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,
 * $$K_i^G(X) = \pi_i(B^+ \operatorname{Coh}^G(X)).$$

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.

Examples
One of the fundamental examples of equivariant K-theory groups are the equivariant K-groups of $$G$$-equivariant coherent sheaves on a points, so $$K^G_i(*)$$. Since $$\text{Coh}^G(*)$$ is equivalent to the category $$\text{Rep}(G)$$ of finite-dimensional representations of $$G$$. Then, the Grothendieck group of $$\text{Rep}(G)$$, denoted $$R(G)$$ is $$K_0^G(*)$$.

Torus ring
Given an algebraic torus $$\mathbb{T}\cong \mathbb{G}_m^k$$ a finite-dimensional representation $$V$$ is given by a direct sum of $$1$$-dimensional $$\mathbb{T}$$-modules called the weights of $$V$$. There is an explicit isomorphism between $$K_\mathbb{T}$$ and $$\mathbb{Z}[t_1,\ldots, t_k]$$ given by sending $$[V]$$ to its associated character.