Lemma 10.13.3. Let $R \to S$ be a ring map. The functors $\text{Mod}_ S \to \text{Mod}_ R$, $N \mapsto N_ R$ (restriction) and $\text{Mod}_ R \to \text{Mod}_ S$, $M \mapsto M \otimes _ R S$ (base change) are adjoint functors. In a formula

$\mathop{\mathrm{Hom}}\nolimits _ R(M, N_ R) = \mathop{\mathrm{Hom}}\nolimits _ S(M \otimes _ R S, N)$

Proof. If $\alpha : M \to N_ R$ is an $R$-module map, then we define $\alpha ' : M \otimes _ R S \to N$ by the rule $\alpha '(m \otimes s) = s\alpha (m)$. If $\beta : M \otimes _ R S \to N$ is an $S$-module map, we define $\beta ' : M \to N_ R$ by the rule $\beta '(m) = \beta (m \otimes 1)$. We omit the verification that these constructions are mutually inverse. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).