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Local Tate Duality
In Galois cohomology, local Tate duality (or simply local duality) is a duality for Galois modules for the absolute Galois group of a non-archimedean local field. It is named after John Tate who first proved it. It shows that the dual of such a Galois module is the Tate twist of usual linear dual. This new dual is called the (local) Tate dual. Local duality combined with Tate's local Euler characteristic formula provide a versatile set of tools for computing the Galois cohomology of local fields. Statement Let ''K'' be a non-archimedean local field, let ''Ks'' denote a separable closure of ''K'', and let ''GK'' = Gal(''Ks''/''K'') be the absolute Galois group of ''K''. Case of finite modules Denote by μ the Galois module of all roots of unity in ''Ks''. Given a finite ''GK''-module ''A'' of order prime to the characteristic of ''K'', the Tate dual of ''A'' is defined as :A^\prime=\mathrm(A,\mu) (i.e. it is the Tate twist of the usual dual ''A''∗). Let ''Hi''(''K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natural way on some abelian groups, for example those constructed directly from ''L'', but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor. History The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of ideal class groups in algebraic number theory was one way to formulate class field theory, at the time it was in the process of ridding itself of connections to L-functions. Galois cohomology makes no assumption that Galois groups are abelian groups, so this was a non-abelian theory. It was formulated abstractly as a theory of class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate Module
In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group ''A''. Often, this construction is made in the following situation: ''G'' is a commutative group scheme over a field ''K'', ''Ks'' is the separable closure of ''K'', and ''A'' = ''G''(''Ks'') (the ''Ks''-valued points of ''G''). In this case, the Tate module of ''A'' is equipped with an action of the absolute Galois group of ''K'', and it is referred to as the Tate module of ''G''. Definition Given an abelian group ''A'' and a prime number ''p'', the ''p''-adic Tate module of ''A'' is :T_p(A)=\underset A ^n/math> where ''A'' 'pn''is the ''pn'' torsion of ''A'' (i.e. the kernel of the multiplication-by-''pn'' map), and the inverse limit is over positive integers ''n'' with transition morphisms given by the multiplication-by-''p'' map ''A'' 'p''''n''+1→ ''A'' 'pn'' Thus, the Tate module encodes all the ''p''-power torsion of ''A''. It is equipped wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Algebraic Number Theory
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' a ... [...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] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, ''The Daily Princetonian'', and later added book publishing to it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Global Field
In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function field: The function field of an algebraic curve over a finite field, equivalently, a finite extension of \mathbb_q(T), the field of rational functions in one variable over the finite field with q=p^n elements. An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s. Formal definitions A ''global field'' is one of the following: ;An algebraic number field An algebraic number field ''F'' is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus ''F'' is a field that contains Q and has finite dimension when considered as a vector space over Q. ;The function field of an algebraic curve over a finite field A function field of a variety is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate Duality
In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by and . Local Tate duality For a ''p''-adic local field k, local Tate duality says there is a perfect pairing of the finite groups arising from Galois cohomology: :\displaystyle H^r(k,M)\times H^(k,M')\rightarrow H^2(k,\mathbb_m)=\Q/ \Z where M is a finite group scheme, M' its dual \operatorname(M,G_m), and \mathbb_m is the multiplicative group. For a local field of characteristic p>0, the statement is similar, except that the pairing takes values in H^2(k, \mu) = \bigcup_ \tfrac \Z/\Z. The statement also holds when k is an Archimedean field, though the definition of the cohomology groups looks somewhat different in this case. Global Tate duality Given a finite group scheme M over a global field k, global Tate duality relates the cohomology of M with that of M' = \operatorname(M,G_m) using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Group Cohomology
Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous game, a generalization of games used in game theory ** Law of Continuity, a heuristic principle of Gottfried Leibniz * Continuous function, in particular: ** Continuity (topology), a generalization to functions between topological spaces ** Scott continuity, for functions between posets ** Continuity (set theory), for functions between ordinals ** Continuity (category theory), for functors ** Graph continuity, for payoff functions in game theory * Continuity theorem may refer to one of two results: ** Lévy's continuity theorem, on random variables ** Kolmogorov continuity theorem, on stochastic processes * In geometry: ** Parametric continuity, for parametrised curves ** Geometric continuity, a concept primarily applied to the conic s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of module (mathematics), modules over a ring (mathematics), ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a (linear) ''endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are Real number, real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Some ... [...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] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say V is if the dimension of V is finite, and if its dimension is infinite. The dimension of the vector space V over the field F can be written as \dim_F(V) or as : F read "dimension of V over F". When F can be inferred from context, \dim(V) is typically written. Examples The vector space \R^3 has \left\ as a standard basis, and therefore \dim_(\R^3) = 3. More generally, \dim_(\R^n) = n, and even more generally, \dim_(F^n) = n for any field F. The complex numbers \Complex are both a real and complex vector space; we have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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