Equivariant algebraic K-theory explained

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 ^{G(X)} _{i} = \pi ^{+} _{i(B} \operatorname{Coh} ^{G(X)).} In particular, K ^{G(C)} _{0} 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,

K ^{G(X)} _{i} may be defined as the K _{i} 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.

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

G -equivariant coherent sheaves on a points, so K ^{G} _{i(*)} . Since Coh ^{G(*)} is equivalent to the category Rep (G) of finite-dimensional representations of G . Then, the Grothendieck group of Rep (G) , denoted R(G) is K ^{G(*)} _{0} .^[4]

Torus ring

Given an algebraic torus

T \cong G ^{k} _{m} a finite-dimensional representation V is given by a direct sum of 1 -dimensional T -modules called the weights of V .^[5] There is an explicit isomorphism between K _{T} and Z [t _{1,\ldots,} t _{k]} given by sending [V] to its associated character.^[6]

See also

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

References

• N. Chris and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, 1997.
• Baum . Paul . Fulton . William . Quart . George . Lefschetz-riemann-roch for singular varieties . Acta Mathematica . 1979 . 143 . 193–211 . 10.1007/BF02392092. free .
• 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

Notes and References

1. Charles A. Weibel, Robert W. Thomason (1952–1995).
2. Adem. Alejandro. Ruan. Yongbin. June 2003. Twisted Orbifold K-Theory. math/0107168. Communications in Mathematical Physics. 237. 3. 533–556. 10.1007/s00220-003-0849-x. 2003CMaPh.237..533A. 12059533. 0010-3616.
3. Krishna. Amalendu. Ravi. Charanya. 2017-08-02. Algebraic K-theory of quotient stacks. 1509.05147. math.AG.
4. Book: Chriss. Neil. Representation theory and complex geometry. Ginzburg. Neil. 243–244.
5. For G _{m} there is a map f: G _{m} \to G _{m} sending t \mapsto t ^{k} . Since G _{m} \subset A ^{1} there is an induced representation \hat{f}: G _{m} \to GL( A ^{1)} of weight k . See Algebraic torus for more info.
6. Okounkov. Andrei. 2017-01-03. Lectures on K-theoretic computations in enumerative geometry. 1512.07363. 13. math.AG.