nLab
Eilenberg-Watts theorem

Contents

Idea

The Eilenberg-Watts theorem idetifies colimit-preserving functors between categories of modules with the operations of forming tensor products with bimodules.

Statement

For ordinary rings and modules

Eilenberg-Watts’ Theorem

Given unital rings RR and SS and an RR-SS-bimodule NN, the tensor product functor

() RN:Mod RMod S (-) \otimes_R N \;\colon\; Mod_R \to Mod_S

between the categories of modules is right exact and preserves small coproducts.

Conversely, if F:Mod RMod SF \colon Mod_R \to Mod_S is right exact and preserves small coproducts, then it is naturally isomorphic to tensoring with a bimodule.

This theorem was more or less simultaneously proved in (Watts) and (Eilenberg).

Remark

Given a cocontinuous functor F:Mod RMod SF \colon Mod_R \to Mod_S, the reconstructed RR-SS-bimodule is given as follows:

  • the underlying right SS-module is F(R)F(R), where RR is regarded as a right module over itself in the canonical way;

  • the left RR-module structure on F(R)F(R) is given for rRr \in R and nNn \in N by

    rnF(r())n, r \cdot n \coloneqq F(r\cdot(-))n \,,

    where r():RRr \cdot (-) \colon R \to R denotes the right RR-module homomorphism given by left multiplication by RR.

Remark

The theorem holds for nonunital rings as well, but then BB reconstructs as F(R 1)F(R_1) where R 1R_1 is the extension of RR by adjoining the unit element (the tensor product is still over the original RR). If FF is a flat functor then F(R 1)F(R_1) is a flat module over RR.

Remark

In the statement of the theorem we can replace “additive, right exact and preserves direct sum” by “additive and left adjoint”.

In this form the theorem is stated for instance in (Hovey, theorem 0.1).

In fact both bimodules and intertwiners as well as functors and natural transformations form a category. In more detail the theorem is:

Theorem

For RR and SS two rings, the functor

RMod SFunc coc(Mod R,Mod S) {}_R Mod_{S} \stackrel{\simeq}{\to} Func_{coc}(Mod_R, Mod_S)

from the category of bimodules to that of colimit-preserving functors between their categories of modules is an equivalence of categories.

For other internal monoids and internal modules

The standard Eilenberg-Watts theorem is a statement about monoids and their actions in Ab. More generally one may ask for generalizations of the theorem to other internalization contexts, and in particular to homotopy theory. See the introduction of (Hovey).

For \infty-algebras and \infty-modules

In particular Eilenberg-Watts theorems hold true in the homotopy theory following model categories (see at model structure on modules over an algebra over an operad)

This is the main theorem in (Hovey).

More generally, we have the first half of the Eilenberg-Watts theorem in (∞,1)-category theory:

Proposition

For (𝒞,)(\mathcal{C}, \otimes) a monoidal (∞,1)-category with geometric realization of simplicial objects in an (∞,1)-category such that the tensor product \otimes preserves this in each variable, then for all A-∞ algebra A,B,CA,B,C in 𝒞\mathcal{C}, the tensor product of ∞-bimodules

() B(): AMod B× BMod C AMod C (-) \otimes_B (-) \;\colon\; {}_A Mod_{B} \times {}_B Mod_{C} \to {}_{A} Mod_{C}

preserves (∞,1)-colimits separately in each argument.

This is (Lurie, cor. 4.3.5.15).

References

The original articles are

  • Charles E. Watts, Intrinsic characterizations of some additive functors, Proc. Amer. Math. Soc. 11, 1960, 5–8, MR0806.0832, doi
  • Samuel Eilenberg, Abstract description of some basic functors, J. Indian Math. Soc. (N.S.) 24, 1960, 231–234 (1961), MR0125148

A generalized statement in which the codomain is not assumed to be a category of modules is discussed in

  • A. Nyman, S. Paul Smith, A generalization of Watts’s Theorem: Right exact functors on module categories, (arxiv/0806.0832).

Generalization to homotopy theory/higher algebra is discussed in

  • Mark Hovey, The Eilenberg-Watts theorem in homotopical algebra (pdf)

and

Revised on February 4, 2013 22:40:35 by Urs Schreiber (89.204.137.52)