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The Stacks project

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


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