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
the functor $\textit{FÉt}_ X \to \textit{FÉt}_ U$ is essentially surjective, i.e., purity holds for $A$,
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. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)