The Stacks project

Lemma 53.20.6. Let $f : X \to S$ be at-worst-nodal of relative dimension $1$. If $Y \to X$ is an ├ętale morphism, then the composition $g : Y \to S$ 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 have at-worst-nodal singularities. This follows from Lemma 53.19.13 (and the fact that the composition of an ├ętale morphism and a smooth morphism is smooth by Morphisms, Lemmas 29.36.5 and 29.34.4). $\square$


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