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

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.