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{{for|the hip hop group|K Theory}}

In [[mathematics]], '''K-theory''' is, roughly speaking, the study of a [[Ring (mathematics)|ring]] generated by [[vector bundles]] over a [[topological space]] or [[scheme (mathematics)|scheme]]. In [[algebraic topology]], it is an [[extraordinary 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 algebra]]s. It can be seen as the study of certain kinds of [[Invariant (mathematics)|invariants]] of large [[Matrix (mathematics)|matrices]].<ref>{{Citation|last = Atiyah |first = Michael |authorlink = Michael Atiyah |year = 2000 |title = K-Theory Past and Present |arxiv = math/0012213 |version = v1 |postscript =  }}</ref>

K-theory involves the construction of families of ''K''-[[functor]]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 (mathematics)|group]]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 [[Grothendieck–Riemann–Roch theorem|Grothendieck-Riemann-Roch]], [[Bott periodicity]], the [[Atiyah-Singer index theorem]], and the [[Adams operation]]s.

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|Ramond–Ramond field strengths]] and also certain [[spinors]] on [[generalized complex structure|generalized complex manifolds]]. In [[condensed matter physics]] K-theory has been used to classify [[topological insulator]]s, [[superconductor]]s and stable [[Fermi surface]]s. For more details, see [[K-theory (physics)]].

==Grothendieck Completion==
The Grothendieck completion is a necessary ingredient for constructing K-theory. Given an abelian monoid <math>(A,+')</math> let <math>\sim</math> be the relation on <math>A^2</math> defined by
:<math>(a_1,a_2) \sim (b_1,b_2)</math> if there exists a <math>c\in A</math> such that <math>a_1 + b_2 + c = a_2 + b_1 + c</math>
Then, the set <math>G(A) = A^2/\sim</math> has the structure of a group <math>(G(A),+)</math> where
:<math> [(a_1,a_2)] + [(b_1,b_2)] = [(a_1+' b_1,a_2+' b_2)]</math>
Equivalence classes in this group should be thought of formal differences of elements in the abelian group.

To get a better understanding of this group, let's look at some equivalence classes of the abelian monoid <math>(A,+)</math>. Here we will denote the identity element by <math>0</math>. First, notice that <math>(0,0) \sim (n,n)</math> for any <math>n\in A</math> since we can set <math>c = 0</math> and apply the equation from the equivalence relation to get <math>n = n</math>. Now, notice that this implies
:<math>[(a,b)] + [(b,a)] = [(a+b,a+b)] = 0</math>
hence we have an additive inverse for each element in in <math>G(A)</math>. This should give us the hint that we should be thinking of the equivalence classes <math>[(a,b)]</math> as formal differences <math>a-b</math>. Another useful observation is the invariance of equivalence classes under scaling:
:<math>(a,b) \sim (a+k,b+k)</math> for any <math>k \in A</math> 

There is a nice universal property as well for the Grothendieck completion: given a morphism <math>\phi:A \to B</math> of an abelian monoid <math>A</math> to the underlying abelian monoid of an abelian group <math>B</math>, there exists a unique abelian group morphism <math>G(A) \to B</math>.

A nice illustrative example to look at is the Grothendieck completion of <math>\mathbb{N}</math>. We can see that <math>G((\mathbb{N},+)) = (\mathbb{Z},+)</math>. Notice that for any pair <math>(a,b)</math> we can find a minimal representative <math>(a',b')</math> by using the invariance under scaling. For example, we can see from the scaling invariance that
:<math>(4,6) \sim (3,5) \sim (2,4) \sim (1,3) \sim (0,2)</math>
In general, if we set <math>k = \min\{a,b\}</math> then we find that
:<math>(a,b) \sim (a-k,b-k)</math> which is of the form <math>(c,0)</math> or <math>(0,d)</math>
This shows that we should think of the <math>(a,0)</math> as positive integers and the <math>(0,b)</math> as negative integers.

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

Given a compact Hausdorff space <math>X</math> consider the set of isomorphism classes of finite dimensional vector bundles over <math>X</math>, denoted <math>\text{Vect}(X)</math> and let the isomorphism class of a vector bundle <math>\pi:E \to X</math> be denoted <math>[E]</math>. Since isomorphism classes of vector bundles behave well with respect to direct sums, we can write these operations on isomorphism classes by
:<math>
[E]\oplus[E'] =[E\oplus E'] 
</math>
It should be clear that <math>(\text{Vect}(X),\oplus)</math> is an abelian monoid where the unit is given by the trivial vector bundle <math>\mathbb{R}^0\times X \to X</math>. We can then apply the Grothendieck-completion to get an abelian group from this abelian monoid. This is called the K-theory of <math>X</math> and is denoted <math>K^0(X)</math>.

