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# Proposed code for K-theory, which you have asked the bot to commit with edit summary Add: doi-broken-date. You can [[WP:UCB|use this bot]] yourself. [[WP:DBUG|Report bugs here]].{{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)]].

==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 $\text{Spec}(\mathbb{F})$ for a field $\mathbb{F}$. 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 $\mathbb{N}$ corresponding to the dimension of the vector space. It is an easy exercise to show that the Grothendieck group is then $\mathbb{Z}$.
* One important property of the Grothendieck group of a [[Noetherian scheme|Noetherian]] scheme $X$ is that $K(X) = K(X_{red})$<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]] $\mathbb{F}$-algebra is $\mathbb{Z}$.
* 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=2017-04-29}}</ref>: given a rank r vector bundle $\mathcal{E}$ over a Noetherian scheme $X$, the Grothendieck group of the projective bundle $\mathbb{P}(\mathcal{E})=\text{Proj}(\text{Sym}^\bullet(\mathcal{E}^\vee))$ is a free $K(X)$-module of rank r with basis $1,\xi,\cdots,\xi^{n-1}$. This formula allows one to compute the Grothendieck group of <chem>\mathbb{P}^n_\mathbb{F}</chem>.

==Applications==

===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

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

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

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

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==
{{main|Equivariant K-theory}}
The [[Equivariant K-theory|equivariant algebraic K-theory]] is an [[algebraic K-theory]] associated to the category $\operatorname{Coh}^G(X)$ 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,
:$K_i^G(X) = \pi_i(B^+ \operatorname{Coh}^G(X)).$
In particular, $K_0^G(C)$ is the [[Grothendieck group]] of $\operatorname{Coh}^G(X)$. 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.

*[[Bott periodicity]]
*[[KK-theory]]
*[[KR-theory]]
*[[List of cohomology theories]]
*[[algebraic K-theory]]
*[[topological K-theory]]
*[[Operator K-theory]]

==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 | 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.

* [http://www.institut.math.jussieu.fr/~karoubi/ Max Karoubi's Page]
* [http://www.math.uiuc.edu/K-theory/ K-theory preprint archive]

{{Topology}}

[[Category:K-theory| ]]