The Stacks project

Lemma 57.73.5. Let $X$ be a Noetherian scheme. Let $E \subset X$ be a subset closed under specialization.

  1. Let $\mathcal{F}$ be a torsion abelian sheaf on $X_{\acute{e}tale}$ whose support is contained in $E$. Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ is a filtered colimit of constructible abelian sheaves $\mathcal{F}_ i$ such that for each $i$ the support of $\mathcal{F}_ i$ is contained in a closed subset contained in $E$.

  2. Let $\Lambda $ be a Noetherian ring and $\mathcal{F}$ a sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$ whose support is contained in $E$. Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ is a filtered colimit of constructible sheaves of $\Lambda $-modules $\mathcal{F}_ i$ such that for each $i$ the support of $\mathcal{F}_ i$ is contained in a closed subset contained in $E$.

Proof. Proof of (1). We can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{F}_ i$ with $\mathcal{F}_ i$ constructible abelian by Lemma 57.72.2. By Proposition 57.73.1 the image $\mathcal{F}'_ i \subset \mathcal{F}$ of the map $\mathcal{F}_ i \to \mathcal{F}$ is constructible. Thus $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}'_ i$ and the support of $\mathcal{F}'_ i$ is contained in $E$. Since the support of $\mathcal{F}'_ i$ is constructible (by our definition of constructible sheaves), we see that its closure is also contained in $E$, see for example Topology, Lemma 5.23.5.

The proof in case (2) is exactly the same and we omit it. $\square$


Comments (0)


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Your email address will not be published.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 0F0N. Beware of the difference between the letter 'O' and the digit '0'.