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Eilenberg-Watts theorem for arbitrary unital rings

Statement

Let $A, B$ be unital rings.

(i) For every $(B, A)$ -bimodule $M$ ( $(S, R)$ -bimodules and commuting left and right scalar actions) the functor $T_{M} = M \otimes_{A} - : A - M o d \to B - M o d$ is additive, right exact, coproduct-preserving and therefore cocontinuous (Additive cocontinuous module functors and their schematic category), and the assignment $M \mapsto T_{M}$ is functorial: a bimodule map $f : M \to M^{'}$ gives the natural transformation with components $f \otimes 1_{X}$ .

(ii) Conversely every additive cocontinuous functor $F : A - M o d \to B - M o d$ is naturally isomorphic to $T_{F (A)}$ , where $F (A)$ carries the $(B, A)$ -bimodule structure $m a = F (r_{a}) (m)$ ( $F (A)$ is a $(B, A)$ -bimodule for every additive functor $F$ ).

Hence, up to natural isomorphism, the additive cocontinuous functors are exactly the tensor functors with bimodule kernels: the quasi-inverse of $M \mapsto T_{M}$ is $F \mapsto F (_{A} A)$ , with categorical language interpreted schematically as in Additive cocontinuous module functors and their schematic category. No commutativity of $A$ or $B$ is assumed and no choice is used.

Facts & Assumptions

Given: Unital rings $A, B$ ; the class of additive cocontinuous functors $A - M o d \to B - M o d$ ; $(B, A)$ -bimodules $M, M^{'}$ ; an additive cocontinuous functor $F$ ; a bimodule map $f : M \to M^{'}$ .

[F1]

A functor is additive cocontinuous when it is additive and preserves every small colimit (Additive cocontinuous module functors and their schematic category).

[F2]

The additive cocontinuous functors with all natural transformations as morphisms satisfy the category laws schematically, with a set of component codes for each fixed Hom-collection, componentwise identities and vertical composition (Natural transformations of additive cocontinuous module functors are determined at the regular module).

[F3]

An additive module functor is cocontinuous if and only if it is right exact and preserves arbitrary coproducts; equivalently if and only if it preserves cokernels and arbitrary direct sums (An additive module functor is cocontinuous exactly when it is right exact and preserves coproducts).

[F4]

$T_{M} = M \otimes_{A} -$ is additive, right exact and preserves arbitrary direct sums, including the empty one; if $M$ is a $(B, A)$ -bimodule then $T_{M}$ takes values in left $B$ -modules and all displayed maps are $B$ -linear (The functor $M \otimes_{A} -$ is additive, right exact, and preserves direct sums over an arbitrary unital ring).

[F5]

Every bimodule map $f : M \to M^{'}$ yields a natural transformation $T_{M} \Rightarrow T_{M^{'}}$ with components $f \otimes 1_{X}$ , and the assignment is compatible with identities and vertical composition (Natural transformations between tensor functors are bimodule maps).

[F6]

If $F$ is additive and $M = F (A)$ , then $m a = F (r_{a}) (m)$ makes $M$ a $(B, A)$ -bimodule ( $F (A)$ is a $(B, A)$ -bimodule for every additive functor $F$ ).

[F7]

If $F$ is additive, right exact and coproduct-preserving and $M = F (A)$ with that bimodule structure, then the canonical comparison $τ : M \otimes_{A} - \Rightarrow F$ is a natural isomorphism (Canonical free presentations force the comparison to be an isomorphism).

[F8]

$ρ_{M} : M \otimes_{A} A \to M$ , $ρ_{M} (m \otimes a) = m a$ , is a group isomorphism (The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ ).

Proof

technique · direct

1.1F1F2F3F4F5

For a $(B, A)$ -bimodule $M$ the functor $T_{M}$ is additive, right exact and coproduct-preserving by [F4], hence cocontinuous by the equivalence [F3]. Given a bimodule map $f : M \to M^{'}$ , the components $f \otimes 1_{X}$ are natural in $X$ and compatible with identities and vertical composition by [F5], so they define a morphism $T_{M} \Rightarrow T_{M^{'}}$ in the category of [F2]; this makes $M \mapsto T_{M}$ a functor from $(B, A)$ -bimodules to the additive cocontinuous functors.

1.2F3F6F7

Let $F$ be additive cocontinuous. By [F3] it is right exact and coproduct-preserving, and $M : = F (A)$ is a $(B, A)$ -bimodule by [F6]. The canonical comparison $τ : M \otimes_{A} - \Rightarrow F$ is then a natural isomorphism by [F7], so $F ≅ T_{F (A)}$ .

2.1F8step 1.1step 1.2

The two assignments are inverse up to natural isomorphism: for a bimodule $M$ one has $T_{M} (A) = M \otimes_{A} A ≅ M$ by the unit isomorphism [F8], and for additive cocontinuous $F$ one has $F ≅ T_{F (A)}$ by step 1.2. Steps 1.1 and 1.2 therefore show that, up to natural isomorphism, the additive cocontinuous functors $A - M o d \to B - M o d$ are exactly the functors $T_{M}$ with $M$ a $(B, A)$ -bimodule, with quasi-inverse $F \mapsto F (_{A} A)$ .

3.1step 1.1step 1.2step 2.1∎

The construction of $τ$ used no presentation of any module and no element selection, and the module category is treated over arbitrary unital rings; hence neither commutativity of $A$ or $B$ nor the axiom of choice is used.

Depends on

Used by

Additive cocontinuous module functors admit right adjoints Corollary
Eilenberg-Watts is a schematic equivalence of Hom categories Corollary
Exact module tensor functors correspond to right-flat bimodules Corollary
A coproduct-preserving left exact module functor is not tensor Counterexample
A right exact module functor without coproduct preservation is not tensor Counterexample
A finite right exact functor needs no infinite-coproduct hypothesis Example
Eilenberg-Watts recovers extension of scalars Example
Tensoring defines a schematic pseudofunctor with interchange Lemma
Morita equivalence is invertibility of a bimodule Theorem

Dependency tree · two levels

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Sources

J. Fuchs, G. Schaumann, C. Schweigert, Eilenberg-Watts calculus for finite categories and a bimodule Radford S^4 theorem, arXiv:1612.04561v3, introduction (classical statement for unital rings) (standard reference, not scraped)
M. Kamensky, Non-Commutative Algebra (BGU course notes, Spring 2017), §5.1, Theorem 5.1.43, Proposition 5.1.40, Lemma 5.1.46, Corollaries 5.1.48-5.1.49 (standard reference, not scraped)
A. Nyman and S. P. Smith, A Generalization of Watts's Theorem: Right Exact Functors on Module Categories, arXiv:0806.0832, Theorem 1.1-1.2, Propositions 3.2-3.3, Lemma 3.4 (standard reference, not scraped)