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. $\square$

Comments (1)

Comment #11029 by Kiran Kedlaya on

This lemma can be strengthened to say that is an epimorphism if and only if is fully faithful. The point is that the full faithfulness implies that the two different -module structures on coincide, and now tag 04VN implies that is epic.

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