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.
[1] Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.
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.)
A version of the Lefschetz fixed-point theorem holds in the setting of equivariant (algebraic) K-theory.
[4] Let X be an equivariant algebraic scheme.
Localization theorem — Given a closed immersion
of equivariant algebraic schemes and an open immersion
, there is a long exact sequence of groups One of the fundamental examples of equivariant K-theory groups are the equivariant K-groups of
-equivariant coherent sheaves on a points, so
is equivalent to the category
of finite-dimensional representations of
Then, the Grothendieck group of
[5] Given an algebraic torus
a finite-dimensional representation
is given by a direct sum of
-modules called the weights of
[6] There is an explicit isomorphism between