Lemma 27.11.4. With hypotheses and notation as in Lemma 27.11.1 above. Assume $A_ d \to B_ d$ is an isomorphism for all $d \gg 0$. Then

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

2. $r_\psi : Y \to X$ is an isomorphism, and

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

Proof. We have (1) by Lemma 27.11.3. Let $f \in A_{+}$ be homogeneous. The assumption on $\psi$ implies that $A_ f \to B_ f$ is an isomorphism (details omitted). Thus it is clear that $r_\psi$ and $\theta$ restrict to isomorphisms over $D_{+}(f)$. The lemma follows. $\square$

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