Lemma 53.19.13 (0C57)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 53: Algebraic Curves
Section 53.19: Nodal curves
Lemma 53.19.13 (cite)

Lemma 53.19.13. Let $k$ be a field. Let $X$ be a locally algebraic $k$ -scheme of dimension $1$ . Let $Y \to X$ be an étale morphism. Let $y \in Y$ be a point with image $x \in X$ . The following are equivalent

$x$ is a closed point of $X$ and a node, and
$y$ is a closed point of $Y$ and a node.

Proof. By Lemma 53.19.12 we may base change to the algebraic closure of $k$ . Then the residue fields of $x$ and $y$ are $k$ . Hence the map $\mathcal{O}_{X, x}^\wedge \to \mathcal{O}_{Y, y}^\wedge$ is an isomorphism (for example by Étale Morphisms, Lemma 41.11.3 or More on Algebra, Lemma 15.43.9). Thus the lemma is clear. $\square$

Comments (2)

Comment #5108 by Tongmu He on May 19, 2020 at 11:31

Typo: in the proof of 53.19.13, the map $\mathcal{O}_{X, x}^\wedge \to \mathcal{O}_{Y, y}$ should be $\mathcal{O}_{X, x}^\wedge \to \mathcal{O}_{Y, y}^\wedge$ .

Comment #5315 by Johan on June 17, 2020 at 10:51

Thanks and fixed here.

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