The Stacks project

Remark 59.29.8. Let $S$ be a scheme and let $\overline{s} : \mathop{\mathrm{Spec}}(k) \to S$ and $\overline{s}' : \mathop{\mathrm{Spec}}(k') \to S$ be two geometric points of $S$. A morphism $a : \overline{s} \to \overline{s}'$ of geometric points is simply a morphism $a : \mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k')$ such that $\overline{s}' \circ a = \overline{s}$. Given such a morphism we obtain a functor from the category of étale neighbourhoods of $\overline{s}'$ to the category of étale neighbourhoods of $\overline{s}$ by the rule $(U, \overline{u}') \mapsto (U, \overline{u}' \circ a)$. Hence we obtain a canonical map

\[ \mathcal{F}_{\overline{s}'} = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u}')} \mathcal{F}(U) \longrightarrow \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} \mathcal{F}(U) = \mathcal{F}_{\overline{s}} \]

from Categories, Lemma 4.14.8. Using the description of elements of stalks as triples this maps the element of $\mathcal{F}_{\overline{s}'}$ represented by the triple $(U, \overline{u}', \sigma )$ to the element of $\mathcal{F}_{\overline{s}}$ represented by the triple $(U, \overline{u}' \circ a, \sigma )$. Since the functor above is clearly an equivalence we conclude that this canonical map is an isomorphism of stalk functors.

Let us make sure we have the map of stalks corresponding to $a$ pointing in the correct direction. Note that the above means, according to Sites, Definition 7.37.2, that $a$ defines a morphism $a : p \to p'$ between the points $p, p'$ of the site $S_{\acute{e}tale}$ associated to $\overline{s}, \overline{s}'$ by Lemma 59.29.7. There are more general morphisms of points (corresponding to specializations of points of $S$) which we will describe later, and which will not be isomorphisms, see Section 59.75.

Comments (4)

Comment #77 by Keenan Kidwell on

There are two places where should be .

Comment #6340 by Will Chen on

I think should be .

There are also:

  • 3 comment(s) on Section 59.29: Neighborhoods, stalks and points

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Your email address will not be published.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