K-теория: математика және физикадағы қолданыстары

Branch of mathematics
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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. 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).

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 One of the main techniques for computing the Grothendieck group for topological spaces comes from the Atiyah–Hirzebruch spectral sequence, which makes it very accessible. The only required computations for understanding the spectral sequences are computing the group for the spheres pg 51 110

There is an analogous construction by considering vector bundles in algebraic geometry. For a Noetherian scheme there is a set of all isomorphism classes of algebraic vector bundles on Then, as before, the direct sum of isomorphisms classes of vector bundles is well defined, giving an abelian monoid Then, the Grothendieck group is defined by the application of the Grothendieck construction on this abelian monoid.

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". 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 Atiyah–Singer 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.)

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.

The easiest example of the Grothendieck group is the Grothendieck group of a point for a field Since a vector bundle over this space is just a finite dimensional vector space, which is a free object in the category of coherent sheaves, hence projective, the monoid of isomorphism classes is corresponding to the dimension of the vector space. It is an easy exercise to show that the Grothendieck group is then .

One important property of the Grothendieck group of a Noetherian scheme is that it is invariant under reduction, hence Hence the Grothendieck group of any Artinian algebra is a direct sum of copies of , one for each connected component of its spectrum. For example,

Another important formula for the Grothendieck group is the projective bundle formula: given a rank r vector bundle over a Noetherian scheme , the Grothendieck group of the projective bundle is a free module of rank r with basis This formula allows one to compute the Grothendieck group of This make it possible to compute the or Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group by observing it is a projective bundle over the field .

One recent technique for computing the Grothendieck group of spaces with minor singularities comes from evaluating the difference between and , which comes from the fact every vector bundle can be equivalently described as a coherent sheaf. This is done using the Grothendieck group of the Singularity category from derived noncommutative algebraic geometry. It gives a long exact sequence starting with

where the higher terms come from higher K theory. Note that vector bundles on a singular are given by vector bundles on the smooth locus This makes it possible to compute the Grothendieck group on weighted projective spaces since they typically have isolated quotient singularities. In particular, if these singularities have isotropy groups then the map

is injective and the cokernel is annihilated by for pg 72 of algebraic K theory. For a regular scheme of finite type over a field, there is a convergent spectral sequence

for the set of codimension points, meaning the set of subschemes of codimension , and the algebraic function field of the subscheme. This spectral sequence has the property