nLab
A Survey of Elliptic Cohomology - equivariant cohomology

This is a sub-entry of

• A Survey of Elliptic Cohomology.

See there for background and context.

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

Previous:

• A Survey of Elliptic Cohomology - cohomology theories

• A Survey of Elliptic Cohomology - formal groups and cohomology

• A Survey of Elliptic Cohomology - E-infinity rings and derived schemes

• A Survey of Elliptic Cohomology - elliptic curves

Next:

• A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations

• A Survey of Elliptic Cohomology - A-equivariant cohomology

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

---

Introduction

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

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

Notions of equivariant cohomology

Borel equivariant cohomology

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

Let $X$ be a $G$-space, form the Borel construction $ℰG{\times }_{G}X$ with $ℰG\to ℬG$ the universal principal bundle.

Then we can define

$$A_G^{\mathrm{Borel}}(X):=A(ℰG{\times }_{G}X).$$A_G^{Borel}(X) := A(\mathcal{E}G \times_G X) \,.

notice that $ℰG{\times }_{G}X$ is the realization of the action groupoid $X//G$. 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 $K_G$ and there is a morphism

$$K_G(X)\to K_G^{\mathrm{Borel}}(X)$$K_G(X) \to K_G^{Borel}(X)

but it is not an isomorphism. Instead, $K_G$ is a completion of $K_G^{\mathrm{Bor}}$.

Here $K_G(X)$ is the Grothendieck group of equivariant vector bundle?s over the G-space? $X$ (say $G$ is a compact Lie group).

Grothendieck ring of equivariant vector bundles

Definition An equivariant vector bundle? over $X$ is

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

• for each $g\in G$ the map $g:E_x\to E_{gx}$ is linear.

Morephism are the obvious $G$-equivariant morphisms of vector bundles.

examples*

1. $X$ a trivial $G$-space,then a $G$-equivariant vector bundle is a family of complex representations;

2. for $E\to X$ a vector bundle the $k$th tensor power of $E$ is a ${\Sigma }_k$-equivariant vector bundle;

3. if $G$ acts smoothly on $X$ then the complexified tangent bundle $TX\otimes ℂ\to X$ is a $G$-equivariant vector bundle.

remark The category of $G$-equivariant vector bundle?, has

• direct sum $\oplus$

• tensor product $\otimes$

And we can pull back ${\mathrm{Vect}}^G$ \to ${\mathrm{Vecg}}^H$ along any group homomorphism $\varphi :H\to G$

So we are entitled to say

definition the Grothendieck group of ${\mathrm{Vect}}^G(X)$ is

$$K_G(X):=\mathrm{Groth}(E\to X,\oplus ).$$K_G(X) := Groth(E \to X, \oplus) \,.

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

proposition

1. if $X=\mathrm{pt}$ then $K_G(X)\simeq \mathrm{Rep}(G)$ is the representation ring of $G$.

   1. in general, $K_G(X)$ is an algebra over $\mathrm{Rep}(G)$.

2. if $G$ acts freely on $X$, then $K_G(X)\simeq K(X/G)$.

So in particular

$$K(ℰG{\times }_{G}X)\simeq K_G(ℰG\times X)$$K(\mathcal{E}G \times_G X) \simeq K_G(\mathcal{E}G \times X)

so we get a map

$$\alpha :K_G(X)\to K_G(ℰG\times X)\simeq K(ℰG{\times }_{G}X):=K_G^{\mathrm{Borel}}(X)$$\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 $\alpha$ induces an isomorphism

$${\hat{K}}_G(X)\simeq K_G^{\mathrm{Borel}}(X)$$\hat K_G(X) \simeq K_G^{Borel}(X)

where

$${\hat{K}}_G(X):=\underleftarrow{\lim }{K}_G(X)/I_G^n{K}_G(X)$$\hat K_G(X) := \lim_\leftarrow K_G(X)/I_G^n K_G(X)

where $I_G=\ker (\mathrm{Rep}(G)\simeq K_G(*)\to K_G(ℰG)\simeq K(ℬG)\stackrel{ϵ}{\to }\mathbb{Z})$

consider $X=*$, $G=S^1$, $ℂP^{\mathrm{\infty }}\simeq ℬS^1$

$$\alpha :\mathrm{Rep}(G)\to K(ℬG)=K(ℂP^{\mathrm{\infty }})\simeq \mathbb{Z}[[t]]$$\alpha : Rep(G) \to K(\mathcal{B}G) = K(\mathbb{C}P^\infty) \simeq \mathbb{Z}[ [ t ] ]

1. since $S^1$ is an abelian group, every irreducible representation is 1-dimensional

   $$\varphi :S^1\to {ℂ}^{\times }$$\phi : S^1 \to \mathbb{C}^\times

2. $\chi :S^1\hookrightarrow {ℂ}^{\mathrm{times}}$

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

algebraic interpretation

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

$$\mathrm{Rep}(S^1)={ℴ}_{G_m}$$Rep(S^1) = \mathcal{o}_{G_m}

and

$$\mathrm{Rep}(ℂP^{\mathrm{\infty }})={ℴ}_{{\hat{G}}_m}$$Rep(\mathbb{C}P^\infty) = \mathcal{o}_{\hat G_m}

adopt the functor of points perspective

$$X:\mathrm{CRings}\to \mathrm{Set}$$X : CRings \to Set

a functor.

