K-theory

In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a 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 . It can be seen as the study of certain kinds of invariants of large matrices.

K-theory involves the construction of families of K- that map from topological spaces or schemes, or to be even more general: any object of a homotopy category 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 .

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 strengths and also certain spinors on generalized complex manifolds. In condensed matter physics K-theory has been used to classify topological insulators, and stable . For more details, see K-theory (physics).

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Grothendieck completion The Grothendieck completion of an 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 $(A,+\text{'})$ let $\backslash sim$ be the relation on $A^2\; =\; A\; \backslash times\; A$ defined by

$(a\_1,a\_2)\; \backslash sim\; (b\_1,b\_2)$

if there exists a $c\backslash in\; A$ such that $a\_1\; +\text{'}\; b\_2\; +\text{'}\; c\; =\; a\_2\; +\text{'}\; b\_1\; +\text{'}\; c.$ Then, the set $G(A)\; =\; A^2/\backslash sim$ has the structure of a group $(G(A),+)$ where:

$(a\_1,a\_2)\; +\; (b\_1,b\_2)\; =\; (a\_1+\text{'}.$

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 $i\; :\; A\; \backslash to\; G(A)$ given by $a\; \backslash mapsto\; (a,,$ 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 $(0,0)$ will be the identity element of $(G(A),+).$ First, $(0,0)\; \backslash sim\; (n,n)$ for any $n\backslash in\; A$ 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 $(b,a)$ for each $(a,b)\; \backslash in\; G(A)$. 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:

$(a,b)\; \backslash sim\; (a+k,b+k)$ for any $k\; \backslash in\; A.$

The Grothendieck completion can be viewed as a functor $G:\backslash mathbf\{AbMon\}\backslash to\backslash mathbf\{AbGrp\},$ and it has the property that it is left adjoint to the corresponding forgetful functor $U:\backslash mathbf\{AbGrp\}\backslash to\backslash mathbf\{AbMon\}.$ That means that, given a morphism $\backslash phi:A\; \backslash to\; U(B)$ of an abelian monoid $A$ to the underlying abelian monoid of an abelian group $B,$ there exists a unique abelian group morphism $G(A)\; \backslash to\; B.$

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Example for natural numbers An illustrative example to look at is the Grothendieck completion of $\backslash N$. We can see that $G((\backslash N,+))\; =\; (\backslash Z,+).$ For any pair $(a,b)$ we can find a minimal representative $(a\text{'},b\text{'})$ by using the invariance under scaling. For example, we can see from the scaling invariance that

$(4,6)\; \backslash sim\; (3,5)\; \backslash sim\; (2,4)\; \backslash sim\; (1,3)\; \backslash sim\; (0,2)$

In general, if $k\; :=\; \backslash min\backslash \{a,b\backslash \}$ then

$(a,b)\; \backslash sim\; (a-k,b-k)$ 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.

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

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Grothendieck group for compact Hausdorff spaces Given a compact Hausdorff space $X$ consider the set of isomorphism classes of finite-dimensional vector bundles over $X$, denoted $\backslash text\{Vect\}(X)$ and let the isomorphism class of a vector bundle $\backslash pi:E\; \backslash to\; X$ be denoted $E$. Since isomorphism classes of vector bundles behave well with respect to , we can write these operations on isomorphism classes by

$E\backslash oplusE\text{'}\; =E\backslash oplus$

It should be clear that $(\backslash text\{Vect\}(X),\backslash oplus)$ is an abelian monoid where the unit is given by the trivial vector bundle $\backslash R^0\backslash times\; X\; \backslash to\; X$. 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 $X$ as projective modules over the ring $C^0(X;\backslash Complex)$ of continuous complex-valued functions. Then, these can be identified with Idempotence matrices in some ring of matrices $M\_\{n\backslash times\; n\}(C^0(X;\backslash Complex))$. We can define equivalence classes of idempotent matrices and form an abelian monoid $\backslash textbf\{Idem\}(X)$. Its Grothendieck completion is also called $K^0(X)$. One of the main techniques for computing the Grothendieck group for topological spaces comes from the Atiyah–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}

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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 $\backslash text\{Vect\}(X)$ of all isomorphism classes of algebraic vector bundles on $X$. Then, as before, the direct sum $\backslash oplus$ of isomorphisms classes of vector bundles is well-defined, giving an abelian monoid $(\backslash text\{Vect\}(X),\backslash oplus)$. Then, the Grothendieck group $K^0(X)$ is defined by the application of the Grothendieck construction on this abelian monoid.

