The Stacks project

Lemma 59.71.4. Let $X$ be a scheme. Checking constructibility of a sheaf of sets, abelian groups, $\Lambda $-modules (with $\Lambda $ Noetherian) can be done Zariski locally on $X$.

Proof. The statement means if $X = \bigcup U_ i$ is an open covering such that $\mathcal{F}|_{U_ i}$ is constructible, then $\mathcal{F}$ is constructible. If $U \subset X$ is affine open, then $U = \bigcup U \cap U_ i$ and $\mathcal{F}|_{U \cap U_ i}$ is constructible (it is trivial that the restriction of a constructible sheaf to an open is constructible). It follows from Lemma 59.71.2 that $\mathcal{F}|_ U$ is constructible, i.e., a suitable partition of $U$ exists. $\square$

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