Lemma 27.18.3 (07ZI)—The Stacks project

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Part 2: Schemes
Chapter 27: Constructions of Schemes
Section 27.18: Functoriality of relative Proj
Lemma 27.18.3 (cite)

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Lemma 27.18.3. With hypotheses and notation as in Lemma 27.18.1 above. Assume $\mathcal{A}_ d \to \mathcal{B}_ d$ is surjective for $d \gg 0$ . Then

$U(\psi ) = Y$ ,
$r_\psi : Y \to X$ is a closed immersion, and
the maps $\theta : r_\psi ^*\mathcal{O}_ X(n) \to \mathcal{O}_ Y(n)$ are surjective but not isomorphisms in general (even if $\mathcal{A} \to \mathcal{B}$ is surjective).

Proof. Follows on combining Lemma 27.18.1 with Lemma 27.11.3. $\square$

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