Suppose $M$ and $N$ are $(A,B)$- and $(B,A)$-bimodules, respectively, where $A$ and $B$ are symmetric $R$-algebras, such that the functors $M\otimes_B - $ and $N\otimes_A - $ are both left and right adjoint via some isomorphisms $M\simeq\hom_B(N,B)$ and $N\simeq \hom_A(M,A)$. (I don't think the specific details are relevant for what follows.)

What does it mean to describe these adjunctions in terms of morphisms of bimodules? I read that these adjunctions can be described by $\epsilon_{M,N}\colon M\otimes_B N\to A$, $\epsilon_{N,M}\colon N\otimes_A M\to B$, $\eta_{M,N}\colon B\to N\otimes_A M$, and $\eta_{N,M}\colon A\to M\otimes_B N$.

This notation seems to suggest various units and counits. However, I'm used to be a counit/unit as being a natural transformation, say $\epsilon\colon M\otimes_B -\circ N\otimes_A -\to \text{Id}_A$, so it would be described in terms of the component morphisms, $\epsilon_X\colon M\otimes_B N\otimes_A X\to X$, where $X$ is an $A$-module. Now $X\simeq A\otimes_A X$, so viewing a component as a map of modules $$ \epsilon_X\colon M\otimes_B N\otimes_A X\to A\otimes_A X $$ seems like it would closely related to the bimodule map $\epsilon_{M,N}\colon M\otimes_B N\to A$. Is it saying something like $\epsilon_X=\epsilon_{M,N}\otimes 1_X$? And likewise for all the other three cases?

  • $\begingroup$ Natural transformations between tensor product functors as in your question are naturally in bijection with module homomorphisms. This is an aspect of the Eilenberg-Watts theorem. $\endgroup$ Oct 2 '17 at 6:49
  • $\begingroup$ @QiaochuYuan Is this natural bijection the one I guessed above, or something else? All instances of Watts Theorem I know of only characterize certain functors as isomorphic to tensor product or hom functors, but say nothing about the natural transformations between them. $\endgroup$ Oct 2 '17 at 19:21

Remark: The classical Eilenberg-Watts theorem says the following: Let $R,S$ be commutative $k$-algebras. If

$$F: Mod(R) \rightarrow Mod(S)$$

is a $k$-linear right exact functor commuting with direct limits it follows there is an $R\otimes_k S$-module $M$ and an isomorphism of functors

$$F(-) \cong -\otimes_R M.$$

Note: This result has "recently" been generalized to schemes and you will find all details in the online paper.


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