The Stacks project

Lemma 15.93.11. 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. Write $B \to C$ as the composition $B \to B \otimes _ A C \to C$. The first map is flat as the base change of the flat ring map $A \to C$. The second is the base change of the flat ring map $B \otimes _ A B \to B$ by the ring map $B \otimes _ A B \to B \otimes _ A C$, hence flat. Thus $B \to C$ is flat. The ring map $C \otimes _ A C \to C \otimes _ B C$ is surjective, hence an epimorphism. Thus Lemma 15.93.2 implies, that since $C$ is flat over $C \otimes _ A C$ it follows that $C$ is flat over $C \otimes _ B C$. $\square$

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