The Stacks project

Proof. There exists an étale ring map $A \to B'$ such that $C = B' \otimes _ A A_ Z^\sim $ as $A_ Z^\sim $-algebras. The image $U' \subset \mathop{\mathrm{Spec}}(A)$ of $\mathop{\mathrm{Spec}}(B') \to \mathop{\mathrm{Spec}}(A)$ is open and contains $V(I)$, hence we can find $f \in I$ such that $\mathop{\mathrm{Spec}}(A) = U' \cup D(f)$. Then $A \to B' \times A_ f$ is étale and faithfully flat. By assumption there is a retraction $B' \times A_ f \to A$. Localizing we obtain the desired retraction $C \to A_ Z^\sim $. $\square$