# Equivariant K-theory

For the topological equivariant K-theory, see topological K-theory.

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,[citation needed] ${\displaystyle K_{i}^{G}(X)}$ may be defined as the ${\displaystyle K_{i}}$ of the category of coherent sheaves on the quotient stack ${\displaystyle [X/G]}$. (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.[2]

## Fundamental theorems

Let X be an equivariant algebraic scheme.

Localization theorem — Given a closed immersion ${\displaystyle Z\hookrightarrow X}$ of equivariant algebraic schemes and an open immersion ${\displaystyle Z-U\hookrightarrow X}$, there is a long exact sequence of groups

${\displaystyle \dots \to K_{i}^{G}(Z)\to K_{i}^{G}(X)\to K_{i}^{G}(U)\to K_{i-1}^{G}(Z)\to \dots }$

## References

• N. Chris and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, 1997.
• Baum, P., Fulton, W., Quart, G.: Lefschetz Riemann Roch for singular varieties. Acta. Math. 143, 193-211 (1979)
• 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.