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 <math>C^0(X;\mathbb{C})</math> as projective modules. Then, these can be identified with idempotent matrices in some ring of matrices <math>M_{n\times n}(C^0(X;\mathbb{C}))</math>. We can define equivalence classes of idempotent matrices and form an abelian monoid <math>\textbf{Idem}(X)</math>. It's Grothendieck completion is also called <math>K^0(X)</math>.

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 <math>X</math>. If we look at the isomorphism classes of coherent sheaves <math>\text{Coh}(X)</math> we can mod out by the relation <math>[\mathcal{E}] = [\mathcal{E}'] + [\mathcal{E}'']</math> if there is a short exact sequence
:<math>0 \to \mathcal{E}' \to \mathcal{E} \to \mathcal{E}'' \to 0</math>
This gives the Grothendieck-group <math>K_0(X)</math> which is isomorphic to <math>K^0(X)</math> if <math>X</math> is smooth. The group <math>K_0(X)</math> is special because there is also a ring structure: we define it as
:<math>[\mathcal{E}]\cdot[\mathcal{E}'] = \sum(-1)^k[\text{Tor}_k^{\mathcal{O}_X}(\mathcal{E}, \mathcal{E}')]</math>
Using Grothendieck-Riemann-Roch we have that
:<math>ch: K_0(X)\otimes \mathbb{Q} \to A(X)\otimes \mathbb{Q}</math>
is an isomorphism of rings. Hence we can use <math>K_0(X)</math> for intersection theory.

==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".<ref>Karoubi, 2006</ref>  Grothendieck needed to work with [[Coherent sheaf|coherent sheaves]] on an [[algebraic variety]] ''X''. Rather than working directly with the sheaves, he defined a group using [[isomorphism class]]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 sheaf|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 [[Cohomology|cohomological]] behavior and ''G(X)'' has [[Homology (mathematics)|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 bundle]]s, [[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|Index Theorem]] (circa 1962). Furthermore, this approach led to a [[noncommutative topology|noncommutative]] K-theory for [[C*-algebra]]s.

Already in 1955, [[Jean-Pierre Serre]] had used the analogy of [[vector bundle]]s with [[projective module]]s to formulate [[Quillen–Suslin theorem|Serre's conjecture]], which states that every finitely generated projective module over a [[polynomial ring]] is [[free module|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 Whitehead and others on what later became known as [[Whitehead torsion]].

There followed a period in which there were various partial definitions of ''[[Algebraic K-theory#Higher K-theory|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.<ref>by Ruben Minasian (http://string.lpthe.jussieu.fr/members.pl?key=7), and [[Greg Moore (physicist)|Gregory Moore]] (http://www.physics.rutgers.edu/~gmoore) in [http://xxx.lanl.gov/abs/hep-th/9710230 K-theory and Ramond–Ramond Charge].</ref>

== Examples ==
* The easiest example of the Grothendieck group is the Grothendieck group of a point <math>\text{Spec}(\mathbb{F})</math> for a field <math>\mathbb{F}</math>. 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 <math>\mathbb{N}</math> corresponding to the dimension of the vector space. It is an easy exercise to show that the Grothendieck group is then <math>\mathbb{Z}</math>.
* One important property of the Grothendieck group of a [[Noetherian scheme|Noetherian]] scheme <math>X</math> is that <math>K(X) = K(X_{red})</math>.<ref>{{Cite web|url=https://mathoverflow.net/questions/77089/grothendieck-group-for-projective-space-over-the-dual-numbers|title=Grothendieck group for projective space over the dual numbers|website=mathoverflow.net|access-date=2017-04-16}}</ref> Hence the Grothendieck group of any [[Artinian ring|Artinian]] <math>\mathbb{F}</math>-algebra is <math>\mathbb{Z}</math>.
* Another important formula for the Grothendieck group is the projective bundle formula:<ref>{{Cite journal|last=Manin|first=Yu I|date=1969-01-01|title=LECTURES ON THE K-FUNCTOR IN ALGEBRAIC GEOMETRY|url=http://iopscience.iop.org/article/10.1070/RM1969v024n05ABEH001357/meta|journal=Russian Mathematical Surveys|language=en|volume=24|issue=5|doi=10.1070/rm1969v024n05abeh001357/meta|issn=0036-0279|doi-broken-date=2017-06-24}}</ref> given a rank r vector bundle <math>\mathcal{E}</math> over a Noetherian scheme <math>X</math>, the Grothendieck group of the projective bundle <math>\mathbb{P}(\mathcal{E})=\text{Proj}(\text{Sym}^\bullet(\mathcal{E}^\vee))</math> is a free <math>K(X)</math>-module of rank r with basis <math>1,\xi,\cdots,\xi^{n-1}</math>. This formula allows one to compute the Grothendieck group of <chem>\mathbb{P}^n_\mathbb{F}</chem>.


