The Stacks project

Lemma 37.62.5. Let $X \to Y \to Z$ be morphisms of schemes.

  1. If $X \to X \times _ Y X$ and $Y \to Y \times _ Z Y$ are flat, then $X \to X \times _ Z X$ is flat.

  2. If $X \to Y$ and $Y \to Z$ are weakly étale, then $X \to Z$ is weakly étale.

Proof. Part (1) follows from the factorization

\[ X \to X \times _ Y X \to X \times _ Z X \]

of the diagonal of $X$ over $Z$, the fact that

\[ X \times _ Y X = (X \times _ Z X) \times _{(Y \times _ Z Y)} Y, \]

the fact that a base change of a flat morphism is flat, and the fact that the composition of flat morphisms is flat (Morphisms, Lemmas 29.25.8 and 29.25.6). Part (2) follows from part (1) and the fact (just used) that the composition of flat morphisms is flat. $\square$

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