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 completionEdit

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   let   be the relation on   defined by

 

if there exists a   such that   Then, the set   has the structure of a group   where:

 

Equivalence classes in this group should be thought of as formal differences of elements in the abelian monoid. This group   is also associated with a monoid homomorphism   given by   which has a certain universal property.

To get a better understanding of this group, consider some equivalence classes of the abelian monoid  . Here we will denote the identity element of   by   so that   will be the identity element of   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:

  for any  

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  

Example for natural numbersEdit

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

 

In general, if   then

  which is of the form   or  

This shows that we should think of the   as positive integers and the   as negative integers.

DefinitionsEdit

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

Grothendieck group for compact Hausdorff spacesEdit

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  [2]pg 51-110.

Grothendieck group of vector bundles in algebraic geometryEdit

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.

Grothendieck group of coherent sheaves in algebraic geometryEdit

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   if   is smooth. The group   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   for intersection theory.[3]

Early historyEdit

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".[4] 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.)

DevelopmentsEdit

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.[5]

Examples and propertiesEdit

K0 of a fieldEdit

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  .

K0 of an Artinian algebra over a fieldEdit

One important property of the Grothendieck group of a Noetherian scheme   is that it is invariant under reduction, hence  .[6] 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,

 

K0 of projective spaceEdit

One of the most commonly used computations of the Grothendieck group is with the computation of   for projective space over a field. This is because the intersection numbers of a projective   can be computed by embedding   and using the push pull formula  . This makes it possible to do concrete calculations with elements in   without having to explicitly know its structure since[7]

 
One technique for determining the Grothendieck group of   comes from its stratification as
 
since the Grothendieck group of coherent sheaves on affine spaces are isomorphic to  , and the intersection of   is generically
 
for  .

K0 of a projective bundleEdit

Another important formula for the Grothendieck group is the projective bundle formula:[8] 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  .

K0 of singular spaces and spaces with isolated quotient singularitiesEdit

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  [9][10] 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  [10]pg 3.

K0 of a smooth projective curveEdit

For a smooth projective curve   the Grothendieck group is

 
for Picard group of  . This follows from the Brown-Gersten-Quillen spectral sequence[11]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[11]pg 80
 
for the Chow ring of  , essentially giving the computation of  . Note that because   has no codimension   points, the only nontrivial parts of the spectral sequence are  , hence
 
The coniveau filtration can then be used to determine   as the desired explicit direct sum since it gives an exact sequence
 
where the left hand term is isomorphic to   and the right hand term is isomorphic to  . Since  , we have the sequence of abelian groups above splits, giving the isomorphism. Note that if   is a smooth projective curve of genus   over  , then
 
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.

ApplicationsEdit

Virtual bundlesEdit

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   in  . 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   as

 

Kontsevich uses this construction in one of his papers.[12]

Chern charactersEdit

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-theoryEdit

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.[13] Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.

See alsoEdit

NotesEdit

  1. ^ Atiyah, Michael (2000). "K-Theory Past and Present". arXiv:math/0012213.
  2. ^ Park, Efton. (2008). Complex topological K-theory. Cambridge: Cambridge University Press. ISBN 978-0-511-38869-9. OCLC 227161674.
  3. ^ Grothendieck. "SGA 6 - Formalisme des intersections sur les schema algebriques propres".
  4. ^ Karoubi, 2006
  5. ^ by Ruben Minasian (http://string.lpthe.jussieu.fr/members.pl?key=7), and Gregory Moore in K-theory and Ramond–Ramond Charge.
  6. ^ "Grothendieck group for projective space over the dual numbers". mathoverflow.net. Retrieved 2017-04-16.
  7. ^ "kt.k theory and homology - Grothendieck group for projective space over the dual numbers". MathOverflow. Retrieved 2020-10-20.
  8. ^ Manin, Yuri I (1969-01-01). "Lectures on the K-functor in algebraic geometry". Russian Mathematical Surveys. 24 (5): 1–89. Bibcode:1969RuMaS..24....1M. doi:10.1070/rm1969v024n05abeh001357. ISSN 0036-0279.
  9. ^ "ag.algebraic geometry - Is the algebraic Grothendieck group of a weighted projective space finitely generated ?". MathOverflow. Retrieved 2020-10-20.
  10. ^ a b Pavic, Nebojsa; Shinder, Evgeny (2021). "K-theory and the singularity category of quotient singularities". Annals of K-Theory. 6 (3): 381–424. arXiv:1809.10919. doi:10.2140/akt.2021.6.381. S2CID 85502709.
  11. ^ a b Srinivas, V. (1991). Algebraic K-theory. Boston: Birkhäuser. ISBN 978-1-4899-6735-0. OCLC 624583210.
  12. ^ Kontsevich, Maxim (1995), "Enumeration of rational curves via torus actions", The moduli space of curves (Texel Island, 1994), Progress in Mathematics, vol. 129, Boston, MA: Birkhäuser Boston, pp. 335–368, arXiv:hep-th/9405035, MR 1363062
  13. ^ Charles A. Weibel, Robert W. Thomason (1952–1995).

ReferencesEdit

External linksEdit