Equivariant K-theory

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For the topological equivariant K-theory, see topological 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,[citation needed] K_i^G(X) may be defined as the K_i of the category of coherent sheaves on the quotient stack [X/G]. (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.[2]

Fundamental theorems[edit]

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

\dots \to K^G_i(Z) \to K^G_i(X) \to K^G_i(U) \to K^G_{i-1}(Z) \to \dots

References[edit]

  • 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[edit]