Equivariant Algebraic K-theory

In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category $\operatorname^G(X)$ 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,
:$K_i^G(X) = \pi_i(B^+ \operatorname^G(X)).$
In particular, $K_0^G(C)$ is the Grothendieck group of $\operatorname^G(X)$. The theory was developed by R. W. Thomason in 1980s. Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.

Equivalently, $K_i^G(X)$ may be defined as the $K_i$ of the category of coherent sheaves on the quotient stack /G/math>. (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.

Fundamental theorems

Let ''X'' be an equivariant algebraic scheme.

Examples

One of the fundamental examples of equivariant K-theory groups are the equivariant K-groups of $G$-equivariant coherent sheaves on a points, so $K^G_i(*)$. Since $\text^G(*)$ is equivalent to the category $\text(G)$ of finite-dimensional representations of $G$. Then, the Grothendieck group of $\text(G)$, denoted $R(G)$ is $K_0^G(*)$.

Torus ring

Given an algebraic torus $\mathbb\cong \mathbb_m^k$ a finite-dimensional representation $V$ is given by a direct sum of $1$-dimensional $\mathbb$-modules called the weights of $V$. There is an explicit isomorphism between $K_\mathbb$ and \mathbb _1,\ldots, t_k/math> given by sending /math> to its associated character.

See also

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

References

*N. Chris and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, 1997.
*{{cite journal , last1=Baum , first1=Paul , last2=Fulton , first2=William , last3=Quart , first3=George , title=Lefschetz-riemann-roch for singular varieties , journal=Acta Mathematica , date=1979 , volume=143 , pages=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.

Further reading

*Dan Edidin
Riemann–Roch for Deligne–Mumford stacks
2012

Algebraic K-theory