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In mathematics, K-theory is, roughly speaking, the study of a ring generated by
vector bundles In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every ...
over a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
or scheme. In
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...
, it is a cohomology theory known as topological K-theory. In
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
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''-
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
s 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 In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for two of the five consistent superstring theories in ten dimensions. Both theories ...
where it has been conjectured that they classify
D-branes In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchi ...
, Ramond–Ramond field strengths and also certain spinors on generalized complex manifolds. In
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the s ...
K-theory has been used to classify topological insulators, superconductors and stable Fermi surfaces. For more details, see
K-theory (physics) In string theory, K-theory classification refers to a conjectured application of K-theory (in abstract algebra and algebraic topology) to superstrings, to classify the allowed Ramond–Ramond field strengths as well as the charges of stable D-bra ...
.


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 (A,+') let \sim be the relation on A^2 = A \times A defined by :(a_1,a_2) \sim (b_1,b_2) if there exists a c\in A such that a_1 +' b_2 +' c = a_2 +' b_1 +' c. Then, the set G(A) = A^2/\sim has the structure of a group (G(A),+) where: : a_1,a_2)+
b_1,b_2) B, or b, is the second letter of the Latin-script alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''bee'' (pronounced ), plural ''bees''. It re ...
= a_1+' b_1,a_2+' b_2) Equivalence classes in this group should be thought of as formal differences of elements in the abelian monoid. This group (G(A),+) is also associated with a monoid homomorphism i : A \to G(A) given by a \mapsto a, 0) which has a certain universal property. To get a better understanding of this group, consider some
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of the abelian monoid (A,+). Here we will denote the identity element of A by 0 so that
0,0) The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> will be the identity element of (G(A),+). First, (0,0) \sim (n,n) for any n\in A since we can set c = 0 and apply the equation from the equivalence relation to get n = n. This implies : a,b)+ b,a)= a+b,a+b)=
0,0) The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> hence we have an additive inverse for each element in G(A). This should give us the hint that we should be thinking of the equivalence classes a,b)/math> as formal differences a-b. Another useful observation is the invariance of equivalence classes under scaling: :(a,b) \sim (a+k,b+k) for any k \in A. The Grothendieck completion can be viewed as a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
G:\mathbf\to\mathbf, and it has the property that it is left adjoint to the corresponding forgetful functor U:\mathbf\to\mathbf. That means that, given a morphism \phi:A \to U(B) of an abelian monoid A to the underlying abelian monoid of an abelian group B, there exists a unique abelian group morphism G(A) \to B.


Example for natural numbers

An illustrative example to look at is the Grothendieck completion of \N. We can see that G((\N,+)) = (\Z,+). For any pair (a,b) we can find a minimal representative (a',b') by using the invariance under scaling. For example, we can see from the scaling invariance that :(4,6) \sim (3,5) \sim (2,4) \sim (1,3) \sim (0,2) In general, if k := \min\ then :(a,b) \sim (a-k,b-k) which is of the form (c,0) or (0,d). This shows that we should think of the (a,0) as positive integers and the (0,b) 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 In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
X consider the set of isomorphism classes of finite-dimensional vector bundles over X, denoted \text(X) and let the isomorphism class of a vector bundle \pi:E \to X be denoted /math>. Since isomorphism classes of vector bundles behave well with respect to
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
s, we can write these operations on isomorphism classes by : oplus '= \oplus E' It should be clear that (\text(X),\oplus) is an abelian monoid where the unit is given by the trivial vector bundle \R^0\times X \to X. We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. This is called the K-theory of X and is denoted K^0(X). 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 C^0(X;\Complex) as projective modules. Then, these can be identified with idempotent matrices in some ring of matrices M_(C^0(X;\Complex)). We can define equivalence classes of idempotent matrices and form an abelian monoid \textbf(X). Its Grothendieck completion is also called K^0(X). One of the main techniques for computing the Grothendieck group for topological spaces comes from the
Atiyah–Hirzebruch spectral sequence In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet ...
, which makes it very accessible. The only required computations for understanding the spectral sequences are computing the group K^0 for the spheres S^n.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 X there is a set \text(X) of all isomorphism classes of algebraic vector bundles on X. Then, as before, the direct sum \oplus of isomorphisms classes of vector bundles is well-defined, giving an abelian monoid (\text(X),\oplus). Then, the Grothendieck group K^0(X) 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 X. If we look at the isomorphism classes of coherent sheaves \operatorname(X) we can mod out by the relation mathcal= mathcal'+ mathcal''/math> if there is a short exact sequence :0 \to \mathcal' \to \mathcal \to \mathcal'' \to 0. This gives the Grothendieck-group K_0(X) which is isomorphic to K^0(X) if X is smooth. The group K_0(X) is special because there is also a ring structure: we define it as : mathcalcdot mathcal'= \sum(-1)^k \left operatorname_k^(\mathcal, \mathcal') \right Using the Grothendieck–Riemann–Roch theorem, we have that :\operatorname : K_0(X)\otimes \Q \to A(X)\otimes \Q is an isomorphism of rings. Hence we can use K_0(X) for
intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
.


