In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category 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, In particular, K_0^G(C) is the Grothendieck group of. 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 is equivalent to the category of finite-dimensional representations of G. Then, the Grothendieck group of, denoted R(G) is K_0^G().

Torus ring

Given an algebraic torus 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 and given by sending [V] to its associated character.