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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 conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.

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rdf:type	Thing yago:WikicatCohomologyTheories yago:WikicatTopologicalMethodsOfAlgebraicGeometry yago:Ability105616246 yago:Abstraction100002137 yago:Cognition100023271 yago:Explanation105793000 yago:HigherCognitiveProcess105770664 yago:Know-how105616786 yago:Method105660268 yago:Process105701363 yago:PsychologicalFeature100023100 yago:Theory105989479 yago:Thinking105770926
rdfs:label	Cohomologie étale (fr) エタール・コホモロジー (ja) 에탈 코호몰로지 (ko) Étale cohomologie (nl) Coomologia etal (pt) Étale cohomology (en) 平展上同调 (zh)
rdfs:comment	La cohomologie étale est la théorie cohomologique des faisceaux associée à la topologie étale. Elle mime le comportement habituel de la cohomologie classique sur des objets mathématiques où celle-ci n'est pas envisageable, en particulier les schémas et les espaces analytiques. (fr) 대수기하학에서 에타일 코호몰로지(영어: étale cohomology)는 스킴 위에서 정의되는 코호몰로지이다.스킴의 경우, 자리스키 위상을 사용하면 표준적인 코호몰로지 이론(특이 코호몰로지, 체흐 코호몰로지)들은 잘 작동하지 않는데, 에탈 코호몰로지는 에탈 위상을 사용하여 이러한 단점들을 보완한다. (ko) 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 conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type. (en) エタール・コホモロジー（étale cohomology）はアレクサンドル・グロタンディークがヴェイユ予想を証明するための道具として考案したコホモロジー理論であり、位相空間上の定数係数コホモロジー、すなわち特異コホモロジーの類似になっている。エタール・コホモロジーはヴェイユ・コホモロジーの一種であるℓ進コホモロジーを構成する枠組みを与える。代数幾何学における基本的な道具の一つで、非常に多くの応用を持ち、ヴェイユ予想への貢献やフェルマーの最終定理の証明の際にも用いられた。 (ja) In de algebraïsche meetkunde en de homologische algebra, deelgebieden van de wiskunde, zijn étale cohomologiegroepen van een algebraïsche variëteit of van een schema algebraïsche analoga van de gebruikelijke cohomologiegroepen met eindige coëfficiënten van een topologische ruimte. Het concept werd geïntroduceerd door Alexander Grothendieck om zo de vermoedens van Weil te bewijzen. Étale cohomologietheorie kan worden gebruikt om ℓ-adische cohomologie te construeren, wat in de algebraïsche meetkunde een voorbeeld is van een . Dit heeft vele toepassingen, zoals het bewijs van de vermoedens van Weil en de constructie van . (nl) Em matemática, a cohomologia etal de grupos de uma variedade algébrica ou esquema são análogos algébricos da usual cohomologia de grupos com finitos coeficientes de um espaço topológico, introduzido por Alexander Grothendieck de maneira a provar as . A teoria da cohomologia etal pode ser usada para construir chomologia ℓ-ádica, a qual é um exemplo de uma em geometria algébrica. Isto tem muitas aplicações, tais como a demonstração das conjecturas de Weil e a construção de conjecturas e construção de . (pt) 在数学中，一个代数簇或概形的平展上同调（Étale cohomology）是一个与一般拓扑空间的有限系数上同调群类似的代数结构。这一概念作为证明的工具由亚历山大·格罗滕迪克引入。平展上同调的理论可以用于构建ℓ进上同调，后者则是代数几何中的一个例子。这一理论有着众多的应用，包括Weil猜想的证明以及的构造。 (zh)
rdfs:seeAlso	Behrend's formula
dcterms:subject	Cohomology theories Topological methods of algebraic geometry Homological algebra
Wikipage page ID	396231 (xsd:integer)
Wikipage revision ID	1095239434 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Princeton University Press Representation theory Weil cohomology theory Motive (mathematics) Non-singular Bernard Dwork David Mumford Betti number Riemann hypothesis Riemann surface Derived functor Jacobian variety L-adic sheaf Torsion sheaf Complex number Mathematics Genus (mathematics) Cohomology theories George Lusztig Constructible sheaf Topological methods of algebraic geometry Local zeta-function Deligne–Lusztig theory Yuri Manin Zermelo–Fraenkel set theory Functor Fundamental group Picard group Stalk (sheaf) Weil conjectures Galois cohomology Galois group Coherent sheaves Hasse's theorem on elliptic curves Locally acyclic morphism Poincaré duality Algebraic curve Algebraic number Algebraic variety Fermat's Last Theorem Finite field Brauer group P-adic integers P-adic number Diophantine geometry Lefschetz fixed-point theorem Complex conjugation Projective line Group cohomology Hilbert's theorem 90 Inverse limit Jean-Louis Verdier Jean-Pierre Serre Homological algebra Abelian group Abelian variety Absolute Galois group Characteristic (algebra) Coherent sheaf Cohomology Homological algebra Tsen's theorem Artin–Schreier theory Pierre Deligne Springer Science+Business Media

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