The Stacks project

Lemma 54.70.6. Let $X$ be a scheme.

  1. The category of constructible sheaves of sets is closed under finite limits and colimits inside $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$.

  2. The category of constructible abelian sheaves is a weak Serre subcategory of $\textit{Ab}(X_{\acute{e}tale})$.

  3. Let $\Lambda $ be a Noetherian ring. The category of constructible sheaves of $\Lambda $-modules on $X_{\acute{e}tale}$ is a weak Serre subcategory of $\textit{Mod}(X_{\acute{e}tale}, \Lambda )$.

Proof. We prove (3). We will use the criterion of Homology, Lemma 12.9.3. Suppose that $\varphi : \mathcal{F} \to \mathcal{G}$ is a map of constructible sheaves of $\Lambda $-modules. We have to show that $\mathcal{K} = \mathop{\mathrm{Ker}}(\varphi )$ and $\mathcal{Q} = \mathop{\mathrm{Coker}}(\varphi )$ are constructible. Similarly, suppose that $0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0$ is a short exact sequence of sheaves of $\Lambda $-modules with $\mathcal{F}$, $\mathcal{G}$ constructible. We have to show that $\mathcal{E}$ is constructible. In both cases we can replace $X$ with the members of an affine open covering. Hence we may assume $X$ is affine. The we may further replace $X$ by the members of a finite partition of $X$ by constructible locally closed subschemes on which $\mathcal{F}$ and $\mathcal{G}$ are of finite type and locally constant. Thus we may apply Lemma 54.63.6 to conclude.

The proofs of (1) and (2) are very similar and are omitted. $\square$


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Your email address will not be published.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 03RZ. Beware of the difference between the letter 'O' and the digit '0'.