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The Stacks project

Lemma 71.12.5. With hypotheses and notation as in Lemma 71.12.1 above. Assume \mathcal{A}_ d \to \mathcal{B}_ d is surjective for d \gg 0 and that \mathcal{A} is generated by \mathcal{A}_1 over \mathcal{A}_0. Then

  1. U(\psi ) = Q,

  2. r_\psi : Q \to P is a closed immersion, and

  3. the maps \theta : r_\psi ^*\mathcal{O}_ P(n) \to \mathcal{O}_ Q(n) are isomorphisms.

Proof. Follows from the case of schemes (Constructions, Lemma 27.18.5) by étale localization. \square


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