In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category ${\displaystyle \operatorname {Coh} ^{G}(X)}$ 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,

${\displaystyle K_{i}^{G}(X)=\pi _{i}(B^{+}\operatorname {Coh} ^{G}(X)).}$

In particular, ${\displaystyle K_{0}^{G}(C)}$ is the Grothendieck group of ${\displaystyle \operatorname {Coh} ^{G}(X)}$. 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, ${\displaystyle K_{i}^{G}(X)}$ may be defined as the ${\displaystyle K_{i))$ of the category of coherent sheaves on the quotient stack ${\displaystyle [X/G]}$.^[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.

One of the fundamental examples of equivariant K-theory groups are the equivariant K-groups of ${\displaystyle G}$-equivariant coherent sheaves on a points, so ${\displaystyle K_{i}^{G}(*)}$. Since ${\displaystyle {\text{Coh))^{G}(*)}$ is equivalent to the category ${\displaystyle {\text{Rep))(G)}$ of finite-dimensional representations of ${\displaystyle G}$. Then, the Grothendieck group of ${\displaystyle {\text{Rep))(G)}$, denoted ${\displaystyle R(G)}$ is ${\displaystyle K_{0}^{G}(*)}$.^[5]

Given an algebraic torus ${\displaystyle \mathbb {T} \cong \mathbb {G} _{m}^{k))$ a finite-dimensional representation ${\displaystyle V}$ is given by a direct sum of ${\displaystyle 1}$-dimensional ${\displaystyle \mathbb {T} }$-modules called the weights of ${\displaystyle V}$.^[6] There is an explicit isomorphism between ${\displaystyle K_{\mathbb {T} ))$ and ${\displaystyle \mathbb {Z} [t_{1},\ldots ,t_{k}]}$ given by sending ${\displaystyle [V]}$ to its associated character.^[7]

• Topological K-theory, the topological equivariant K-theory

• N. Chris and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, 1997.
• Baum, Paul; Fulton, William; Quart, George (1979). "Lefschetz-riemann-roch for singular varieties". Acta Mathematica. 143: 193–211. doi:10.1007/BF02392092.
• 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
• Thomason, R.W.: Lefschetz–Riemann–Roch theorem and coherent trace formula. Invent. Math. 85, 515–543 (1986)
• Thomason, R.W., Trobaugh, T.: Higher algebraic K-theory of schemes and of derived categories. In: Cartier, P., Illusie, L., Katz, N.M., Laumon, G., Manin, Y., Ribet, K.A. (eds.) The Grothendieck Festschrift, vol. III. (Prog. Math. vol. 88, pp. 247 435) Boston Basel Berlin: Birkhfiuser 1990
• Thomason, R.W., Une formule de Lefschetz en K-théorie équivariante algébrique, Duke Math. J. 68 (1992), 447–462.

• Dan Edidin, Riemann–Roch for Deligne–Mumford stacks, 2012