The Stacks project

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

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

  2. 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 35.28.1, 35.27.1, and 35.33.8). 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 35.30.1). $\square$

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