K-theory

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

${\displaystyle [E]\oplus [E^{\prime }]=[E\oplus E^{\prime }]}$

It should be clear that ${\displaystyle (\text{Vect}(X),\oplus )}$ is an abelian monoid where the unit is given by the trivial vector bundle ${\displaystyle {\mathbb{R}}^0\times X\to X}$. We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. This is called the K-theory of ${\displaystyle X}$ and is denoted ${\displaystyle K^0(X)}$.

We can use the Serre–Swan theorem and some algebra to get an alternative description of vector bundles over ${\displaystyle X}$ as projective modules over the ring ${\displaystyle C^0(X;\mathbb{C})}$ of continuous complex-valued functions. Then, these can be identified with idempotent matrices in some ring of matrices ${\displaystyle M_{n\times n}(C^0(X;\mathbb{C}))}$. We can define equivalence classes of idempotent matrices and form an abelian monoid ${\displaystyle \mathbf{\text{Idem}}(X)}$. Its Grothendieck completion is also called ${\displaystyle K^0(X)}$. 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 ${\displaystyle K^0}$ for the spheres ${\displaystyle S^n}$.^{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 ${\displaystyle X}$ there is a set ${\displaystyle \text{Vect}(X)}$ of all isomorphism classes of algebraic vector bundles on ${\displaystyle X}$. Then, as before, the direct sum ${\displaystyle \oplus }$ of isomorphisms classes of vector bundles is well-defined, giving an abelian monoid ${\displaystyle (\text{Vect}(X),\oplus )}$. Then, the Grothendieck group ${\displaystyle K^0(X)}$ 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 ${\displaystyle X}$. If we look at the isomorphism classes of coherent sheaves ${\displaystyle \mathrm{Coh}(X)}$ we can mod out by the relation ${\displaystyle [\mathcal{E}]=[{\mathcal{E}}^{\prime }]+[{\mathcal{E}}^{″}]}$ if there is a short exact sequence

${\displaystyle 0\to {\mathcal{E}}^{\prime }\to \mathcal{E}\to {\mathcal{E}}^{″}\to 0.}$

This gives the Grothendieck-group ${\displaystyle K_0(X)}$ which is isomorphic to ${\displaystyle K^0(X)}$ if ${\displaystyle X}$ is smooth. The group ${\displaystyle K_0(X)}$ is special because there is also a ring structure: we define it as

${\displaystyle [\mathcal{E}]\cdot [{\mathcal{E}}^{\prime }]=\sum (-1{)}^k\left[{\mathrm{Tor}}_k^{{\mathcal{O}}_X}(\mathcal{E},{\mathcal{E}}^{\prime })\right].}$

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

${\displaystyle \mathrm{ch}:K_0(X)\otimes \mathbb{Q}\to A(X)\otimes \mathbb{Q}}$

is an isomorphism of rings. Hence we can use ${\displaystyle K_0(X)}$ 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". Grothendieck needed to work with coherent sheaves 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 cohomological 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 vector bundles, 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 vector bundles with projective modules to formulate Serre's conjecture, which states that every finitely generated projective module over a polynomial ring 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 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.

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

Examples and properties

K0 of a field

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

K0 of an Artinian algebra over a field

One important property of the Grothendieck group of a Noetherian scheme ${\displaystyle X}$ is that it is invariant under reduction, hence ${\displaystyle K(X)=K(X_{\text{red}})}$. Hence the Grothendieck group of any Artinian ${\displaystyle \mathbb{F}}$-algebra is a direct sum of copies of ${\displaystyle \mathbb{Z}}$, one for each connected component of its spectrum. For example, $${\displaystyle K_0\left(\text{Spec}\left(\frac{\mathbb{F}[x]}{(x^9)}\times \mathbb{F}\right)\right)=\mathbb{Z}\oplus \mathbb{Z}}$$

