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Crystal (mathematics)
In mathematics, crystals are Cartesian sections of certain fibered categories. They were introduced by , who named them crystals because in some sense they are "rigid" and "grow". In particular quasicoherent crystals over the crystalline site are analogous to quasicoherent modules over a scheme. An isocrystal is a crystal up to isogeny. They are p-adic analogues of \mathbf_l-adic étale sheaves, introduced by and (though the definition of isocrystal only appears in part II of this paper by ). Convergent isocrystals are a variation of isocrystals that work better over non-perfect fields, and overconvergent isocrystals are another variation related to overconvergent cohomology theories. A Dieudonné crystal is a crystal with Verschiebung and Frobenius maps. An F-crystal is a structure in semilinear algebra somewhat related to crystals. Crystals over the infinitesimal and crystalline sites The infinitesimal site \text(X/S) has as objects the infinitesimal extensions of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Section
Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which ''inverse images'' (or ''pull-backs'') of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space ''X'' to another topological space ''Y'' is associated the pullback bundle, pullback functor taking bundles on ''Y'' to bundles on ''X''. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stack (mathematics), stacks, which are fibered categories (over a site) with "descent". Fibrations also play an impor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibered Categories
Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which ''inverse images'' (or ''pull-backs'') of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space ''X'' to another topological space ''Y'' is associated the pullback functor taking bundles on ''Y'' to bundles on ''X''. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories (over a site) with "descent". Fibrations also play an important role in categorical semantics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystalline Site
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H''''n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed by . Crystalline cohomology is partly inspired by the ''p''-adic proof in of part of the Weil conjectures and is closely related to the algebraic version of de Rham cohomology that was introduced by Grothendieck (1963). Roughly speaking, crystalline cohomology of a variety ''X'' in characteristic ''p'' is the de Rham cohomology of a smooth lift of ''X'' to characteristic 0, while de Rham cohomology of ''X'' is the crystalline cohomology reduced mod ''p'' (after taking into account higher ''Tor''s). The idea of crystalline cohomology, roughly, is to replace the Zariski open sets of a scheme by infinitesimal thickenings of Zariski open sets with divided power structures. The motivation for this is that it can then be calculated by taking ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dieudonné Crystal
Dieudonné is a French name meaning "Gift of God", and thus similar to the Greek-derived Theodore or the Spanish Diosdado. It may refer to: People Given name * Dieudonné Cédor (1925–2010), Haitian painter * Dieudonné Costes (1892–1973), French aviator * Dieudonné Disi (born 1980), Rwandan long-distance and cross county runner * Dieudonne Dolassem (born 1979), Cameroonian judoka * Dieudonné Sylvain Guy Tancrède de Dolomieu or Déodat Gratet de Dolomieu (1750–1801), French geologist * Dieudonné Ganga (born c. 1946), Congolese politician and diplomat * Dieudonné Gnammankou, Beninean historian * Dieudonné de Gozon ( 1346–53), French knight * Dieudonné-Félix Godefroid or Félix Godefroid (1818–1897), Belgian harpist * Dieudonné Jamar (1878 – after 1905), Belgian racing cyclist * Dieudonné Kabongo (1950–2011), Congolese-born Belgian humorist and actor * Dieudonné Kalilulika (born 1981), Congolese football player * Dieudonné Kayembe Mbandakulu (born 1945 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Verschiebung Operator
In mathematics, the Verschiebung or Verschiebung operator ''V'' is a homomorphism between affine commutative group schemes over a field of nonzero characteristic ''p''. For finite group schemes it is the Cartier dual In mathematics, Cartier duality is an analogue of Pontryagin duality for commutative group schemes. It was introduced by . Definition using characters Given any finite flat commutative group scheme ''G'' over ''S'', its Cartier dual is the group o ... of the Frobenius homomorphism. It was introduced by as the shift operator on Witt vectors taking (''a''0, ''a''1, ''a''2, ...) to (0, ''a''0, ''a''1, ...). ("Verschiebung" is German for "shift", but the term "Verschiebung" is often used for this operator even in other languages.) The Verschiebung operator ''V'' and the Frobenius operator ''F'' are related by ''FV'' = ''VF'' = 'p'' where 'p''is the ''p''th power homomorphism of an abelian group scheme. Examples *If ''G'' is the discrete group with ''n'' elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F-crystal
In algebraic geometry, F-crystals are objects introduced by that capture some of the structure of crystalline cohomology groups. The letter ''F'' stands for Frobenius, indicating that ''F''-crystals have an action of Frobenius on them. F-isocrystals are crystals "up to isogeny". F-crystals and F-isocrystals over perfect fields Suppose that ''k'' is a perfect field, with ring of Witt vectors ''W'' and let ''K'' be the quotient field of ''W'', with Frobenius automorphism σ. Over the field ''k'', an ''F''-crystal is a free module ''M'' of finite rank over the ring ''W'' of Witt vectors of ''k'', together with a σ-linear injective endomorphism of ''M''. An ''F''-isocrystal is defined in the same way, except that ''M'' is a module for the quotient field ''K'' of ''W'' rather than ''W''. Dieudonné–Manin classification theorem The Dieudonné–Manin classification theorem was proved by and . It describes the structure of ''F''-isocrystals over an algebraically closed fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasicoherent Sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. Definitions A quasi-coherent sheaf on a ringed space (X, \mathcal O_X) is a sheaf \mathcal F of \mathcal O_X-modules which has a local presentation, that is, every point in X has an open neighborhood U in which there is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
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] |