The Stacks project

Lemma 27.2.7. A partition $X = \coprod _{i \in I} X_ i$ of a scheme $X$ with retrocompact parts is locally finite if and only if the parts are locally constructible.

Proof. See Topology, Definitions 5.12.1, 5.28.1, and 5.28.4 for the definitions of retrocompact, partition, and locally finite.

If the partition is locally finite and $U \subset X$ is an affine open, then we see that $U = \coprod _{i \in I} U \cap X_ i$ is a finite partition (more precisely, all but a finite number of its parts are empty). Hence $U \cap X_ i$ is quasi-compact and its complement is retrocompact in $U$ as a finite union of retrocompact parts. Thus $U \cap X_ i$ is constructible by Topology, Lemma 5.15.13. It follows that $X_ i$ is locally constructible by Lemma 27.2.1.

Assume the parts are locally constructible. Then for any affine open $U \subset X$ we obtain a covering $U = \coprod X_ i \cap U$ by constructible subsets. Since the constructible topology is quasi-compact, see Topology, Lemma 5.23.2, this covering has a finite refinement, i.e., the partition is locally finite. $\square$


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