The Stacks project

Lemma 61.28.5. Let $X$ be a Noetherian scheme. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. Let $\mathcal{F}$ be a constructible $\Lambda $-sheaf on $X_{pro\text{-}\acute{e}tale}$. Then $\mathcal{F}$ is adic constructible.

Proof. This is a consequence of Lemmas 61.28.2 and 61.28.4, the fact that a Noetherian scheme is locally connected (Topology, Lemma 5.9.6), and the definitions. $\square$

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