Lemma 35.28.3 (036P)—The Stacks project

Loading [MathJax]/extensions/tex2jax.js

Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.28: Properties of morphisms local in the fppf topology on the source
Lemma 35.28.3 (cite)

previous lemma
next lemma

numberstags

history
statistics
2 tags refer to this tag

Lemma 35.28.3. The property $\mathcal{P}(f)=$“$f$ is open” is fppf local on the source.

Proof. Being an open morphism is clearly Zariski local on the source and the target. It is a property which is preserved under composition, see Morphisms, Lemma 29.23.3, and a flat morphism of finite presentation is open, see Morphisms, Lemma 29.25.10 This proves (1), (2) and (3) of Lemma 35.26.4. The final condition (4) follows from Morphisms, Lemma 29.25.12. Hence we win. $\square$

Comments (0)

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).

Comments (0)

Post a comment