# Equivariant algebraic K-theory

## From Wikipedia, the free encyclopedia

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,

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

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.[1] 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 $[X/G]$.[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]