K-theory

For the hip hop group, seeK Theory.

Inmathematics,K-theoryis, roughly speaking, the study of aringgenerated byvector bundlesover atopological spaceorscheme.Inalgebraic topology,it is acohomology theoryknown astopological K-theory.Inalgebraandalgebraic geometry,it is referred to asalgebraic K-theory.It is also a fundamental tool in the field ofoperator algebras.It can be seen as the study of certain kinds ofinvariantsof largematrices.

K-theory involves the construction of families ofK-functorsthat 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 togroupsin 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 theGrothendieck–Riemann–Roch theorem,Bott periodicity,theAtiyah–Singer index theorem,and theAdams operations.

Inhigh energy physics,K-theory and in particulartwisted K-theoryhave appeared inType II string theorywhere it has been conjectured that they classifyD-branes,Ramond–Ramond field strengthsand also certainspinorsongeneralized complex manifolds.Incondensed matter physicsK-theory has been used to classifytopological insulators,superconductorsand stableFermi surfaces.For more details, seeK-theory (physics).

Grothendieck completion

Main article:Grothendieck group

The Grothendieck completion of anabelian monoidinto 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 agroup$(G(A),+)$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 acertain universal property.

To get a better understanding of this group, consider someequivalence classesof the abelian monoid$(A,+)$.Here we will denote the identity element of$A$by$0$so that$[(0,0)]$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

$[(a,b)]+[(b,a)]=[(a+b,a+b)]=[(0,0)]$

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$[(a,b)]$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 afunctor$G:\mathbf {AbMon} \to \mathbf {AbGrp},$and it has the property that it is left adjoint to the correspondingforgetful functor$U:\mathbf {AbGrp} \to \mathbf {AbMon}.$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$\mathbb {N}$.We can see that$G((\mathbb {N},+))=(\mathbb {Z},+).$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$k:=\min\{a,b\}$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

This section is about K₀,the most basic K-theory group (see alsoGrothendieck group). For definitions of higher K-groups K_i,seeAlgebraic K-theoryandTopological K-theory.

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 compactHausdorff space$X$consider the set of isomorphism classes of finite-dimensional vector bundles over$X$,denoted${\text{Vect}}(X)$and let the isomorphism class of a vector bundle$\pi:E\to X$be denoted$[E]$.Since isomorphism classes of vector bundles behave well with respect todirect sums,we can write these operations on isomorphism classes by

$[E]\oplus [E']=[E\oplus E']$

It should be clear that$({\text{Vect}}(X),\oplus )$is an abelian monoid where the unit is given by the trivial vector bundle$\mathbb {R} ^{0}\times X\to X$.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 theSerre–Swan theoremand some algebra to get an alternative description of vector bundles over the ring of continuous complex-valued functions$C^{0}(X;\mathbb {C} )$asprojective modules.Then, these can be identified withidempotentmatrices in some ring of matrices$M_{n\times n}(C^{0}(X;\mathbb {C} ))$.We can define equivalence classes of idempotent matrices and form an abelian monoid${\textbf {Idem}}(X)$.Its Grothendieck completion is also called$K^{0}(X)$.One of the main techniques for computing the Grothendieck group for topological spaces comes from theAtiyah–Hirzebruch spectral sequence,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 inalgebraic geometry.For aNoetherian scheme$X$there is a set${\text{Vect}}(X)$of all isomorphism classes ofalgebraic vector bundleson$X$.Then, as before, the direct sum$\oplus$of isomorphisms classes of vector bundles is well-defined, giving an abelian monoid$({\text{Vect}}(X),\oplus )$.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 ofcoherent sheaves$\operatorname {Coh} (X)$we can mod out by the relation$[{\mathcal {E}}]=[{\mathcal {E}}']+[{\mathcal {E}}'']$if there is ashort exact sequence

$0\to {\mathcal {E}}'\to {\mathcal {E}}\to {\mathcal {E}}''\to 0.$

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

$[{\mathcal {E}}]\cdot [{\mathcal {E}}']=\sum (-1)^{k}\left[\operatorname {Tor} _{k}^{{\mathcal {O}}_{X}}({\mathcal {E}},{\mathcal {E}}')\right].$

Using theGrothendieck–Riemann–Roch theorem,we have that

$\operatorname {ch}:K_{0}(X)\otimes \mathbb {Q} \to A(X)\otimes \mathbb {Q}$

is an isomorphism of rings. Hence we can use$K_{0}(X)$forintersection theory.

