In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme.
[1] K-theory involves the construction of families of K-functors that map from topological spaces or schemes, or to be even more general: any object of a homotopy category 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.
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.
where: Equivalence classes in this group should be thought of as formal differences of elements in the abelian monoid.
Another useful observation is the invariance of equivalence classes under scaling: The Grothendieck completion can be viewed as a functor
There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry.
We can define equivalence classes of idempotent matrices and form an abelian monoid
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.
[2]pg 51-110 There is an analogous construction by considering vector bundles in algebraic geometry.
of isomorphisms classes of vector bundles is well-defined, giving an abelian monoid
[4] Grothendieck needed to work with coherent sheaves on an algebraic variety X.
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).
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.
Finally, two useful and equivalent definitions were given by Daniel Quillen using homotopy theory in 1969 and 1972.
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.
In string theory, the K-theory classification of Ramond–Ramond field strengths and the charges of stable D-branes was first proposed in 1997.
since the Grothendieck group of coherent sheaves on affine spaces are isomorphic to
One recent technique for computing the Grothendieck group of spaces with minor singularities comes from evaluating the difference between
, which comes from the fact every vector bundle can be equivalently described as a coherent sheaf.
This makes it possible to compute the Grothendieck group on weighted projective spaces since they typically have isolated quotient singularities.
For a regular scheme of finite type over a field, there is a convergent spectral sequence
as the desired explicit direct sum since it gives an exact sequence
This is because reduction gives a generically smooth curve, and all singularities are Cohen-Macaulay.
[12] 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