Lemma 37.61.5. Let $X \to Y \to Z$ be morphisms of schemes.

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.

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

Lemma 37.61.5. Let $X \to Y \to Z$ be morphisms of schemes.

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.

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. $\square$

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)