2024-10-14

This following is a very brief summary of what happened in class on 2024-10-14.

We started with a simple idea: given an $S$-module $N$ and a subring $R\subseteq S$, we can "restrict the the action of the scalars" to $R$ to allow us to view $N$ "as an $R$-modules." (We later noted that we are actually describing a forgetful functor from the category or $S$-modules to the category of $R$-modules.)

We then asked if we could reverse this process. In other words, given an $R$-module $M$ could we "extend the action of the scalars" to somehow give $M$ the structure of an $S$-module. We considered an example (with $M=\mathbf{Z}$, $R=\mathbf{Z}$ and $S=\mathbf{Q}$) that showed this is sometimes impossible.

Then we decided to relax our idea of "extending scalars" to ask whether $M$ could be "embedded" into a larger $R$-module $N$ that itself had the structure of an $S$-module. An example quickly showed us that this can sometimes also be impossible.

Finally we let category theory guide us, and instead asked whether there is a left adjoint to the forgetful functor $U:S\mathbf{\text{-Mod}}\to R\mathbf{\text{-Mod}}$. The answer to that question is YES. There is functor, called tensor product over $R$ and denoted $S{\otimes }_R-$, that is left adjoint to that forgetful functor.

We spent the remainder of class mainly outlining the construction of $S{\otimes }_{R}M$ for each $R$-module $M$.

We will spend the remainder of this week fleshing out (and vastly generalizing) this idea.

Concepts

• Tensor Products I - Extending scalars

References

• Dummit & Foote: Section 10.4