Lemma 29.13.2. A quasi-affine morphism is separated and quasi-compact.

Proof. Let $f : X \to S$ be quasi-affine. Quasi-compactness is immediate from Schemes, Lemma 26.19.2. Let $U \subset S$ be an affine open. If we can show that $f^{-1}(U)$ is a separated scheme, then $f$ is separated (Schemes, Lemma 26.21.7 shows that being separated is local on the base). By assumption $f^{-1}(U)$ is isomorphic to an open subscheme of an affine scheme. An affine scheme is separated and hence every open subscheme of an affine scheme is separated as desired. $\square$

## Comments (2)

Comment #4294 by Xuande Liu on

Without loss of generality we can assume that $S=\mathit{Spec}(B)$ is affine. Let $A=\Gamma(X,\mathcal O_X)$. And hence we can factorize $f$ as $X\to \mathit{Spec}(A)\to\mathit{Spec(B)}$. The former morphism is an open immersion and the latter is morphism between affine schemes. Therefore $f$ is separated.

## 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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01SL. Beware of the difference between the letter 'O' and the digit '0'.