Lemma 15.104.11. Let $A$ be a ring. Let $B \to C$ be an $A$-algebra map of weakly étale $A$-algebras. Then $B \to C$ is weakly étale.

**Proof.**
The ring map $B \to C$ is flat by Lemma 15.104.2. The ring map $C \otimes _ A C \to C \otimes _ B C$ is surjective, hence an epimorphism. Thus Lemma 15.104.2 implies, that since $C$ is flat over $C \otimes _ A C$ also $C$ is flat over $C \otimes _ B C$.
$\square$

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## Comments (2)

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