étale Topology
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In
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, the étale topology is a
Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is ca ...
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
Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 â€“ 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
to define
étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...
, and this is still the étale topology's most well-known use.


Definitions

For any scheme ''X'', let Ét(''X'') be the
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
of all
étale morphism In algebraic geometry, an étale morphism () is a morphism of Scheme (mathematics), schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topol ...
s 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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
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 is an object of Ét(''X'') and that 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 , which is loosely speaking the inclusion of the intersection of ''U''''i'' and ''U''''j'' in ''U''''i'', induces a restriction map . If for all ''i'' and ''j'' the restrictions of ''x''''i'' and ''x''''j'' to are equal, then there must exist a unique section ''x'' of ''F'' over ''U'' which restricts to ''x''''i'' for all ''i''. Suppose that ''X'' is a
Noetherian scheme In algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, where each A_i is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noe ...
. An abelian étale sheaf ''F'' on ''X'' is called finite locally constant if it is a
representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets an ...
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''. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves. Grothendieck originally introduced the machinery of Grothendieck topologies and
topoi In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notion ...
to define the étale topology. In this language, the definition of the étale topology is succinct but abstract: It is the topology generated by the pretopology whose covering families are jointly surjective families of étale morphisms. The small étale site of ''X'' is the category ''O''(''X''ét) whose objects are schemes ''U'' with a fixed étale morphism ''U'' → ''X''. The morphisms are morphisms of schemes compatible with the fixed maps to ''X''. The big étale site of ''X'' is the category Ét/''X'', that is, the category of schemes with a fixed map to ''X'', considered with the étale topology. The étale topology can be defined using slightly less data. First, notice that the étale topology is finer than the Zariski topology. Consequently, to define an étale cover of a scheme ''X'', it suffices to first cover ''X'' by open affine subschemes, that is, to take a Zariski cover, and then to define an étale cover of an affine scheme. An étale cover of an affine scheme ''X'' can be defined as a jointly surjective family such that the set of all ''α'' is finite, each ''X''''α'' is affine, and each ''u''''α'' is étale. Then an étale cover of ''X'' is a family which becomes an étale cover after base changing to any open affine subscheme of ''X''.


Local rings

Let ''X'' be a scheme with its étale topology, and fix a point ''x'' of ''X''. In the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...
, the stalk of ''X'' at ''x'' is computed by taking a
direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
of the sections of the structure sheaf over all the Zariski open neighborhoods of ''x''. In the étale topology, there are strictly more open neighborhoods of ''x'', so the correct analog of the local ring at ''x'' is formed by taking the limit over a strictly larger family. The correct analog of the local ring at ''x'' for the étale topology turns out to be the strict henselization of the local ring \mathcal_. It is usually denoted \mathcal_^\text.


Examples

*For each étale morphism U \to X, let \mathbb_m(U) = \mathcal_U(U)^. Then U \mapsto \mathbb_m(U) is a presheaf on ''X''; it is a sheaf since it can be represented by the scheme \operatorname_X (\mathcal_X , t^.


Étale topos

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 In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is ca ...
, 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 In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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.


See also

* Nisnevich topology * Smooth topology * ℓ-adic sheaf * Étale spectrum Topos theory Sheaf theory


References

* * * * * * *J. S. Milne (2008).
Lectures on Étale Cohomology
' {{DEFAULTSORT:Etale Topology Algebraic geometry