----
> [!definition] Definition. ([[extension of scalars]])
> If $M$ is an $R$-[[module]] and $N$ is an $(R,S)$-[[bimodule]],[^1] then their [[tensor product of modules|tensor product]] over $R$, $M \otimes_{R} N$, naturally carries an *$S$*-[[module]] structure: define the action of $s \in S$ on pure tensors $m \otimes n$ by $s( m\otimes n):=m \otimes (sn)$
and extend to all tensors by linearity. This gives $M \otimes_{R} N$ an $(R,S)$-[[bimodule]] structure.
>
**Extension of scalars** is a [[covariant functor|functor]] $R$-$\mathsf{Mod} \to {S}$-$\mathsf{Mod}$ defined by associating to an $R$-[[module]] $M$ the [[tensor product of modules|tensor product]] $f^{*}(M):= M \otimes_{R}S$, viewed as an $S$-[[module]]. If $R^{\oplus B} \to R^{\oplus A} \to M \to 0$
is a [[finitely presented module|presentation]] of $M$, tensoring by $S$ gives (recall [[adjointness and exactness|the tensor functor is right-exact]]) $S^{\oplus B} \to S^{\oplus A} \to M \otimes_{R} S \to 0.$
Intuitively, this says that $f^{*}(M)$ is the [[module]] defined by 'the same generators (from $A$) and relations (from $B$)' as $M$, but now with coefficients coming from $S$.
>
More generally, if $N$ is any $S$-[[module]], we might call $M \otimes_{R} N$ extension of scalars. $f^{*}$ acts on morphisms by sending $\varphi:M \to M'$ to the [[tensor functor|tensor product of linear maps]] $f^{*}(\varphi):=\varphi \otimes \id_{N}.$
- [ ] left-[[adjoint functor|adjoint]] to restriction of scalars (presumably that's somewhere because it gets referenced below)
> [!basicnonexample] Warning.
> The class notes sometimes swap the roles of $M$ and $N$ from the discussion here.
^nonexample
> [!basicexample]
>
> - We already know the $R$-[[linear map|module isomorphism]] $S \otimes_{R}R \xrightarrow{\sim}S$ determined by $s \otimes r \mapsto sr$. It is in fact an $S$-module isomorphism since $s'(s \otimes r)=(s's) \otimes r \mapsto (s's)r=s'(sr)$. In particular, $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{R} \xrightarrow{\sim} \mathbb{C}$ as $\mathbb{C}$-[[module|modules]].
> - For an $S$-[[module]] $M$ and $R$-[[module|modules]] $N_{i}$, $i \in I$, we know an $R$-[[module]] [[isomorphism]] $M \otimes_{R} \bigoplus_{i \in I} N_{i} \mapsto \bigoplus_{i \in I}(M \otimes_{R} N_{i})$. Again it can be verified this is also an $S$-linear isomorphism. In particular, $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{R}^{n} \xrightarrow{\sim} \mathbb{C}^{n}$ as $\mathbb{C}$-modules.
> - **Restrict and then extend.** Take the $\mathbb{C}$-[[module]] $\mathbb{C}^{n}$. [[restriction of scalars|Restrict scalars]] to $\mathbb{R}$ and obtain $\mathbb{R}^{2n}$. Extend scalars to $\mathbb{C}$ and obtain $\mathbb{C} \otimes_{\mathbb{R}}\mathbb{R}^{2n} \cong \mathbb{C}^{2n}$ (using also the above example).
> - **Extend and then restrict.** Take the $\mathbb{R}$-[[module]] $\mathbb{R}^{n}$. Extend scalars to $\mathbb{C}$ and obtain $\mathbb{C} \otimes_{\mathbb{R}}\mathbb{R}^{n} \cong \mathbb{C}^{n}$. [[restriction of scalars|Restrict]] scalars to $\mathbb{R}$ and obtain $\mathbb{R}^{2n}$.
> - The $\mathbb{C}$-module structure on $\mathbb{C}^{n} \otimes_{\mathbb{R}} \mathbb{R}^{\ell}$? First we have an $\mathbb{R}$-module isomorphism $\mathbb{C}^{n} \otimes_{\mathbb{R}} \mathbb{R}^{\ell} \cong \mathbb{R}^{2n} \otimes_{\mathbb{R}} \mathbb{R}^{\ell} \cong \mathbb{C}^{n\ell}$ where the second '$\cong