. We are especially interested in torsion theories on finitely generated $FI$-modules and the category of what we call "shift finitely generated" $FI$-modules. We also apply these methods to study inductive descriptions of $FI$-modules over $\mathcal R$.