The Stacks project

Lemma 59.64.6. Let $X$ be a scheme.

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

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

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

Proof. This holds on any site, see Modules on Sites, Lemma 18.43.5. $\square$

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