Equivariant algebraic K-theory

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

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

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

Fundamental theorems

Let X be an equivariant algebraic scheme.

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

\cdots \to K_{i}^{G}(Z)\to K_{i}^{G}(X)\to K_{i}^{G}(U)\to K_{i-1}^{G}(Z)\to \cdots

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 $K_{i}^{G}(*)$ . Since ${\text{Coh}}^{G}(*)$ is equivalent to the category ${\text{Rep}}(G)$ of finite-dimensional representations of $G$ . Then, the Grothendieck group of ${\text{Rep}}(G)$ , denoted $R(G)$ is $K_{0}^{G}(*)$ .^[5]

Torus ring

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

References

^ Charles A. Weibel, Robert W. Thomason (1952–1995).
^ Adem, Alejandro; Ruan, Yongbin (June 2003). "Twisted Orbifold K-Theory". Communications in Mathematical Physics. 237 (3): 533–556. arXiv:math/0107168. Bibcode:2003CMaPh.237..533A. doi:10.1007/s00220-003-0849-x. ISSN 0010-3616. S2CID 12059533.
^ Krishna, Amalendu; Ravi, Charanya (2017-08-02). "Algebraic K-theory of quotient stacks". arXiv:1509.05147 [math.AG].
^ Baum, Fulton & Quart 1979
^ Chriss, Neil; Ginzburg, Neil. Representation theory and complex geometry. pp. 243–244.
^ For $\mathbb {G} _{m}$ there is a map $f:\mathbb {G} _{m}\to \mathbb {G} _{m}$ sending $t\mapsto t^{k}$ . Since $\mathbb {G} _{m}\subset \mathbb {A} ^{1}$ there is an induced representation ${\hat {f}}:\mathbb {G} _{m}\to GL(\mathbb {A} ^{1})$ of weight $k$ . See Algebraic torus for more info.
^ Okounkov, Andrei (2017-01-03). "Lectures on K-theoretic computations in enumerative geometry". p. 13. arXiv:1512.07363 [math.AG].

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.