Incarnations of a theorem of Eilenberg

Question

Let $R$ be any ring, let $\text{Mod}_R$ be the category of right $R$-modules and let $\text{Ab}$ be the category of abelian groups. There is a classical theorem of Eilenberg (I think) which says that for any right exact functor $F:\text{Mod}_R \to \text{Ab}$ which preserves direct sums, there exists a left module structure on $F(R)$ making $F$ naturally isomorphic to the functor $- \otimes_R F(R)$.

Does anyone know any nice "incarnations" of this theorem? By this I mean "nice", simple and concrete right exact functors from $\text{Mod}_R$ to $\text{Ab}$ (for some $R$) preserving direct sums, for which it is not immediately clear that they are given by a tensor product (= isomorphic to a tensor functor) with a left $R$-module, but for which this left $R$-module can still be constructed in a concrete way.

Answer 1

More generally, the Theorem of Eilenberg-Watts says the following: The category of cocontinuous functors $\mathrm{Mod}(R) \to \mathrm{Mod}(S)$ is equivalent to the category of $(R,S)$-bimodules. A bimodule ${}_R M _{S}$ corresponds to the functor $- \otimes_R M$. There is a recent paper by A. Nyman which deals more generally with cocontinuous functors $\mathrm{Qcoh}(X) \to \mathrm{Qcoh}(Y)$ for nice schemes $X,Y$.

The Eilenberg-Watts theorem over schemes, J. Pure Appl. Algebra, 214 (2010), 1922-1954; online.

As for your question, take a finitely generated projective module $P$. Then $\mathrm{Hom}(P,-) : \mathrm{Mod}(R) \to \mathrm{Mod}(R)$ is right exact (since $P$ is projective) and preserves infinite direct sums (since $P$ is finitely generated), thus cocontinuous. In this case the Theorem shows that $\mathrm{Hom}(P,-) \cong (-) \otimes P^*$, where $P^* = \mathrm{Hom}(P,R)$ is the dual of $P$. More generally, for every vector bundle $V$ on some scheme/manifold we have $\underline{\mathrm{Hom}}(V,-) \cong (-) \otimes V^*$.

Answer 2

There is a paper by Mark Hovey which discusses incarnations of the Eilenberg-Watts theorem in homotopy theory. First he reviews the algebraic version and provides an equivalent formulation wherein the theorem says that the category of abelian groups is left self-contained (note that Eilenberg also proved Ab is right self-contained, though it seems Watts did not prove this). Hovey then finds the correct definition of this concept in homotopy theory and proves a very general theorem about when a model category has this property. His theorem holds for topological spaces, simplicial sets, chain complexes, and all the models of spectra. Note, there is a more recent version of this paper on Hovey's website, but it does not compare the new results to the classical ones as fully.

On page 2 of Hovey's paper above is an open problem which you might be interested in. It asks for conditions such that a closed symmetric monoidal category is self-contained (rather than homotopically self-contained as Hovey's paper answers).