K0 of projective space

One of the most commonly used computations of the Grothendieck group is with the computation of ${\displaystyle K({\mathbb{P}}^n)}$ for projective space over a field. This is because the intersection numbers of a projective ${\displaystyle X}$ can be computed by embedding ${\displaystyle i:X\hookrightarrow {\mathbb{P}}^n}$ and using the push pull formula ${\displaystyle i^{\ast }([i_{\ast }\mathcal{E}]\cdot [i_{\ast }\mathcal{F}])}$. This makes it possible to do concrete calculations with elements in ${\displaystyle K(X)}$ without having to explicitly know its structure since $${\displaystyle K({\mathbb{P}}^n)=\frac{\mathbb{Z}[T]}{(T^{n+1})}}$$ One technique for determining the Grothendieck group of ${\displaystyle {\mathbb{P}}^n}$ comes from its stratification as $${\displaystyle {\mathbb{P}}^n={\mathbb{A}}^n\coprod {\mathbb{A}}^{n-1}\coprod \cdots \coprod {\mathbb{A}}^0}$$ since the Grothendieck group of coherent sheaves on affine spaces are isomorphic to ${\displaystyle \mathbb{Z}}$, and the intersection of ${\displaystyle {\mathbb{A}}^{n-k_1},{\mathbb{A}}^{n-k_2}}$ is generically $${\displaystyle {\mathbb{A}}^{n-k_1}\cap {\mathbb{A}}^{n-k_2}={\mathbb{A}}^{n-k_1-k_2}}$$ for ${\displaystyle k_1+k_2\le n}$.

K0 of a projective bundle

Another important formula for the Grothendieck group is the projective bundle formula: given a rank r vector bundle ${\displaystyle \mathcal{E}}$ over a Noetherian scheme ${\displaystyle X}$, the Grothendieck group of the projective bundle ${\displaystyle \mathbb{P}(\mathcal{E})=\mathrm{Proj}({\mathrm{Sym}}^{\bullet }({\mathcal{E}}^{\vee }))}$ is a free ${\displaystyle K(X)}$-module of rank r with basis ${\displaystyle 1,\xi ,\dots ,{\xi }^{n-1}}$. This formula allows one to compute the Grothendieck group of ${\displaystyle {\mathbb{P}}_{\mathbb{F}}^n}$. This make it possible to compute the ${\displaystyle K_0}$ or Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group ${\displaystyle K({\mathbb{P}}^n)}$ by observing it is a projective bundle over the field ${\displaystyle \mathbb{F}}$.

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 ${\displaystyle K^0(X)}$ and ${\displaystyle K_0(X)}$, 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 ${\displaystyle D_{sg}(X)}$ from derived noncommutative algebraic geometry. It gives a long exact sequence starting with $${\displaystyle \cdots \to K^0(X)\to K_0(X)\to K_{sg}(X)\to 0}$$ where the higher terms come from higher K-theory. Note that vector bundles on a singular ${\displaystyle X}$ are given by vector bundles ${\displaystyle E\to X_{sm}}$ on the smooth locus ${\displaystyle X_{sm}\hookrightarrow X}$. 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 ${\displaystyle G_i}$ then the map $${\displaystyle K^0(X)\to K_0(X)}$$ is injective and the cokernel is annihilated by ${\displaystyle \text{lcm}(|G_1|,\dots ,|G_k|{)}^{n-1}}$ for ${\displaystyle n=\dim X}$.^{pg 3}

