The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 54.70.2. 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 27.2.1) and $U = \coprod U \cap X_ i$ is a partition of $U$ as in Definition 54.70.1. Thus (1) holds. $\square$


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