Lemma 58.17.1. Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Assume the completion functor
is fully faithful on the full subcategory of finite locally free objects (see above). Then the restriction functor $\textit{FÉt}_ X \to \textit{FÉt}_ Y$ is fully faithful.
Proof. Since the category of finite étale coverings has an internal hom (Lemma 58.5.4) it suffices to prove the following: Given $U$ finite étale over $X$ and a morphism $t : Y \to U$ over $X$ there exists a unique section $s : X \to U$ such that $t = s|_ Y$. Picture
Finding the dotted arrow $s$ is the same thing as finding an $\mathcal{O}_ X$-algebra map
which reduces modulo the ideal sheaf of $Y$ to the given algebra map $t^\sharp : f_*\mathcal{O}_ U \to \mathcal{O}_ Y$. By Lemma 58.8.3 we can lift $t$ uniquely to a compatible system of maps $t_ n : Y_ n \to U$ and hence a map
of sheaves of algebras on $X$. Since $f_*\mathcal{O}_ U$ is a finite locally free $\mathcal{O}_ X$-module, we conclude that we get a unique $\mathcal{O}_ X$-module map $\sigma : f_*\mathcal{O}_ U \to \mathcal{O}_ X$ whose completion is $\mathop{\mathrm{lim}}\nolimits t_ n^\sharp $. To see that $\sigma $ is an algebra homomorphism, we need to check that the diagram
commutes. For every $n$ we know this diagram commutes after restricting to $Y_ n$, i.e., the diagram commutes after applying the completion functor. Hence by faithfulness of the completion functor we conclude. $\square$
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