\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*}

The Stacks project

Lemma 54.70.8. Let $X$ be a quasi-compact and quasi-separated scheme.

  1. Let $\mathcal{F} \to \mathcal{G}$ be a map of constructible sheaves of sets on $X_{\acute{e}tale}$. Then the set of points $x \in X$ where $\mathcal{F}_{\overline{x}} \to \mathcal{F}_{\overline{x}}$ is surjective, resp. injective, resp. is isomorphic to a given map of sets, is constructible in $X$.

  2. Let $\mathcal{F}$ be a constructible abelian sheaf on $X_{\acute{e}tale}$. The support of $\mathcal{F}$ is constructible.

  3. Let $\Lambda $ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$. The support of $\mathcal{F}$ is constructible.

Proof. Proof of (1). Let $X = \coprod X_ i$ be a partition of $X$ by locally closed constructible subschemes such that both $\mathcal{F}$ and $\mathcal{G}$ are finite locally constant over the parts (use Lemma 54.70.2 for both $\mathcal{F}$ and $\mathcal{G}$ and choose a common refinement). Then apply Lemma 54.63.5 to the restriction of the map to each part.

The proof of (2) and (3) is omitted. $\square$


Comments (2)

Comment #2357 by Simon Pepin Lehalleur on

Suggested slogan: Notions of constructibility for étale sheaves and for subsets of schemes are compatible

Comment #3387 by Dario on

Typo in (1): the map on stalks should also go from F to G

There are also:

  • 2 comment(s) on Section 54.70: Constructible sheaves

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