Lemma 58.31.3 (0EYE)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.31: Tame ramification
Lemma 58.31.3 (cite)

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Lemma 58.31.3. Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be an effective Cartier divisor such that $D$ is a regular scheme. Let $Y \to X \setminus D$ be a finite étale morphism. If $Y$ is unramified over $X$ in codimension $1$, then there exists a finite étale morphism $Y' \to X$ whose restriction to $X \setminus D$ is $Y$.

Proof. This is a special case of Lemma 58.31.2. First, $D$ is nowhere dense in $X$ (see discussion in Divisors, Section 31.13) and hence $X \setminus D$ is dense in $X$. Second, the ring $\mathcal{O}_{X, x}$ is a regular local ring for all $x \in D$ by Algebra, Lemma 10.106.7 and our assumption that $\mathcal{O}_{D, x}$ is regular. $\square$

Comments (2)

Comment #8104 by Laurent Moret-Bailly on February 09, 2023 at 03:32

The assumptions on $D$ are very restrictive. A more general (and natural) statement, with essentially the same proof, would be:

Let $X$ be a locally Noetherian scheme. Let $U\subset X$ be open and dense. Assume that $X\setminus U\subset \mathrm{Reg}(X)$ . (This holds in particular if $X\setminus U$ is the support of an effective Cartier divisor which is regular, by Algebra, Lemma 10.106.7.) Let $Y\to U$ be a finite étale morphism, unramified over $X$ in codimension $1$ . Then (...)

Comment #8215 by Stacks Project on March 03, 2023 at 12:40

Sure. I made it a separate lemma. But I think this original one is what we always use. See this.

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