K-theory
The mathematical subfield of K- theory deals with the study of vector bundles on topological spaces ( topological K-theory ) or rings or schemes ( algebraic K- theory). The name K-theory was created by Alexander Grothendieck; the K stands for " class " in a very general sense.
- 2.1 Low Dimensions 2.1.1 K0 2.1.1.1 properties
- 2.1.1.2 Examples
- 2.2.1 Examples
- 2.2.2 Milnorvermutung
- 2.3.1 Classifying spaces of categories
- 2.3.2 Quillens Q-Construction
- 2.3.3 The K- groups 2.3.3.1 Examples 2.3.3.1.1 Finite Fields
- 2.3.3.1.2 The integers
- 2.3.3.1.3 group rings
- 2.3.3.1.4 Number Fields and wholeness rings
- 3.1 K0
- 3.2 K1
- 3.3 Cyclic sequence
- 3.4 Further properties 3.4.1 functoriality
- 3.4.2 homotopy
- 3.4.3 stability
Topological K-theory
Definitions
Let X be a fixed compact Hausdorff space.
Then K ( X) is the quotient of the free abelian group on the isomorphism classes of complex vector bundles over X by the subgroup of elements of the form
For vector bundles E, F is generated. This construction, which is modeled on the structure of the integers from the natural numbers is, Grothendieck group (after Alexander Grothendieck ). If we consider instead real vector bundle, you get the real K-theory KO (X).
Two vector bundles E and F on X if and only define the same element in K ( X), if they are equivalent stable, ie if there is a trivial vector bundles G, so that
With the tensor product of vector bundles K (X ) to a commutative ring with unit element.
The concept of the rank of a vector bundle carries over to elements of K- theory. The reduced K-theory is the subgroup of elements of rank 0 Next one introduces the designation; while S denotes the reduced device to attach.
Properties
- K is a contravariant functor on the category of compact Hausdorff spaces.
- There is a topological space BU, so that elements of K (X ) correspond to homotopy classes of maps X → BU.
- There is a natural ring homomorphism K ( X) → H * (X, Q), the Chern character.
Bott periodicity
Named after Raoul Bott Periodizitätsphänomen can be formulated in the following ways:
- And this is the class of the tautological bundle.
- .
In the real K-theory there is a similar periodicity with period 8
Complex and Real K-theory
The above-defined functor is also referred to as complexes K theory. When performing the analog structures with real vector bundles obtained the real K-theory. For these Bott periodicity applies with period, ie.
Calculation
The ( complex or real ) topological K-theory can often be computed using the Atiyah - Hirzebruch spectral sequence.
Algebraic K-theory
A is always a unitary ring.
Low dimensions
K0
The functor K0 is a covariant functor from the category of rings with unit element in the category of groups; he orders a ring to the Grothendieck group of isomorphism classes of finitely generated projective modules. Occasionally one considers the reduced K- group, this is the quotient of by the free - module generated by the cyclic group.
Properties
- ( Morita invariance )
For each ring, and there is a canonical isomorphism.
- ( Serre- Swan theorem)
Be a compact Hausdorff space and the ring of continuous functions. Then there is an isomorphism between topological K- theory of space and algebraic K-theory of the ring.
Examples
- If A is a Dedekind ring, then
- For Body, principal ideal rings and local rings all projective modules are free, the K-theory is therefore isomorphic to.
K1
Hyman Bass proposed the following definition of a functor K1 ago: K1 ( A) is the Abelisierung the infinite general linear group:
It is
Where GLn (A) shall embedded in the upper left corner of GLn 1 ( A).
See also the lemma by Whitehead. For a field k K1 ( k) the unit group.
K2
J. Milnor found the right candidate for K2: It is the Steinberg group ( according to Robert Steinberg ) St ( A) of a ring A is defined as the group with the generators xij (r) for positive integers i ≠ j and ring elements r and the relations
These relations are also valid for the elementary matrices, so there is a group homomorphism
K2 ( A) is then, by definition, the core of this figure. It can be shown to coincide with the center of St (A). K1 and K2 are defined by the exact sequence
Connected.
For a ( commutative ) field k the set of Matsumoto applies
Milnor K-theory
J. Milnor defined for a field k "higher" K- groups by
Ie as the quotient of the tensor algebra graduated components on the abelian group k × by the two-sided ideal, which of the elements of the form
Is generated for a ≠ 0.1. For n = 0,1,2 the vote milnorschen K- groups coincide with those defined above. The motivation for this definition comes from the theory of quadratic forms. There is a natural homomorphism, its cokernel is by definition the irreducible K- theory. For number fields.
Examples
For a finite field k and n ≠ 0.1
For an algebraic number field k and n ≠ 0,1,2 applies
Wherein the number of the real points of k.
Milnorvermutung
There are isomorphisms
Awarded between milnorschen K- groups of a field k of characteristic not equal to two and the Galoiskohomologie or the graduated Witt ring of k was among other things for the proof of this known as Milnorvermutung outcome Vladimir Wojewodski on the international Congress of Mathematicians 2002, the Field Medal. The proof is based on the homotopy theory of algebraic varieties developed by Wojewodski and designed by Beilinson and Lichtenbaum motivic cohomology.
Quillens K-theory
The most comprehensive definition of a K- theory was given by D. Quillen.
Classifying spaces of categories
For a small category C, the nerve NC is defined as the simplicial set whose p- simplices the diagrams
Are. The geometric realization is called classifying space BC of NC of C.
Quillens Q-Construction
Let P be an exact category, that is, an additive category together with a class E of "exact" charts
Apply to certain axioms that are the properties of short exact sequences in an abelian category modeled.
