2025-10-14

This following is a very brief summary of what happened in class on 2025-10-14.

We began by noting that there are many forgetful functors from categories of bimodules, such as the functors $(R,S)\text{-}\mathbf{\text{Bimod}}\to (R,\mathbf{Z})\text{-}\mathbf{\text{Bimod}}\simeq R\text{-}\mathbf{\text{Mod}}$ and $(R,S)\text{-}\mathbf{\text{Bimod}}\to (\mathbf{Z},S)\text{-}\mathbf{\text{Bimod}}\simeq \mathbf{\text{Mod}}\text{-}S$, and even all the way to $(\mathbf{Z},\mathbf{Z})\text{-}\mathbf{\text{Bimod}}\simeq \mathbf{A}\mathbf{b}$. We then observed how we could use these forgetful functors to talk about morphisms between $(R,S)$-bimodules and $(R,S^{\prime })$-bimodules, namely by first forgetting the right actions on each.

Suppressing the forgetful functor notation for simplicity, we then looked at the set ${\mathrm{Hom}}_{R\text{-}\mathbf{\text{Mod}}}(M,N)$. We then saw how this set has a "canonical" structure of an $(S,S^{\prime })$-bimodule, where:

• the additive operation is "addition of outputs"
• the left $S$-action is via "right $S$-action on inputs"
• the right $S^{\prime }$-action is via "right $S$'-action on outputs"

We then noted ominously how this unlocks a certain "triple hom-set" construction, later to be seen as key for showing the general tensor product construction is a left adjoint functor.

We then briefly outlined a new, more general tensor product construction. Beginning with an $(R,S)$-bimodule $M$ and an $(S,T)$-bimodule $N$, we constructed an $(R,T)$-bimodule denoted $M{\otimes }_{S}N$, called the tensor product of $M$ and $N$ over $S$.

More next class!

Concepts

Bimodules
Bimodule morphisms
Tensor Products II - Tensor products of bimodules

References

• Dummit & Foote, Abstract Algebra: Section 10.4