In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients...
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the adjective étale refers to several closely related concepts: Étale morphism Formally étale morphism Étale cohomology Étale topology Étale fundamental...
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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...
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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...
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In algebraic geometry, an étale morphism (French: [etal]) is a morphism of schemes that is formally étale and locally of finite presentation. This is...
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characteristic. The étale topology was originally introduced by Alexander Grothendieck to define étale cohomology, and this is still the étale topology's most...
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Grothendieck topology (redirect from Small étale site)
their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme...
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Sheaf (mathematics) (redirect from Étale space)
notion of covering. This allowed Grothendieck to define étale cohomology and ℓ-adic cohomology, which eventually were used to prove the Weil conjectures...
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étale cohomology states that the higher direct images of a constructible sheaf are constructible. Here we use the definition of constructible étale sheaves...
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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...
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Alexander Grothendieck (section Cohomology theories)
Topoi Étale cohomology and l-adic cohomology Motives and the motivic Galois group (Grothendieck ⊗-categories) Crystals and crystalline cohomology, yoga...
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the étale fundamental group of the scheme and the étale cohomology of locally constant étale sheaves. Artin, Michael; Mazur, Barry (1969). Etale homotopy...
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inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts...
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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...
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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...
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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...
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ℓ-adic sheaf (section ℓ-adic cohomology)
}{\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...
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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...
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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...
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Weil conjectures (section Weil cohomology)
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...
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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...
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Duality (mathematics) (section Homology and cohomology)
encountered in arithmetics: étale cohomology of finite, local and global fields (also known as Galois cohomology, since étale cohomology over a field is equivalent...
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Motive (algebraic geometry) (redirect from Universal Weil cohomology)
similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically,...
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isomorphisms. There is yet another projection formula in the setting of étale cohomology. Integration along fibers § Projection formula Hartshorne, Robin (1977)...
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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...
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Flat topology (redirect from Flat cohomology)
except in cases where they reduce to other theories, such as the étale cohomology. The following example shows why the "faithfully flat topology" without...
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turned the position around. Firstly, Galois cohomology appeared as the foundational layer of étale cohomology theory (roughly speaking, the theory as it...
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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...
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Base change theorems (redirect from Cohomology base change theorem)
James S. (1980), Étale cohomology, Princeton University Press, ISBN 978-0-691-08238-7 Milne, James S. (2012), Lectures on Étale Cohomology (PDF) Mumford...
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vector cohomology Monsky–Washnitzer cohomology Infinitesimal cohomology Crystalline cohomology Rigid cohomology p-adic Hodge theory Étale cohomology, taking...
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