The Stacks project

Lemma 64.18.2. Let $\{ \mathcal{G}_ n\} _{n\geq 1}$ be an inverse system of constructible $\mathbf{Z}/\ell ^ n\mathbf{Z}$-modules. Suppose that for all $k\geq 1$, the maps

\[ \mathcal{G}_{n+1}/\ell ^ k \mathcal{G}_{n+1}\to \mathcal{G}_ n /\ell ^ k \mathcal{G}_ n \]

are isomorphisms for all $n\gg 0$ (where the bound possibly depends on $k$). In other words, assume that the system $\{ \mathcal{G}_ n/\ell ^ k\mathcal{G}_ n\} _{n\geq 1}$ is eventually constant, and call $\mathcal{F}_ k$ the corresponding sheaf. Then the system $\left\{ \mathcal{F}_ k\right\} _{k\geq 1}$ forms a $\mathbf{Z}_\ell $-sheaf on $X$.

Proof. The proof is obvious. $\square$

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