The Stacks project

Lemma 28.18.4. Let $X$ be a scheme. Then $X$ is quasi-affine if and only if the canonical morphism

\[ X \longrightarrow \mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X)) \]

from Schemes, Lemma 26.6.4 is a quasi-compact open immersion.

Proof. If the displayed morphism is a quasi-compact open immersion then $X$ is isomorphic to a quasi-compact open subscheme of $\mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X))$ and clearly $X$ is quasi-affine.

Assume $X$ is quasi-affine, say $X \subset \mathop{\mathrm{Spec}}(R)$ is quasi-compact open. This in particular implies that $X$ is separated, see Schemes, Lemma 26.23.9. Let $A = \Gamma (X, \mathcal{O}_ X)$. Consider the ring map $R \to A$ coming from $R = \Gamma (\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)})$ and the restriction mapping of the sheaf $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$. By Schemes, Lemma 26.6.4 we obtain a factorization:

\[ X \longrightarrow \mathop{\mathrm{Spec}}(A) \longrightarrow \mathop{\mathrm{Spec}}(R) \]

of the inclusion morphism. Let $x \in X$. Choose $r \in R$ such that $x \in D(r)$ and $D(r) \subset X$. Denote $f \in A$ the image of $r$ in $A$. The open $X_ f$ of Lemma 28.17.1 above is equal to $D(r) \subset X$ and hence $A_ f \cong R_ r$ by the conclusion of that lemma. Hence $D(r) \to \mathop{\mathrm{Spec}}(A)$ is an isomorphism onto the standard affine open $D(f)$ of $\mathop{\mathrm{Spec}}(A)$. Since $X$ can be covered by such affine opens $D(f)$ we win. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 28.18: Quasi-affine schemes

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Your email address will not be published.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 01P9. Beware of the difference between the letter 'O' and the digit '0'.