In mathematicstopological K -theory how to make an inductor for a welder a branch of algebraic topology. It was founded to study vector bundles on topological spacesby means of ideas now recognised as general K-theory that were introduced by Alexander Grothendieck. Tensor product of bundles gives K -theory a commutative ring structure. The remaining discussion is focused on complex K -theory.
As a first example, note that the K -theory of a point are the integers. This is because vector bundles over a point are trivial and thus classified by their rank and the Grothendieck group of the natural numbers is the integers. This reduced theory is intuitively K X modulo trivial bundles.
It is defined as the group of stable equivalence classes of bundles. This equivalence relation results in a group since every vector bundle can be completed to a trivial bundle by summing with its orthogonal complement. K -theory forms a multiplicative generalized cohomology theory as follows. The short exact sequence of a pair of pointed spaces XA. Let S n be the n -th reduced suspension of a space and then define. Negative indices are chosen so that the coboundary maps increase dimension.
Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers. The phenomenon of periodicity named after Raoul Bott see Bott periodicity theorem can be formulated this way:. In real K -theory there is a similar periodicity, but modulo 8. The two most famous applications of topological K -theory are both due to Frank Adams.
First he solved the Hopf invariant one problem by doing a computation with his Adams operations. Then he proved an upper bound for the number of linearly independent vector fields on spheres.
In particular, they showed that there exists a homomorphism. From Wikipedia, the free encyclopedia. Retrieved 27 July Categories : K-theory. Namespaces Article Talk.
Views Read Edit View history. Help Community portal Recent changes Upload file. Download as PDF Printable version.In mathematicsthe Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over Kwith addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer. The Brauer group arose out of attempts to classify division algebras over a field.
It can also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebrasor equivalently using projective bundles. If we look just at Dthat is, if we impose an equivalence relation identifying M mD with M nD for all positive integers m and nwe get the Brauer equivalence relation on CSAs over K. It turns out that this is always central simple.
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A slick way to see this is to use a characterization: a central simple algebra A over K is a K -algebra that becomes a matrix ring when we extend the field of scalars to an algebraic closure of K. This result also shows that the dimension of a central simple algebra A as a K -vector space is always a square. The degree of A is defined to be the square root of its dimension. The Brauer group of any field is a torsion group. In more detail, define the period of a central simple algebra A over K to be its order as an element of the Brauer group.
Define the index of A to be the degree of the division algebra that is Brauer equivalent to A. Then the period of A divides the index of A and hence is finite. Another important interpretation of the Brauer group of a field K is that it classifies the projective varieties over K that become isomorphic to projective space over an algebraic closure of K.From algebraic K-theory to motivic cohomology and back - Marc Levine - Лекториум
For example, the Severi—Brauer varieties of dimension 1 are exactly the smooth conics in the projective plane over K. The corresponding central simple algebra is the quaternion algebra . The conic is isomorphic to the projective line P 1 over K if and only if the corresponding quaternion algebra is isomorphic to the matrix algebra M 2, K.
For nonzero elements a and b of Kthe associated cyclic algebra is the central simple algebra of degree n over K defined by. Cyclic algebras are the best-understood central simple algebras. The Merkurjev—Suslin theorem in algebraic K-theory has a strong consequence about the Brauer group. Namely, for a positive integer nlet K be a field in which n is invertible such that K contains a primitive n th root of unity.
Then the subgroup of the Brauer group of K killed by n is generated by cyclic algebras of degree n. Even for a prime number pthere are examples showing that a division algebra of period p need not be actually isomorphic to a tensor product of cyclic algebras of degree p.
It is a major open problem raised by Albert whether every division algebra of prime degree over a field is cyclic. This is true if the degree is 2 or 3, but the problem is wide open for primes at least 5. The known results are only for special classes of fields. For example, if K is a global field or local fieldthen a division algebra of any degree over K is cyclic, by Albert— Brauer — Hasse — Noether. For any central simple algebra A over a field Kthe period of A divides the index of Aand the two numbers have the same prime factors.
For example, if A is a central simple algebra over a local field or global field, then Albert—Brauer—Hasse—Noether showed that the index of A is equal to the period of A. The Brauer group plays an important role in the modern formulation of class field theory. The case of a global field K such as a number field is addressed by global class field theory.
