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 Abstract Hilbert Schemes. The present article is a natural extension of some of the ideas in the first part of their article.

Key Word:

• Watts's Theorem

2010 Mathematics Subject Classification:

• Primary: 18F99
• Secondary: 14A22
• 16D90
• 18A25

Notes

An additive category is cocomplete if it has arbitrary direct sums. This is Grothendieck's condition Ab3.

It is essential that be cocomplete for − ⊗_Rℱ to exist. For example, if R = ℤ and consists of finitely generated abelian groups and ℱ = ℤ, there is no adjoint. But the hypothesis of cocompleteness is absent from [6, p. 108] and parts of [1].

After we finished writing this article, we learned that a version of this result had already been proved by Brzezinski and Wisbauer [2, 39.3, p. 410] under the hypothesis that the objects of are abelian groups.

The argument in the last part of the proof is a result of B. Mitchell. See [2, 39.1, p. 409] for more details.