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

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

2. $y$ 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. $\square$

Comment #5108 by Tongmu He on

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

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