Lemma 76.15.2. Let $S$ be a scheme. Let $Z \to Y \to X$ be morphisms of algebraic spaces over $S$. If $Z \subset Z'$ is a universal first order thickening of $Z$ over $Y$ and $Y \to X$ is formally étale, then $Z \subset Z'$ is a universal first order thickening of $Z$ over $X$.

**Proof.**
This is formal. Namely, by Lemma 76.15.1 it suffices to consider solid commutative diagrams (76.15.0.1) with $T'$ an affine scheme. The composition $T \to Z \to Y$ lifts uniquely to $T' \to Y$ as $Y \to X$ is assumed formally étale. Hence the fact that $Z \subset Z'$ is a universal first order thickening over $Y$ produces the desired morphism $a' : T' \to Z'$.
$\square$

## Post a comment

Your email address will not be published.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)

There are also: