Lemma 61.8.1. Given a ring A there exists a faithfully flat ind-étale A-algebra C such that every faithfully flat étale ring map C \to B has a retraction.
Proof. Set T^1(A) = T(A) and T^{n + 1}(A) = T(T^ n(A)). Let
This algebra is faithfully flat over each T^ n(A) and in particular over A, see Algebra, Lemma 10.39.20. Moreover, C is ind-étale over A by Lemma 61.7.4. If C \to B is étale, then there exists an n and an étale ring map T^ n(A) \to B' such that B = C \otimes _{T^ n(A)} B', see Algebra, Lemma 10.143.3. If C \to B is faithfully flat, then \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(C) \to \mathop{\mathrm{Spec}}(T^ n(A)) is surjective, hence \mathop{\mathrm{Spec}}(B') \to \mathop{\mathrm{Spec}}(T^ n(A)) is surjective. In other words, T^ n(A) \to B' is faithfully flat. By our construction, there is a T^ n(A)-algebra map B' \to T^{n + 1}(A). This induces a C-algebra map B \to C which finishes the proof. \square
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