Equivariant algebraic K-theory


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, 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, may be defined as the of the category of coherent sheaves on the quotient stack.
A version of the Lefschetz fixed point theorem holds in the setting of equivariant 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 -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.

Torus ring

Given an algebraic torus a finite-dimensional representation is given by a direct sum of -dimensional -modules called the weights of. There is an explicit isomorphism between and given by sending to its associated character.