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.
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 associate 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:
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 numbers
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.
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 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.
Grothendieck group of vector bundles in algebraic geometry
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 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 . 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
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 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 field
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,
K0 of projective space
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
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
K0 of a projective bundle
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 .
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 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 3.
K0 of a smooth projective curve
For a smooth projective curve the Grothendieck group is
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 propertypg 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.
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.
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.
In particular, is the Grothendieck group of . The theory was developed by R. W. Thomason in 1980s. Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.
^by Ruben Minasian (http://string.lpthe.jussieu.fr/members.pl?key=7), and Gregory Moore in K-theory and Ramond–Ramond Charge.
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^"kt.k theory and homology - Grothendieck group for projective space over the dual numbers". MathOverflow. Retrieved 2020-10-20.
^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.
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^ abPavic, Nebojsa; Shinder, Evgeny (2019-03-25). "K-theory and the singularity category of quotient singularities". arXiv:1809.10919 [math.AG].
^ abSrinivas, V. (1991). Algebraic K-theory. Boston: Birkhäuser. ISBN 978-1-4899-6735-0. OCLC 624583210.
^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
^Charles A. Weibel, Robert W. Thomason (1952–1995).