The Stacks project

Lemma 58.21.3. Let $(A, \mathfrak m)$ be a regular local ring of dimension $d \geq 2$. Set $X = \mathop{\mathrm{Spec}}(A)$ and $U = X \setminus \{ \mathfrak m\} $. Then

  1. the functor $\textit{FÉt}_ X \to \textit{FÉt}_ U$ is essentially surjective, i.e., purity holds for $A$,

  2. any finite $A \to B$ with $B$ normal which induces a finite étale morphism on punctured spectra is étale.

Proof. Recall that a regular local ring is normal by Algebra, Lemma 10.157.5. Hence (1) and (2) are equivalent by Lemma 58.20.3. We prove the lemma by induction on $d$.

The case $d = 2$. In this case $A \to B$ is flat. Namely, we have going down for $A \to B$ by Algebra, Proposition 10.38.7. Then $\dim (B_{\mathfrak m'}) = 2$ for all maximal ideals $\mathfrak m' \subset B$ by Algebra, Lemma 10.112.7. Then $B_{\mathfrak m'}$ is Cohen-Macaulay by Algebra, Lemma 10.157.4. Hence and this is the important step Algebra, Lemma 10.128.1 applies to show $A \to B_{\mathfrak m'}$ is flat. Then Algebra, Lemma 10.39.18 shows $A \to B$ is flat. Thus we can apply Lemma 58.21.2 (or you can directly argue using the easier Discriminants, Lemma 49.3.1) to see that $A \to B$ is étale.

The case $d \geq 3$. Let $V \to U$ be finite étale. Let $f \in \mathfrak m_ A$, $f \not\in \mathfrak m_ A^2$. Then $A/fA$ is a regular local ring of dimension $d - 1 \geq 2$, see Algebra, Lemma 10.106.3. Let $U_0$ be the punctured spectrum of $A/fA$ and let $V_0 = V \times _ U U_0$. By Lemma 58.20.7 it suffices to show that $V_0$ is in the essential image of $\textit{FÉt}_{\mathop{\mathrm{Spec}}(A/fA)} \to \textit{FÉt}_{U_0}$. This follows from the induction hypothesis. $\square$

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