Lemma 37.62.6. Let $X \to Y$ and $Y' \to Y$ be morphisms of schemes and let $X' = Y' \times _ Y X$ be the base change of $X$.

1. If $X \to X \times _ Y X$ is flat, then $X' \to X' \times _{Y'} X'$ is flat.

2. If $X \to Y$ is weakly étale, then $X' \to Y'$ is weakly étale.

Proof. Assume $X \to X \times _ Y X$ is flat. The morphism $X' \to X' \times _{Y'} X'$ is the base change of $X \to X \times _ Y X$ by $Y' \to Y$. Hence it is flat by Morphisms, Lemmas 29.25.8. This proves (1). Part (2) follows from (1) and the fact (just used) that the base change of a flat morphism is flat. $\square$

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