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 [math]\displaystyle{ \{ \varphi_i: U_i \to X \}_{i\in I} }[/math], where each [math]\displaystyle{ \varphi_i }[/math] is an étale morphism of schemes, such that the family is jointly surjective that is [math]\displaystyle{ X = \bigcup_{i \in I} \varphi_i(U_i) }[/math].

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 [math]\displaystyle{ X^\text{ét} }[/math] 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 [math]\displaystyle{ \mathcal F }[/math] 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 [math]\displaystyle{ \{ \varphi_i: U_i \to U \} }[/math] of U the sequence

[math]\displaystyle{ 0 \to \mathcal F(U) \to \prod_{i \in I} \mathcal F(U_i) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \prod_{i,j \in I} \mathcal F(U_{ij}) }[/math]

is exact, where [math]\displaystyle{ U_{ij} = U_i \times_U U_j }[/math].