The Stacks project

Lemma 69.5.3. 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 69.4.1). 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 ├ętale 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. $\square$

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