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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 sheaf $\backslash operatorname\{Coh\}(X)$ we can mod out by the relation $\backslash mathcal\{E\}\; =\; \backslash mathcal\{E\}\text{'}\; +\; \backslash mathcal\{E\}\text{'}\text{'}$ if there is a Exact sequence

$0\; \backslash to\; \backslash mathcal\{E\}\text{'}\; \backslash to\; \backslash mathcal\{E\}\; \backslash to\; \backslash mathcal\{E\}\text{'}\text{'}\; \backslash to\; 0.$

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

$\backslash mathcal\{E\}\backslash cdot\backslash mathcal\{E\}\text{'}\; =\; \backslash sum(-1)^k\; \backslash left\; \backslash operatorname\{Tor\}\_k^\{\backslash mathcal\{O\}\_X\}(\backslash mathcal\{E\},.$

Using the Grothendieck–Riemann–Roch theorem, we have that

$\backslash operatorname\{ch\}\; :\; K\_0(X)\backslash otimes\; \backslash Q\; \backslash to\; A(X)\backslash otimes\; \backslash Q$

is an isomorphism of rings. Hence we can use $K\_0(X)$ for intersection theory.

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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".Karoubi, 2006 Grothendieck needed to work with Coherent sheaf on an algebraic variety 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 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 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, by applying the same construction to , 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 (circa 1962). Furthermore, this approach led to a noncommutative K-theory for C*-algebras.

Already in 1955, Jean-Pierre Serre had used the analogy of with projective modules to formulate Serre's conjecture, which states that every finitely generated projective module over a polynomial ring is free module; this assertion is correct, but was not settled until 20 years later. (Swan's theorem is another aspect of this analogy.)

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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 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.by Ruben Minasian (http://string.lpthe.jussieu.fr/members.pl?key=7), and Gregory Moore in K-theory and Ramond–Ramond Charge.

In 2022, Russian mathematician Alexander Ivanovich Efimov constructed a significant generalization of algebraic K-theory, particularly applied to dualizable $(\backslash infty,1)$-categories

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Examples and properties

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K₀ of a field The easiest example of the Grothendieck group is the Grothendieck group of a point $\backslash text\{Spec\}(\backslash mathbb\{F\})$ for a field $\backslash mathbb\{F\}$. 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 $\backslash N$ corresponding to the dimension of the vector space. It is an easy exercise to show that the Grothendieck group is then $\backslash Z$.

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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 $K(X)\; =\; K(X\_\{\backslash text\{red\}\})$. Hence the Grothendieck group of any Artinian ring $\backslash mathbb\{F\}$-algebra is a direct sum of copies of $\backslash Z$, one for each connected component of its spectrum. For example, $$K\_0\; \backslash left(\backslash text\{Spec\}\backslash left(\backslash frac\{\backslash mathbb\{F\}x\}\{(x^9)\}\backslash times\backslash mathbb\{F\}\backslash right)\backslash right)\; =\; \backslash mathbb\{Z\}\backslash oplus\backslash mathbb\{Z\}$$

