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,
In particular, is the Grothendieck group of . The theory was developed by R. W. Thomason in 1980s.Charles A. Weibel, Robert W. Thomason (1952–1995). 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 . (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.
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 .
Torus ring Given an algebraic torus a finite-dimensional representation is given by a direct sum of -dimensional -modules called the weights of .For there is a map sending . Since there is an induced representation of weight . See Algebraic torus for more info. There is an explicit isomorphism between and given by sending to its associated character.
See also Topological K-theory, the topological equivariant K-theory
References
N. Chris and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, 1997.
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