This defines a homomorphism from the Brauer group of K into the Brauer group of K v. A given central simple algebra D splits for all but finitely many vso that the image of D under almost all such homomorphisms is 0. The Brauer group Br K fits into an exact sequence constructed by Hasse:  . The injectivity of the left arrow is the content of the Albert—Brauer—Hasse—Noether theorem.
The fact that the sum of all local invariants of a central simple algebra over K is zero is a typical reciprocity law. For example, applying this to a quaternion algebra ab over Q gives the quadratic reciprocity law. For an arbitrary field Kthe Brauer group can be expressed in terms of Galois cohomology as follows: .
The isomorphism of the Brauer group with a Galois cohomology group can be described as follows.In mathematics, twisted K-theory also called K-theory with local coefficients is a variation on K-theorya mathematical theory from the s that spans algebraic topologyabstract algebra and operator theory.
More specifically, twisted K-theory with twist H is a particular variant of K-theory, in which the twist is given by an integral 3-dimensional cohomology class. It is special among the various twists that K-theory admits for two reasons. First, it admits a geometric formulation. This was provided in two steps; the first one was done in Publ.
In physics, it has been conjectured to classify D-branesRamond-Ramond field strengths and in some cases even spinors in type II string theory. For more information on twisted K-theory in string theorysee K-theory physics. In the broader context of K-theory, in each subject it has numerous isomorphic formulations and, in many cases, isomorphisms relating definitions in various subjects have been proven.
It also has numerous deformations, for example, in abstract algebra K-theory may be twisted by any integral cohomology class.
A slightly more complicated way of saying the same thing is as follows. Then the group of maps. This more complicated construction of ordinary K-theory is naturally generalized to the twisted case.
It has an addition, but it is no longer closed under multiplication. In particular twisted K-theory is a module over classical K-theory. Physicist typically want to calculate twisted K-theory using the Atiyah—Hirzebruch spectral sequence.
Thus the even and odd cohomologies are both isomorphic to the integers. Thus its entire domain is its kernel, and nothing is in its image. This leaves the original odd cohomology, which is again the integers.
In conclusion. There is an extension of this calculation to the group manifold of SU 3. From Wikipedia, the free encyclopedia. String theory. Strings History of string theory First superstring revolution Second superstring revolution String theory landscape. T-duality S-duality U-duality Montonen—Olive duality.
Kaluza—Klein theory Compactification Why 10 dimensions?
For a detailed discussion of an algorithm that proceeds essentially along these lines 'Tate's methodsee the paper. For the even K-groups I don't know of any such description. I think Kolster's survey is a good introduction to questions related to arithmetic interpretations of higher K-groups. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.
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Andy Putman. Andy Putman Andy Putman You should look it up, especially as it is now probably a theorem after the work of Voedvodsky and Rost. Active Oldest Votes. Minhyong Kim Minhyong Kim I agree. Nevertheless I thought Tate's theorem might be good to see for those not familiar with the subject. As other answers and comments here indicate, the subject hasn't reached a definitive state yet. Franz Lemmermeyer Franz Lemmermeyer Charles Matthews 12k 28 28 silver badges 61 61 bronze badges.
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Dirichlet's unit theorem
These exceptional sequences yield lower bounds for Lenstra's constant which are large enough to prove the existence of 42 new Euclidean number fields of degree 8 to The aim of the second part of this thesis is proving upper bounds for the torsion part of the K-groups of a number field ring of integers.
A method due to C. Firstly we describe some properties of rank one hermitian lattices, especially of ideal lattices. Secondly we apply these properties to arbitrary rank hermitian lattices and this implies a significant improvement of the upper bounds for their invariants and accordingly for the torsion of K-groups.
The progress mainly achieves much lower contributions of the number field attributes, particularly the degree and the absolute discriminant. Information Usage statistics Files. Back to search. Record createdlast modified Rate this document: Rate this document: 1 2 3 4 5.In mathematicsK-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme.
In algebraic topologyit is a cohomology theory known as topological K-theory. In algebra and algebraic geometryit is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.
K-theory involves the construction of families of K - functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes.
As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes.
Examples of results gleaned from the K-theory approach include the Grothendieck—Riemann—Roch theoremBott periodicitythe Atiyah—Singer index theoremand the Adams operations. In high energy physicsK-theory and in particular twisted K-theory have appeared in Type II string theory where it has been conjectured that they classify D-branesRamond—Ramond field strengths and also certain spinors on generalized complex manifolds. In condensed matter physics K-theory has been used to classify topological insulatorssuperconductors and stable Fermi surfaces.
