Lemma 25.6.6. Let $Y$ be an affine scheme. Let $f \in \Gamma (Y, \mathcal{O}_ Y)$. The open subspace $D(f)$ is an affine scheme.

**Proof.**
We may assume that $Y = \mathop{\mathrm{Spec}}(R)$ and $f \in R$. Consider the morphism of affine schemes $\phi : U = \mathop{\mathrm{Spec}}(R_ f) \to \mathop{\mathrm{Spec}}(R) = Y$ induced by the ring map $R \to R_ f$. By Algebra, Lemma 10.16.6 we know that it is a homeomorphism onto $D(f)$. On the other hand, the map $\phi ^{-1}\mathcal{O}_ Y \to \mathcal{O}_ U$ is an isomorphism on stalks, hence an isomorphism. Thus we see that $\phi $ is an open immersion. We conclude that $D(f)$ is isomorphic to $U$ by Lemma 25.3.4.
$\square$

## 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Your email address will not be published.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 (1)

Comment #677 by Anfang Zhou on