Lemma 53.19.14 (0CD6)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 53: Algebraic Curves
Section 53.19: Nodal curves
Lemma 53.19.14 (cite)

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Lemma 53.19.14. Let $k'/k$ be a finite separable field extension. Let $X$ be a locally algebraic $k'$ -scheme of dimension $1$ . Let $x \in X$ be a closed point. The following are equivalent

$x$ is a node, and
$x$ is a node when $X$ viewed as a locally algebraic $k$ -scheme.

Proof. Follows immediately from the characterization of nodes in Lemma 53.19.7. $\square$

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