Bimodule morphisms

Definition

If you had to guess the definition of a bimodule morphism, you'd guess correctly:

Definition of bimodule morphism

If $M$ and $N$ are $(R, S)$ -bimodules, then a set map $f : M \to N$ is a bimodule morphism if it is both a left $R$ -module morphism and a right $S$ -module morphism.

This is cheating slightly, of course. We should really say that a bimodule morphism $f : M \to N$ consists of the data of a set map $f : U (M) \to U (N)$ such that the following conditions hold:

$ϕ (m_{1} + m_{2}) = ϕ (m_{1}) + ϕ (m_{2})$ , for all $m_{1}, m_{2} \in M$
$ϕ (r m) = r ϕ (m)$ , for all $r \in R$ and $m \in M$
$ϕ (m s) = ϕ (m) s$ , for all $s \in S$ and $m \in M$

In any case, we can now talk about the category of $(R, S)$ -bimodules. What should we denote this category? It's not completely agreed upon. Some people denote it $(R, S) -Bimod$ . Others denote is $R -Bimod- S$ , or even $R -Mod- S$ . Choose your favorite, make sure it's clear, and stick with it.

More than just hom-sets

As with $R$ -modules, for any pair of $(R, S)$ -bimodules $M$ and $N$ , the set of bimodule morphisms between them has the structure of an abelian group (using the addition in $N$ ).

There's a lot more to the story about bimodule morphisms, though. First suppose $M$ is an $(R, S)$ -bimodule and $N$ is an $(R, S^{'})$ -bimodule. If we forget the right-actions and consider the left $R$ -modules $M$ and $N$ , we can consider the set of left $R$ -module morphisms, ${Hom}_{R} (M, N)$ . This set actually has the structure of an $(S, S^{'})$ -bimodule, as follows.

The addition in ${Hom}_{R} (M, N)$ is defined through the addition in $N$ . In other words, given $R$ -module morphisms $f, g : M \to N$ we define $f + g : M \to N$ by $(f + g) (m) = f (m) + g (m)$ . Observe that $f + g$ is indeed an $R$ -module morphism. First, it is additive since $f$ and $g$ are additive; second, it is compatible with the $R$ -actions since $f$ and $g$ are compatible with the $R$ -actions.

The left $S$ -action on ${Hom}_{R} (M, N)$ is defined through the right $S$ -action on $M$ . In detail, for each $R$ -module morphism $f : M \to N$ and $s \in S$ we define $s \cdot f : M \to N$ by $(s \cdot f) (m) = f (m s)$ . Again, it is straightforward to verify that $s \cdot f$ is an $R$ -module morphism. Moreover, this really does define a left $S$ -action on ${Hom}_{R} (M, N)$ , since

(s_{1} s_{2} \cdot f) (m) = f (m (s_{1} s_{2})) = f ((m s_{1}) s_{2}) = (s_{2} \cdot f) (m s_{1}) = (s_{1} \cdot (s_{2} \cdot f)) (m)

In other words, $s_{1} s_{2} \cdot f = s_{1} \cdot (s_{2} \cdot f)$ .

The right $S^{'}$ -action on ${Hom}_{R} (M, N)$ is defined through the right $S^{'}$ -action on $N$ . In detail, for each $R$ -module morphism $f : M \to N$ and $s^{'} \in S^{'}$ we define $f \cdot s^{'} : M \to N$ by $(f \cdot s^{'}) (m) = f (m) s^{'}$ . It is once more simple to verify that $f \cdot s^{'}$ is an $R$ -module morphism, and that this really does define a right $S^{'}$ -action on ${Hom}_{R} (M, N)$ .

In summary:

Hom-bimodules

For each $(R, S)$ -bimodule $M$ and $(R, S^{'})$ -bimodule $N$ , the set of ${Hom}_{R} (M, N)$ of left $R$ -module morphisms between $M$ and $N$ (viewed as left $R$ -modules) has the structure of an $(S, S^{'})$ -bimodule.

Similarly, for each $(R, S)$ -bimodule $M$ and $(R^{'}, S)$ -bimodule $N$ , the set ${Hom}_{S} (M, N)$ of right $S$ -module morphisms between $M$ and $N$ (viewed as right $S$ -modules) has the structure of an $(R^{'}, R)$ -bimodule.

A more careful approach?

We should really be careful here and use forgetful functors to move $M$ and $N$ into the category of left $R$ -modules. Can you fill in the details?

Triples of bimodules and hom-sets

Suppose $M$ is an $(R, S)$ -bimodule, $N$ is an $(S, T)$ -bimodule, and $P$ is an $(R, T)$ -bimodule. By the above construction, the set ${Hom}_{T} (N, P)$ has the structure of an $(R, S)$ -bimodule. We can then consider the set of $(R, S)$ -bimodule morphisms between $M$ and ${Hom}_{T} (N, P)$ . This is the set

{Hom}_{(R, S)} (M, {Hom}_{T} (N, P)) .

This set will play a critical role with the tensor product construction on bimodules.

Suggested next notes

The 2-category of bimodules
Tensor Products II - Tensor products of bimodules