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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