Equivariant algebraic K-theory

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In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category CohG(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,

KiG(X)=πi(B+CohG(X)).

In particular, K0G(C) is the Grothendieck group of CohG(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, KiG(X) may be defined as the Ki 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]

## Contents

* 1 Fundamental theorems
* 2 Examples
* 2.1 Torus ring
* 3 See also
* 4 References
* 5 Further reading

## Fundamental theorems

Let X be an equivariant algebraic scheme.

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

⋯->KiG(Z)->KiG(X)->KiG(U)->Ki−1G(Z)->⋯

## 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 KiG(*). Since CohG(*) is equivalent to the category Rep(G) of finite-dimensional representations of G. Then, the Grothendieck group of Rep(G), denoted R(G) is K0G(*).[5]

### Torus ring

Given an algebraic torus 𝕋≅𝔾mk a finite-dimensional representation V is given by a direct sum of 1-dimensional 𝕋-modules called the weights of V.[6] There is an explicit isomorphism between K𝕋 and ℤ[t1,…,tk] given by sending [V] to its associated character.[7]

## See also

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

## 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. doi:10.1007/s00220-003-0849-x. ISSN 0010-3616. Bibcode: 2003CMaPh.237..533A.
3. ↑ Krishna, Amalendu; Ravi, Charanya (2017-08-02). "Algebraic K-theory of quotient stacks". arXiv:1509.05147 [math.AG].
4. ↑ Baum, Fulton & Quart 1979
5. ↑ Chriss, Neil; Ginzburg, Neil. Representation theory and complex geometry. pp. 243–244.
6. ↑ For 𝔾m there is a map f:𝔾m->𝔾m sending t↦tk. Since 𝔾m⊂𝔸1 there is an induced representation f^:𝔾m->GL(𝔸1) of weight 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, 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

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