For an exact category P the category QP now be defined as the category whose objects are the same as those of P and whose morphisms between two objects M 'and M " isomorphism classes of exact graphs
Are.
The K- groups
The i-th group of K- P is then defined by
With a fixed chosen zero object 0, where are the (higher) homotopy groups.
Coincides with the Grothendieck group of consistent, ie the quotient of the free abelian group on the isomorphism classes in according to the subgroup of
For charts
Is produced in E.
A unitary ring A, the K groups Ki ( A), the newly defined class of K groups of the finitely generated projective O modules.
For Noetherian rings also the unitary groups K'i (A) be defined as the K- groups of the category of all finitely generated A -modules.
For schemas defined Quillen, the category of vector bundles on.
Examples
Be the body with elements. Then
For the groups of
If so is a finite group and, then the direct sum of a finite group and. Using the grid - Voevodsky theorem we can determine for even the Torsionsanteil. For if is the Kummer- Vandiver conjecture is correct.
The Farrell -Jones Conjecture describes the algebraic K- theory of the group ring, if you know the algebraic K-theory of the ring. It is proven in several special cases, for example, CAT (0 ) groups.
The algebraic K- theory of the group ring of fundamental groups has applications in algebraic topology. Walls Endlichkeits obstruction for CW complexes is an element in. The obstruction for the simplicity of a homotopy equivalence is the Whitehead torsion in.
Be a number field with real and complex embeddings. Be the wholeness of the ring. Then for all:
The isomorphisms are realized by the Borel regulator.
For is.
K-theory for Banach algebras
The topological K-theory can be extended to general Banach algebras, where the C *-algebras play an important role. The topological K-theory of compact spaces can be reformulated as a K- theory of Banach algebras of continuous functions, and then transferred to any Banach algebras, even on the unit element of the algebra can be dispensed with. Since the assignment is a kontravianter functor from the category of compact Hausdorff spaces into the category of Banach algebras and since the topological K-theory is also contravariant, we get a total of a covariant functor from the category of Banach algebras in the category of abelian groups.
Since there can also non-commutative algebras occur, it is called non- commutative topology. The K-theory is an important object of study in the theory of C * - algebras. Below is a - Banach algebra, go out by the adjunction of a unit element out.
K0
The vector bundles of topological K-theory correspond to the algebraic side of the finitely generated, projective modules and these are direct summands in free modules, so can be described by idempotents over a sufficiently large matrix algebra. For the idempotents, there are several suitable equivalence terms, all together fall when you go to the inductive limit, with equivalent idempotents belong to stable - isomorphic projective modules. One possible definition is that two idempotents and equivalent means if there is a so with that exist and elements. The equivalence class of is denoted by. Did you replace two idempotents and so you can for instance by an equivalent idempotents, so then is a idempotents again. Substituting so by a well-defined semigroup link on the set of equivalence classes is given by idempotents of. Of these, one might again form the corresponding Grothendieck group, but the definition of the group shall be a small technical change prior to even algebras without identity, about ideals in Banach algebras, to be able to adequately treat. One defines a subset of the Grothendieck group of, as a set of all differences, which are idempotent, so that.
If a two -sided closed ideal, so is obtained from the short exact sequence
Exact sequence
Which are generally neither to the left nor can continue exactly with 0 to the right.
The definition is designed to apply for compact spaces. In the case of C *-algebras can in the above construction, the idempotents by projections, that is, by self-adjoint idempotents, replace and get the same result, since each idempotents is equivalent to a projection. As an important application can be made using the K0 AF C *-algebras classified.
K1
To define, we define as a set of all invertible matrices whose image in the quotient algebra is equal to the unit matrix. by means of
We take as a subset of, and provided the so resulting inductive limit with the final topology. The connected component of the identity is a normal subgroup and we define
Despite the non- commutativity of the Matrizenalgebren as defined group turns out to be commutative. While in algebraic K-theory to define the K1 - group Kommutatoruntergruppe is divided out ( Abelisierung ), is used in the topological K-theory for Banach algebras the connected component of the identity. In the case of con C *-algebras one can replace in the above construction, the invertible elements by unitary elements and get the same result.
If a two -sided closed ideal, so is obtained from the short exact sequence
Exact sequence
Which are generally neither to the left nor can continue exactly with 0 to the right.
Again, the definition is designed to apply for compact spaces. If we denote by the Banach algebra of all continuous functions which vanish at infinity, equipped with the supremum norm, it can be shown. This is called the suspension of; it is the Banachachalgebrenversion the suspension or reduction device to attach topological spaces. Iteration means of the suspension could be defined high K groups, for example, but due to the valid here Bott periodicity is not required.
Cyclic sequence
As in the topological K-theory one can construct an index image and a Bott isomorphism, so that together above exact sequences to the following cyclic exact sequence:
This sequence is very useful for the calculation of K groups. Are some groups of the sequence is known, this can be due to the accuracy of conclusions about the unknown.
Other properties
Functoriality
It is a continuous homomorphism between Banach algebras. This defines homomorphisms that are compatible with the above constructions of the K- groups and thus lead to group homomorphisms and. This will and covariant functors between the category of Banach algebras and the category of abelian groups.
Homotopy
Two continuous homomorphisms between Banach algebras are called homotopic if there is a family of homomorphisms such that for each is and is steadily increasing. Homotopic morphisms induce the same group homomorphisms between K- groups.
Stability
If a Banach algebra, then for all and. If an inductive limit in the category of Banach algebras, the following applies
The compatibility with the formation of the inductive limit follows directly from the construction of the K - groups by means of inductive limits.
Is specifically for C * - algebras and the inductive limit in the category of C * is isomorphic to the tensor Algreben, where the C * - algebra of compact operators is a separable Hilbert space. This applies to.