K Theory
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In mathematics, K-theory is, roughly speaking, the study of a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
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 p ...
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 classify ...
, it is a
cohomology theory In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
known as
topological K-theory In mathematics, topological -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 ...
. In
algebra Algebra () is one of the 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 mathematics. Elementary ...
and algebraic geometry, it is referred to as
algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
. It is also a fundamental tool in the field of
operator algebra In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study of ...
s. It can be seen as the study of certain kinds of invariants of large
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. 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, and m ...
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
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
s 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 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 ...
,
Bott periodicity 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 comp ...
, the Atiyah–Singer index theorem, and the
Adams operation In mathematics, an Adams operation, denoted ψ''k'' for natural numbers ''k'', is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduce ...
s. In
high energy physics Particle physics or high energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standa ...
, K-theory and in particular
twisted K-theory In mathematics, twisted K-theory (also called K-theory with local coefficients) is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory. More specifically, twisted K-th ...
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 In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
on generalized complex manifolds. In condensed matter physics K-theory has been used to classify
topological insulator A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material. A topological insulator is an ...
s, superconductors and stable
Fermi surface In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the crys ...
s. 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 In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ar ...
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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
(G(A),+) where: : a_1,a_2)+ b_1,b_2)= 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 classes 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, and m ...
G:\mathbf\to\mathbf, and it has the property that it is left adjoint to the corresponding
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
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 m ...
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 sums, 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 In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout ...
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 Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
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 In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Thu ...
X there is a set \text(X) of all isomorphism classes of
algebraic vector bundle 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 ref ...
s 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 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 ...
\operatorname(X) we can mod out by the relation mathcal= mathcal'+ mathcal''/math> if there is a
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
: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 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 ...
, 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 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 ...
. It takes its name from the German ''Klasse'', meaning "class". Grothendieck needed to work with
coherent 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 ...
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 class In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other. Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the stru ...
es 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 Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
behavior. If ''X'' is a
smooth variety In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smo ...
, the two groups are the same. If it is a smooth
affine variety In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. ...
, 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 ...
, 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 every p ...
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 th ...
and
Friedrich Hirzebruch Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...
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 comp ...
they made it the basis of an
extraordinary cohomology theory In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
. It played a major role in the second proof of the Atiyah–Singer index theorem (circa 1962). Furthermore, this approach led to a
noncommutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
K-theory for C*-algebras. 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 ina ...
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 every p ...
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 variables ...
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 Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 197 ...
using homotopy theory in 1969 and 1972. A variant was also given by
Friedhelm Waldhausen Friedhelm Waldhausen (born 1938 in Millich, Hückelhoven, Rhine Province) is a German mathematician known for his work in algebraic topology. He made fundamental contributions in the fields of 3-manifolds and (algebraic) K-theory. Career Wald ...
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 Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geo ...
. The corresponding constructions involving an auxiliary quadratic form received the general name L-theory. It is a major tool of
surgery theory In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while And ...
. 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 \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 In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Thu ...
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 D_(X) from derived noncommutative algebraic geometry. 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 In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
of C. This follows from the Brown-Gersten-Quillen spectral sequencepg 72 of
algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
. For a
regular scheme In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth.. For an example of a regul ...
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 letter, capital: Ω, lower case, 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 numerals, Greek numeric system/isopsephy ...
/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 In mathematics, topological -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 ...
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 In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebra ...
.


Equivariant K-theory

The
equivariant algebraic K-theory In mathematics, 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 ...
is an
algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
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 Gr ...
; thus, by definition, :K_i^G(X) = \pi_i(B^+ \operatorname^G(X)). In particular, K_0^G(C) is the Grothendieck group 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 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 comp ...
* KK-theory * KR-theory *
List of cohomology theories This is a list of some of the ordinary and generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectra. For other sorts of homology theories see the links at ...
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Algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
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Topological K-theory In mathematics, topological -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 ...
* Operator K-theory *
Grothendieck–Riemann–Roch theorem 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 ...


Notes


References

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External links


Grothendieck-Riemann-Roch

Max Karoubi's Page

K-theory preprint archive
{{Topology