Equivariant algebraic K-theory

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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]