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The Stacks project

Lemma 5.24.4. Let \mathcal{I} be a cofiltered category. Let i \mapsto X_ i be a diagram of spectral spaces such that for a : j \to i in \mathcal{I} the corresponding map f_ a : X_ j \to X_ i is spectral. Let X = \mathop{\mathrm{lim}}\nolimits X_ i with projections p_ i : X \to X_ i. Let E \subset X be a constructible subset. Then there exists an i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}) and a constructible subset E_ i \subset X_ i such that p_ i^{-1}(E_ i) = E. If E is open, resp. closed, we may choose E_ i open, resp. closed.

Proof. Assume E is a quasi-compact open of X. By Lemma 5.14.2 we can write E = p_ i^{-1}(U_ i) for some i and some open U_ i \subset X_ i. Write U_ i = \bigcup U_{i, \alpha } as a union of quasi-compact opens. As E is quasi-compact we can find \alpha _1, \ldots , \alpha _ n such that E = p_ i^{-1}(U_{i, \alpha _1} \cup \ldots \cup U_{i, \alpha _ n}). Hence E_ i = U_{i, \alpha _1} \cup \ldots \cup U_{i, \alpha _ n} works.

Assume E is a constructible closed subset. Then E^ c is quasi-compact open. So E^ c = p_ i^{-1}(F_ i) for some i and quasi-compact open F_ i \subset X_ i by the result of the previous paragraph. Then E = p_ i^{-1}(F_ i^ c) as desired.

If E is general we can write E = \bigcup _{l = 1, \ldots , n} U_ l \cap Z_ l with U_ l constructible open and Z_ l constructible closed. By the result of the previous paragraphs we may write U_ l = p_{i_ l}^{-1}(U_{l, i_ l}) and Z_ l = p_{j_ l}^{-1}(Z_{l, j_ l}) with U_{l, i_ l} \subset X_{i_ l} constructible open and Z_{l, j_ l} \subset X_{j_ l} constructible closed. As \mathcal{I} is cofiltered we may choose an object k of \mathcal{I} and morphism a_ l : k \to i_ l and b_ l : k \to j_ l. Then taking E_ k = \bigcup _{l = 1, \ldots , n} f_{a_ l}^{-1}(U_{l, i_ l}) \cap f_{b_ l}^{-1}(Z_{l, j_ l}) we obtain a constructible subset of X_ k whose inverse image in X is E. \square


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