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. 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 ${\displaystyle [X/G]}$. (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.

Let X be an equivariant algebraic scheme.

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 ${\displaystyle K_{0}^{G}(*)}$.

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 ${\displaystyle V}$. 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.

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

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• Dan Edidin, Riemann–Roch for Deligne–Mumford stacks, 2012