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The Stacks project

Lemma 5.24.1. Let \mathcal{I} be a 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.

  1. Given subsets Z_ i \subset X_ i closed in the constructible topology with f_ a(Z_ j) \subset Z_ i for all a : j \to i in \mathcal{I}, then \mathop{\mathrm{lim}}\nolimits Z_ i is quasi-compact.

  2. The space X = \mathop{\mathrm{lim}}\nolimits X_ i is quasi-compact.

Proof. The limit Z = \mathop{\mathrm{lim}}\nolimits Z_ i exists by Lemma 5.14.1. Denote X'_ i the space X_ i endowed with the constructible topology and Z'_ i the corresponding subspace of X'_ i. Let a : j \to i in \mathcal{I} be a morphism. As f_ a is spectral it defines a continuous map f_ a : X'_ j \to X'_ i. Thus f_ a|_{Z_ j} : Z'_ j \to Z'_ i is a continuous map of quasi-compact Hausdorff spaces (by Lemmas 5.23.2 and 5.12.3). Thus Z' = \mathop{\mathrm{lim}}\nolimits Z_ i is quasi-compact by Lemma 5.14.5. The maps Z'_ i \to Z_ i are continuous, hence Z' \to Z is continuous and a bijection on underlying sets. Hence Z is quasi-compact as the image of the surjective continuous map Z' \to Z (Lemma 5.12.7). \square


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