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

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

2008-06-04

arxiv.org/abs/0806.0832v1

Mathematics

Rings and Algebras

9 pages

Scientific paper

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.

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