The Stacks project

Lemma 58.10.1. Let $f : X \to Y$ be a morphism of schemes. If $f(X)$ is dense in $Y$ then the base change functor $\textit{FÉt}_ Y \to \textit{FÉt}_ X$ is faithful.

Proof. Since the category of finite étale coverings has an internal hom (Lemma 58.5.4) it suffices to prove the following: Given $W$ finite étale over $Y$ and a morphism $s : X \to W$ over $X$ there is at most one section $t : Y \to W$ such that $s = t \circ f$. Consider two sections $t_1, t_2 : Y \to W$ such that $s = t_1 \circ f = t_2 \circ f$. Since the equalizer of $t_1$ and $t_2$ is closed in $Y$ (Schemes, Lemma 26.21.5) and since $f(X)$ is dense in $Y$ we see that $t_1$ and $t_2$ agree on $Y_{red}$. Then it follows that $t_1$ and $t_2$ have the same image which is an open and closed subscheme of $W$ mapping isomorphically to $Y$ (Étale Morphisms, Proposition 41.6.1) hence they are equal. $\square$

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