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
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