Early history

The subject can be said to begin withAlexander Grothendieck(1957), who used it to formulate hisGrothendieck–Riemann–Roch theorem.It takes its name from the GermanKlasse,meaning "class".Grothendieck needed to work withcoherent sheaveson analgebraic varietyX.Rather than working directly with the sheaves, he defined a group usingisomorphism classesof sheaves as generators of the group, subject to a relation that identifies any extension of two sheaves with their sum. The resulting group is calledK(X) when onlylocally free sheavesare used, orG(X) when all are coherent sheaves. Either of these two constructions is referred to as theGrothendieck group;K(X) hascohomologicalbehavior andG(X) hashomologicalbehavior.

IfXis asmooth variety,the two groups are the same. If it is a smoothaffine variety,then all extensions of locally free sheaves split, so the group has an alternative definition.

Intopology,by applying the same construction tovector bundles,Michael AtiyahandFriedrich HirzebruchdefinedK(X) for atopological spaceXin 1959, and using theBott periodicity theoremthey made it the basis of anextraordinary cohomology theory.It played a major role in the second proof of theAtiyah–Singer index theorem(circa 1962). Furthermore, this approach led to anoncommutativeK-theory forC*-algebras.

Already in 1955,Jean-Pierre Serrehad used the analogy ofvector bundleswithprojective modulesto formulateSerre's conjecture,which states that every finitely generated projective module over apolynomial ringisfree;this assertion is correct, but was not settled until 20 years later. (Swan's theoremis another aspect of this analogy.)

Developments

The other historical origin of algebraic K-theory was the work ofJ. H. C. Whiteheadand others on what later became known asWhitehead torsion.

There followed a period in which there were various partial definitions ofhigher K-theory functors.Finally, two useful and equivalent definitions were given byDaniel Quillenusinghomotopy theoryin 1969 and 1972. A variant was also given byFriedhelm Waldhausenin order to study thealgebraic 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 ofmotivic cohomology.

The corresponding constructions involving an auxiliaryquadratic formreceived the general nameL-theory.It is a major tool ofsurgery theory.

Instring theory,the K-theory classification ofRamond–Ramond fieldstrengths and the charges of stableD-braneswas first proposed in 1997.

Examples and properties

K₀of a field

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$\mathbb {Z}$.

K₀of an Artinian algebra over a field

One important property of the Grothendieck group of aNoetherian scheme$X$is that it is invariant under reduction, hence$K(X)=K(X_{\text{red}})$.Hence the Grothendieck group of anyArtinian$\mathbb {F}$-algebra is a direct sum of copies of$\mathbb {Z}$,one for each connected component of its spectrum. For example, $$K_{0}\left({\text{Spec}}\left({\frac {\mathbb {F} [x]}{(x^{9})}}\times \mathbb {F} \right)\right)=\mathbb {Z} \oplus \mathbb {Z}$$

K₀of projective space

One of the most commonly used computations of the Grothendieck group is with the computation of$K(\mathbb {P} ^{n})$for projective space over a field. This is because the intersection numbers of a projective$X$can be computed by embedding$i:X\hookrightarrow \mathbb {P} ^{n}$and using the push pull formula$i^{*}([i_{*}{\mathcal {E}}]\cdot [i_{*}{\mathcal {F}}])$.This makes it possible to do concrete calculations with elements in$K(X)$without having to explicitly know its structure since $$K(\mathbb {P} ^{n})={\frac {\mathbb {Z} [T]}{(T^{n+1})}}$$ One technique for determining the Grothendieck group of$\mathbb {P} ^{n}$comes from its stratification as $$\mathbb {P} ^{n}=\mathbb {A} ^{n}\coprod \mathbb {A} ^{n-1}\coprod \cdots \coprod \mathbb {A} ^{0}$$ since the Grothendieck group of coherent sheaves on affine spaces are isomorphic to$\mathbb {Z}$,and the intersection of$\mathbb {A} ^{n-k_{1}},\mathbb {A} ^{n-k_{2}}$is generically $$\mathbb {A} ^{n-k_{1}}\cap \mathbb {A} ^{n-k_{2}}=\mathbb {A} ^{n-k_{1}-k_{2}}$$ for$k_{1}+k_{2}\leq n$.

K₀of a projective bundle

Another important formula for the Grothendieck group is the projective bundle formula:given a rank r vector bundle${\mathcal {E}}$over a Noetherian scheme$X$,the Grothendieck group of the projective bundle$\mathbb {P} ({\mathcal {E}})=\operatorname {Proj} (\operatorname {Sym} ^{\bullet }({\mathcal {E}}^{\vee }))$is a free$K(X)$-module of rankrwith basis$1,\xi,\dots,\xi ^{n-1}$.This formula allows one to compute the Grothendieck group of$\mathbb {P} _{\mathbb {F} }^{n}$.This make it possible to compute the$K_{0}$or Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group$K(\mathbb {P} ^{n})$by observing it is a projective bundle over the field$\mathbb {F}$.