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 Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
''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 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 refer ...
are used, or ''G''(''X'') when all are coherent sheaves. Either of these two constructions is referred to as the
Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic ...
; ''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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
, by applying the same construction to
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
s,
Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded t ...
and Friedrich Hirzebruch defined ''K''(''X'') for a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' in 1959, and using the
Bott periodicity theorem In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable co ...
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*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...
s. Already in 1955,
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ...
had used the analogy of
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
s with projective modules to formulate Serre's conjecture, which states that every finitely generated projective module over a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...
is
free Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procur ...
; 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 In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau(f) which is an element in the Whitehead group \o ...
. 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 mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topol ...
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 In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
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 In theoretical physics, Ramond–Ramond fields are differential form fields in the 10-dimensional spacetime of type II supergravity theories, which are the classical limits of type II string theory. The ranks of the fields depend on which type II th ...
strengths and the charges of stable
D-branes In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchi ...
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 \text(\mathbb) for a field \mathbb. 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 \N corresponding to the dimension of the vector space. It is an easy exercise to show that the Grothendieck group is then \Z.


K0 of an Artinian algebra over a field

One important property of the Grothendieck group of a Noetherian scheme X is that it is invariant under reduction, hence K(X) = K(X_). Hence the Grothendieck group of any Artinian \mathbb-algebra is a direct sum of copies of \Z, one for each connected component of its spectrum. For example, K_0 \left(\text\left(\frac\times\mathbb\right)\right) = \mathbb\oplus\mathbb


K0 of projective space

One of the most commonly used computations of the Grothendieck group is with the computation of K(\mathbb^n) for projective space over a field. This is because the intersection numbers of a projective X can be computed by embedding i:X \hookrightarrow \mathbb^n and using the push pull formula i^*( _*\mathcalcdot _*\mathcal. This makes it possible to do concrete calculations with elements in K(X) without having to explicitly know its structure since K(\mathbb^n) = \frac One technique for determining the Grothendieck group of \mathbb^n comes from its stratification as \mathbb^n = \mathbb^n \coprod \mathbb^ \coprod \cdots \coprod \mathbb^0 since the Grothendieck group of coherent sheaves on affine spaces are isomorphic to \mathbb, and the intersection of \mathbb^,\mathbb^ is generically \mathbb^ \cap \mathbb^ = \mathbb^ for k_1 + k_2 \leq n.


K0 of a projective bundle

Another important formula for the Grothendieck group is the projective bundle formula: given a rank r vector bundle \mathcal over a Noetherian scheme X, the Grothendieck group of the projective bundle \mathbb(\mathcal)=\operatorname(\operatorname^\bullet(\mathcal^\vee)) is a free K(X)-module of rank ''r'' with basis 1,\xi,\dots,\xi^. This formula allows one to compute the Grothendieck group of \mathbb^n_\mathbb. This make it possible to compute the K_0 or Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group K(\mathbb^n) by observing it is a projective bundle over the field \mathbb.


