“K-theory”的意思、由来-开放百科全书

In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it 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.^[1]

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 theorem, Bott periodicity, the Atiyah–Singer index theorem, and the Adams operations.

In high energy physics, K-theory and in particular twisted K-theory have appeared in Type II string theory where it has been conjectured that they classify D-branes, Ramond–Ramond field strengths and also certain spinors on generalized complex manifolds. In condensed matter physics K-theory has been used to classify topological insulators, superconductors and stable Fermi surfaces. For more details, see K-theory (physics).

Grothendieck completion

The Grothendieck completion is a necessary ingredient for constructing K-theory. Given an abelian monoid

let

be the relation on

defined by

Equivalence classes in this group should be thought of as formal differences of elements in the abelian monoid.

To get a better understanding of this group, consider some equivalence classes of the abelian monoid

. Here we will denote the identity element by

. First,

for any

since we can set

and apply the equation from the equivalence relation to get

. This implies

hence we have an additive inverse for each element in

. This should give us the hint that we should be thinking of the equivalence classes

as formal differences

. Another useful observation is the invariance of equivalence classes under scaling:

The Grothendieck completion can be viewed as a functor

, and it has the property that it is left adjoint to the corresponding forgetful functor

. That means that, given a morphism

of an abelian monoid

to the underlying abelian monoid of an abelian group

, there exists a unique abelian group morphism

An illustrative example to look at is the Grothendieck completion of

. We can see that

. For any pair

we can find a minimal representative

by using the invariance under scaling. For example, we can see from the scaling invariance that

This shows that we should think of the

as positive integers and the

as negative integers.

Definitions

There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry.

Given a compact Hausdorff space

consider the set of isomorphism classes of finite-dimensional vector bundles over

, denoted

and let the isomorphism class of a vector bundle

be denoted

. Since isomorphism classes of vector bundles behave well with respect to direct sums, we can write these operations on isomorphism classes by

It should be clear that

is an abelian monoid where the unit is given by the trivial vector bundle

. We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. This is called the K-theory of

and is denoted

We can use the Serre-Swan theorem and some algebra to get an alternative description of vector bundles over the ring of continuous complex valued functions

as projective modules. Then, these can be identified with idempotent matrices in some ring of matrices

. We can define equivalence classes of idempotent matrices and form an abelian monoid

. Its Grothendieck completion is also called

In algebraic geometry, the same construction can be applied to algebraic vector bundles over a smooth scheme. But, there is an alternative construction for any Noetherian scheme

. If we look at the isomorphism classes of coherent sheaves

we can mod out by the relation

if there is a short exact sequence

This gives the Grothendieck-group

which is isomorphic to

is smooth. The group

is special because there is also a ring structure: we define it as

Early history

The subject can be said to begin with Alexander Grothendieck (1957), who used it to formulate his Grothendieck–Riemann–Roch theorem. It takes its name from the German Klasse, meaning "class".^[2] Grothendieck needed to work with coherent sheaves on an algebraic variety X. 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.

If X is a smooth variety, the two groups are the same. If it is a smooth affine variety, then all extensions of locally free sheaves split, so the group has an alternative definition.

In topology, by applying the same construction to vector bundles, Michael Atiyah and Friedrich Hirzebruch defined K(X) for a topological space X in 1959, and using the Bott periodicity theorem they made it the basis of an extraordinary cohomology theory. It played a major role in the second proof of the Index Theorem (circa 1962). Furthermore, this approach led to a noncommutative K-theory for C*-algebras.

Already in 1955, Jean-Pierre Serre had used the analogy of vector bundles with projective modules to formulate Serre's conjecture, which states that every finitely generated projective module over a polynomial ring is free; this assertion is correct, but was not settled until 20 years later. (Swan's theorem is another aspect of this analogy.)

Developments

The other historical origin of algebraic K-theory was the work of J. H. C. Whitehead and others on what later became known as Whitehead torsion.

There followed a period in which there were various partial definitions of higher K-theory functors. Finally, two useful and equivalent definitions were given by Daniel Quillen using homotopy theory in 1969 and 1972. A variant was also given by Friedhelm Waldhausen in order to study the algebraic K-theory of spaces, which is related to the study of pseudo-isotopies. Much modern research on higher K-theory is related to algebraic geometry and the study of motivic cohomology.

The corresponding constructions involving an auxiliary quadratic form received the general name L-theory. It is a major tool of surgery theory.

In string theory, the K-theory classification of Ramond–Ramond field strengths and the charges of stable D-branes was first proposed in 1997.^[3]

Examples

Applications

Virtual bundles

One useful application of the Grothendieck-group is to define virtual vector bundles. For example, if we have an embedding of smooth spaces

then there is a short exact sequence

where

is the conormal bundle of

. If we have a singular space

embedded into a smooth space

we define the virtual conormal bundle as

Another useful application of virtual bundles is with the definition of a virtual tangent bundle of an intersection of spaces: Let

be projective subvarieties of a smooth projective variety. Then, we can define the virtual tangent bundle of their intersection

Chern characters

Chern classes can be used to construct a homomorphism of rings from the topological K-theory of a space to (the completion of) its rational cohomology. For a line bundle L, the Chern character ch is defined by

More generally, if

is a direct sum of line bundles, with first Chern classes

the Chern character is defined additively

The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product. The Chern character is used in the Hirzebruch–Riemann–Roch theorem.

Equivariant K-theory

The equivariant algebraic K-theory is an algebraic K-theory associated to the category

of equivariant coherent sheaves on an algebraic scheme

with action of a linear algebraic group

, via Quillen's Q-construction; thus, by definition,

In particular,

is the Grothendieck group of

. The theory was developed by R. W. Thomason in 1980s.^[7] Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.

Grothendieck completion

Definitions

Early history

Developments

Examples

Applications

Virtual bundles

Chern characters

Equivariant K-theory

See also

Notes

References

External links