Lemma 26.9.2 (01IK)—The Stacks project

Processing math: 100%

Lemma 26.9.2. Let $X$ be a scheme. Let $j : U \to X$ be an open immersion of locally ringed spaces. Then $U$ is a scheme. In particular, any open subspace of $X$ is a scheme.

Proof. Let $U \subset X$ . Let $u \in U$ . Pick an affine open neighbourhood $u \in V \subset X$ . Because standard opens of $V$ form a basis of the topology on $V$ we see that there exists a $f\in \mathcal{O}_ V(V)$ such that $u \in D(f) \subset U$ . And $D(f)$ is an affine scheme by Lemma 26.6.6. This proves that every point of $U$ has an open neighbourhood which is affine. $\square$

Comments (0)

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).

Comments (0)

Post a comment