The Stacks project

Lemma 58.26.3. Let $f : X \to S$ be a morphism of schemes. Let $U \subset X$ be an open subscheme. Assume

  1. $f$ is smooth,

  2. $S$ is Noetherian,

  3. for $s \in S$ with $\text{depth}(\mathcal{O}_{S, s}) \leq 1$ we have $X_ s = U_ s$,

  4. $U_ s \subset X_ s$ is dense for all $s \in S$.

Then $\textit{FÉt}_ X \to \textit{FÉt}_ U$ is an equivalence.

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 $Y \to X$. $\square$

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