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

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

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

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