Lemma 35.20.21. The property $\mathcal{P}(f) =$“$f$ is a quasi-compact immersion” is fpqc local on the base.

**Proof.**
Let $f : X \to Y$ be a morphism of schemes. Let $\{ Y_ i \to Y\} $ be an fpqc covering. Write $X_ i = Y_ i \times _ Y X$ and $f_ i : X_ i \to Y_ i$ the base change of $f$. Also denote $q_ i : Y_ i \to Y$ the given flat morphisms. Assume each $f_ i$ is a quasi-compact immersion. By Schemes, Lemma 26.23.8 each $f_ i$ is separated. By Lemmas 35.20.1 and 35.20.6 this implies that $f$ is quasi-compact and separated. Let $X \to Z \to Y$ be the factorization of $f$ through its scheme theoretic image. By Morphisms, Lemma 29.6.3 the closed subscheme $Z \subset Y$ is cut out by the quasi-coherent sheaf of ideals $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to f_*\mathcal{O}_ X)$ as $f$ is quasi-compact. By flat base change (see for example Cohomology of Schemes, Lemma 30.5.2; here we use $f$ is separated) we see $f_{i, *}\mathcal{O}_{X_ i}$ is the pullback $q_ i^*f_*\mathcal{O}_ X$. Hence $Y_ i \times _ Y Z$ is cut out by the quasi-coherent sheaf of ideals $q_ i^*\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_{Y_ i} \to f_{i, *}\mathcal{O}_{X_ i})$. By Morphisms, Lemma 29.7.7 the morphisms $X_ i \to Y_ i \times _ Y Z$ are open immersions. Hence by Lemma 35.20.16 we see that $X \to Z$ is an open immersion and hence $f$ is a immersion as desired (we already saw it was quasi-compact).
$\square$

## Post a comment

Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: