# 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,

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

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.

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.[2]

## Fundamental theorems

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

## References

- Charles A. Weibel, Robert W. Thomason (1952–1995).
- BFQ 1979

- 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.