Lemma 15.9.4. Let $A \to B$ be a ring map and $J \subset B$ an ideal. If $A \to B$ is étale at every prime of $V(J)$, then there exists a $g \in B$ mapping to an invertible element of $B/J$ such that $A' = B_ g$ is étale over $A$.

**Proof.**
The set of points of $\mathop{\mathrm{Spec}}(B)$ where $A \to B$ is not étale is a closed subset of $\mathop{\mathrm{Spec}}(B)$, see Algebra, Definition 10.143.1. Write this as $V(J')$ for some ideal $J' \subset B$. Then $V(J') \cap V(J) = \emptyset $ hence $J + J' = B$ by Algebra, Lemma 10.17.2. Write $1 = f + g$ with $f \in J$ and $g \in J'$. Then $g$ works.
$\square$

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