Etale topology

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The most important example of a Grothendieck topology (see Topologized category), making it possible to define cohomology and homotopy invariants for abstract algebraic varieties and schemes. Let be a scheme. The étale topology on is the name for the category of étale -schemes the objects of which are étale morphisms (cf. Etale morphism) and the morphisms of which are those of the -schemes. Finite families such that are taken as coverings and so in a topology is introduced.

A pre-sheaf of sets (groups, Abelian groups, etc.) on is defined as a contravariant functor from the category into that of sets (groups, etc.). A pre-sheaf is called a sheaf if for any covering a section is determined by its restriction to and if for any compatible collection of sections there exists a unique section such that . Many standard concepts of sheaf theory carry over to étale sheaves (that is, sheaves on ). For example, if is a morphism of schemes and is an étale sheaf on , then by putting

one obtains the so-called direct image of for the morphism . The functor adjoint to on the left is called the inverse-image functor. In particular, the stalk of at a geometric point (where is an algebraically closed field) is defined as the set .

An important example of a sheaf on is , representable by a certain -scheme ; for it . If is a finite étale -scheme, then the sheaf is called locally constant. A sheaf is said to be constructible if there exists a finite partition of into locally closed subschemes such that the restriction is locally constant on every .

See also Etale cohomology; Homotopy type of a topological category.


[1] Yu.I. Manin, "Algebraic topology of algebraic varieties" Russian Math. Surveys , 20 : 5/6 (1965) pp. 183–192 Uspekhi Mat. Nauk , 20 : 6 (1965) pp. 3–12
[2] J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980)
[3] M. Artin (ed.) A. Grothendieck (ed.) J.-L. Verdier (ed.) , Théorie des topos et cohomologie étale des schémas (SGA 4) , Lect. notes in math. , 269; 270; 305 , Springer (1972–1973)
[4] P. Deligne, "Cohomologie étale (SGA 4 1/2)" , Lect. notes in math. , 569 , Springer (1977)
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Etale topology. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article