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Bloch-Kato Conjecture
In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor ''K''-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic forms, algebraic K-theory and the theory of motives. The theorem asserts that a certain statement holds true for any prime \ell and any natural number n. John MilnorMilnor (1970) speculated that this theorem might be true for \ell=2 and all n, and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato and became known as the Bloch–Kato conjecture or the motivic Bloch–Kato conjecture to distinguish it from the Bloch–Kato conjecture on values of ''L''-functions.Bloch, Spencer and Kato, Kazuya, "L-functions and Tamagawa numbers of motives", The Grothendieck Festschrift, Vol. I, 333–400, Progr. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Symbol
In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of reals or p-adic numbers . It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields. The Hilbert symbol has been generalized to higher local fields. Quadratic Hilbert symbol Over a local field ''K'' whose multiplicative group of non-zero elements is ''K''×, the quadratic Hilbert symbol is the function (–, –) from ''K''× × ''K''× to defined by :(a,b)=\begin+1,&\mboxz^2=ax^2+by^2\mbox(x,y,z)\in K^3;\\-1,&\mbox\end Equivalently, (a, b) = 1 if and only if b is equal to the norm of an element of the quadratic extension Ksqrt/math> page 110. Properties The follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steenrod Algebra
In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology. For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p, consisting of all stable cohomology operations for mod p cohomology. It is generated by the Steenrod squares introduced by for p=2, and by the Steenrod reduced pth powers introduced in and the Bockstein homomorphism for p>2. The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory. Cohomology operations A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring R, the cup product squaring operation yields a family of cohomology operations: :H^n(X;R) \to H^(X;R) :x \mapsto x \smile x. Cohomology operations need not be homomorphisms of graded rings; see the Cartan formula below. Thes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thom Space
In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. Construction of the Thom space One way to construct this space is as follows. Let :p: E \to B be a rank ''n'' real vector bundle over the paracompact space ''B''. Then for each point ''b'' in ''B'', the fiber E_b is an n-dimensional real vector space. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity. Let D(E) be the unit ball bundle with respect to our orthogonal structure, and let S(E) be the unit sphere bundle, then the Thom space T(E) is the quotient T(E) := D(E)/S(E) of topological spaces. T(E) is a pointed space with the image of S(E) in the quotient as basepoint. If ''B'' is compact, then T(E) is the one-point compactification of ''E''. For example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spanier–Whitehead Duality
In mathematics, Spanier–Whitehead duality is a duality theory in homotopy theory, based on a geometrical idea that a topological space ''X'' may be considered as dual to its complement in the ''n''-sphere, where ''n'' is large enough. Its origins lie in Alexander duality theory, in homology theory, concerning complements in manifolds. The theory is also referred to as ''S-duality'', but this can now cause possible confusion with the S-duality of string theory. It is named for Edwin Spanier and J. H. C. Whitehead, who developed it in papers from 1955. The basic point is that sphere complements determine the homology, but not the homotopy type, in general. What is determined, however, is the stable homotopy type, which was conceived as a first approximation to homotopy type. Thus Spanier–Whitehead duality fits into stable homotopy theory. Statement Let ''X'' be a compact neighborhood retract in \R^n. Then X^+ and \Sigma^\Sigma'(\R^n \setminus X) are dual obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motivic Homotopy Theory
In music, a motif IPA: ( /moʊˈtiːf/) (also motive) is a short musical phrase, a salient recurring figure, musical fragment or succession of notes that has some special importance in or is characteristic of a composition: "The motive is the smallest structural unit possessing thematic identity". The '' Encyclopédie de la Pléiade'' regards it as a "melodic, rhythmic, or harmonic cell", whereas the 1958 ''Encyclopédie Fasquelle'' maintains that it may contain one or more cells, though it remains the smallest analyzable element or phrase within a subject. It is commonly regarded as the shortest subdivision of a theme or phrase that still maintains its identity as a musical idea. "The smallest structural unit possessing thematic identity". Grove and Larousse also agree that the motif may have harmonic, melodic and/or rhythmic aspects, Grove adding that it "is most often thought of in melodic terms, and it is this aspect of the motif that is connoted by the term 'f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Cobordism
