Lemma 53.20.4 (0C5B)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 53: Algebraic Curves
Section 53.20: Families of nodal curves
Lemma 53.20.4 (cite)

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Lemma 53.20.4. Let $f : X \to S$ be at-worst-nodal of relative dimension $1$ . Then the same is true for any base change of $f$ .

Proof. This is true because the base change of a syntomic morphism is syntomic (Morphisms, Lemma 29.30.4), the base change of a morphism of relative dimension $1$ has relative dimension $1$ (Morphisms, Lemma 29.29.2), the formation of $\text{Sing}(f)$ commutes with base change (Divisors, Lemma 31.10.1), and the base change of an unramified morphism is unramified (Morphisms, Lemma 29.35.5). $\square$

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