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K₀ of projective space One of the most commonly used computations of the Grothendieck group is with the computation of $K(\backslash mathbb\{P\}^n)$ for projective space over a field. This is because the intersection numbers of a projective $X$ can be computed by embedding $i:X\; \backslash hookrightarrow\; \backslash mathbb\{P\}^n$ and using the push pull formula $i^*(i\_*\backslash mathcal\{E\}\backslash cdot\; i\_*\backslash mathcal\{F\})$. This makes it possible to do concrete calculations with elements in $K(X)$ without having to explicitly know its structure since $$K(\backslash mathbb\{P\}^n)\; =\; \backslash frac\{\backslash mathbb\{Z\}T\}\{(T^\{n+1\})\}$$ One technique for determining the Grothendieck group of $\backslash mathbb\{P\}^n$ comes from its stratification as $$\backslash mathbb\{P\}^n\; =\; \backslash mathbb\{A\}^n\; \backslash coprod\; \backslash mathbb\{A\}^\{n-1\}\; \backslash coprod\; \backslash cdots\; \backslash coprod\; \backslash mathbb\{A\}^0$$ since the Grothendieck group of coherent sheaves on affine spaces are isomorphic to $\backslash mathbb\{Z\}$, and the intersection of $\backslash mathbb\{A\}^\{n-k\_1\},\backslash mathbb\{A\}^\{n-k\_2\}$ is generically $$\backslash mathbb\{A\}^\{n-k\_1\}\; \backslash cap\; \backslash mathbb\{A\}^\{n-k\_2\}\; =\; \backslash mathbb\{A\}^\{n-k\_1-k\_2\}$$ for $k\_1\; +\; k\_2\; \backslash leq\; n$.

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K₀ of a projective bundle Another important formula for the Grothendieck group is the projective bundle formula: given a rank r vector bundle $\backslash mathcal\{E\}$ over a Noetherian scheme $X$, the Grothendieck group of the projective bundle $\backslash mathbb\{P\}(\backslash mathcal\{E\})=\backslash operatorname\{Proj\}(\backslash operatorname\{Sym\}^\backslash bullet(\backslash mathcal\{E\}^\backslash vee))$ is a free $K(X)$-module of rank r with basis $1,\backslash xi,\backslash dots,\backslash xi^\{n-1\}$. This formula allows one to compute the Grothendieck group of $\backslash mathbb\{P\}^n\_\backslash mathbb\{F\}$. This make it possible to compute the $K\_0$ or Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group $K(\backslash mathbb\{P\}^n)$ by observing it is a projective bundle over the field $\backslash mathbb\{F\}$.

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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 $D\_\{sg\}(X)$ from derived noncommutative algebraic geometry. It gives a long exact sequence starting with $$\backslash cdots\; \backslash to\; K^0(X)\; \backslash to\; K\_0(X)\; \backslash to\; K\_\{sg\}(X)\; \backslash to\; 0$$ where the higher terms come from higher K-theory. Note that vector bundles on a singular $X$ are given by vector bundles $E\; \backslash to\; X\_\{sm\}$ on the smooth locus $X\_\{sm\}\; \backslash hookrightarrow\; X$. 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 $$K^0(X)\; \backslash to\; K\_0(X)$$ is injective and the cokernel is annihilated by $\backslash text\{lcm\}(|G\_1|,\backslash ldots,\; |G\_k|)^\{n-1\}$ for $n\; =\; \backslash dim\; X$.^{pg 3}

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K₀ of a smooth projective curve

For a smooth projective curve $C$ the Grothendieck group is $$K\_0(C)\; =\; \backslash mathbb\{Z\}\backslash oplus\backslash text\{Pic\}(C)$$ for Picard group of $C$. This follows from the Brown-Gersten-Quillen spectral sequence

