K-theory

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).
Grothendieck completion.
The Grothendieck completion of an abelian monoid into an abelian group is a necessary ingredient for defining K-theory since all definitions start by constructing an abelian monoid from a suitable category and turning it into an abelian group through this universal construction. Given an abelian monoid formula_1 let formula_2 be the relation on formula_3 defined by
if there exists a formula_5 such that formula_6 Then, the set formula_7 has the structure of a group formula_8 where:
Equivalence classes in this group should be thought of as formal differences of elements in the abelian monoid. This group formula_8 is also associate with a monoid homomorphism formula_11 given by formula_12 which has a certain universal property. 
To get a better understanding of this group, consider some equivalence classes of the abelian monoid formula_13. Here we will denote the identity element of formula_14 by formula_15 so that formula_16 will be the identity element of formula_17 First, formula_18 for any formula_19 since we can set formula_20 and apply the equation from the equivalence relation to get formula_21 This implies
hence we have an additive inverse for each element in formula_23. This should give us the hint that we should be thinking of the equivalence classes formula_24 as formal differences formula_25 Another useful observation is the invariance of equivalence classes under scaling:
The Grothendieck completion can be viewed as a functor formula_28 and it has the property that it is left adjoint to the corresponding forgetful functor formula_29 That means that, given a morphism formula_30 of an abelian monoid formula_14 to the underlying abelian monoid of an abelian group formula_32 there exists a unique abelian group morphism formula_33
Example for natural numbers.
An illustrative example to look at is the Grothendieck completion of formula_34. We can see that formula_35 For any pair formula_36 we can find a minimal representative formula_37 by using the invariance under scaling. For example, we can see from the scaling invariance that
In general, if formula_39 then 
This shows that we should think of the formula_43 as positive integers and the formula_44 as negative integers.
Definitions.
There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry.
Grothendieck group for compact Hausdorff spaces.
Given a compact Hausdorff space formula_45 consider the set of isomorphism classes of finite-dimensional vector bundles over formula_45, denoted formula_47 and let the isomorphism class of a vector bundle formula_48 be denoted formula_49. 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 formula_51 is an abelian monoid where the unit is given by the trivial vector bundle formula_52. We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. This is called the K-theory of formula_45 and is denoted formula_54.
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 formula_55 as projective modules. Then, these can be identified with idempotent matrices in some ring of matrices formula_56. We can define equivalence classes of idempotent matrices and form an abelian monoid formula_57. Its Grothendieck completion is also called formula_54. 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 formula_59 for the spheres formula_60pg 51-110.
Grothendieck group of vector bundles in algebraic geometry.
There is an analogous construction by considering vector bundles in algebraic geometry. For a Noetherian scheme formula_45 there is a set formula_47 of all isomorphism classes of algebraic vector bundles on formula_45. Then, as before, the direct sum formula_64 of isomorphisms classes of vector bundles is well-defined, giving an abelian monoid formula_51. Then, the Grothendieck group formula_54 is defined by the application of the Grothendieck construction on this abelian monoid.
Grothendieck group of coherent sheaves in algebraic geometry.
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 formula_45. If we look at the isomorphism classes of coherent sheaves formula_68 we can mod out by the relation formula_69 if there is a short exact sequence
This gives the Grothendieck-group formula_71 which is isomorphic to formula_54 if formula_45 is smooth. The group formula_71 is special because there is also a ring structure: we define it as
Using the Grothendieck–Riemann–Roch theorem, we have that
is an isomorphism of rings. Hence we can use formula_71 for intersection theory.
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". 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.)
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.
Examples and properties.
K0 of a field.
The easiest example of the Grothendieck group is the Grothendieck group of a point formula_78 for a field formula_79. 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 formula_34 corresponding to the dimension of the vector space. It is an easy exercise to show that the Grothendieck group is then formula_81.
K0 of an Artinian algebra over a field.
