A Generalization of Watts's Theorem: Right Exact Functors on Module Categories

Abstract

Watts's Theorem says that a right exact functor that commutes with direct sums is isomorphic to − ⊗RB, where B is the R-S-bimodule FR. The main result in this article is the following one: If is a cocomplete category and is a right exact functor commuting with direct sums, then F is isomorphic to − ⊗Rℱ, where ℱ is a suitable R-module in , i.e., a pair (ℱ, ρ) consisting of an object and a ring homomorphism . Part of the point is to give meaning to the notation − ⊗Rℱ. That is done in the article by Artin and Zhang [1] on Hilbert Schemes. The present article is a natural extension of some of the ideas in the first part of their article.