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

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