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Rigidity (K-theory)
In mathematics, rigidity of ''K''-theory encompasses results relating algebraic ''K''-theory of different rings. Suslin rigidity ''Suslin rigidity'', named after Andrei Suslin, refers to the invariance of mod-''n'' algebraic ''K''-theory under the base change between two algebraically closed fields: showed that for an extension :E / F of algebraically closed fields, and an algebraic variety ''X'' / ''F'', there is an isomorphism :K_*(X, \mathbf Z/n) \cong K_*(X \times_F E, \mathbf Z/n), \ i \ge 0 between the mod-''n'' ''K''-theory of coherent sheaves on ''X'', respectively its base change to ''E''. A textbook account of this fact in the case ''X'' = ''F'', including the resulting computation of ''K''-theory of algebraically closed fields in characteristic ''p'', is in . This result has stimulated various other papers. For example show that the base change functor for the mod-''n'' stable A1-homotopy category :\mathrm(F, \mathbf Z/n) \to \mathrm(E, \mathbf Z/n) is fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the ''K''-groups of the integers. ''K''-theory was discovered in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only ''K''0, the zeroth ''K''-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic ''K''-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrei Suslin
Andrei Suslin (russian: Андре́й Алекса́ндрович Су́слин, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee Chair and Professor of mathematics at Northwestern University. He was born on 27 December 1950 in St. Petersburg, Russia. As a youth, he was an "all Leningrad" gymnast. He received his PhD from Leningrad University in 1974; his thesis was titled ''Projective modules over polynomial rings''. In 1976 he and Daniel Quillen independently proved Serre's conjecture about the triviality of algebraic vector bundles on affine space. In 1982 he and Alexander Merkurjev proved the Merkurjev–Suslin theorem on the norm residue homomorphism in Milnor K2-theory, with applications to the Brauer group. Suslin was an invited speaker at the International Congress of Mathematicians in 1978 and 1994, and he gave a plenary invited address at the Congre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed Field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation ''x''2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field ''F'' is algebraically closed, because if ''a''1, ''a''2, ..., ''an'' are the elements of ''F'', then the polynomial (''x'' − ''a''1)(''x'' − ''a''2) ⋯ (''x'' − ''a''''n'') + 1 has no zero in ''F''. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraicall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ''F''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L have the same zero eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A1 Homotopy Theory
A1, A-1, A01 or A.1. may refer to: Education * A1, the Basic Language Certificate of the Common European Framework of Reference for Languages * Language A1, the former name for "Language A: literature", one of the IB Group 1 subjects * A1, a secondary school subdivision in the Education in the Republic of the Congo, Congolese education system * A1, a baccalauréat series in the education system of some parts of France * A1, a baccalaureate in the Gabonese education system, see Education in Gabon * A1, the highest category of Qualified Flying Instructor in the Central Flying School of the UK Royal Air Force Grades * A1, a grade for the Irish Leaving Certificate, Leaving Certificate, a qualification in the education system of Ireland * A1, the highest obtainable grade for the Sijil Pelajaran Malaysia examination in Malaysia * A1, a grade for the Senior Secondary Certificate Examination in Nigeria, see Education in Nigeria * A1, a grade for the Singaporean GCE 'O' Level, an ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henselian Ring
In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now restrict them to be commutative. Some standard references for Hensel rings are , , and . Definitions In this article rings will be assumed to be commutative, though there is also a theory of non-commutative Henselian rings. * A local ring ''R'' with maximal ideal ''m'' is called Henselian if Hensel's lemma holds. This means that if ''P'' is a monic polynomial in ''R'' 'x'' then any factorization of its image ''P'' in (''R''/''m'') 'x''into a product of coprime monic polynomials can be lifted to a factorization in ''R'' 'x'' * A local ring is Henselian if and only if every finite ring extension is a product of local rings. * A Henselian local ring is called strictly Henselian if its residue field is separably closed. * By abuse of termino ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue Field
In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a local ring and ''m'' is then its unique maximal ideal. This construction is applied in algebraic geometry, where to every point ''x'' of a scheme ''X'' one associates its residue field ''k''(''x''). One can say a little loosely that the residue field of a point of an abstract algebraic variety is the 'natural domain' for the coordinates of the point. Definition Suppose that ''R'' is a commutative local ring, with maximal ideal ''m''. Then the residue field is the quotient ring ''R''/''m''. Now suppose that ''X'' is a scheme and ''x'' is a point of ''X''. By the definition of scheme, we may find an affine neighbourhood ''U'' = Spec(''A''), with ''A'' some commutative ring. Considered in the neighbourhood ''U'', the point ''x'' correspond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Map
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Homology
In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize the de Rham (co)homology of manifolds. These notions were independently introduced by Boris Tsygan (homology) and Alain Connes (cohomology) in the 1980s. These invariants have many interesting relationships with several older branches of mathematics, including de Rham theory, Hochschild (co)homology, group cohomology, and the K-theory. Contributors to the development of the theory include Max Karoubi, Yuri L. Daletskii, Boris Feigin, Jean-Luc Brylinski, Mariusz Wodzicki, Jean-Louis Loday, Victor Nistor, Daniel Quillen, Joachim Cuntz, Ryszard Nest, Ralf Meyer, and Michael Puschnigg. Hints about definition The first definition of the cyclic homology of a ring ''A'' over a field of characteristic zero, denoted :''HC''''n''(''A'') or ''H''''n''λ(''A''), proceeded by the means of the following explicit c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-groups Of A Field
In mathematics, especially in algebraic ''K''-theory, the algebraic ''K''-group of a field is important to compute. For a finite field, the complete calculation was given by Daniel Quillen. Low degrees The map sending a finite-dimensional ''F''-vector space to its dimension induces an isomorphism :K_0(F) \cong \mathbf Z for any field ''F''. Next, :K_1(F) = F^\times, the multiplicative group of ''F''. The second K-group of a field is described in terms of generators and relations by Matsumoto's theorem. Finite fields The K-groups of finite fields are one of the few cases where the K-theory is known completely: for n \ge 1, :K_n(\mathbb_q) = \pi_n(BGL(\mathbb_q)^+) \simeq \begin \mathbb/, & \textn = 2i - 1 \\ 0, & \textn\text \end For ''n''=2, this can be seen from Matsumoto's theorem, in higher degrees it was computed by Quillen in conjunction with his work on the Adams conjecture. A different proof was given by . Local and global fields surveys the computations of K-the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
