ℓ-adic sheaf


In algebraic geometry, an ℓ-adic sheaf on a Noetherian scheme X is an inverse system consisting of -modules in the étale topology and inducing.
Bhatt–Scholze's pro-étale topology gives an alternative approach.

Constructible and lisse ℓ-adic sheaves

An ℓ-adic sheaf is said to be
Some authors assume an ℓ-adic sheaf to be constructible.
Given a connected scheme X with a geometric point x, SGA 1 defines the étale fundamental group of X at x to be the group classifying Galois coverings of X. Then the category of lisse ℓ-adic sheaves on X is equivalent to the category of continuous representations of on finite free -modules. This is an analog of the correspondence between local systems and continuous representations of the fundament group in algebraic topology.

ℓ-adic cohomology

An ℓ-adic cohomology groups is an inverse limit of étale cohomology groups with certain torsion coefficients.

The "derived category" of constructible \overline{\mathbb{Q}_{\ell}}-sheaves

In a way similar to that for ℓ-adic cohomology, the derived category of constructible -sheaves is defined essentially as
writes "in daily life, one pretends that is simply the full subcategory of some hypothetical derived category ..."