The Stacks project

Lemma 61.28.4. Let $X$ be a connected scheme. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. If $\mathcal{F}$ is a lisse constructible $\Lambda $-sheaf on $X_{pro\text{-}\acute{e}tale}$, then $\mathcal{F}$ is adic lisse.

Proof. By Lemma 61.19.9 we have $\mathcal{F}/I^ n\mathcal{F} = \epsilon ^{-1}\mathcal{G}_ n$ for some locally constant sheaf $\mathcal{G}_ n$ of $\Lambda /I^ n$-modules. By √Čtale Cohomology, Lemma 59.64.8 there exists a finite $\Lambda /I^ n$-module $M_ n$ such that $\mathcal{G}_ n$ is locally isomorphic to $\underline{M_ n}$. Choose a covering $\{ W_ t \to X\} _{t \in T}$ with each $W_ t$ affine and weakly contractible. Then $\mathcal{F}|_{W_ t}$ satisfies the assumptions of Lemma 61.28.3 and hence $\mathcal{F}|_{W_ t} \cong \underline{N_ t}^\wedge $ for some finite $\Lambda ^\wedge $-module $N_ t$. Note that $N_ t/I^ nN_ t \cong M_ n$ for all $t$ and $n$. Hence $N_ t \cong N_{t'}$ for all $t, t' \in T$, see More on Algebra, Lemma 15.100.5. This proves that $\mathcal{F}$ is adic lisse. $\square$


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