Equivariant algebraic 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.Charles A. Weibel, [https://www.ams.org/notices/199608/comm-thomason.pdf Robert W. Thomason (1952–1995)]. Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.

Equivalently, $K_i^G(X)$ may be defined as the $K_i$ of the category of coherent sheaves on the quotient stack $[X/G]$ .{{Cite journal|last1=Adem|first1=Alejandro|last2=Ruan|first2=Yongbin|date=June 2003|title=Twisted Orbifold K-Theory|arxiv=math/0107168|journal=Communications in Mathematical Physics|volume=237|issue=3|pages=533–556|doi=10.1007/s00220-003-0849-x|bibcode=2003CMaPh.237..533A|s2cid=12059533|issn=0010-3616}}{{cite arXiv|last1=Krishna|first1=Amalendu|last2=Ravi|first2=Charanya|date=2017-08-02|title=Algebraic K-theory of quotient stacks|eprint=1509.05147|class=math.AG}} (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.{{harvnb|Baum|Fulton|Quart|1979}}

Fundamental theorems

Let X be an equivariant algebraic scheme.

{{math_theorem|name=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

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

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 $\text{Coh}^G(*)$ is equivalent to the category $\text{Rep}(G)$ of finite-dimensional representations of $G$ . Then, the Grothendieck group of $\text{Rep}(G)$ , denoted $R(G)$ is $K_0^G(*)$ .{{Cite book|last1=Chriss|first1=Neil|title=Representation theory and complex geometry|last2=Ginzburg|first2=Neil|pages=243–244}}

= Torus ring =

Given an algebraic torus $\mathbb{T}\cong \mathbb{G}_m^k$ a finite-dimensional representation $V$ is given by a direct sum of $1$ -dimensional $\mathbb{T}$ -modules called the weights of $V$ .For $\mathbb{G}_m$ there is a map $f:\mathbb{G}_m \to \mathbb{G}_m$ sending $t \mapsto t^k$ . Since $\mathbb{G}_m \subset \mathbb{A}^1$ there is an induced representation $\hat{f}:\mathbb{G}_m \to GL(\mathbb{A}^1)$ of weight $k$ . See Algebraic torus for more info. There is an explicit isomorphism between $K_\mathbb{T}$ and $\mathbb{Z}[t_1,\ldots, t_k]$ given by sending $[V]$ to its associated character.{{cite arXiv|last=Okounkov|first=Andrei|date=2017-01-03|title=Lectures on K-theoretic computations in enumerative geometry|eprint=1512.07363|pages=13|class=math.AG}}

References

N. Chris and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, 1997.
{{cite journal |last1=Baum |first1=Paul |last2=Fulton |first2=William |last3=Quart |first3=George |title=Lefschetz-riemann-roch for singular varieties |journal=Acta Mathematica |date=1979 |volume=143 |pages=193–211 |doi=10.1007/BF02392092|doi-access=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.

Equivariant algebraic K-theory

Fundamental theorems

Examples

= Torus ring =

See also

References

Further reading