Lemma 15.91.9. Let $A \to B$ and $B \to C$ be ring maps.

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

If $A \to B$ and $B \to C$ are weakly étale, then $A \to C$ is weakly étale.

Lemma 15.91.9. Let $A \to B$ and $B \to C$ be ring maps.

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

If $A \to B$ and $B \to C$ are weakly étale, then $A \to C$ is weakly étale.

**Proof.**
Part (1) follows from the factorization

\[ C \otimes _ A C \longrightarrow C \otimes _ B C \longrightarrow C \]

of the multiplication map, the fact that

\[ C \otimes _ B C = (C \otimes _ A C) \otimes _{B \otimes _ A B} B, \]

the fact that a base change of a flat map is flat, and the fact that the composition of flat ring maps is flat. See Algebra, Lemmas 10.38.7 and 10.38.4. Part (2) follows from (1) and the fact (just used) that the composition of flat ring maps is flat. $\square$

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