In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients...

and used sheaf cohomology over the étale topology to define the cohomology theory for varieties over a finite field. Using the étale topology for a variety...

the adjective étale refers to several closely related concepts: Étale morphism Formally étale morphism Étale cohomology Étale topology Étale fundamental...

In algebraic geometry, an étale morphism (French: [etal]) is a morphism of schemes that is formally étale and locally of finite presentation. This is...

motivic cohomology, whereas étale cohomology is often easier to understand. For example, if the base field k is the complex numbers, then étale cohomology coincides...

characteristic. The étale topology was originally introduced by Alexander Grothendieck to define étale cohomology, and this is still the étale topology's most...

their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme...

class formations. In turn, this led to the notion of Galois cohomology and étale cohomology (which builds on it) (Weibel 1999, p. 822). Some refinements...

inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts...

notion of covering. This allowed Grothendieck to define étale cohomology and ℓ-adic cohomology, which eventually were used to prove the Weil conjectures...

the étale fundamental group of the scheme and the étale cohomology of locally constant étale sheaves. Artin, Michael; Mazur, Barry (1969). Etale homotopy...

in étale cohomology that arise from a morphism of schemes f : X → Y. The basic insight was that many of the elementary facts relating cohomology on X...

étale cohomology states that the higher direct images of a constructible sheaf are constructible. Here we use the definition of constructible étale sheaves...

Topoi Étale cohomology and l-adic cohomology Motives and the motivic Galois group (Grothendieck ⊗-categories) Crystals and crystalline cohomology, yoga...

scheme, Čech and sheaf cohomology agree for any quasi-coherent sheaf. For the étale topology, the two cohomologies agree for any étale sheaf on X, provided...

cohomologie étale des schémas, 1963–1964 (Topos theory and étale cohomology), Lecture Notes in Mathematics 269, 270 and 305, 1972/3 SGA4½ Cohomologie étale (Étale...

}{\to }}F_{n}} . Bhatt–Scholze's pro-étale topology gives an alternative approach. The development of étale cohomology as a whole was fueled by the desire...

and the link to Betti numbers by using the properties of étale cohomology, a new cohomology theory developed by Grothendieck and Michael Artin for attacking...

characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of F with coefficients in Z/2Z. It was proved by Vladimir Voevodsky (1996...

except in cases where they reduce to other theories, such as the étale cohomology. The following example shows why the "faithfully flat topology" without...

James S. (1980), Étale cohomology, Princeton University Press, ISBN 978-0-691-08238-7 Milne, James S. (2012), Lectures on Étale Cohomology (PDF) Mumford...

similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically,...

étale cohomology is an isomorphism. The case ℓ = 2 is the Milnor conjecture, and the case n = 2 is the Merkurjev–Suslin theorem. The étale cohomology...

For the étale topology, this gives the notion of étale cohomology, which led to the proof of the Weil conjectures. Crystalline cohomology and many other...

consider the entire category of finite étale coverings of X {\displaystyle X} . One can then define the étale fundamental group as an inverse limit of...

quasi-compact scheme X is defined to be the torsion subgroup of the étale cohomology group H 2(X, Gm). (The whole group H 2(X, Gm) need not be torsion,...

encountered in arithmetics: étale cohomology of finite, local and global fields (also known as Galois cohomology, since étale cohomology over a field is equivalent...

isomorphisms. There is yet another projection formula in the setting of étale cohomology. Integration along fibers § Projection formula Hartshorne, Robin (1977)...

there is a cohomological classification of Azumaya algebras using Étale cohomology. In fact, this group, called the Brauer group, can be also defined...

towards their proof, and other advances, lay in the construction of étale cohomology. With the benefit of hindsight, it can be said that algebraic geometry...