Equivariant algebraic K-theory

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

[math]\displaystyle{ K_i^G(X) = \pi_i(B^+ \operatorname{Coh}^G(X)). }[/math]

In particular, [math]\displaystyle{ K_0^G(C) }[/math] is the Grothendieck group of [math]\displaystyle{ \operatorname{Coh}^G(X) }[/math]. The theory was developed by R. W. Thomason in 1980s. Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.

Equivalently, [math]\displaystyle{ K_i^G(X) }[/math] may be defined as the [math]\displaystyle{ K_i }[/math] of the category of coherent sheaves on the quotient stack [math]\displaystyle{ [X/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.

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

[math]\displaystyle{ \cdots \to K^G_i(Z) \to K^G_i(X) \to K^G_i(U) \to K^G_{i-1}(Z) \to \cdots }[/math]

Examples

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

Torus ring

Given an algebraic torus [math]\displaystyle{ \mathbb{T}\cong \mathbb{G}_m^k }[/math] a finite-dimensional representation [math]\displaystyle{ V }[/math] is given by a direct sum of [math]\displaystyle{ 1 }[/math]-dimensional [math]\displaystyle{ \mathbb{T} }[/math]-modules called the weights of [math]\displaystyle{ V }[/math]. There is an explicit isomorphism between [math]\displaystyle{ K_\mathbb{T} }[/math] and [math]\displaystyle{ \mathbb{Z}[t_1,\ldots, t_k] }[/math] given by sending [math]\displaystyle{ [V] }[/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.
• 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.

Further reading

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