Lemma 15.104.9. Let $A \to B$ and $B \to C$ be ring maps.

1. 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.

2. 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 10.39.7 and 10.39.4. Part (2) follows from (1) and the fact (just used) that the composition of flat ring maps is flat. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).