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The Stacks project

Lemma 5.24.3. 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 i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}) and let E, F \subset X_ i be subsets with E closed in the constructible topology and F open in the constructible topology. Then p_ i^{-1}(E) \subset p_ i^{-1}(F) if and only if there is a morphism a : j \to i in \mathcal{I} such that f_ a^{-1}(E) \subset f_ a^{-1}(F).

Proof. Observe that

p_ i^{-1}(E) \setminus p_ i^{-1}(F) = \mathop{\mathrm{lim}}\nolimits _{a : j \to i} f_ a^{-1}(E) \setminus f_ a^{-1}(F)

Since f_ a is a spectral map, it is continuous in the constructible topology hence the set f_ a^{-1}(E) \setminus f_ a^{-1}(F) is closed in the constructible topology. Hence Lemma 5.24.2 applies to show that the LHS is nonempty if and only if each of the spaces of the RHS is nonempty. \square


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