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The Stacks project

Lemma 70.5.9. Notation and assumptions as in Situation 70.5.5. If X is separated, then X_ i is separated for some i \in I.

Proof. Choose an affine scheme U_0 and a surjective étale morphism U_0 \to X_0. For i \geq 0 set U_ i = U_0 \times _{X_0} X_ i and set U = U_0 \times _{X_0} X. Note that U_ i and U are affine schemes which come equipped with surjective étale morphisms U_ i \to X_ i and U \to X. Set R_ i = U_ i \times _{X_ i} U_ i and R = U \times _ X U with projections s_ i, t_ i : R_ i \to U_ i and s, t : R \to U. Note that R_ i and R are quasi-compact separated schemes (as the algebraic spaces X_ i and X are quasi-separated). The maps s_ i : R_ i \to U_ i and s : R \to U are of finite type. By definition X_ i is separated if and only if (t_ i, s_ i) : R_ i \to U_ i \times U_ i is a closed immersion, and since X is separated by assumption, the morphism (t, s) : R \to U \times U is a closed immersion. Since R \to U is of finite type, there exists an i such that the morphism R \to U_ i \times U is a closed immersion (Limits, Lemma 32.4.16). Fix such an i \in I. Apply Limits, Lemma 32.8.5 to the system of morphisms R_{i'} \to U_ i \times U_{i'} for i' \geq i (this is permissible as indeed R_{i'} = R_ i \times _{U_ i \times U_ i} U_ i \times U_{i'}) to see that R_{i'} \to U_ i \times U_{i'} is a closed immersion for i' sufficiently large. This implies immediately that R_{i'} \to U_{i'} \times U_{i'} is a closed immersion finishing the proof of the lemma. \square


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