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] }

- Grothendieck completion
- Example for natural numbers
- Definitions
- Grothendieck group for compact Hausdorff spaces
- Grothendieck group of vector bundles in algebraic geometry
- Grothendieck group of coherent sheaves in algebraic geometry
- Early history
- Developments
- Examples and properties
- K0 of a field
- K0 of an Artinian algebra over a field
- K0 of projective space
- K0 of a projective bundle
- K0 of singular spaces and spaces with isolated quotient singularities
- K0 of a smooth projective curve
- Applications
- Virtual bundles
- Chern characters
- Equivariant K-theory
- See also
- Notes
- References
- External links

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).

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.

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

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 we set then we find that

- 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.

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}.

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.

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] }

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.)

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] }

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 .^{ [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,

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 .

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 .

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 annilihated by for ^{ [10] }^{pg 3}.

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.

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 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.

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.

- ↑ Atiyah, Michael (2000). "K-Theory Past and Present". arXiv: math/0012213 .
- ↑ Park, Efton. (2008).
*Complex topological K-theory*. Cambridge: Cambridge University Press. ISBN 978-0-511-38869-9. OCLC 227161674. - ↑ Grothendieck. "SGA 6 - Formalisme des intersections sur les schema algebriques propres".
- ↑ Karoubi, 2006
- ↑ by Ruben Minasian (http://string.lpthe.jussieu.fr/members.pl?key=7), and Gregory Moore in K-theory and Ramond–Ramond Charge.
- ↑ "Grothendieck group for projective space over the dual numbers".
*mathoverflow.net*. Retrieved 2017-04-16. - ↑ "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. - ↑ "ag.algebraic geometry - Is the algebraic Grothendieck group of a weighted projective space finitely generated ?".
*MathOverflow*. Retrieved 2020-10-20. - 1 2 Pavic, Nebojsa; Shinder, Evgeny (2019-03-25). "K-theory and the singularity category of quotient singularities". arXiv: 1809.10919 [math.AG].
- 1 2 Srinivas, 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,**129**, Boston, MA: Birkhäuser Boston, pp. 335–368, arXiv: hep-th/9405035 , MR 1363062 - ↑ Charles A. Weibel, Robert W. Thomason (1952–1995).

In mathematics, a **sheaf** is a tool for systematically tracking data attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set.

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the **Chern classes** are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc.

In algebraic geometry, a **projective variety** over an algebraically closed field *k* is a subset of some projective *n*-space over *k* that is the zero-locus of some finite family of homogeneous polynomials of *n* + 1 variables with coefficients in *k*, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .

In mathematics, especially in algebraic geometry and the theory of complex manifolds, **coherent sheaves** are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.

In mathematics, **Kähler differentials** provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available.

In mathematics, a **gerbe** is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group. Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them.

In mathematics, specifically in algebraic geometry, the **Grothendieck–Riemann–Roch theorem** is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces.

In mathematics, the **Grothendieck group** construction constructs an abelian group from a commutative monoid *M* in the most universal way, in the sense that any abelian group containing a homomorphic image of *M* will also contain a homomorphic image of the Grothendieck group of *M*. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.

In mathematics, **coherent duality** is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory.

In mathematics, the **Picard group** of a ringed space *X*, denoted by Pic(*X*), is the group of isomorphism classes of invertible sheaves on *X*, with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds.

In mathematics, **topological K-theory** is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological K-theory is due to Michael Atiyah and Friedrich Hirzebruch.

In mathematics, the **Leray spectral sequence** was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence.

In category theory, a branch of mathematics, the **center** is a variant of the notion of the center of a monoid, group, or ring to a category.

In mathematics a **stack** or **2-sheaf** is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, **coherent sheaf cohomology** is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.

This is a **glossary of algebraic geometry**.

In algebraic geometry, the **Quot scheme** is a scheme parametrizing locally free sheaves on a projective scheme. More specifically, if *X* is a projective scheme over a Noetherian scheme *S* and if *F* is a coherent sheaf on *X*, then there is a scheme whose set of *T*-points is the set of isomorphism classes of the quotients of that are flat over *T*. The notion was introduced by Alexander Grothendieck.

In mathematics, a **sheaf of O-modules** or simply an

In mathematics, **derived noncommutative algebraic geometry**, the derived version of noncommutative algebraic geometry, is the geometric study of derived categories and related constructions of triangulated categories using categorical tools. Some basic examples include the bounded derived category of coherent sheaves on a smooth variety, , called its derived category, or the derived category of perfect complexes on an algebraic variety, denoted . For instance, the derived category of coherent sheaves on a smooth projective variety can be used as an invariant of the underlying variety for many cases. Unfortunately, studying derived categories as geometric objects of themselves does not have a standardized name.

In Mathematics, an **Abelian 2-group** is a higher dimensional analogue of an Abelian group, in the sense of higher algebra, which were originally introduced by Alexander Grothendieck while studying abstract structures surrounding Abelian varieties and Picard groups. More concretely, they are given by groupoids which have a bifunctor which acts formally like the addition an Abelian group. Namely, the bifunctor has a notion of commutativity, associativity, and an identity structure. Although this seems like a rather lofty and abstract structure, there are several examples of Abelian 2-groups. In fact, some of which provide prototypes for more complex examples of higher algebraic structures, such as Abelian n-groups.

- Atiyah, Michael Francis (1989).
*K-theory*. Advanced Book Classics (2nd ed.). Addison-Wesley. ISBN 978-0-201-09394-0. MR 1043170. - Friedlander, Eric; Grayson, Daniel, eds. (2005).
*Handbook of K-Theory*. Berlin, New York: Springer-Verlag. doi:10.1007/978-3-540-27855-9. ISBN 978-3-540-30436-4. MR 2182598. - Park, Efton (2008).
*Complex Topological K-Theory*. Cambridge Studies in Advanced Mathematics.**111**. Cambridge University Press. ISBN 978-0-521-85634-8. - Swan, R. G. (1968).
*Algebraic K-Theory*. Lecture Notes in Mathematics.**76**. Springer. ISBN 3-540-04245-8. - Karoubi, Max (1978).
*K-theory: an introduction*. Classics in Mathematics. Springer-Verlag. doi:10.1007/978-3-540-79890-3. ISBN 0-387-08090-2. - Karoubi, Max (2006). "K-theory. An elementary introduction". arXiv: math/0602082 .
- Hatcher, Allen (2003). "Vector Bundles & K-Theory".
- Weibel, Charles (2013).
*The K-book: an introduction to algebraic K-theory*. Grad. Studies in Math.**145**. American Math Society. ISBN 978-0-8218-9132-2.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.