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. $\square$

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