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The Stacks project

Lemma 5.24.6. Let \mathcal{I} be a cofiltered index 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. Set X = \mathop{\mathrm{lim}}\nolimits X_ i and denote p_ i : X \to X_ i the projection.

  1. Given any quasi-compact open U \subset X there exists an i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}) and a quasi-compact open U_ i \subset X_ i such that p_ i^{-1}(U_ i) = U.

  2. Given U_ i \subset X_ i and U_ j \subset X_ j quasi-compact opens such that p_ i^{-1}(U_ i) \subset p_ j^{-1}(U_ j) there exist k \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}) and morphisms a : k \to i and b : k \to j such that f_ a^{-1}(U_ i) \subset f_ b^{-1}(U_ j).

  3. If U_ i, U_{1, i}, \ldots , U_{n, i} \subset X_ i are quasi-compact opens and p_ i^{-1}(U_ i) = p_ i^{-1}(U_{1, i}) \cup \ldots \cup p_ i^{-1}(U_{n, i}) then f_ a^{-1}(U_ i) = f_ a^{-1}(U_{1, i}) \cup \ldots \cup f_ a^{-1}(U_{n, i}) for some morphism a : j \to i in \mathcal{I}.

  4. Same statement as in (3) but for intersections.

Proof. Part (1) is a special case of Lemma 5.24.4. Part (2) is a special case of Lemma 5.24.3 as quasi-compact opens are both open and closed in the constructible topology. Parts (3) and (4) follow formally from (1) and (2) and the fact that taking inverse images of subsets commutes with taking unions and intersections. \square


Comments (2)

Comment #7564 by nkym on

I was wondering how (1) was used in the proof of (3) and (4).

Comment #7680 by on

Well, it never hurts to use more than is needed in mathematics (it doesn't cost money!). But sure we don't use (1). Going to leave as is, but others should feel free to remove it.


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