Lemma 58.21.5. Let $j : U \to X$ be an open immersion of locally Noetherian schemes such that $\text{depth}(\mathcal{O}_{X, x}) \geq 2$ for $x \not\in U$. Let $\pi : V \to U$ be finite étale. Then
$\mathcal{B} = j_*\pi _*\mathcal{O}_ V$ is a reflexive coherent $\mathcal{O}_ X$-algebra, set $Y = \underline{\mathop{\mathrm{Spec}}}_ X(\mathcal{B})$,
$Y \to X$ is the unique finite morphism such that $V = Y \times _ X U$ and $\text{depth}(\mathcal{O}_{Y, y}) \geq 2$ for $y \in Y \setminus V$,
$Y \to X$ is étale at $y$ if and only if $Y \to X$ is flat at $y$, and
$Y \to X$ is étale if and only if $\mathcal{B}$ is finite locally free as an $\mathcal{O}_ X$-module.
Moreover, (a) the construction of $\mathcal{B}$ and $Y \to X$ commutes with base change by flat morphisms $X' \to X$ of locally Noetherian schemes, and (b) if $V' \to U'$ is a finite étale morphism with $U \subset U' \subset X$ open which restricts to $V \to U$ over $U$, then there is a unique isomorphism $Y' \times _ X U' = V'$ over $U'$.
Proof.
Observe that $\pi _*\mathcal{O}_ V$ is a finite locally free $\mathcal{O}_ U$-module, in particular reflexive. By Divisors, Lemma 31.12.12 the module $j_*\pi _*\mathcal{O}_ V$ is the unique reflexive coherent module on $X$ restricting to $\pi _*\mathcal{O}_ V$ over $U$. This proves (1).
By construction $Y \times _ X U = V$. Since $\mathcal{B}$ is coherent, we see that $Y \to X$ is finite. We have $\text{depth}(\mathcal{B}_ x) \geq 2$ for $x \in X \setminus U$ by Divisors, Lemma 31.12.11. Hence $\text{depth}(\mathcal{O}_{Y, y}) \geq 2$ for $y \in Y \setminus V$ by Algebra, Lemma 10.72.11. Conversely, suppose that $\pi ' : Y' \to X$ is a finite morphism such that $V = Y' \times _ X U$ and $\text{depth}(\mathcal{O}_{Y', y'}) \geq 2$ for $y' \in Y' \setminus V$. Then $\pi '_*\mathcal{O}_{Y'}$ restricts to $\pi _*\mathcal{O}_ V$ over $U$ and satisfies $\text{depth}((\pi '_*\mathcal{O}_{Y'})_ x) \geq 2$ for $x \in X \setminus U$ by Algebra, Lemma 10.72.11. Then $\pi '_*\mathcal{O}_{Y'}$ is canonically isomorphic to $j_*\pi _*\mathcal{O}_ V$ for example by Divisors, Lemma 31.5.11. This proves (2).
If $Y \to X$ is étale at $y$, then $Y \to X$ is flat at $y$. Conversely, suppose that $Y \to X$ is flat at $y$. If $y \in V$, then $Y \to X$ is étale at $y$. If $y \not\in V$, then we check (1), (2), (3), and (4) of Lemma 58.21.2 hold to see that $Y \to X$ is étale at $y$. Parts (1) and (2) are clear and so is (3) since $\text{depth}(\mathcal{O}_{Y, y}) \geq 2$. If $y' \leadsto y$ is a specialization and $\dim (\mathcal{O}_{Y, y'}) = 1$, then $y' \in V$ since otherwise the depth of this local ring would be $2$ a contradiction by Algebra, Lemma 10.72.3. Hence $Y \to X$ is étale at $y'$ and we conclude (4) of Lemma 58.21.2 holds too. This finishes the proof of (3).
Part (4) follows from (3) and the fact that $((Y \to X)_*\mathcal{O}_ Y)_ x$ is a flat $\mathcal{O}_{X, x}$-module if and only if $\mathcal{O}_{Y, y}$ is a flat $\mathcal{O}_{X, x}$-module for all $y \in Y$ mapping to $x$, see Algebra, Lemma 10.39.18. Here we also use that a finite flat module over a Noetherian ring is finite locally free, see Algebra, Lemma 10.78.2 (and Algebra, Lemma 10.31.4).
As to the final assertions of the lemma, part (a) follows from flat base change, see Cohomology of Schemes, Lemma 30.5.2 and part (b) follows from the uniqueness in (2) applied to the restriction $Y \times _ X U'$.
$\square$
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