The Stacks project

Lemma 54.70.3. Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F}$ be a sheaf of sets, abelian groups, $\Lambda $-modules (with $\Lambda $ Noetherian) on $X_{\acute{e}tale}$. If there exist constructible locally closed subschemes $T_ i \subset X$ such that (a) $X = \bigcup T_ j$ and (b) $\mathcal{F}|_{T_ j}$ is constructible, then $\mathcal{F}$ is constructible.

Proof. First, we can assume the covering is finite as $X$ is quasi-compact in the spectral topology (Topology, Lemma 5.23.2 and Properties, Lemma 27.2.4). Observe that each $T_ i$ is a quasi-compact and quasi-separated scheme in its own right (because it is constructible in $X$; details omitted). Thus we can find a finite partition $T_ i = \coprod T_{i, j}$ into locally closed constructible parts of $T_ i$ such that $\mathcal{F}|_{T_{i, j}}$ is finite locally constant (Lemma 54.70.2). By Topology, Lemma 5.15.12 we see that $T_{i, j}$ is a constructible locally closed subscheme of $X$. Then we can apply Topology, Lemma 5.28.7 to $X = \bigcup T_{i, j}$ to find the desired partition of $X$. $\square$

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