Lemma 15.9.1. Let $A$ be a ring, let $I \subset A$ be an ideal, let $\overline{u} \in A/I$ be an invertible element. There exists an étale ring map $A \to A'$ which induces an isomorphism $A/I \to A'/IA'$ and an invertible element $u' \in A'$ lifting $\overline{u}$.

**Proof.**
Choose any lift $f \in A$ of $\overline{u}$ and set $A' = A_ f$ and $u$ the image of $f$ in $A'$.
$\square$

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