Lemma 53.20.8. Let $f : X \to S$ be a morphism of schemes. Let $\{ U_ i \to X\} $ be an étale covering. The following are equivalent

$f$ is at-worst-nodal of relative dimension $1$,

each $U_ i \to S$ is at-worst-nodal of relative dimension $1$.

In other words, being at-worst-nodal of relative dimension $1$ is étale local on the source.

**Proof.**
One direction we have seen in Lemma 53.20.6. For the other direction, observe that being locally of finite presentation, flat, or to have relative dimension $1$ is étale local on the source (Descent, Lemmas 35.25.1, 35.24.1, and 35.30.8). Taking fibres we reduce to the case where $S$ is the spectrum of a field. In this case the result follows from Lemma 53.19.13 (and the fact that being smooth is étale local on the source by Descent, Lemma 35.27.1).
$\square$

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