One important property of the Grothendieck group of a Noetherian scheme formula_45 is that it is invariant under reduction, hence formula_83. Hence the Grothendieck group of any Artinian formula_79-algebra is a direct sum of copies of formula_81, one for each connected component of its spectrum. For example,
formula_86
K0 of projective space.
One of the most commonly used computations of the Grothendieck group is with the computation of formula_87 for projective space over a field. This is because the intersection numbers of a projective formula_45 can be computed by embedding formula_89 and using the push pull formula formula_90. This makes it possible to do concrete calculations with elements in formula_91 without having to explicitly know its structure since
formula_92
One technique for determining the grothendieck group of formula_93 comes from its stratification as
formula_94
since the grothendieck group of coherent sheaves on affine spaces are isomorphic to formula_95, and the intersection of formula_96 is generically
formula_97
for formula_98.
K0 of a projective bundle.
Another important formula for the Grothendieck group is the projective bundle formula: given a rank r vector bundle formula_99 over a Noetherian scheme formula_45, the Grothendieck group of the projective bundle formula_101 is a free formula_91-module of rank "r" with basis formula_103. This formula allows one to compute the Grothendieck group of formula_104. This make it possible to compute the formula_105 or Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group formula_87 by observing it is a projective bundle over the field formula_79.
K0 of singular spaces and spaces with isolated quotient singularities.
One recent technique for computing the Grothendieck group of spaces with minor singularities comes from evaluating the difference between formula_54 and formula_71, 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 formula_110 from derived noncommutative algebraic geometry. It gives a long exact sequence starting with
formula_111
where the higher terms come from higher K-theory. Note that vector bundles on a singular formula_45 are given by vector bundles formula_113 on the smooth locus formula_114. 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 formula_115 then the map
formula_116
is injective and the cokernel is annihilated by formula_117 for formula_118pg 3.
K0 of a smooth projective curve.
For a smooth projective curve formula_119 the Grothendieck group is
formula_120
for Picard group of formula_119. This follows from the Brown-Gersten-Quillen spectral sequencepg 72 of algebraic K-theory. For a regular scheme of finite type over a field, there is a convergent spectral sequence
formula_122
for formula_123 the set of codimension formula_124 points, meaning the set of subschemes formula_125 of codimension formula_124, and formula_127 the algebraic function field of the subscheme. This spectral sequence has the propertypg 80
formula_128
for the Chow ring of formula_45, essentially giving the computation of formula_130. Note that because formula_119 has no codimension formula_132 points, the only nontrivial parts of the spectral sequence are formula_133, hence
formula_134
The coniveau filtration can then be used to determine formula_130 as the desired explicit direct sum since it gives an exact sequence
formula_136
where the left hand term is isomorphic to formula_137 and the right hand term is isomorphic to formula_138. Since formula_139, we have the sequence of abelian groups above splits, giving the isomorphism. Note that if formula_119 is a smooth projective curve of genus formula_141 over formula_142, then
formula_143
Moreover, the techniques above using the derived category of singularities for isolated singularities can be extended to isolated Cohen-Macaulay singularities, giving techniques for computing the Grothendieck group of any singular algebraic curve. This is because reduction gives a generically smooth curve, and all singularities are Cohen-Macaulay.
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 formula_144 then there is a short exact sequence
where formula_146 is the conormal bundle of formula_147 in formula_45. If we have a singular space formula_147 embedded into a smooth space formula_45 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 formula_152 be projective subvarieties of a smooth projective variety. Then, we can define the virtual tangent bundle of their intersection formula_153 as
Kontsevich uses this construction in one of his papers.
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 formula_156 is a direct sum of line bundles, with first Chern classes formula_157 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 formula_159 of equivariant coherent sheaves on an algebraic scheme formula_45 with action of a linear algebraic group formula_161, via Quillen's Q-construction; thus, by definition,
In particular, formula_163 is the Grothendieck group of formula_159. The theory was developed by R. W. Thomason in 1980s. Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.