# Equivariant algebraic K-theory

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

$K_{i}^{G}(X)=\pi _{i}(B^{+}\operatorname {Coh} ^{G}(X)).$ For the topological equivariant K-theory, see topological K-theory.

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

Equivalently, $K_{i}^{G}(X)$ may be defined as the $K_{i}$ of the category of coherent sheaves on the quotient stack $[X/G]$ . (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.

## Fundamental theorems

Let X be an equivariant algebraic scheme.

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

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