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 $\{\varphi _{i}:U_{i}\to X\}_{i\in I}$, where each $\varphi _{i}$ is an étale morphism of schemes, such that the family is jointly surjective that is $X=\bigcup _{i\in I}\varphi _{i}(U_{i})$.

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 $X^{\text{ét}}$ 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 ${\mathcal {F}}$ 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 $\{\varphi _{i}:U_{i}\to U\}$ of U the sequence

$0\to {\mathcal {F}}(U)\to \prod _{i\in I}{\mathcal {F}}(U_{i}){{{} \atop \longrightarrow } \atop {\longrightarrow \atop {}}}\prod _{i,j\in I}{\mathcal {F}}(U_{ij})$

is exact, where $U_{ij}=U_{i}\times _{U}U_{j}$.