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A Survey of Elliptic Cohomology - equivariant cohomology

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This is a sub-entry of

See there for background and context.

This entry considers equivariant cohomology from the perspective of algebraic geometry.

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the following are rough unpolished notes taken more or less verbatim from some seminar talk – needs attention, meaning: somebody should go through this and polish


Contents

Introduction

The slogan is: for AA a cohomology theory, finding equivariant cohomology AA-theory corresponds to finding group scheme over A(*)A({*}).

The main example we’ll be looking at here is complex K-theory.

Notions of equivariant cohomology

Borel equivariant cohomology

Remark if A GA_G is an equivariant cohomology theory and if YXY \to X is a GG-principal bundle then we want that A G(Y)A(X)A_G(Y) \simeq A(X). So whenever we have a GG-space where the GG-action is free enough.

Let XX be a GG-space, form the Borel construction G× GX\mathcal{E}G \times_G X with GG\mathcal{E}G \to \mathcal{B}G the universal principal bundle.

Then we can define

A G Borel(X):=A(G× GX). A_G^{Borel}(X) := A(\mathcal{E}G \times_G X) \,.

notice that G× GX\mathcal{E}G \times_G X is the realization of the action groupoid X//GX//G. This Borel equivariant cohomology theory is what is discussed currently at the entry equivariant cohomology. The following will actually define a refinement of the discussion currently at equivariant cohomology.

Problem If the cohomology theory is given by a geometric model, such as topological K-theory in terms of vector bundles or elliptic cohomology potentially in a geometric model for elliptic cohomology, then the above notion of equivariant cohomology need not coincide with the cohomology theory given by the equivariant version of these geometric models. In particular, equivariant vector bundles are geometric cocycles of equivariant K-theory K GK_G and there is a morphism

K G(X)K G Borel(X) K_G(X) \to K_G^{Borel}(X)

but it is not an isomorphism. Instead, K GK_G is a completion of K G BorK_G^{Bor}.

Here K G(X)K_G(X) is the Grothendieck group of equivariant vector bundles over the G-space XX (say GG is a compact Lie group).

Grothendieck ring of equivariant vector bundles

Definition An equivariant vector bundle over XX is

  • a G-space EE and GG-equivariant map p:EXp : E \to X such that this is a (complex, here) vector bundle of finite rank

  • for each gGg \in G the map g:E xE gxg : E_x \to E_{g x} is linear.

Morephism are the obvious GG-equivariant morphisms of vector bundles.

examples*

  1. XX a trivial GG-space,then a GG-equivariant vector bundle is a family of complex representations;

  2. for EXE \to X a vector bundle the kkth tensor power of EE is a Σ k\Sigma_k-equivariant vector bundle;

  3. if GG acts smoothly on XX then the complexified tangent bundle TXXT X \otimes \mathbb{C} \to X is a GG-equivariant vector bundle.

remark The category of GG-equivariant vector bundle, has

And we can pull back Vect GVect^G \to Vecg HVecg^H along any group homomorphism ϕ:HG\phi : H \to G

So we are entitled to say

definition the Grothendieck group of Vect G(X)Vect^G(X) is

K G(X):=Groth(EX,). K_G(X) := Groth(E \to X, \oplus) \,.

With the remaining tensor product \otimes this yields a commutative ring.

proposition

  1. if X=ptX = pt then K G(X)Rep(G)K_G(X) \simeq Rep(G) is the representation ring of GG.

    1. in general, K G(X)K_G(X) is an algebra over Rep(G)Rep(G).
  2. if GG acts freely on XX, then K G(X)K(X/G)K_G(X) \simeq K(X/G).