Srinivas, V. (1991). 9781489967350, Birkhäuser. ISBN 9781489967350

^{pg 72} of algebraic K-theory. For a regular scheme of finite type over a field, there is a convergent spectral sequence $$E\_1^\{p,q\}\; =\; \backslash coprod\_\{x\; \backslash in\; X^\{(p)\}\}K^\{-p-q\}(k(x))\; \backslash Rightarrow\; K\_\{-p-q\}(X)$$ for $X^\{(p)\}$ the set of codimension $p$ points, meaning the set of subschemes $x:\; Y\; \backslash to\; X$ of codimension $p$, and $k(x)$ the algebraic function field of the subscheme. This spectral sequence has the property^{pg 80} $$E\_2^\{p,-p\}\; \backslash cong\; \backslash text\{CH\}^p(X)$$ 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 The coniveau filtration can then be used to determine $K\_0(C)$ as the desired explicit direct sum since it gives an exact sequence $$0\; \backslash to\; F^1(K\_0(X))\; \backslash to\; K\_0(X)\; \backslash to\; K\_0(X)/F^1(K\_0(X))\; \backslash to\; 0$$ where the left hand term is isomorphic to $\backslash text\{CH\}^1\; (C)\; \backslash cong\; \backslash text\{Pic\}(C)$ and the right hand term is isomorphic to $CH^0(C)\; \backslash cong\; \backslash mathbb\{Z\}$. Since $\backslash text\{Ext\}^1\_\{\backslash text\{Ab\}\}(\backslash mathbb\{Z\},G)\; =\; 0$, 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 $\backslash mathbb\{C\}$, then $$K\_0(C)\; \backslash cong\; \backslash mathbb\{Z\}\backslash oplus(\backslash mathbb\{C\}^g/\backslash mathbb\{Z\}^\{2g\})$$ 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.

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Applications

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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 $Y\; \backslash hookrightarrow\; X$ then there is a short exact sequence

$0\; \backslash to\; \backslash Omega\_Y\; \backslash to\; \backslash Omega\_X|\_Y\; \backslash to\; C\_\{Y/X\}\; \backslash to\; 0$

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

$\backslash Omega\_X|\_Y\; -\; \backslash Omega\_Y$

Another useful application of virtual bundles is with the definition of a virtual tangent bundle of an intersection of spaces: Let $Y\_1,Y\_2\backslash subset\; X$ be projective subvarieties of a smooth projective variety. Then, we can define the virtual tangent bundle of their intersection $Z\; =\; Y\_1\backslash cap\; Y\_2$ as

$T\_Z^\{vir\}\; =\; T\_\{Y\_1\}|\_Z\; +\; T\_\{Y\_2\}|\_Z\; -\; T\_\{X\}|\_Z.$

Kontsevich uses this construction in one of his papers.

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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

$\backslash operatorname\{ch\}(L)\; =\; \backslash exp(c\_\{1\}(L))\; :=\; \backslash sum\_\{m=0\}^\backslash infty\; \backslash frac\{c\_1(L)^m\}\{m!\}.$

More generally, if $V\; =\; L\_1\; \backslash oplus\; \backslash dots\; \backslash oplus\; L\_n$ is a direct sum of line bundles, with first Chern classes $x\_i\; =\; c\_1(L\_i),$ the Chern character is defined additively

$\backslash operatorname\{ch\}(V)\; =\; e^\{x\_1\}\; +\; \backslash dots\; +\; e^\{x\_n\}\; :=\backslash sum\_\{m=0\}^\backslash infty\; \backslash frac\{1\}\{m!\}(x\_1^m\; +\; \backslash dots\; +\; x\_n^m).$

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.

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Equivariant K-theory

The equivariant algebraic K-theory is an algebraic K-theory associated to the category $\backslash operatorname\{Coh\}^G(X)$ of equivariant coherent sheaves on an algebraic scheme $X$ with action of a linear algebraic group $G$, via Quillen's Q-construction; thus, by definition,

$K\_i^G(X)\; =\; \backslash pi\_i(B^+\; \backslash operatorname\{Coh\}^G(X)).$

In particular, $K\_0^G(C)$ is the Grothendieck group of $\backslash operatorname\{Coh\}^G(X)$. 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.

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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

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Notes

• (1989). 9780201093940, Addison-Wesley. ISBN 9780201093940
• (2026). 9783540304364, Springer-Verlag. ISBN 9783540304364
• (2026). 9780521856348, Cambridge University Press. ISBN 9780521856348
• (1968). 9783540042457, Springer. ISBN 9783540042457
• (1978). 9780387080901, Springer-Verlag. ISBN 9780387080901
• (2026). 9780821891322, American Math Society. ISBN 9780821891322

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External links

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

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