Étale sheaf

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 fibered 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 U_i → U is a jointly surjective family of étale morphisms over X. For each i, choose a section x_i of F over U_i. The projection map U_i × U_j → U_i, which is loosely speaking the inclusion of the intersection of U_i and U_j in U_i, induces a restriction map F(U_i) → F(U_i × U_j). If for all i and j the restrictions of x_i and x_j to U_i × U_j are equal, then there must exist a unique section x of F over U which restricts to x_i for all i.

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