Étale topos explained

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} , where each \varphi _{i} is an étale morphism of schemes, such that the family is jointly surjective that is X = c up _{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 ^{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 l 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 sequence0 \to l F(U) \to \prod _{i} l F(U _{i)} {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \prod _{i,j} l F(U _{ij} )

is exact, where

U _{ij} = U _{i} x _{U} U _{j} .