Torsion theory of coherent functors

TL;DR

This paper develops a torsion theory for coherent functors over additive categories with cokernels, exploring properties of pseudo-kernels and the structure of mod($C$) as an abelian category, extending to Mod($C$).

Contribution

It introduces a torsion theory framework for coherent functors, analyzing their properties and extending results to the broader category of all additive functors.

Findings

mod($C$) is abelian when $C$ has pseudo-kernels

Characterization of radical, half exact, and left exact functors in mod($C$)

Extension of results from mod($C$) to Mod($C$)

Abstract

Let $C$ be an additive category with cokernels and let Mod($C$) be the category of additive functors from $C^{op}$ to the category Ab of abelian groups. Let mod($C$) be the full subcategory of Mod($C$) consisting of coherent functors. In this paper, we first study some basic properties of pseudo-kernel of morphisms in $C$. When $C$ has pseudo-kernels, mod($C$) is abelian and then, in this case, we study radical functors, half exact functors, left exact functors and injective objects in mod($C$). At last, we extend the results for Mod($C$).