The Stacks project

Definition 61.28.1. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. Let $X$ be a scheme. Let $\mathcal{F}$ be a sheaf of $\Lambda $-modules on $X_{pro\text{-}\acute{e}tale}$.

1. We say $\mathcal{F}$ is a constructible $\Lambda $-sheaf if $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}$ and each $\mathcal{F}/I^ n\mathcal{F}$ is a constructible sheaf of $\Lambda /I^ n$-modules.

2. If $\mathcal{F}$ is a constructible $\Lambda $-sheaf, then we say $\mathcal{F}$ is lisse if each $\mathcal{F}/I^ n\mathcal{F}$ is locally constant.

3. We say $\mathcal{F}$ is adic lisse¹ if there exists a $I$-adically complete $\Lambda $-module $M$ with $M/IM$ finite such that $\mathcal{F}$ is locally isomorphic to

   \[ \underline{M}^\wedge = \mathop{\mathrm{lim}}\nolimits \underline{M/I^ nM}. \]

4. We say $\mathcal{F}$ is adic constructible² if for every affine open $U \subset X$ there exists a decomposition $U = \coprod U_ i$ into constructible locally closed subschemes such that $\mathcal{F}|_{U_ i}$ is adic lisse.