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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