Étale topos

Definition

Let X be a scheme. An étale covering of X is a family { φ i : U i → X } i ∈ I , where each φ i is an étale morphism of schemes, such that the family is jointly surjective that is X = ⋃ i ∈ I φ i ( U i ) .

The category Ét(X) is the category of all étale schemes over X. The collection of all étale coverings of a étale scheme U over X i.e. an object in Ét(X) defines a Grothendieck pretopology on Ét(X) which in turn induces a Grothendieck topology, the étale topology on X. The category together with the étale topology on it is called the étale site on X.

The étale topos X ét of a scheme X is then the category of all sheaves of sets on the site Ét(X). Such sheaves are called étale sheaves on X. In other words, an étale sheaf F is a (contravariant) functor from the category Ét(X) to the category of sets satisfying the following sheaf axiom:

For each étale U over X and each étale covering { φ i : U i → U } of U the sequence

is exact, where U i j = U i × U U j .