The Stacks project

Lemma 27.11.7. With hypotheses and notation as in Lemma 27.11.1 above. Assume there exists a $g \in A_0$ such that $\psi $ induces an isomorphism $A_ g \to B$. Then $U(\psi ) = Y$, $r_\psi : Y \to X$ is an open immersion which induces an isomorphism of $Y$ with the inverse image of $D(g) \subset \mathop{\mathrm{Spec}}(A_0)$. Moreover the map $\theta $ is an isomorphism.

Proof. This is a special case of Lemma 27.11.6 above. $\square$

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