The Stacks project

Lemma 15.104.10. Let $A \to B \to C$ be ring maps.

  1. If $B \to C$ is faithfully flat and $C \otimes _ A C \to C$ is flat, then $B \otimes _ A B \to B$ is flat.

  2. If $B \to C$ is faithfully flat and $A \to C$ is weakly étale, then $A \to B$ is weakly étale.

Proof. Assume $B \to C$ is faithfully flat and $C \otimes _ A C \to C$ is flat. Consider the commutative diagram

\[ \xymatrix{ C \otimes _ A C \ar[r] & C \\ B \otimes _ A B \ar[r] \ar[u] & B \ar[u] } \]

The vertical arrows are flat, the top horizontal arrow is flat. Hence $C$ is flat as a $B \otimes _ A B$-module. The map $B \to C$ is faithfully flat and $C = B \otimes _ B C$. Hence $B$ is flat as a $B \otimes _ A B$-module by Algebra, Lemma 10.39.9. This proves (1). Part (2) follows from (1) and the fact that $A \to B$ is flat if $A \to C$ is flat and $B \to C$ is faithfully flat (Algebra, Lemma 10.39.9). $\square$

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