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 . (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.