Lemma 53.20.9. Let $f : X \to S$ be a morphism of schemes. Let $\{ U_ i \to S\} $ be an fpqc covering. The following are equivalent
$f$ is at-worst-nodal of relative dimension $1$,
each $X \times _ S U_ i \to U_ i$ is at-worst-nodal of relative dimension $1$.
In other words, being at-worst-nodal of relative dimension $1$ is fpqc local on the target.
Proof. One direction we have seen in Lemma 53.20.4. For the other direction, observe that being locally of finite presentation, flat, or to have relative dimension $1$ is fpqc local on the target (Descent, Lemmas 35.23.11, 35.23.15, and Morphisms, Lemma 29.28.3). Taking fibres we reduce to the case where $S$ is the spectrum of a field. In this case the result follows from Lemma 53.19.12 (and the fact that being smooth is fpqc local on the target by Descent, Lemma 35.23.27). $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)