The Stacks project

Lemma 10.107.14. Let $R \to S$ be an epimorphism of rings. Let $N_1, N_2$ be $S$-modules. Then $\mathop{\mathrm{Hom}}\nolimits _ S(N_1, N_2) = \mathop{\mathrm{Hom}}\nolimits _ R(N_1, N_2)$. In other words, the restriction functor $\text{Mod}_ S \to \text{Mod}_ R$ is fully faithful.

Proof. Let $\varphi : N_1 \to N_2$ be an $R$-linear map. For any $x \in N_1$ consider the map $S \otimes _ R S \to N_2$ defined by the rule $g \otimes g' \mapsto g\varphi (g'x)$. Since both maps $S \to S \otimes _ R S$ are isomorphisms (Lemma 10.107.1), we conclude that $g \varphi (g'x) = gg'\varphi (x) = \varphi (gg' x)$. Thus $\varphi $ is $S$-linear. $\square$

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