Definition 63.18.1. Let $X$ be a Noetherian scheme. A *$\mathbf{Z}_\ell $-sheaf* on $X$, or simply an *$\ell $-adic sheaf* $\mathcal{F}$ is an inverse system $\left\{ \mathcal{F}_ n\right\} _{n\geq 1}$ where

$\mathcal{F}_ n$ is a constructible $\mathbf{Z}/\ell ^ n\mathbf{Z}$-module on $X_{\acute{e}tale}$, and

the transition maps $\mathcal{F}_{n+1}\to \mathcal{F}_ n$ induce isomorphisms $\mathcal{F}_{n+1} \otimes _{\mathbf{Z}/\ell ^{n+1}\mathbf{Z}} \mathbf{Z}/\ell ^ n\mathbf{Z} \cong \mathcal{F}_ n$.

We say that $\mathcal{F}$ is *lisse* if each $\mathcal{F}_ n$ is locally constant. A *morphism* of such is merely a morphism of inverse systems.

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