Equivariant algebraic K-theory

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

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

In particular, is the Grothendieck group of . 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, may be defined as the of the category of coherent sheaves on the quotient stack .[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 of equivariant algebraic schemes and an open immersion , there is a long exact sequence of groups

Examples

One of the fundamental examples of equivariant K-theory groups are the equivariant K-groups of -equivariant coherent sheaves on a points, so . Since is equivalent to the category of finite-dimensional representations of . Then, the Grothendieck group of , denoted is .[5]

Torus ring

Given an algebraic torus a finite-dimensional representation is given by a direct sum of -dimensional -modules called the weights of .[6] There is an explicit isomorphism between and given by sending 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
  5. Chriss, Neil; Ginzburg, Neil. Representation theory and complex geometry. pp. 243–244.
  6. For there is a map sending . Since there is an induced representation of weight . 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

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