K₀of 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 theSingularity category$D_{sg}(X)$fromderived noncommutative algebraic geometry.It gives a long exact sequence starting with $$\cdots \to K^{0}(X)\to K_{0}(X)\to K_{sg}(X)\to 0$$ where the higher terms come fromhigher K-theory.Note that vector bundles on a singular$X$are given by vector bundles$E\to X_{sm}$on the smooth locus$X_{sm}\hookrightarrow X$.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${\text{lcm}}(|G_{1}|,\ldots,|G_{k}|)^{n-1}$for$n=\dim X$.^{pg 3}

K₀of a smooth projective curve

For a smooth projective curve$C$the Grothendieck group is $$K_{0}(C)=\mathbb {Z} \oplus {\text{Pic}}(C)$$ forPicard groupof$C$.This follows from theBrown-Gersten-Quillen spectral sequence^{pg 72}ofalgebraic K-theory.For aregular schemeof finite type over a field, there is a convergent spectral sequence $$E_{1}^{p,q}=\coprod _{x\in X^{(p)}}K^{-p-q}(k(x))\Rightarrow K_{-p-q}(X)$$ for$X^{(p)}$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 property^{pg 80} $$E_{2}^{p,-p}\cong {\text{CH}}^{p}(X)$$ 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$E_{1}^{0,q},E_{1}^{1,q}$,hence $${\begin{aligned}E_{\infty }^{1,-1}\cong E_{2}^{1,-1}&\cong {\text{CH}}^{1}(C)\\E_{\infty }^{0,0}\cong E_{2}^{0,0}&\cong {\text{CH}}^{0}(C)\end{aligned}}$$ Theconiveau filtrationcan 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${\text{CH}}^{1}(C)\cong {\text{Pic}}(C)$and the right hand term is isomorphic to$CH^{0}(C)\cong \mathbb {Z}$.Since${\text{Ext}}_{\text{Ab}}^{1}(\mathbb {Z},G)=0$,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$\mathbb {C}$,then $$K_{0}(C)\cong \mathbb {Z} \oplus (\mathbb {C} ^{g}/\mathbb {Z} ^{2g})$$ Moreover, the techniques above using the derived category of singularities for isolated singularities can be extended to isolatedCohen-Macaulaysingularities, 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

$0\to \Omega _{Y}\to \Omega _{X}|_{Y}\to C_{Y/X}\to 0$

where$C_{Y/X}$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

$[\Omega _{X}|_{Y}]-[\Omega _{Y}]$

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

$[T_{Z}]^{vir}=[T_{Y_{1}}]|_{Z}+[T_{Y_{2}}]|_{Z}-[T_{X}]|_{Z}.$

Kontsevich uses this construction in one of his papers.

Chern characters

Main article:Chern character

Chern classescan be used to construct a homomorphism of rings from thetopological K-theoryof a space to (the completion of) its rational cohomology. For a line bundleL,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 theHirzebruch–Riemann–Roch theorem.

Equivariant K-theory

Theequivariant algebraic K-theoryis analgebraic K-theoryassociated to the category$\operatorname {Coh} ^{G}(X)$ofequivariant coherent sheaveson an algebraic scheme$X$withaction of a linear algebraic group$G$,via Quillen'sQ-construction;thus, by definition,

$K_{i}^{G}(X)=\pi _{i}(B^{+}\operatorname {Coh} ^{G}(X)).$

In particular,$K_{0}^{G}(C)$is theGrothendieck groupof$\operatorname {Coh} ^{G}(X)$.The theory was developed by R. W. Thomason in 1980s.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

References

• Atiyah, Michael Francis(1989).K-theory.Advanced Book Classics (2nd ed.).Addison-Wesley.ISBN978-0-201-09394-0.MR1043170.
• Friedlander, Eric; Grayson, Daniel, eds. (2005).Handbook of K-Theory.Berlin, New York:Springer-Verlag.doi:10.1007/978-3-540-27855-9.ISBN978-3-540-30436-4.MR2182598.
• Park, Efton (2008).Complex Topological K-Theory.Cambridge Studies in Advanced Mathematics. Vol. 111. Cambridge University Press.ISBN978-0-521-85634-8.
• Swan, R. G.(1968).Algebraic K-Theory.Lecture Notes in Mathematics. Vol. 76.Springer.ISBN3-540-04245-8.
• Karoubi, Max(1978).K-theory: an introduction.Classics in Mathematics. Springer-Verlag.doi:10.1007/978-3-540-79890-3.ISBN0-387-08090-2.
• 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. Vol. 145. American Math Society.ISBN978-0-8218-9132-2.

External links

• Grothendieck-Riemann-Roch
• Max Karoubi's Page
• K-theory preprint archive