$FI$-modules over preadditive categories and torsion theories

Abstract

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