Étale topology

In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale topology was originally introduced by Grothendieck to define étale cohomology, and this is still the étale topology's most well-known use. For any scheme X, let Ét(X) be the category of all étale morphisms from a scheme to X. This is the analog of the category of open subsets of X (that is, the category whose objects are varieties and whose morphisms are open immersions). Its objects can be informally thought of as étale open subsets of X. The intersection of two objects corresponds to their fiber product over X. Ét(X) is a large category, meaning that its objects do not form a set. An étale presheaf on X is a contravariant functor from Ét(X) to the category of sets. A presheaf F is called an étale sheaf if it satisfies the analog of the usual gluing condition for sheaves on topological spaces. That is, F is an étale sheaf if and only if the following condition is true. Suppose that U → X is an object of Ét(X) and that Ui → U is a jointly surjective family of étale morphisms over X. For each i, choose a section xi of F over Ui. The projection map Ui × Uj → Ui, which is loosely speaking the inclusion of the intersection of Ui and Uj in Ui, induces a restriction map F(Ui) → F(Ui × Uj). If for all i and j the restrictions of xi and xj to Ui × Uj are equal, then there must exist a unique section x of F over U which restricts to xi for all i. Suppose that X is a Noetherian scheme. An abelian étale sheaf F on X is called finite locally constant if it is a representable functor which can be represented by an étale cover of X. It is called constructible if X can be covered by a finite family of subschemes on each of which the restriction of F is finite locally constant. It is called torsion if F(U) is a torsion group for all étale covers U of X.