So in particular

K(G× GX)K G(G×X) K(\mathcal{E}G \times_G X) \simeq K_G(\mathcal{E}G \times X)

so we get a map

α:K G(X)K G(G×X)K(G× GX):=K G Borel(X) \alpha : K_G(X) \to K_G(\mathcal{E}G \times X) \simeq K(\mathcal{E}G \times_G X) := K_G^{Borel}(X)

theorem (Atiyah-Segal) This α\alpha induces an isomorphism

K^ G(X)K G Borel(X) \hat K_G(X) \simeq K_G^{Borel}(X)

where

K^ G(X):=lim K G(X)/I G nK G(X) \hat K_G(X) := \lim_\leftarrow K_G(X)/I_G^n K_G(X)

where I G=ker(Rep(G)K G(*)K G(G)K(G)ϵ)I_G = ker(Rep(G) \simeq K_G({*}) \to K_G(\mathcal{E}G) \simeq K(\mathcal{B}G) \stackrel{\epsilon}{\to} \mathbb{Z})

consider X=*X = {*}, G=S 1G = S^1, P S 1\mathbb{C}P^\infty \simeq \mathcal{B}S^1

α:Rep(G)K(G)=K(P )[[t]] \alpha : Rep(G) \to K(\mathcal{B}G) = K(\mathbb{C}P^\infty) \simeq \mathbb{Z}[ [ t ] ]
  1. since S 1S^1 is an abelian group, every irreducible representation is 1-dimensional

    ϕ:S 1 × \phi : S^1 \to \mathbb{C}^\times
  2. χ:S 1 times\chi : S^1 \hookrightarrow \mathbb{C}^times

Rep(G)[χ,χ 1]. Rep(G) \simeq \mathbb{Z}[\chi, \chi^{-1}] \,.

algebraic interpretation

goal now find an algebraic interpretation of α\alpha such that

Rep(S 1)= G m Rep(S^1) = \mathcal{o}_{G_m}

and

Rep(P )= G^ m Rep(\mathbb{C}P^\infty) = \mathcal{o}_{\hat G_m}

adopt the functor of points perspective

X:CRingsSet X : CRings \to Set

a functor.

For AA \in CRing get a spectrum

SpecA:RCRing(A,R) Spec A : R \mapsto CRing(A,R)

for XX a functor, it is an affine scheme if it is a representable functor in that there is AA with XSpecAX \simeq Spec A.

examples

  1. A n(R):=R n\mathbf{A}^n(R) := R^n

  2. A^ n(R):=Nil(R) n\hat \mathbf{A}^n(R) := Nil(R)^n

  3. G m(R):=R ×A 1(R)G_m(R) := R^\times \hookrightarrow \mathbf{A}^1(R)

  4. n(R):=R n+1/\mathbb{P}^n(R) := R^{n+1}/\sim, where \sim is multiplication by R ×R^\times

proposition

G mG_m is affine.

proof A=[x,x 1]A = \mathbb{Z}[x,x^{-1}], let uSpec(A(R))u \in Spec(\mathbf{A}(R)) a map uARu A \to R, then define

ϕ:SpecAG m \phi : Spec A \to G_m

by

uu(x) u \mapsto u(x)

Conversely, given vG m(R)=R ×v \in G_m(R) = R^\times, define

Ψ:G mSpecA \Psi : G_m \to Spec A

by

vΨ(v) v \mapsto \Psi(v)

with

Ψ(v)( ka kx k):= ka kv k \Psi(v)(\sum_k a_k x^k) := \sum_k a_k v^k

endofproof

similarly, A nSpec[x 1,,x n]\mathbf{A}^n \simeq Spec \mathbb{Z}[x_1, \cdots, x_n]

group schemes

Given a functor X:CRingSetX : CRing \to Set define the ring of funtions X\mathcal{o}_X as

X:=Hom Func(CRing,Set)(X,A 1) \mathcal{o}_X := Hom_{Func(CRing,Set)}(X, \mathbf{A}^1)

in the functor category.

notice notation: this is global sectins of the structure sheaf, not the structure sheaf itself, properly speaking

we have

SpecAA \mathcal{o}_{Spec A} \simeq A

so that in particular

G m[x,x 1]. \mathcal{o}_{G_m} \simeq \mathbb{Z}[x,x^{-1}] \,.

definition

Let XX be an affine scheme and Y:CRingSetY : CRing \to Set a functor with a natural transformation p:YXp : Y \to X. A system of formal coordinates is a sequence of maps

X i:YA^ 1 X_i : Y \to \hat \mathbf{A}^1

such that

a(x 1(a),,x n(a),p(a))A^ n×X a \mapsto (x_1(a), \cdots, x_n(a), p(a)) \in \hat \mathbf{A}^n \times X

is an isomorphism. A YY that admits a system of formal coordinates is a formal scheme over XX.

warning very restrictive definition. See formal scheme

A formal group GG over a scheme XX is a one-dimensional formal scheme with specified abelian group structure on each fiber p 1{x}p^{-1}\{x\}.

