Lemma 35.23.2. The property $\mathcal{P}(f) =$“$f$ is quasi-separated” is fpqc local on the base.

**Proof.**
Any base change of a quasi-separated morphism is quasi-separated, see Schemes, Lemma 26.21.12. Being quasi-separated is Zariski local on the base (from the definition or by Schemes, Lemma 26.21.6). Finally, let $S' \to S$ be a flat surjective morphism of affine schemes, and let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is quasi-separated. This means that $\Delta ' : X' \to X'\times _{S'} X'$ is quasi-compact. Note that $\Delta '$ is the base change of $\Delta : X \to X \times _ S X$ via $S' \to S$. By Lemma 35.23.1 this implies $\Delta $ is quasi-compact, and hence $f$ is quasi-separated. Therefore Lemma 35.22.4 applies and we win.
$\square$

## 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Your email address will not be published.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: