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Rost Invariant
In mathematics, the Rost invariant is a cohomological invariant of an absolutely simple simply connected algebraic group ''G'' over a field ''k'', which associates an element of the Galois cohomology group H3(''k'', Q/Z(2)) to a principal homogeneous space for ''G''. Here the coefficient group Q/Z(2) is the tensor product of the group of roots of unity of an algebraic closure of ''k'' with itself. first introduced the invariant for groups of type F4 and later extended it to more general groups in unpublished work that was summarized by . The Rost invariant is a generalization of the Arason invariant. Definition Suppose that ''G'' is an absolutely almost simple simply connected algebraic group over a field ''k''. The Rost invariant associates an element ''a''(''P'') of the Galois cohomology group H3(''k'',Q/Z(2)) to a ''G''-torsor ''P''. The element ''a''(''P'') is constructed as follows. For any extension ''K'' of ''k'' there is an exact sequence :0\rightarrow H^3(K,\mathbf/\mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomological Invariant
In mathematics, a cohomological invariant of an algebraic group ''G'' over a field is an invariant of forms of ''G'' taking values in a Galois cohomology group. Definition Suppose that ''G'' is an algebraic group defined over a field ''K'', and choose a separably closed field containing ''K''. For a finite extension ''L'' of ''K'' in let Γ''L'' be the absolute Galois group of ''L''. The first cohomology H1(''L'', ''G'') = H1(Γ''L'', ''G'') is a set classifying the ``G''-torsors over ''L'', and is a functor of ''L''. A cohomological invariant of ''G'' of dimension ''d'' taking values in a Γ''K''-module ''M'' is a natural transformation of functors (of ''L'') from H1(L, ''G'') to H''d''(L, ''M''). In other words a cohomological invariant associates an element of an abelian cohomology group to elements of a non-abelian cohomology set. More generally, if ''A'' is any functor from finitely generated extensions of a field to sets, then a cohomological invariant of ''A'' of dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Simple Group
In mathematics, in the field of group theory, a group (mathematics), group is said to be absolutely simple if it has no proper nontrivial serial subgroups.. That is, G is an absolutely simple group if the only serial subgroups of G are \ (the trivial subgroup), and G itself (the whole group). In the finite case, a group is absolutely simple if and only if it is simple group, simple. However, in the infinite case, absolutely simple is a stronger property than simple. The property of being strictly simple is somewhere in between. See also * Ascendant subgroup * Strictly simple group References Properties of groups {{group-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem states ... [...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] |
Tensor Product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted v \otimes w. An element of the form v \otimes w is called the tensor product of and . An element of V \otimes W is a tensor, and the tensor product of two vectors is sometimes called an ''elementary tensor'' or a ''decomposable tensor''. The elementary tensors span V \otimes W in the sense that every element of V \otimes W is a sum of elementary tensors. If bases are given for and , a basis of V \otimes W is formed by all tensor products of a basis element of and a basis element of . The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V\times W into another vector space factors uniquely through a linear map V\otimes W\to Z (see Universal property). Tenso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of Unity
In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. Roots of unity can be defined in any field (mathematics), field. If the characteristic of a field, characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, converse (logic), conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a number satisfying the equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arason Invariant
In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial Discriminant#Discriminant of a quadratic form, discriminant and Clifford invariant over a Field (mathematics), field ''k'' of Characteristic (algebra), characteristic not 2, taking values in H3(''k'',Z/2Z). It was introduced by . The Rost invariant is a generalization of the Arason invariant to other algebraic groups. Definition Suppose that ''W''(''k'') is the Witt ring (forms), Witt ring of quadratic forms over a field ''k'' and ''I'' is the ideal of forms of even dimension. The Arason invariant is a group homomorphism from ''I''3 to the Galois cohomology group H3(''k'',Z/2Z). It is determined by the property that on the 8-dimensional diagonal form with entries 1, –''a'', –''b'', ''ab'', -''c'', ''ac'', ''bc'', -''abc'' (the 3-fold Pfister form«''a'',''b'',''c''») it is given by the cup product of the classes of ''a'', ''b'', ''c'' in H1(''k'',Z/2Z) = '' ... [...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 of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
