2008/06/04 by A. Nyman, Nyman, A., S. Paul Smith +1 · 1 citation
Mathematics · #14A22 #16D90 #18A25 #18F99 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:14A22 #msc:16D90 #msc:18A25 #msc:18F99
paper · pdf · doi:10.48550/arxiv.0806.0832
9 pages
arxiv created 2008/06/04 · arxiv updated 2009/12/01
Watts's Theorem says that a right exact functor F:Mod R-->Mod S 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 abelian category and F:Mod R --> A is a right exact functor commuting with direct sums, then F is isomorphic to - ⊗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 --> HomA(B,B). Part of the point is to give meaning to the notation -⊗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.