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Equivariant algebraic K-theory

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

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

In particular, K 0 G ( C ) {\displaystyle K_{0}^{G}(C)} is the Grothendieck group of Coh G ⁡ ( X ) {\displaystyle \operatorname {Coh} ^{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 ) {\displaystyle K_{i}^{G}(X)} may be defined as the K i {\displaystyle K_{i}} of the category of coherent sheaves on the quotient stack [ X / G ] {\displaystyle } . (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 Z ↪ X {\displaystyle Z\hookrightarrow X} of equivariant algebraic schemes and an open immersion Z − U ↪ X {\displaystyle Z-U\hookrightarrow X} , there is a long exact sequence of groups

⋯ → K i G ( Z ) → K i G ( X ) → K i G ( U ) → K i − 1 G ( Z ) → ⋯ {\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 G {\displaystyle G} -equivariant coherent sheaves on a points, so K i G ( ∗ ) {\displaystyle K_{i}^{G}(*)} . Since Coh G ( ∗ ) {\displaystyle {\text{Coh}}^{G}(*)} is equivalent to the category Rep ( G ) {\displaystyle {\text{Rep}}(G)} of finite-dimensional representations of G {\displaystyle G} . Then, the Grothendieck group of Rep ( G ) {\displaystyle {\text{Rep}}(G)} , denoted R ( G ) {\displaystyle R(G)} is K 0 G ( ∗ ) {\displaystyle K_{0}^{G}(*)} .

Torus ring

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

See also

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. S2CID 12059533.
  3. Krishna, Amalendu; Ravi, Charanya (2017-08-02). "Algebraic K-theory of quotient stacks". arXiv:1509.05147 .
  4. Baum, Fulton & Quart 1979
  5. Chriss, Neil; Ginzburg, Neil. Representation theory and complex geometry. pp. 243–244.
  6. For G m {\displaystyle \mathbb {G} _{m}} there is a map f : G m → G m {\displaystyle f:\mathbb {G} _{m}\to \mathbb {G} _{m}} sending t ↦ t k {\displaystyle t\mapsto t^{k}} . Since G m ⊂ A 1 {\displaystyle \mathbb {G} _{m}\subset \mathbb {A} ^{1}} there is an induced representation f ^ : G m → G L ( A 1 ) {\displaystyle {\hat {f}}:\mathbb {G} _{m}\to GL(\mathbb {A} ^{1})} of weight k {\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 .
  • 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

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