The Stacks project

Remark 66.19.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. We claim that for any pair of geometric points $\overline{x}$ and $\overline{x}'$ lying over $x$ the stalk functors are isomorphic. By definition of $|X|$ we can find a third geometric point $\overline{x}''$ so that there exists a commutative diagram

\[ \xymatrix{ \overline{x}'' \ar[r] \ar[d] \ar[rd]^{\overline{x}''} & \overline{x}' \ar[d]^{\overline{x}'} \\ \overline{x} \ar[r]^{\overline{x}} & X. } \]

Since the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is given by pullback along the morphism $\overline{x}$ (and similarly for the others) we conclude by functoriality of pullbacks.

Comments (0)

There are also:

  • 4 comment(s) on Section 66.19: Points of the small ├ętale site

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Your email address will not be published.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 04K4. Beware of the difference between the letter 'O' and the digit '0'.