K-theory

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

${\displaystyle (4,6)\sim (3,5)\sim (2,4)\sim (1,3)\sim (0,2)}$

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

${\displaystyle (a,b)\sim (a-k,b-k)}$ which is of the form ${\displaystyle (c,0)}$ or ${\displaystyle (0,d).}$

This shows that we should think of the ${\displaystyle (a,0)}$ as positive integers and the ${\displaystyle (0,b)}$ as negative integers.

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

${\displaystyle [E]\oplus [E']=[E\oplus E']}$

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 the ring of continuous complex-valued functions ${\displaystyle C^{0}(X;\mathbb {C} )}$ as projective modules. 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 ${\displaystyle {\textbf {Idem}}(X)}$. 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 ${\displaystyle S^{n}}$.[2]^{pg 51-110}

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.

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 ${\displaystyle \operatorname {Coh} (X)}$ we can mod out by the relation ${\displaystyle [{\mathcal {E}}]=[{\mathcal {E}}']+[{\mathcal {E}}'']}$ if there is a short exact sequence

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

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

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

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

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

is an isomorphism of rings. Hence we can use ${\displaystyle K_{0}(X)}$ for intersection theory.[3]

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 ${\displaystyle \mathbb {Z} }$.

One important property of the Grothendieck group of a Noetherian scheme ${\displaystyle X}$ is that it is invariant under reduction, hence ${\displaystyle K(X)=K(X_{\text{red}})}$.[6] 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,

$${\displaystyle K_{0}\left({\text{Spec}}\left({\frac {\mathbb {F} [x]}{(x^{9})}}\times \mathbb {F} \right)\right)=\mathbb {Z} \oplus \mathbb {Z} }$$

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 ${\displaystyle i^{*}([i_{*}{\mathcal {E}}]\cdot [i_{*}{\mathcal {F}}])}$. This makes it possible to do concrete calculations with elements in ${\displaystyle K(X)}$ without having to explicitly know its structure since[7]

$${\displaystyle K(\mathbb {P} ^{n})={\frac {\mathbb {Z} [T]}{(T^{n+1})}}}$$

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 ${\displaystyle k_{1}+k_{2}\leq n}$.

Another important formula for the Grothendieck group is the projective bundle formula:[8] given a rank r vector bundle ${\displaystyle {\mathcal {E}}}$ over a Noetherian scheme ${\displaystyle X}$, the Grothendieck group of the projective bundle ${\displaystyle \mathbb {P} ({\mathcal {E}})=\operatorname {Proj} (\operatorname {Sym} ^{\bullet }({\mathcal {E}}^{\vee }))}$ 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 ${\displaystyle \mathbb {F} }$.

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 ${\displaystyle D_{sg}(X)}$[9][10] 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 ${\displaystyle {\text{lcm}}(|G_{1}|,\ldots ,|G_{k}|)^{n-1}}$ for ${\displaystyle n=\dim X}$.[10]^{pg 3}

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[11]^{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[11]^{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

$${\displaystyle {\begin{aligned}E_{\infty }^{1,-1}\cong E_{2}^{1,-1}&\cong {\text{CH}}^{1}(C)\\E_{\infty }^{0,0}\cong E_{2}^{0,0}&\cong {\text{CH}}^{0}(C)\end{aligned}}}$$

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 ${\displaystyle CH^{0}(C)\cong \mathbb {Z} }$. 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.

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

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

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

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

Another useful application of virtual bundles is with the definition of a virtual tangent bundle of an intersection of spaces: Let ${\displaystyle Y_{1},Y_{2}\subset X}$ 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.[12]

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

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

More generally, if ${\displaystyle V=L_{1}\oplus \dots \oplus L_{n}}$ is a direct sum of line bundles, with first Chern classes ${\displaystyle x_{i}=c_{1}(L_{i}),}$ the Chern character is defined additively

${\displaystyle \operatorname {ch} (V)=e^{x_{1}}+\dots +e^{x_{n}}:=\sum _{m=0}^{\infty }{\frac {1}{m!}}(x_{1}^{m}+\dots +x_{n}^{m}).}$

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.