Equivariant algebraic K-theory

In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category ${\displaystyle \operatorname {Coh} ^{G}(X)}$ 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,

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

In particular, ${\displaystyle K_{0}^{G}(C)}$ is the Grothendieck group of ${\displaystyle \operatorname {Coh} ^{G}(X)}$. The theory was developed by R. W. Thomason in 1980s.^[1] Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.

Equivalently, ${\displaystyle K_{i}^{G}(X)}$ may be defined as the ${\displaystyle K_{i}}$ of the category of coherent sheaves on the quotient stack ${\displaystyle [X/G]}$.^[2][3] (Hence, the equivariant K-theory is a specific case of the K-theory of a stack.)

A version of the Lefschetz fixed-point theorem holds in the setting of equivariant (algebraic) K-theory.^[4]

Fundamental theorems

Let X be an equivariant algebraic scheme.

Localization theorem—Given a closed immersion ${\displaystyle Z\hookrightarrow X}$ of equivariant algebraic schemes and an open immersion ${\displaystyle Z-U\hookrightarrow X}$, there is a long exact sequence of groups

${\displaystyle \cdots \to K_{i}^{G}(Z)\to K_{i}^{G}(X)\to K_{i}^{G}(U)\to K_{i-1}^{G}(Z)\to \cdots }$

Examples

One of the fundamental examples of equivariant K-theory groups are the equivariant K-groups of ${\displaystyle G}$-equivariant coherent sheaves on a points, so ${\displaystyle K_{i}^{G}(*)}$. Since ${\displaystyle {\text{Coh}}^{G}(*)}$ is equivalent to the category ${\displaystyle {\text{Rep}}(G)}$ of finite-dimensional representations of ${\displaystyle G}$. Then, the Grothendieck group of ${\displaystyle {\text{Rep}}(G)}$, denoted ${\displaystyle R(G)}$ is ${\displaystyle K_{0}^{G}(*)}$.^[5]

Torus ring

Given an algebraic torus ${\displaystyle \mathbb {T} \cong \mathbb {G} _{m}^{k}}$ a finite-dimensional representation ${\displaystyle V}$ is given by a direct sum of ${\displaystyle 1}$-dimensional ${\displaystyle \mathbb {T} }$-modules called the weights of ${\displaystyle V}$.^[6] There is an explicit isomorphism between ${\displaystyle K_{\mathbb {T} }}$ and ${\displaystyle \mathbb {Z} [t_{1},\ldots ,t_{k}]}$ given by sending ${\displaystyle [V]}$ to its associated character.^[7]

See also

• Topological K-theory, the topological equivariant K-theory