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The Stacks project

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

  1. U(\psi ) = Y,

  2. r_\psi : Y \to X is a closed immersion, and

  3. 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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