The Stacks project

94.12 The stack in groupoids of finite type algebraic spaces

Let $p : \mathcal{S}\! \mathit{paces}_{ft} \to (\mathit{Sch}/S)_{fppf}$ be the stack introduced in Section 94.8 (using the abuse of notation introduced there). We can turn this into a stack in groupoids $p' : \mathcal{S}\! \mathit{paces}_{ft}' \to (\mathit{Sch}/S)_{fppf}$ by the procedure of Categories, Lemma 4.35.3, see Stacks, Lemma 8.5.3. In this particular case this simply means $\mathcal{S}\! \mathit{paces}_{ft}'$ has the same objects as $\mathcal{S}\! \mathit{paces}_{ft}$, i.e., finite type morphisms $X \to U$ where $X$ is an algebraic space over $S$ and $U$ is a scheme over $S$. But the morphisms $(f, g) : X/U \to Y/V$ are now commutative diagrams

\[ \xymatrix{ X \ar[d] \ar[r]_ f & Y \ar[d] \\ U \ar[r]^ g & V } \]

which are cartesian.


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Your email address will not be published.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 04UH. Beware of the difference between the letter 'O' and the digit '0'.