K0 of a smooth projective curve

For a smooth projective curve ${\displaystyle C}$ the Grothendieck group is $${\displaystyle K_0(C)=\mathbb{Z}\oplus \text{Pic}(C)}$$ for Picard group of ${\displaystyle C}$. This follows from the Brown-Gersten-Quillen spectral sequence^{pg 72} of algebraic K-theory. For a regular scheme of finite type over a field, there is a convergent spectral sequence $${\displaystyle E_1^{p,q}=\coprod _{x\in X^{(p)}}{K}^{-p-q}(k(x))\Rightarrow K_{-p-q}(X)}$$ for ${\displaystyle X^{(p)}}$ the set of codimension ${\displaystyle p}$ points, meaning the set of subschemes ${\displaystyle x:Y\to X}$ of codimension ${\displaystyle p}$, and ${\displaystyle k(x)}$ the algebraic function field of the subscheme. This spectral sequence has the property^{pg 80} $${\displaystyle E_2^{p,-p}\cong {\text{CH}}^p(X)}$$ for the Chow ring of ${\displaystyle X}$, essentially giving the computation of ${\displaystyle K_0(C)}$. Note that because ${\displaystyle C}$ has no codimension ${\displaystyle 2}$ points, the only nontrivial parts of the spectral sequence are ${\displaystyle E_1^{0,q},E_1^{1,q}}$, hence $${\displaystyle \begin{array}{cc}{E}_{\mathrm{\infty }}^{1,-1}\cong E_2^{1,-1}& \cong {\text{CH}}^1(C)\\ E_{\mathrm{\infty }}^{0,0}\cong E_2^{0,0}& \cong {\text{CH}}^0(C)\end{array}}$$ The coniveau filtration can then be used to determine ${\displaystyle K_0(C)}$ as the desired explicit direct sum since it gives an exact sequence $${\displaystyle 0\to F^1(K_0(X))\to K_0(X)\to K_0(X)/F^1(K_0(X))\to 0}$$ where the left hand term is isomorphic to ${\displaystyle {\text{CH}}^1(C)\cong \text{Pic}(C)}$ and the right hand term is isomorphic to ${\displaystyle C{H}^0(C)\cong \mathbb{Z}}$. Since ${\displaystyle {\text{Ext}}_{\text{Ab}}^1(\mathbb{Z},G)=0}$, we have the sequence of abelian groups above splits, giving the isomorphism. Note that if ${\displaystyle C}$ is a smooth projective curve of genus ${\displaystyle g}$ over ${\displaystyle \mathbb{C}}$, then $${\displaystyle K_0(C)\cong \mathbb{Z}\oplus ({\mathbb{C}}^g/{\mathbb{Z}}^{2g})}$$ Moreover, the techniques above using the derived category of singularities for isolated singularities can be extended to 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 ${\displaystyle Y\hookrightarrow X}$ then there is a short exact sequence

${\displaystyle 0\to {\mathrm{\Omega }}_Y\to {\mathrm{\Omega }}_X{|}_Y\to C_{Y/X}\to 0}$

where ${\displaystyle C_{Y/X}}$ is the conormal bundle of ${\displaystyle Y}$ in ${\displaystyle X}$. If we have a singular space ${\displaystyle Y}$ embedded into a smooth space ${\displaystyle X}$ we define the virtual conormal bundle as

${\displaystyle [{\mathrm{\Omega }}_X{|}_Y]-[{\mathrm{\Omega }}_Y]}$

Another useful application of virtual bundles is with the definition of a virtual tangent bundle of an intersection of spaces: Let ${\displaystyle Y_1,Y_2\subset X}$ be projective subvarieties of a smooth projective variety. Then, we can define the virtual tangent bundle of their intersection ${\displaystyle Z=Y_1\cap Y_2}$ as

${\displaystyle [T_Z{]}^{vir}=[T_{Y_1}]{|}_Z+[T_{Y_2}]{|}_Z-[T_X]{|}_Z.}$

Kontsevich uses this construction in one of his papers.

Chern characters

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

${\displaystyle \mathrm{ch}(L)=\exp (c_1(L)):=\sum _{m=0}^{\mathrm{\infty }}\frac{c_1(L{)}^m}{m!}.}$

More generally, if ${\displaystyle V=L_1\oplus \cdots \oplus L_n}$ is a direct sum of line bundles, with first Chern classes ${\displaystyle x_i=c_1(L_i),}$ the Chern character is defined additively

${\displaystyle \mathrm{ch}(V)=e^{x_1}+\cdots +e^{x_n}:=\sum _{m=0}^{\mathrm{\infty }}\frac{1}{m!}(x_1^m+\cdots +x_n^m).}$

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

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

${\displaystyle K_i^G(X)={\pi }_i(B^{+}{\mathrm{Coh}}^G(X)).}$

In particular, ${\displaystyle K_0^G(C)}$ is the Grothendieck group of ${\displaystyle {\mathrm{Coh}}^G(X)}$. The theory was developed by R. W. Thomason in 1980s. Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.