A generalization of Watts's Theorem: Right exact functors on module categories

TL;DR

This paper extends Watts's Theorem to a broader context, showing that right exact functors from module categories to cocomplete abelian categories are representable via tensor products with suitable modules, generalizing classical results.

Contribution

It generalizes Watts's Theorem to functors into cocomplete abelian categories, defining tensor products with modules in these categories.

Findings

Right exact functors are representable as tensor products with modules in the target category.

The paper provides a framework to interpret tensor products in abstract abelian categories.

It extends classical module theory results to more general categorical settings.

Abstract

Watts's Theorem says that a right exact functor F:Mod R-->Mod S that commutes with direct sums is isomorphic to -\otimes_R B where B is the R-S-bimodule FR. The main result in this paper is the following: if A is a cocomplete abelian category and F:Mod R --> A is a right exact functor commuting with direct sums, then F is isomorphic to - \otimes_R B where B is a suitable R-module in A, i.e., a pair (B,f) consisting of an object B in A and a ring homomorphism f:R --> Hom_A(B,B). Part of the point is to give meaning to the notation -\otimes_R B. That is done in the paper by Artin and Zhang on Abstract Hilbert Schemes. The present paper is a natural extension of some of the ideas in the first part of their paper.