ℓ-adic Sheaf

	ℓ-adic Sheaf In algebraic geometry, an ℓ-adic sheaf on a Noetherian scheme ''X'' is an inverse system consisting of \mathbb/\ell^n-modules F_n in the étale topology and F_ \to F_n inducing F_ \otimes_ \mathbb/\ell^n \overset\to F_n.. Bhatt–Scholze's pro-étale topology gives an alternative approach. Constructible and lisse ℓ-adic sheaves An ℓ-adic sheaf \_ is said to be * ''constructible'' if each F_n is constructible. * ''lisse'' if each F_n is constructible and locally constant. Some authors (e.g., those of SGA 4½) assume an ℓ-adic sheaf to be constructible. Given a connected scheme ''X'' with a geometric point ''x'', SGA 1 defines the étale fundamental group \pi^_1(X, x) of ''X'' at ''x'' to be the group classifying Galois coverings of ''X''. Then the category of lisse ℓ-adic sheaves on ''X'' is equivalent to the category of continuous representations of \pi^_1(X, x) on finite free \mathbb_l-modules. This is an analog of the correspondence between local systems and conti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Inverse System In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can be defined in any category although their existence depends on the category that is considered. They are a special case of the concept of limit in category theory. By working in the dual category, that is by reverting the arrows, an inverse limit becomes a direct limit or ''inductive limit'', and a ''limit'' becomes a colimit. Formal definition Algebraic objects We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let (I, \leq) be a directed poset (not all authors require ''I'' to be directed). Let (''A''''i'')''i''∈''I'' be a family of groups and suppose we have a family of homomorphisms f_: A_j \to A_i for all i \leq j (note the order) with the following properties: # f_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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 presh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Constructible Sheaf In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space ''X'', such that ''X'' is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origins in algebraic geometry, where in 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 conject ... constructible sheaves are defined in a similar way . For the derived category of constructible sheaves, see a section in ℓ-adic sheaf. The finiteness theorem in étale cohomology states that the higher direct images of a constructible sheaf are constructible. Definition of étale constructible sheaves on a scheme ''X'' Here we use the definition of constructible étale sheaves from the book by Freitag and Kiehl referenced below. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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 ''π''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 are the appropriate analogue of covering 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''. One can then defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Fourier–Deligne Transform In algebraic geometry, the Fourier–Deligne transform, or ℓ-adic Fourier transform, or geometric Fourier transform, is an operation on objects of the derived category of ''ℓ''-adic sheaves over the affine line. It was introduced by Pierre Deligne on November 29, 1976 in a letter to David Kazhdan as an analogue of the usual Fourier transform. It was used by Gérard Laumon to simplify Deligne's proof of the Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. .... References * * * Algebraic geometry {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Springer Science+Business Media Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]