Proof. By Schemes, Lemma 25.21.6 we have to show that the intersection $U \cap V$ of two affine opens of $X$ is quasi-compact. This follows from Lemma 27.5.3 above on considering the open immersion $U \cap V \to U$ for example. (But really it is just because any open of the spectrum of a Noetherian ring is quasi-compact.) $\square$

There are also:

• 2 comment(s) on Section 27.5: Noetherian schemes

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).