The Stacks project

Lemma 61.19.10. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring. Let $D_{flc}(X_{\acute{e}tale}, \Lambda )$, resp. $D_{flc}(X_{pro\text{-}\acute{e}tale}, \Lambda )$ be the full subcategory of $D(X_{\acute{e}tale}, \Lambda )$, resp. $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$ consisting of those complexes whose cohomology sheaves are locally constant sheaves of $\Lambda $-modules of finite type. Then

\[ \epsilon ^{-1} : D_{flc}^+(X_{\acute{e}tale}, \Lambda ) \longrightarrow D_{flc}^+(X_{pro\text{-}\acute{e}tale}, \Lambda ) \]

is an equivalence of categories.

Proof. The categories $D_{flc}(X_{\acute{e}tale}, \Lambda )$ and $D_{flc}(X_{pro\text{-}\acute{e}tale}, \Lambda )$ are strictly full, saturated, triangulated subcategories of $D(X_{\acute{e}tale}, \Lambda )$ and $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$ by Modules on Sites, Lemma 18.43.5 and Derived Categories, Section 13.17. The statement of the lemma follows by combining Lemmas 61.19.8 and 61.19.9. $\square$

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