Lisse ell-adic sheaf

Definition

Let $X$ be a connected scheme on which the prime $\ell$ is invertible. A lisse $\mathbb Z_\ell$ -sheaf is an inverse system

\mathcal F=(\mathcal F_n)_{n\ge1}

in which each $\mathcal F_n$ is a locally constant constructible sheaf of finite free $\mathbb Z/\ell^n\mathbb Z$ -modules on the étale site and $\mathcal F_{n+1}/\ell^n\simeq\mathcal F_n$ . A lisse $\mathbb Q_\ell$ -sheaf is obtained by inverting $\ell$ , or equivalently by using a finite-dimensional $\mathbb Q_\ell$ -local system in the pro-étale formalism.

Monodromy representation

After choosing a geometric point $\bar x$ , taking the fiber gives an equivalence between lisse $\mathbb Q_\ell$ -sheaves of rank $n$ and continuous representations

\pi_1^{\mathrm{ét}}(X,\bar x) \longrightarrow\operatorname{GL}_n(\mathbb Q_\ell).

Changing $\bar x$ changes the representation by the usual inner identification. On a variety over a finite field, arithmetic or geometric Frobenius acts on each closed-point fiber.

Lisse versus constructible

A constructible sheaf need only be lisse on each piece of a finite stratification. Thus “lisse” is the algebro-geometric analogue of a local system, while constructible sheaves allow singularities and extension across strata.

References

Alexander Grothendieck, SGA 4½: Cohomologie étale, Lecture Notes in Mathematics 569, Springer, 1977.
The Stacks Project Authors, “Pro-étale Cohomology,” §61.28, “Constructible adic sheaves.” Stacks Project.