Lemma 70.12.4. With hypotheses and notation as in Lemma 70.12.1 above. Assume $\mathcal{A}_ d \to \mathcal{B}_ d$ is an isomorphism for all $d \gg 0$. Then

1. $U(\psi ) = Q$,

2. $r_\psi : Q \to P$ is an isomorphism, 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 27.18.4) by étale localization. $\square$

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