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 K^0(X) and K_0(X), 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 Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
D_(X) from
derived noncommutative algebraic geometry 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 bas ...
. It gives a long exact sequence starting with \cdots \to K^0(X) \to K_0(X) \to K_(X) \to 0 where the higher terms come from higher K-theory. Note that vector bundles on a singular X are given by vector bundles E \to X_ on the smooth locus X_ \hookrightarrow X. 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 G_i then the map K^0(X) \to K_0(X) is injective and the cokernel is annihilated by \text(, G_1, ,\ldots, , G_k, )^ for n = \dim X.pg 3


K0 of a smooth projective curve

For a smooth projective curve C the Grothendieck group is K_0(C) = \mathbb\oplus\text(C) for Picard group of C. 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 E_1^ = \coprod_K^(k(x)) \Rightarrow K_(X) for X^ the set of codimension p points, meaning the set of subschemes x: Y \to X of codimension p, and k(x) the algebraic function field of the subscheme. This spectral sequence has the propertypg 80 E_2^ \cong \text^p(X) for the Chow ring of X, essentially giving the computation of K_0(C). Note that because C has no codimension 2 points, the only nontrivial parts of the spectral sequence are E_1^,E_1^, hence \begin E_\infty^\cong E_2^ &\cong \text^1(C) \\ E_\infty^ \cong E_2^ &\cong \text^0(C) \end The coniveau filtration can then be used to determine K_0(C) as the desired explicit direct sum since it gives an exact sequence 0 \to F^1(K_0(X)) \to K_0(X) \to K_0(X)/F^1(K_0(X)) \to 0 where the left hand term is isomorphic to \text^1 (C) \cong \text(C) and the right hand term is isomorphic to CH^0(C) \cong \mathbb. Since \text^1_(\mathbb,G) = 0, we have the sequence of abelian groups above splits, giving the isomorphism. Note that if C is a smooth projective curve of genus g over \mathbb, then K_0(C) \cong \mathbb\oplus(\mathbb^g/\mathbb^) 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 Y \hookrightarrow X then there is a short exact sequence : 0 \to \Omega_Y \to \Omega_X, _Y \to C_ \to 0 where C_ is the conormal bundle of Y in X. If we have a singular space Y embedded into a smooth space X we define the virtual conormal bundle as : _Y-
Omega_Y Omega (; capital: Ω, lowercase: ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and final letter in the Greek alphabet. In the Greek numeric system/ isopsephy ( gematria), it has a value of 800. The w ...
/math> Another useful application of virtual bundles is with the definition of a virtual tangent bundle of an intersection of spaces: Let Y_1,Y_2\subset X be projective subvarieties of a smooth projective variety. Then, we can define the virtual tangent bundle of their intersection Z = Y_1\cap Y_2 as : _Z = __Z + __Z - __Z. 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 :\operatorname(L) = \exp(c_(L)) := \sum_^\infty \frac. More generally, if V = L_1 \oplus \dots \oplus L_n is a direct sum of line bundles, with first Chern classes x_i = c_1(L_i), the Chern character is defined additively : \operatorname(V) = e^ + \dots + e^ :=\sum_^\infty \frac(x_1^m + \dots + x_n^m). 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 \operatorname^G(X) of equivariant coherent sheaves on an algebraic scheme X with action of a linear algebraic group G, via Quillen's
Q-construction In algebra, Quillen's Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory. More precisely, given an exact category ''C'', the construction creates a topological space B^+C so that \pi_0 (B^+C) is the ...
; thus, by definition, :K_i^G(X) = \pi_i(B^+ \operatorname^G(X)). In particular, K_0^G(C) is the
Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic ...
of \operatorname^G(X). The theory was developed by R. W. Thomason in 1980s.Charles A. Weibel
Robert W. Thomason (1952–1995)
Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.


See also

* Bott periodicity *
KK-theory In mathematics, ''KK''-theory is a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician Gennadi Kasparov in 1980. It was influ ...
* KR-theory * List of cohomology theories * Algebraic K-theory * Topological K-theory * Operator K-theory * Grothendieck–Riemann–Roch theorem


Notes


References

* * * * * * * *


External links


Grothendieck-Riemann-Roch

Max Karoubi's Page

K-theory preprint archive
{{Topology