This means that there is a natural map

σ:G× XGG \sigma : G \times_X G \to G

and a natural map ζ:XG\zeta : X \to G which maps x0p 1{x}x \mapsto 0 \in p^{-1}\{x\}.

definition (formal multiplicative group)

define G^ m\hat G_m on each RCRingR \in CRing by

G^ m(R)={1+n|nNil(R)} \hat G_m(R) = \left\{ 1+n | n \in Nil(R) \right\}

which is a group under multiplication.

there is an isomorphism of underlying formal schemes

G^ mA^ 1 \hat G_m \simeq \hat \mathbf{A}^1

We compute A^ 1 G^ m\mathcal{o}_{\hat \mathbf{A}^1} \simeq \mathcal{o}_{\hat G_m} in two ways:

  1. Recall that A^ 1\hat \mathbf{A}^1 can be defined as Spf[t]Spf \; \mathbb{Z} [t], so the global sections of the structure sheaf (which is what we have been calling \mathcal{o}) is

    A^ 1=lim [t]/(t n)=[[t]].\mathcal{o}_{\hat \mathbf{A}^1} = \lim_\rightarrow \mathbb{Z}[t]/ (t^n) = \mathbb{Z} [[t]] .
  2. We can also see this in the functor of points perspective. Consider the functor Spec[t]/(t n)\mathrm{Spec} \; \mathbb{Z} [t]/ (t^n), then for any ring RR $A^ 1(R)=lim Spec[t]/(t n)(R).\hat \mathbf{A}^1 (R) = \lim_\rightarrow \mathrm{Spec} \; \mathbb{Z} [t]/ (t^n) (R).$ By the universal property of colimits we have

    Nat(A^ 1,A 1)lim Nat(Spec[t]/(t n),A 1)[[t]].\mathrm{Nat} (\hat \mathbf{A}^1 , \mathbf{A}^1 ) \simeq \lim_\leftarrow \mathrm{Nat} (\mathrm{Spec} \; \mathbb{Z}[t]/ (t^n) , \mathbf{A}^1) \simeq \mathbb{Z} [[t]].
G^ m[[t]]. \mathcal{o}_{\hat G_m} \simeq \mathbb{Z}[ [ t] ] \,.

recall we have a morphism α:Rep(S 1)K(P )\alpha : Rep(S^1) \to K(\mathbb{C}P^\infty)

such that

G m[x,x 1]Rep(S 1)K(P )[[t]] G^ m \mathcal{o}_{G_m} \simeq \mathbb{Z}[x,x^{-1}] \simeq Rep(S^1) \to K(\mathbb{C}P^\infty) \simeq \mathbb{Z}[ [ t ] ] \simeq \mathcal{o}_{\hat G_m}

is the canonical inclusion

G m G^ m \mathcal{o}_{G_m} \to \mathcal{o}_{\hat G_m}

exercise

x[x,x 1] G m x \in \mathbb{Z}[x,x^{-1}] \simeq \mathcal{o}_{G_m}

is the natural transformation R ×RR^\times \to R

and

t[[t]] G^ m t \in \mathbb{Z}[ [t] ] \simeq \mathcal{o}_{\hat G_m}

is the natural transformation

{1+Nil(R)}Nil(R)R \{1 + Nil(R)\} \to Nil(R) \to R

so that we get a map

G m G^ m \mathcal{o}_{G_m} \to \mathcal{o}_{\hat G_m}

by sending xx to 1+t1 + t, and this corresponds to taking the germ of functions at 1G m1 \in G_m

lesson

given GG an algebraic group such that the formal spectrum SpfA(P )Spf A(\mathbb{C}P^\infty) is the completion G^\hat G, define A S 1(*):= GA_{S^1}({*}) := \mathcal{o}_{G} then passing to germs gives a completion map

A S 1(*)A(P )=A S 1 Bor(*) A_{S^1}({*}) \to A(\mathbb{C}P^\infty) = A^{Bor}_{S^1}({*})
Revised on November 3, 2009 15:05:38 by Urs Schreiber (131.211.235.223)