The Stacks project

Lemma 18.42.5. Let $\mathcal{C}$ be a site.

  1. The category of finite locally constant sheaves of sets is closed under finite limits and colimits inside $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

  2. The category of finite locally constant abelian sheaves is a weak Serre subcategory of $\textit{Ab}(\mathcal{C})$.

  3. Let $\Lambda $ be a Noetherian ring. The category of finite type, locally constant sheaves of $\Lambda $-modules on $\mathcal{C}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{C}, \Lambda )$.

Proof. Proof of (1). We may work locally on $\mathcal{C}$. Hence by Lemma 18.42.3 we may assume we are given a finite diagram of finite sets such that our diagram of sheaves is the associated diagram of constant sheaves. Then we just take the limit or colimit in the category of sets and take the associated constant sheaf. Some details omitted.

To prove (2) and (3) we use the criterion of Homology, Lemma 12.9.3. Existence of kernels and cokernels is argued in the same way as above. Of course, the reason for using a Noetherian ring in (3) is to assure us that the kernel of a map of finite $\Lambda $-modules is a finite $\Lambda $-module. To see that the category is closed under extensions (in the case of sheaves $\Lambda $-modules), assume given an extension of sheaves of $\Lambda $-modules

\[ 0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0 \]

on $\mathcal{C}$ with $\mathcal{F}$, $\mathcal{G}$ finite type and locally constant. Localizing on $\mathcal{C}$ we may assume $\mathcal{F}$ and $\mathcal{G}$ are constant, i.e., we get

\[ 0 \to \underline{M} \to \mathcal{E} \to \underline{N} \to 0 \]

for some $\Lambda $-modules $M, N$. Choose generators $y_1, \ldots , y_ m$ of $N$, so that we get a short exact sequence $0 \to K \to \Lambda ^{\oplus m} \to N \to 0$ of $\Lambda $-modules. Localizing further we may assume $y_ j$ lifts to a section $s_ j$ of $\mathcal{E}$. Thus we see that $\mathcal{E}$ is a pushout as in the following diagram

\[ \xymatrix{ 0 \ar[r] & \underline{K} \ar[d] \ar[r] & \underline{\Lambda ^{\oplus m}} \ar[d] \ar[r] & \underline{N} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \underline{M} \ar[r] & \mathcal{E} \ar[r] & \underline{N} \ar[r] & 0 } \]

By Lemma 18.42.3 again (and the fact that $K$ is a finite $\Lambda $-module as $\Lambda $ is Noetherian) we see that the map $\underline{K} \to \underline{M}$ is locally constant, hence we conclude. $\square$


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 093U. Beware of the difference between the letter 'O' and the digit '0'.