The Stacks project

Lemma 85.17.7. Let $S$ be a scheme. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$ which is stable under base change. Let $f : X \to Y$ and $g : Z \to Y$ be morphisms of locally countably indexed formal algebraic spaces over $S$. If $f$ satisfies the equivalent conditions of Lemma 85.17.3 then so does $\text{pr}_2 : X \times _ Y Z \to Z$.

Proof. Choose a covering $\{ Y_ j \to Y\} $ as in Definition 85.7.1. For each $j$ choose a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Definition 85.7.1. For each $j$ choose a covering $\{ Z_{jk} \to Y_ j \times _ Y Z\} $ as in Definition 85.7.1. Observe that $X_{ji} \times _{Y_ j} Z_{jk}$ is an affine formal algebraic space which is countably indexed, see Lemma 85.16.8. Then we see that

\[ \{ X_{ji} \times _{Y_ j} Z_{jk} \to X \times _ Y Z\} \]

is a covering as in Definition 85.7.1. Moreover, the morphisms $X_{ji} \times _{Y_ j} Z_{jk} \to Z$ factor through $Z_{jk}$. By assumption we know that $X_{ji} \to Y_ j$ corresponds to a morphism $B_ j \to A_{ji}$ of $\text{WAdm}^{count}$ having property $P$. The morphisms $Z_{jk} \to Y_ j$ correspond to morphisms $B_ j \to C_{jk}$ in $\text{WAdm}^{count}$. Since $X_{ji} \times _{Y_ j} Z_{jk} = \text{Spf}(A_{ji} \widehat{\otimes }_{B_ j} C_{jk})$ by Lemma 85.12.4 we see that it suffices to show that $C_{jk} \to A_{ji} \widehat{\otimes }_{B_ j} C_{jk}$ has property $P$ which is exactly what the condition that $P$ is stable under base change guarantees. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GBD. Beware of the difference between the letter 'O' and the digit '0'.