Loading [MathJax]/extensions/tex2jax.js

The Stacks project

Lemma 70.23.2. In Situation 70.23.1. Let $X \to B$ be a quasi-separated and finite type morphism of algebraic spaces. Then there exists an $i \in I$ and a diagram

70.23.2.1
\begin{equation} \label{spaces-limits-equation-good-diagram} \vcenter { \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ B \ar[r] & B_ i } } \end{equation}

such that $W \to B_ i$ is of finite type and such that the induced morphism $X \to B \times _{B_ i} W$ is a closed immersion.

Proof. By Lemma 70.11.6 we can find a closed immersion $X \to X'$ over $B$ where $X'$ is an algebraic space of finite presentation over $B$. By Lemma 70.7.1 we can find an $i$ and a morphism of finite presentation $X'_ i \to B_ i$ whose pull back is $X'$. Set $W = X'_ i$. $\square$

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Your email address will not be published.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 0CP7. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 70.23.2 as pdf