Processing math: 100%

The Stacks project

Lemma 70.5.2. With same notation and assumptions as in Lemma 70.5.1 we have |X| = \mathop{\mathrm{lim}}\nolimits _ i |X_ i| as topological spaces.

Proof. We will use the criterion of Topology, Lemma 5.14.3. We have seen that |X| = \mathop{\mathrm{lim}}\nolimits _ i |X_ i| as sets in Lemma 70.5.1. The maps f_ i : X \to X_ i are morphisms of algebraic spaces hence determine continuous maps |X| \to |X_ i|. Thus f_ i^{-1}(U_ i) is open for each open U_ i \subset |X_ i|. Finally, let x \in |X| and let x \in V \subset |X| be an open neighbourhood. We have to find an i and an open neighbourhood W_ i \subset |X_ i| of the image x with f_ i^{-1}(W_ i) \subset V. Choose 0 \in I. Choose a scheme U_0 and a surjective étale morphism U_0 \to X_0. Set U = X \times _{X_0} U_0 and U_ i = X_ i \times _{X_0} U_0 for i \geq 0. Then U = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} U_ i in the category of schemes by Lemma 70.4.1. Choose u \in U mapping to x. By the result for schemes (Limits, Lemma 32.4.2) we can find an i \geq 0 and an open neighbourhood E_ i \subset U_ i of the image of u whose inverse image in U is contained in the inverse image of V in U. Then we can set W_ i \subset |X_ i| equal to the image of E_ i. This works because |U_ i| \to |X_ i| is open. \square


Comments (0)


Your email address will not be published.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.