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.