The Stacks project

Lemma 59.71.2. Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F}$ be a sheaf of sets on $X_{\acute{e}tale}$. The following are equivalent

  1. $\mathcal{F}$ is constructible,

  2. there exists an open covering $X = \bigcup U_ i$ such that $\mathcal{F}|_{U_ i}$ is constructible, and

  3. there exists a partition $X = \bigcup X_ i$ by constructible locally closed subschemes such that $\mathcal{F}|_{X_ i}$ is finite locally constant.

A similar statement holds for abelian sheaves and sheaves of $\Lambda $-modules if $\Lambda $ is Noetherian.

Proof. It is clear that (1) implies (2).

Assume (2). For every $x \in X$ we can find an $i$ and an affine open neighbourhood $V_ x \subset U_ i$ of $x$. Hence we can find a finite affine open covering $X = \bigcup V_ j$ such that for each $j$ there exists a finite decomposition $V_ j = \coprod V_{j, k}$ by locally closed constructible subsets such that $\mathcal{F}|_{V_{j, k}}$ is finite locally constant. By Topology, Lemma 5.15.5 each $V_{j, k}$ is constructible as a subset of $X$. By Topology, Lemma 5.28.7 we can find a finite stratification $X = \coprod X_ l$ with constructible locally closed strata such that each $V_{j, k}$ is a union of $X_ l$. Thus (3) holds.

Assume (3) holds. Let $U \subset X$ be an affine open. Then $U \cap X_ i$ is a constructible locally closed subset of $U$ (for example by Properties, Lemma 28.2.1) and $U = \coprod U \cap X_ i$ is a partition of $U$ as in Definition 59.71.1. Thus (1) holds. $\square$

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