Lemma 59.73.11. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be a limit of a directed system of schemes with affine transition morphisms. We assume that $X_ i$ is quasi-compact and quasi-separated for all $i \in I$.

The category of finite locally constant sheaves on $X_{\acute{e}tale}$ is the colimit of the categories of finite locally constant sheaves on $(X_ i)_{\acute{e}tale}$.

The category of finite locally constant abelian sheaves on $X_{\acute{e}tale}$ is the colimit of the categories of finite locally constant abelian sheaves on $(X_ i)_{\acute{e}tale}$.

Let $\Lambda $ be a Noetherian ring. The category of finite type, locally constant sheaves of $\Lambda $-modules on $X_{\acute{e}tale}$ is the colimit of the categories of finite type, locally constant sheaves of $\Lambda $-modules on $(X_ i)_{\acute{e}tale}$.

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