The Stacks project

Proof. The functor is fully faithful by Lemma 58.10.3 combined with Local Cohomology, Lemma 51.3.1 (plus an application of Algebra, Lemma 10.163.2 to check the depth condition).

Let $\pi : V \to U$ be a finite étale morphism. Let $Y \to X$ be the finite morphism constructed in Lemma 58.21.5. We have to show that $Y \to X$ is finite étale. To show that this is true for all points $x \in X$ mapping to a given point $s \in S$ we may perform a base change by a flat morphism $S' \to S$ of Noetherian schemes such that $s$ is in the image. This follows from the compatibility of the construction in Lemma 58.21.5 with flat base change.

After enlarging $U$ we may assume $U \subset X$ is the maximal open over which $Y \to X$ is finite étale. Let $Z \subset X$ be the complement of $U$. To get a contradiction, assume $Z \not= \emptyset $. Let $s \in S$ be a point in the image of $Z \to S$ such that no strict generalization of $s$ is in the image. Then after base change to $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ we see that $S = \mathop{\mathrm{Spec}}(A)$ with $(A, \mathfrak m, \kappa )$ a local Noetherian ring of depth $\geq 2$ and $Z$ contained in the closed fibre $X_ s$ and nowhere dense in $X_ s$. Choose a closed point $z \in Z$. Then $\kappa (z)/\kappa $ is finite (by the Hilbert Nullstellensatz, see Algebra, Theorem 10.34.1). Choose a finite flat morphism $(S', s') \to (S, s)$ of local schemes realizing the residue field extension $\kappa (z)/\kappa $, see Algebra, Lemma 10.159.3. After doing a base change by $S' \to S$ we reduce to the case where $\kappa (z) = \kappa $.

By More on Morphisms, Lemma 37.38.5 there exists a locally closed subscheme $S' \subset X$ passing through $z$ such that $S' \to S$ is étale at $z$. After performing the base change by $S' \to S$, we may assume there is a section $\sigma : S \to X$ such that $\sigma (s) = z$. Choose an affine neighbourhood $\mathop{\mathrm{Spec}}(B) \subset X$ of $s$. Then $A \to B$ is a smooth ring map which has a section $\sigma : B \to A$. Denote $I = \mathop{\mathrm{Ker}}(\sigma )$ and denote $B^\wedge $ the $I$-adic completion of $B$. Then $B^\wedge \cong A[[x_1, \ldots , x_ d]]$ for some $d \geq 0$, see Algebra, Lemma 10.139.4. Observe that $d > 0$ since otherwise we see that $X \to S$ is étale at $z$ which would imply that $z$ is a generic point of $X_ s$ and hence $z \in U$ by assumption (4). Similarly, if $d > 0$, then $\mathfrak m B^\wedge $ maps into $U$ via the morphism $\mathop{\mathrm{Spec}}(B^\wedge ) \to X$. It suffices prove $Y \to X$ is finite étale after base change to $\mathop{\mathrm{Spec}}(B^\wedge )$. Since $B \to B^\wedge $ is flat (Algebra, Lemma 10.97.2) this follows from Lemma 58.26.2 and the uniqueness in the construction of $Y \to X$. $\square$