The Stacks project

Lemma 53.20.7. Let $S' \to S$ be an ├ętale morphism of schemes. Let $f : X \to S'$ be at-worst-nodal of relative dimension $1$. Then the composition $g : X \to S$ is at-worst-nodal of relative dimension $1$.

Proof. Observe that $g$ is flat and locally of finite presentation as a composition of morphisms which are flat and locally of finite presentation (use Morphisms, Lemmas 29.36.11, 29.36.12, 29.21.3, and 29.25.6). Thus it suffices to prove the fibres of $g$ have at-worst-nodal singularities. This follows from Lemma 53.19.14 and the analogous result for smooth points. $\square$

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