For more details, see K-theory physics. The Grothendieck completion is a necessary ingredient for constructing K-theory.
Equivalence classes in this group should be thought of as formal differences of elements in the abelian monoid. This implies. Another useful observation is the invariance of equivalence classes under scaling:. For example, we can see from the scaling invariance that. There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry. Since isomorphism classes of vector bundles behave well with respect to direct sumswe can write these operations on isomorphism classes by.
We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. In algebraic geometry, the same construction can be applied to algebraic vector bundles over a smooth scheme. Using the Grothendieck—Riemann—Roch theoremwe have that. The subject can be said to begin with Alexander Grothendieckwho used it to formulate his Grothendieck—Riemann—Roch theorem. It takes its name from the German Klassemeaning "class".
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Rather than working directly with the sheaves, he defined a group using isomorphism classes of sheaves as generators of the group, subject to a relation that identifies any extension of two sheaves with their sum. The resulting group is called K X when only locally free sheaves are used, or G X when all are coherent sheaves. Either of these two constructions is referred to as the Grothendieck group ; K X has cohomological behavior and G X has homological behavior.In mathematicsalgebraic K-theory is an important part of homological algebra concerned with defining and applying a sequence.
Indeed, the lower groups are more accessible, and have more applications, than the higher groups. The theory of the higher K-groups is noticeably deeper, and certainly much harder to compute even when R is the ring of integers.
The group K 0 R generalises the construction of the ideal class group of a ring, using projective modules. Its development in the s and s was linked to attempts to solve a conjecture of Serre on projective modules that now is the Quillen—Suslin theorem ; numerous other connections with classical algebraic problems were found in this era. Similarly, K 1 R is a modification of the group of units in a ring, using elementary matrix theory.
The group K 1 R is important in topologyespecially when R is a group ringbecause its quotient the Whitehead group contains the Whitehead torsion used to study problems in simple homotopy theory and surgery theory ; the group K 0 R also contains other invariants such as the finiteness invariant. Since the s, algebraic K -theory has increasingly had applications to algebraic geometry. For example, motivic cohomology is closely related to algebraic K -theory.
Alexander Grothendieck discovered K-theory in the mids as a framework to state his far-reaching generalization of the Riemann—Roch theorem. Within a few years, its topological counterpart was considered by Michael Atiyah and Friedrich Hirzebruch and is now known as topological K-theory. Applications of K -groups were found from onwards in surgery theory for manifoldsin particular; and numerous other connections with classical algebraic problems were brought out. A little later a branch of the theory for operator algebras was fruitfully developed, resulting in operator K-theory and KK-theory.
It also became clear that K -theory could play a role in algebraic cycle theory in algebraic geometry Gersten's conjecture :  here the higher K-groups become connected with the higher codimension phenomena, which are exactly those that are harder to access. The problem was that the definitions were lacking or, too many and not obviously consistent. Using Robert Steinberg 's work on universal central extensions of classical algebraic groups, John Milnor defined the group K 2 A of a ring A as the center, isomorphic to H 2 E AZof the universal central extension of the group E A of infinite elementary matrices over A.
Definitions below. Eventually the foundational difficulties were resolved leaving a deep and difficult theory by Template:Harvswho gave several definitions of K n A for arbitrary non-negative nvia the plus construction and the Q -construction. The lower K-groups were discovered first, and given various ad hoc descriptions, which remain useful. Throughout, let A be a ring. The functor K 0 takes a ring A to the Grothendieck group of the set of isomorphism classes of its finitely generated projective modulesregarded as a monoid under direct sum.
If the ring A is commutative, we can define a subgroup of K 0 A as the set. If B is a ring without an identity elementwe can extend the definition of K 0 as follows. An algebro-geometric variant of this construction is applied to the category of algebraic varieties ; it associates with a given algebraic variety X the Grothendieck's K-group of the category of locally free sheaves or coherent sheaves on X. Given a compact topological space Xthe topological K-theory K top X of real vector bundles over X coincides with K 0 of the ring of continuous real-valued functions on X.
The relative K-group is defined in terms of the "double" . The independence from A is an analogue of the Excision theorem in homology. If A is a commutative ring, then the tensor product of projective modules is again projective, and so tensor product induces a multiplication turning K 0 into a commutative ring with the class [ A ] as identity.