For $A\in$ CRing get a spectrum

$$\mathrm{Spec}A:R\mapsto \mathrm{CRing}(A,R)$$Spec A : R \mapsto CRing(A,R)

for $X$ a functor, it is an affine scheme if it is a representable functor in that there is $A$ with $X\simeq \mathrm{Spec}A$.

examples

1. $A^n(R):=R^n$

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

3. $G_m(R):=R^{\times }\hookrightarrow A^1(R)$

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

proposition

$G_m$ is affine.

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

$$\varphi :\mathrm{Spec}A\to G_m$$\phi : Spec A \to G_m

by

$$u\mapsto u(x)$$u \mapsto u(x)

Conversely, given $v\in G_m(R)=R^{\times }$, define

$$\Psi :G_m\to \mathrm{Spec}A$$\Psi : G_m \to Spec A

by

$$v\mapsto \Psi (v)$$v \mapsto \Psi(v)

with

$$\Psi (v)(\sum _k{a}_k{x}^k):=\sum _k{a}_k{v}^k$$\Psi(v)(\sum_k a_k x^k) := \sum_k a_k v^k

endofproof

similarly, $A^n\simeq \mathrm{Spec}\mathbb{Z}[x_1,\cdots ,x_n]$

group schemes

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

$${ℴ}_X:={\mathrm{Hom}}_{\mathrm{Func}(\mathrm{CRing},\mathrm{Set})}(X,A^1)$$\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

$${ℴ}_{\mathrm{Spec}A}\simeq A$$\mathcal{o}_{Spec A} \simeq A

so that in particular

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

definition

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

$$X_i:Y\to {\hat{A}}^1$$X_i : Y \to \hat \mathbf{A}^1

such that

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

is an isomorphism. A $Y$ that admits a system of formal coordinates is a formal scheme over $X$.

warning very restrictive definition. See formal scheme

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

This means that there is a natural map

$$\sigma :G{\times }_{X}G\to G$$\sigma : G \times_X G \to G

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

definition (formal multiplicative group)

define ${\hat{G}}_m$ on each $R\in \mathrm{CRing}$ by

$${\hat{G}}_m(R)=\{1+n\mid n\in \mathrm{Nil}(R)\}$$\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

$${\hat{G}}_m\simeq {\hat{A}}^1$$\hat G_m \simeq \hat \mathbf{A}^1

We compute ${ℴ}_{{\hat{A}}^1}\simeq {ℴ}_{{\hat{G}}_m}$ in two ways:

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

   $${ℴ}_{{\hat{A}}^1}=\underrightarrow{\lim }\mathbb{Z}[t]/(t^n)=\mathbb{Z}[[t]].$$\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 $\mathrm{Spec}\mathbb{Z}[t]/(t^n)$, then for any ring $R$ $${\hat{A}}^1(R)={\lim }_{\to }\mathrm{Spec}\mathbb{Z}[t]/(t^n)(R).$$ By the universal property of colimits we have

   $$\mathrm{Nat}({\hat{A}}^1,A^1)\simeq \underleftarrow{\lim }\mathrm{Nat}(\mathrm{Spec}\mathbb{Z}[t]/(t^n),A^1)\simeq \mathbb{Z}[[t]].$$\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]].

$${ℴ}_{{\hat{G}}_m}\simeq \mathbb{Z}[[t]].$$\mathcal{o}_{\hat G_m} \simeq \mathbb{Z}[ [ t] ] \,.

recall we have a morphism $\alpha :\mathrm{Rep}(S^1)\to K(ℂP^{\mathrm{\infty }})$

such that

$${ℴ}_{G_m}\simeq \mathbb{Z}[x,x^{-1}]\simeq \mathrm{Rep}(S^1)\to K(ℂP^{\mathrm{\infty }})\simeq \mathbb{Z}[[t]]\simeq {ℴ}_{{\hat{G}}_m}$$\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

$${ℴ}_{G_m}\to {ℴ}_{{\hat{G}}_m}$$\mathcal{o}_{G_m} \to \mathcal{o}_{\hat G_m}

exercise

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

is the natural transformation $R^{\times }\to R$

and

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

is the natural transformation

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

so that we get a map

$${ℴ}_{G_m}\to {ℴ}_{{\hat{G}}_m}$$\mathcal{o}_{G_m} \to \mathcal{o}_{\hat G_m}

by sending $x$ to $1+t$, and this corresponds to taking the germ of functions at $1\in G_m$

lesson

given $G$ an algebraic group such that the formal spectrum $\mathrm{Spf}A(ℂP^{\mathrm{\infty }})$ is the completion $\hat{G}$, define $A_{S^1}(*):={ℴ}_G$ then passing to germs gives a completion map

$$A_{S^1}(*)\to A(ℂP^{\mathrm{\infty }})=A_{S^1}^{\mathrm{Bor}}(*)$$A_{S^1}({*}) \to A(\mathbb{C}P^\infty) = A^{Bor}_{S^1}({*})