The Stacks project

Lemma 27.26.11. 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 27.26.9 is an open immersion with dense image.

Proof. By Lemma 27.26.7 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 27.26.9. By Lemma 27.17.2 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 27.26.10. $\square$


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