The Stacks project

Lemma 15.9.3. Let $A$ be a ring, let $I \subset A$ be an ideal. Let $\mathop{\mathrm{Spec}}(A/I) = \coprod _{j \in J} \overline{U}_ j$ be a finite disjoint open covering. Then there exists an ├ętale ring map $A \to A'$ which induces an isomorphism $A/I \to A'/IA'$ and a finite disjoint open covering $\mathop{\mathrm{Spec}}(A') = \coprod _{j \in J} U'_ j$ lifting the given covering.

Proof. This follows from Lemma 15.9.2 and the fact that open and closed subsets of Spectra correspond to idempotents, see Algebra, Lemma 10.21.3. $\square$

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