The Stacks project

Lemma 58.17.5. Let $X$ be a scheme and let $Y \subset X$ be a closed subscheme. If every connected component of $X$ meets $Y$, then the restriction functor $\textit{FÉt}_ X \to \textit{FÉt}_ Y$ is faithful.

Proof. Let $a, b : U \to U'$ be two morphisms of schemes finite étale over $X$ whose restriction to $Y$ are the same. The image of a connected component of $U$ is an connected component of $X$; this follows from Topology, Lemma 5.7.7 applied to the restriction of $U \to X$ to a connected component of $X$. Hence the image of every connected component of $U$ meets $Y$ by assumption. We conclude that $a = b$ after restriction to each connected component of $U$ by Étale Morphisms, Proposition 41.6.3. Since the equalizer of $a$ and $b$ is an open subscheme of $U$ (as the diagonal of $U'$ over $X$ is open) we conclude. $\square$

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