Lemma 61.8.4. Let $A$ be a ring such that every faithfully flat étale ring map $A \to B$ has a retraction. Then the same is true for every quotient ring $A/I$.

Proof. Let $A/I \to \overline{B}$ be faithfully flat étale. By Algebra, Lemma 10.143.10 we can write $\overline{B} = B/IB$ for some étale ring map $A \to B$. The image $U$ of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is open and contains $V(I)$. Hence the complement $Z = \mathop{\mathrm{Spec}}(A) \setminus U$ is quasi-compact and disjoint from $V(I)$. Hence $Z \subset D(f_1) \cup \ldots \cup D(f_ r)$ for some $r \geq 0$ and $f_ i \in I$. Then $A \to B' = B \times \prod A_{f_ i}$ is faithfully flat étale and $\overline{B} = B'/IB'$. Hence the retraction $B' \to A$ to $A \to B'$, induces a retraction to $A/I \to \overline{B}$. $\square$

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