Lemma 70.5.3. Let $S$ be a scheme. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be the limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma 70.4.1). If each $X_ i$ is quasi-compact and nonempty, then $|X|$ is nonempty.
Proof. Choose $0 \in I$. Choose an affine scheme $U_0$ and a surjective étale morphism $U_0 \to X_0$. Set $U_ i = X_ i \times _{X_0} U_0$ and $U = X \times _{X_0} U_0$. Then each $U_ i$ is a nonempty affine scheme. Hence $U = \mathop{\mathrm{lim}}\nolimits U_ i$ is nonempty (Limits, Lemma 32.4.3) and thus $X$ is nonempty. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)