Lemma 27.11.5 (01N1)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 27: Constructions of Schemes
Section 27.11: Functoriality of Proj
Lemma 27.11.5 (cite)

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Lemma 27.11.5. With hypotheses and notation as in Lemma 27.11.1 above. Assume $A_ d \to B_ d$ is surjective for $d \gg 0$ and that $A$ is generated by $A_1$ over $A_0$ . 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 isomorphisms.

Proof. By Lemmas 27.11.4 and 27.11.2 we may replace $B$ by the image of $A \to B$ without changing $X$ or the sheaves $\mathcal{O}_ X(n)$ . Thus we may assume that $A \to B$ is surjective. By Lemma 27.11.3 we get (1) and (2) and surjectivity in (3). By Lemma 27.10.3 we see that both $\mathcal{O}_ X(n)$ and $\mathcal{O}_ Y(n)$ are invertible. Hence $\theta$ is an isomorphism. $\square$

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