In mathematics, algebraic cobordism is an analogue of complex cobordism for smooth quasi-projective schemes over a field. It was introduced by . An oriented cohomology theory on the category of smooth quasi-projective schemes Sm over a field ''k'' consists of a contravariant functor ''A''* from Sm to commutative graded rings, together with push-forward maps ''f''* whenever ''f'':''Y''→''X'' has relative dimension ''d'' for some ''d''. These maps have to satisfy various conditions similar to those satisfied by complex cobordism. In particular they are "oriented", which means roughly that they behave well on vector bundles; this is closely related to the condition that a generalized cohomology theory has a complex orientation. Over a field of characteristic 0, algebraic cobordism is the universal oriented cohomology theory for smooth varieties. In other words there is a unique morphism of oriented cohomology theories from algebraic cobordism to any other oriented cohomology th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simone Borghesi
Simone may refer to: * Simone (given name), a feminine (or Italian masculine) given name of Hebrew origin * Simone (surname), an Italian surname Simone may also refer to: * ''Simone'' (1918 film), a French silent drama film * ''Simone'' (1926 film), a French silent drama film * ''Simone'' (2002 film), a 2002 science-fiction drama film * ''Simone'' (2013 film), a 2013 Brazilian drama * Simone (actress) (born 1962), stage name of Lisa Celeste Stroud, daughter of Nina Simone * Nina Simone (1933–2003), stage name of Eunice Kathleen Waymon, singer, songwriter, musician, arranger, and civil rights activist * Simone (born 1966), Egyptian singer and actress * Simone (character), a fictional character in the ABC Family show ''The Nine Lives of Chloe King'' * Simone Bittencourt de Oliveira (born 1949), Brazilian singer and performer, better known by her mononym Simone * Simone Egeriis (born 1992), Danish singer, better known by her mononym Simone * Tropical Storm Simone (disambiguat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Morava K-theories
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a datatype in computer programming each of whose values is data from other datatypes wrapped in one of the constructors of the datatype * Algebraic numbers, a complex number that is a root of a non-zero polynomial in one variable with integer coefficients * Algebraic functions, functions satisfying certain polynomials * Algebraic element, an element of a field extension which is a root of some polynomial over the base field * Algebraic extension, a field extension such that every element is an algebraic element over the base field * Algebraic definition, a definition in mathematical logic which is given using only equalities between terms * Algebraic structure, a set with one or more finitary operations defined on it * Algebraic, the order of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morava K-theory
In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number ''p'' (which is suppressed in the notation), it consists of theories ''K''(''n'') for each nonnegative integer ''n'', each a ring spectrum in the sense of homotopy theory. published the first account of the theories. Details The theory ''K''(0) agrees with singular homology with rational coefficients, whereas ''K''(1) is a summand of mod-''p'' complex K-theory. The theory ''K''(''n'') has coefficient ring :F''p'' 'v''''n'',''v''''n''−1 where ''v''''n'' has degree 2(''p''''n'' − 1). In particular, Morava K-theory is periodic with this period, in much the same way that complex K-theory has period 2. These theories have several remarkable properties. * They have Künneth isomorphisms for arbitrary pairs of spaces: that is, for ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motivic Cohomology
Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology. Motivic homology and cohomology Let ''X'' be a scheme of finite type over a field ''k''. A key goal of algebraic geometry is to compute the Chow groups of ''X'', because they give strong information about all subvarieties of ''X''. The Chow groups of ''X'' have some of the formal properties of Borel–Moore homology in topology, but some things are missing. For example, for a closed subscheme ''Z'' of ''X'', there is an exact sequence of Chow groups, the localization sequence :CH_i(Z) \rightarrow CH_i(X) \rightarrow CH_i(X-Z) \rightarrow 0, whereas in topology this would be part of a long exact sequence. This problem was resolved by generalizing Chow groups to a bigrad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm Residue Isomorphism Theorem
In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor ''K''-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic forms, algebraic K-theory and the theory of motives. The theorem asserts that a certain statement holds true for any prime \ell and any natural number n. John MilnorMilnor (1970) speculated that this theorem might be true for \ell=2 and all n, and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato and became known as the Bloch–Kato conjecture or the motivic Bloch–Kato conjecture to distinguish it from the Bloch–Kato conjecture on values of ''L''-functions.Bloch, Spencer and Kato, Kazuya, "L-functions and Tamagawa numbers of motives", The Grothendieck Festschrift, Vol. I, 333–400, Progr. M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
