The Stacks project

Lemma 32.4.6. In Situation 32.4.5.

  1. We have $S_{set} = \mathop{\mathrm{lim}}\nolimits _ i S_{i, set}$ where $S_{set}$ indicates the underlying set of the scheme $S$.

  2. We have $S_{top} = \mathop{\mathrm{lim}}\nolimits _ i S_{i, top}$ where $S_{top}$ indicates the underlying topological space of the scheme $S$.

  3. If $s, s' \in S$ and $s'$ is not a specialization of $s$ then for some $i \in I$ the image $s'_ i \in S_ i$ of $s'$ is not a specialization of the image $s_ i \in S_ i$ of $s$.

  4. Add more easy facts on topology of $S$ here. (Requirement: whatever is added should be easy in the affine case.)

Proof. Part (1) is a special case of Lemma 32.4.1.

Part (2) is a special case of Lemma 32.4.2.

Part (3) is a special case of Lemma 32.4.4. $\square$


Comments (2)

Comment #1254 by Michael on

there is a typo in the proof of (2): deisred should be desired


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Your email address will not be published.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 01YY. Beware of the difference between the letter 'O' and the digit '0'.