==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 <math>Y \hookrightarrow X</math> then there is a short exact sequence
:<math> 0 \to \Omega_Y \to \Omega_X|_Y \to C_{Y/X} \to 0</math> where <math>C_{Y/X}</math> is the conormal bundle of <math>Y</math> in <math>X</math>
If we have a singular space <math>Y</math> embedded into a smooth space <math>X</math> we define the virtual conormal bundle as
:<math>[\Omega_X|_Y] - [\Omega_Y]</math>

===Chern characters===
{{main|Chern character}}
[[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

:<math>\operatorname{ch}(L) = \exp(c_{1}(L)) := \sum_{m=0}^\infty \frac{c_1(L)^m}{m!}.</math>

More generally, if <math>V = L_1 \oplus \dots \oplus L_n</math> is a direct sum of line bundles, with first Chern classes <math>x_i = c_1(L_i),</math> the Chern character is defined additively

:<math> \operatorname{ch}(V)  = e^{x_1} + \dots + e^{x_n} :=\sum_{m=0}^\infty \frac{1}{m!}(x_1^m + \dots + x_n^m). </math>

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==
{{main|Equivariant K-theory}}
The [[Equivariant K-theory|equivariant algebraic K-theory]] is an [[algebraic K-theory]] associated to the category <math>\operatorname{Coh}^G(X)</math> of [[equivariant sheaf|equivariant coherent sheaves]] on an algebraic scheme ''X'' with [[linear algebraic group action|action of a linear algebraic group]] ''G'',  via Quillen's [[Q-construction]]; thus, by definition,
:<math>K_i^G(X) = \pi_i(B^+ \operatorname{Coh}^G(X)).</math>
In particular, <math>K_0^G(C)</math> is the [[Grothendieck group]] of <math>\operatorname{Coh}^G(X)</math>. The theory was developed by R. W. Thomason in 1980s.<ref>Charles A. Weibel, [http://www.ams.org/notices/199608/comm-thomason.pdf Robert W. Thomason (1952–1995)].</ref> Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.

==See also==
*[[Bott periodicity]]
*[[KK-theory]]
*[[KR-theory]]
*[[List of cohomology theories]]
*[[algebraic K-theory]]
*[[topological K-theory]]
*[[Operator K-theory]]
*[[Grothendieck–Riemann–Roch theorem]]

==Notes==
{{Reflist}}

==References==
* {{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | title=K-theory | publisher=[[Addison-Wesley]] | edition=2nd | series=Advanced Book Classics | isbn=978-0-201-09394-0 | mr=1043170 | year=1989}}
* {{Citation | editor1-last=Friedlander | editor1-first=Eric | editor2-last=Grayson | editor2-first=Daniel | title=Handbook of K-Theory | url=http://www.springerlink.com/content/978-3-540-23019-9/ | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-30436-4 | mr=2182598 | year=2005}}
* {{Citation | last=Park | first=Efton | title=Complex Topological K-Theory | publisher=Cambridge University Press | series=Cambridge Studies in Advanced Mathematics 111 | year=2008}}
* {{Citation | last1=Swan | first1=R. G. | title=Algebraic K-Theory | publisher=[[Springer Science+Business Media|Springer]] | series=Lecture Notes in Mathematics No. 76 | year=1968}}
* [[Max Karoubi]] (1978), [http://www.springer.com/gp/book/9783540798897 K-theory, an introduction] Springer-Verlag
* Max Karoubi (2006), "K-theory. An elementary introduction", {{arxiv|math|0602082}}
* [[Allen Hatcher]], ''[http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html Vector Bundles & K-Theory]'', (2003)
* [[Charles Weibel]] (2013), "The K-book: an introduction to algebraic K-theory," Grad. Studies in Math. 145, American Math Society.

== External links ==
* [http://abel.harvard.edu/theses/senior/patrick/patrick.pdf Grothendieck-Riemann-Roch]
* [https://webusers.imj-prg.fr/~max.karoubi/ Max Karoubi's Page]
* [http://www.math.uiuc.edu/K-theory/ K-theory preprint archive]

{{Topology}}

[[Category:K-theory| ]]