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 $\varphi_i$ is an étale morphism of schemes, such that the family is jointly surjective that is $X = \bigcup_ \varphi_i(U_i)$.

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'' $X^\text$ 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 $\$ of ''U'' the sequence

:$0 \to \mathcal F(U) \to \prod_ \mathcal F(U_i) \prod_ \mathcal F(U_)$

is exact, where $U_ = U_i \times_U U_j$.

{{DEFAULTSORT:Etale topos
Topos theory
Sheaf theory