The Stacks project

Lemma 28.26.11. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Set $S = \Gamma _*(X, \mathcal{L})$. Assume $\mathcal{L}$ is ample. Then the canonical morphism of schemes $f : X \longrightarrow \text{Proj}(S)$ of Lemma 28.26.9 is an open immersion with dense image.

Proof. By Lemma 28.26.7 we see that $X$ is quasi-separated. Choose finitely many $s_1, \ldots , s_ n \in S_{+}$ homogeneous such that $X_{s_ i}$ are affine, and $X = \bigcup X_{s_ i}$. Say $s_ i$ has degree $d_ i$. The inverse image of $D_{+}(s_ i)$ under $f$ is $X_{s_ i}$, see Lemma 28.26.9. By Lemma 28.17.2 the ring map

\[ (S^{(d_ i)})_{(s_ i)} = \Gamma (D_{+}(s_ i), \mathcal{O}_{\text{Proj}(S)}) \longrightarrow \Gamma (X_{s_ i}, \mathcal{O}_ X) \]

is an isomorphism. Hence $f$ induces an isomorphism $X_{s_ i} \to D_{+}(s_ i)$. Thus $f$ is an isomorphism of $X$ onto the open subscheme $\bigcup _{i = 1, \ldots , n} D_{+}(s_ i)$ of $\text{Proj}(S)$. The image is dense by Lemma 28.26.10. $\square$

Comments (0)

There are also:

  • 5 comment(s) on Section 28.26: Ample invertible sheaves

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Your email address will not be published.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 01Q1. Beware of the difference between the letter 'O' and the digit '0'.