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.
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.
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).
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}.
Same statement as in (3) but for intersections.
Comments (2)
Comment #7564 by nkym on
Comment #7680 by Stacks Project on