Equivariant algebraic 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,

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

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

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, ${\displaystyle K_{i}^{G}(X)}$ may be defined as the ${\displaystyle K_{i}}$ of the category of coherent sheaves on the quotient stack ${\displaystyle [X/G]}$.[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 ${\displaystyle Z\hookrightarrow X}$ of equivariant algebraic schemes and an open immersion ${\displaystyle Z-U\hookrightarrow X}$, there is a long exact sequence of groups

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

Examples

One of the fundamental examples of equivariant K-theory groups are the equivariant K-groups of ${\displaystyle G}$-equivariant coherent sheaves on a points, so ${\displaystyle K_{i}^{G}(*)}$. Since ${\displaystyle {\text{Coh}}^{G}(*)}$ is equivalent to the category ${\displaystyle {\text{Rep}}(G)}$ of finite-dimensional representations of ${\displaystyle G}$. Then, the Grothendieck group of ${\displaystyle {\text{Rep}}(G)}$, denoted ${\displaystyle R(G)}$ is ${\displaystyle K_{0}^{G}(*)}$.[5]

Torus ring

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

References

1. Charles A. Weibel, Robert W. Thomason (1952–1995).
2. 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.
3. Krishna, Amalendu; Ravi, Charanya (2017-08-02). "Algebraic K-theory of quotient stacks". arXiv:1509.05147 [math.AG].
4. BFQ 1979 harvnb error: no target: CITEREFBFQ1979 (help)
5. Chriss, Neil; Ginzburg, Neil. Representation theory and complex geometry. pp. 243–244.
6. For ${\displaystyle \mathbb {G} _{m}}$ there is a map ${\displaystyle f:\mathbb {G} _{m}\to \mathbb {G} _{m}}$ sending ${\displaystyle t\mapsto t^{k}}$. Since ${\displaystyle \mathbb {G} _{m}\subset \mathbb {A} ^{1}}$ there is an induced representation ${\displaystyle {\hat {f}}:\mathbb {G} _{m}\to GL(\mathbb {A} ^{1})}$ of weight ${\displaystyle k}$. See Algebraic torus for more info.
7. 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, 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.

Further reading

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