étale Sheaf
In mathematics, more specifically in algebra, the adjective étale refers to several closely related concepts: * Étale morphism ** Formally étale morphism * Étale cohomology * Étale topology * Étale fundamental group * Étale group scheme * Étale algebra Other * Étale (mountain) in Savoie and Haute-Savoie, France See also * Étalé space * Etail Online shopping is a form of electronic commerce which allows consumers to directly buy goods or services from a seller over the Internet using a web browser or a mobile app. Consumers find a product of interest by visiting the website of the ...

Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ...
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Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''algebra'' is ...
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Étale Morphism
In algebraic geometry, an étale morphism () is a morphism of 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 topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology. The word ''étale'' is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle. Definition Let \phi : R \to S be a ring homomorphism. This makes S an R-algebra. Choose a monic polynomial f in R /math> and a polynomial g in R /math> such that the derivative f' of f is a unit in (R fR _g. We say that \phi is ''standard étale'' if f and g can be chose ...
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Formally étale Morphism
In commutative algebra and algebraic geometry, a morphism is called formally étale if it has a lifting property that is analogous to being a local diffeomorphism. Formally étale homomorphisms of rings Let ''A'' be a topological ring, and let ''B'' be a topological ''A''-algebra. Then ''B'' is formally étale if for all discrete ''A''-algebras ''C'', all nilpotent ideals ''J'' of ''C'', and all continuous ''A''-homomorphisms , there exists a unique continuous ''A''-algebra map such that , where is the canonical projection. Formally étale is equivalent to formally smooth plus formally unramified. Formally étale morphisms of schemes Since the structure sheaf of a scheme naturally carries only the discrete topology, the notion of formally étale for schemes is analogous to formally étale for the discrete topology for rings. That is, a morphism of schemes is formally étale if for every affine ''Y''-scheme ''Z'', every nilpotent sheaf of ideals ''J'' on ''Z'' with be the ...
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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 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. History Étale cohomology was introduced by , using some suggestions by Jean-Pierre Serre, and was motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as and SGA 4. Grothendieck used étale cohomology to prove some of the Weil conjectures (Bernard Dwork had already managed to prove the rationality part of ...
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Étale Topology
In algebraic geometry, the étale topology is a Grothendieck topology 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 Grothendieck to define étale cohomology, and this is still the étale topology's most well-known use. Definitions For any scheme ''X'', let Ét(''X'') be the category of all étale morphisms 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 from Ét(''X'') to the category of sets. A presheaf '' ...
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Étale Fundamental Group
The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces. Topological analogue/informal discussion In algebraic topology, the fundamental group \pi_1(X,x) of a pointed topological space (X, x) is defined as the group of homotopy classes of loops based at x. This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the Zariski topology. In the classification of covering spaces, it is shown that the fundamental group is exactly the group of deck transformations of the universal covering space. This is more promising: finite étale morphisms of algebraic varieties are the appropriate analogue of covering spaces of topological spaces. Unfortunately, an algebraic variety X often fails to have a "universal cover" that is finite over ''X'', so one must consider the entire category of finite étale coverings of ''X''. On ...
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Étale Group Scheme
In mathematics, more specifically in algebra, the adjective étale refers to several closely related concepts: * Étale morphism ** Formally étale morphism * Étale cohomology * Étale topology * Étale fundamental group * Étale group scheme * Étale algebra Other * Étale (mountain) in Savoie and Haute-Savoie, France See also * Étalé space * Etail Online shopping is a form of electronic commerce which allows consumers to directly buy goods or services from a seller over the Internet using a web browser or a mobile app. Consumers find a product of interest by visiting the website of the ...

Étale Algebra
In commutative algebra, an étale algebra over a field is a special type of algebra, one that is isomorphic to a finite product of finite separable field extensions. An étale algebra is a special sort of commmutative separable algebra. Definitions Let be a field. Let be a commutative unital associative -algebra. Then is called an ''étale -algebra'' if any one of the following equivalent conditions holds: Examples The \mathbb-algebra \mathbb(i) is étale because it is a finite separable field extension. The \mathbb-algebra \mathbb (x^2) is not étale, since \mathbb (x^2)\otimes_\mathbb\mathbb \simeq \mathbb (x^2). Properties Let denote the absolute Galois group of . Then the category of étale -algebras is equivalent to the category of finite -sets with continuous -action. In particular, étale algebras of dimension are classified by conjugacy classes of continuous homomorphisms from to the symmetric group . These globalize to e.g. the definition of étale funda ...
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Étale (mountain)
Étale is a mountain of Savoie and Haute-Savoie, France. It lies in the Aravis Range The Aravis (french: Chaîne des Aravis) is a mountain range in Savoie and Haute-Savoie, eastern France. It is part of the French Prealps, a lower chain of mountain ranges west of the main chain of the Alps. Its highest summit is the Pointe Percé ... of the French Prealps and has an elevation of 2,484 metres above sea level. References Mountains of the Alps Mountains of Savoie Mountains of Haute-Savoie {{HauteSavoie-geo-stub ...
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Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphisms) from one ...
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