In particular, is the Grothendieck group of . 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, may be defined as the of the category of coherent sheaves on the quotient stack.[2][3] (Hence, the equivariant K-theory is a specific case of the K-theory of a stack.)
Localization theorem — Given a closed immersion of equivariant algebraic schemes and an open immersion , there is a long exact sequence of groups
Examples
One of the fundamental examples of equivariant K-theory groups are the equivariant K-groups of -equivariant coherent sheaves on a points, so . Since is equivalent to the category of finite-dimensional representations of . Then, the Grothendieck group of , denoted is .[5]
Torus ring
Given an algebraic torus a finite-dimensional representation is given by a direct sum of -dimensional -modules called the weights of .[6] There is an explicit isomorphism between and given by sending to its associated character.[7]
Thomason, R.W.:Algebraic K-theory of group scheme actions. In: Browder, W. (ed.) Algebraic topology and algebraic K-theory. (Ann. Math. Stud., vol. 113, pp. 539 563) Princeton: Princeton University Press 1987