Lemma 61.8.6. Let $A$ be a ring such that every faithfully flat étale ring map $A \to B$ has a retraction. Let $Z \subset \mathop{\mathrm{Spec}}(A)$ be a closed subscheme. Let $A \to A_ Z^\sim$ be as constructed in Lemma 61.5.1. Then every faithfully flat étale ring map $A_ Z^\sim \to C$ has a retraction.

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 $C \to A_ Z^\sim$. $\square$

Comment #5932 by Mingchen on

According to the proof, in the statement of the lemma $D(f)\cap V(I)$ should be replaced by $V(I)$.

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