Étale topos


In mathematics, the étale topos of a scheme X is the category of all étale sheaves on X. An étale sheaf is a sheaf on the étale site of X.

Definition

Let X be a scheme. An étale covering of X is a family, where each is an étale morphism of schemes, such that the family is jointly surjective that is.
The category Ét 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 defines a Grothendieck pretopology on Ét 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 of a scheme X is then the category of all sheaves of sets on the site Ét. Such sheaves are called étale sheaves on X. In other words, an étale sheaf is a functor from the category Ét to the category of sets satisfying the following sheaf axiom:
For each étale U over X and each étale covering of U the sequence
is exact, where.