2024-10-15

This following is a very brief summary of what happened in class on 2024-10-15.

We picked up where we left off last class, recalling the construction of the $S$ -module $S \otimes_{R} M$ (for a fixed $R$ -module $M$ and rings $R \subseteq S$ ). We then briefly outlined:

The connection between $M$ and $S \otimes_{R} M$ , namely the $R$ -module morphism $j : M \to U (S \otimes_{R} M$ ) defined by $m \mapsto 1_{S} \otimes m$ . (This should really be denoted $η_{M}$ , since its the component of the unit of the adjunction.)
How we have an actual functor $S \otimes_{R} - : R -Mod \to S -Mod$ , by describing the arrow map. On simple tensors, given $f : M_{1} \to M_{2}$ the corresponding $S$ -module morphism $S \otimes_{R} f : S \otimes_{R} M_{1} \to S \otimes_{R} M_{2}$ is given by $s \otimes m_{1} \mapsto s \otimes f (m_{1})$ .
We proved a tiny lemma and then used it to show that for any finite abelian group $A$ we (unfortunately) have $Q \otimes_{Z} A ≃ 0$ .
We outlined the hom-set bijection through which the functors $(S \otimes_{R} -, U)$ are adjoints.
We noted that since $S \otimes_{R} -$ is a left adjoint, it commutes with colimits, and hence in particular commutes with direct sums.
We used that to quickly deduce that $S \otimes_{R} R^{n} ≃ S^{n}$ .

We ended by asking the:

Question

Can we do something like this for two modules? In other words, if $M$ and $N$ are modules (maybe even of different types), can we modify our construction to produce a new module worthy of the notation $M \otimes N$ ? What properties would we want such a module to have?

We'll answer this next class, when we introduce the notion of bimodules.

Concepts

Tensor Products I - Extending scalars

References

Dummit & Foote: Section 10.4