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 54.70.3. 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 $X$. $\square$


Comments (0)

There are also:

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

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09YR. Beware of the difference between the letter 'O' and the digit '0'.