Lemma 10.14.4. Let $R \to S$ be a ring map. The functors $\text{Mod}_ S \to \text{Mod}_ R$, $N \mapsto N_ R$ (restriction) and $\text{Mod}_ R \to \text{Mod}_ S$, $M \mapsto \mathop{\mathrm{Hom}}\nolimits _ R(S, M)$ are adjoint functors. In a formula

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

Proof. If $\alpha : N_ R \to M$ is an $R$-module map, then we define $\alpha ' : N \to \mathop{\mathrm{Hom}}\nolimits _ R(S, M)$ by the rule $\alpha '(n) = (s \mapsto \alpha (sn))$. If $\beta : N \to \mathop{\mathrm{Hom}}\nolimits _ R(S, M)$ is an $S$-module map, we define $\beta ' : N_ R \to M$ by the rule $\beta '(n) = \beta (n)(1)$. We omit the verification that these constructions are mutually inverse. $\square$

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