nLab A Survey of Elliptic Cohomology - A-equivariant cohomology

cohomology

Theorems

higher algebra

universal algebra

Theorems

This is a sub-entry of

see there for background and context.

This entry contains a basic introduction to getting equivariant cohomology from derived group schemes.

Previous:

Next:

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

Derived Elliptic Curves

Definition A derived elliptic curve over an (affine) derived scheme $\mathrm{Spec}A$ is a commutative derived group scheme (CDGS) $E/A$ such that $\overline{E}\to \mathrm{Spec}{\pi }_{0}A$ is an elliptic curve.

Let $A$ be an ${E}_{\infty }$-ring. Let $E\left(A\right)$ denote the $\infty$-groupoid of oriented elliptic curves over $\mathrm{Spec}A$. Note that $E\left(A\right)$ is in particular a space (we will return to this point later).

The point is to prove the following due to Lurie.

Theorem The functor $A↦E\left(A\right)$ is representable by a derived Deligne-Mumford stack ${ℳ}^{\mathrm{Der}}=\left(ℳ,{O}_{ℳ}\right)$. Further, $ℳ$ is equivalent to the topos underlying ${ℳ}_{1,1}$ and ${\pi }_{0}{O}_{ℳ}={O}_{{ℳ}_{1,1}}$. Also, restricting to discrete rings, ${O}_{ℳ}$ provides a lift in sense of Hopkins and Miller.

$G$-Equivariant $A$-cohomology

The Strategy

1. Define ${A}_{{S}^{1}}\left(*\right)$;

2. Extend to ${A}_{{S}^{1}}\left(X\right)$ where $X$ is a trivial ${S}^{1}$-space;

3. Define ${A}_{T}\left(X\right)$ where $T$ is a compact abelian Lie group where $X$ is again a trivial $T$-space;

4. Extend to ${A}_{T}\left(X\right)$ for any (finite enough) $T$-space;

5. Define ${A}_{G}\left(X\right)$ for $G$ any compact Lie group.

${S}^{1}$-equivariance

To accomplish (1) we need a map

$\sigma :\mathrm{Spf}{A}^{ℂ{P}^{\infty }}\to G$\sigma : \mathrm{Spf} A^{\mathbb{C} P^\infty} \to \mathbf{G}

over $\mathrm{Spec}A$. Then we can define ${A}_{{S}^{1}}\left(*\right)=O\left(G\right)$. Such a map arises from a completion map

${A}_{{S}^{1}}\to {A}^{ℂ{P}^{\infty }}$A_{S^1} \to A^{\mathbb{C}P^\infty}

which we may interpret as a preorientation $\sigma \in \mathrm{Map}\left({\mathrm{BS}}^{1},G\left(A\right)\right)$. Recall that such a map $\sigma$ is an orientation if the induced map to the formal completion of $G$ is an isomorphism.

Recall two facts:

1. There is a bijection $\left\{{\mathrm{BS}}^{1}\to G\left(A\right)\right\}↔\left\{\mathrm{Spf}{A}^{{\mathrm{BS}}^{1}}\to G\right\}$;

2. Orientations of the multiplicative group ${G}_{m}$ associated to $A$ are in bijection with maps of ${E}_{\infty }$-rings $\left\{K\to A\right\}$, where $K$ is the K-theory spectrum.

Theorem We can define equivariant $A$-cohomology using ${G}_{m}$ if and only if $A$ is a $K$-algebra.

The Abelian Lie Group Case for a Point

Fix $G/A$ oriented. Now let $T$ be a compact abelian Lie group. We construct a commutative derived group scheme ${M}_{T}$ over $A$ whose global sections give ${A}_{T}$ which is equipped with an appropriate completion map.

Definition Define the Pontryagin dual, $\stackrel{^}{T}$ of $T$ by $\stackrel{^}{T}:={\mathrm{Hom}}_{\mathrm{Lie}}\left(T,{S}^{1}\right)$.

Examples

1. $T={T}^{n}$, the $n$-fold torus. Then $\stackrel{^}{T}={ℤ}^{n}$ as

$\stackrel{^}{T}\ni \left({\theta }_{1},\dots ,{\theta }_{n}\right)↦\left({k}_{1}{\theta }_{1},\dots ,{k}_{n}{\theta }_{n}\right).$\hat T \ni ( \theta_1 , \dots , \theta_n ) \mapsto (k_1 \theta_1 , \dots , k_n \theta_n ).
2. If $T=\left\{e\right\}$, then $\stackrel{^}{T}=T$.

Pontryagin Duality If $T$ is an abelian, locally compact topological group then $\stackrel{^}{\stackrel{^}{T}}\simeq T$.

Definition Let $B$ be an $A$-algebra. Define ${M}_{T}$ by

${M}_{T}\left(B\right):={\mathrm{Hom}}_{\mathrm{AbTop}}\left(\stackrel{^}{T},G\left(B\right)\right).$M_T (B) := \mathrm{Hom}_\mathrm{AbTop} (\hat T , \mathbf{G} (B)).

Further, ${M}_{T}$ is representable.

Examples

1. ${M}_{{S}^{1}}\left(B\right)=\mathrm{Hom}\left(ℤ,G\left(B\right)\right)=G\left(B\right).$

2. ${M}_{{T}^{n}}=G{×}_{\mathrm{Spec}A}\dots {×}_{\mathrm{Spec}A}G.$

3. ${M}_{ℤ/n}=\mathrm{hker}\left(×n:G\to G\right).$

4. ${M}_{\left\{e\right\}}\left(B\right)=\mathrm{Hom}\left(\left\{e\right\},G\left(B\right)=\left\{e\right\}$, so ${M}_{\left\{e\right\}}$ is final over $\mathrm{Spec}A$, hence it is isomorphic to $\mathrm{Spec}A$.

How do we get a completion map ${\sigma }_{T}:\mathrm{BT}\to {M}_{T}\left(A\right)$ for all $T$ given an orientation ${\sigma }_{{S}^{1}}:{\mathrm{BS}}^{1}\to G\left(A\right)$? By a composition: define

$\mathrm{Bev}:\mathrm{BT}\to \mathrm{Hom}\left(\stackrel{^}{T},{\mathrm{BS}}^{1}\right),\phantom{\rule{thickmathspace}{0ex}}p↦\left(f↦\mathrm{Bf}\left(p\right)\right)$Bev: BT \to \mathrm{Hom}(\hat T , BS^1), \; p \mapsto (f \mapsto Bf(p))

then define

${\sigma }_{T}:={\sigma }_{{S}^{1}}\circ \mathrm{Bev}.$\sigma_T := \sigma_{S^1} \circ Bev.

Proposition There exists a map $\stackrel{^}{M}$ such that the assignment $T↦{M}_{T}$ factors as $T↦\stackrel{^}{M}\left(\mathrm{BT}\right)$. That is the functor $M$ factors through the category of classifying spaces of compact Abelian Lie groups $B\left(\mathrm{CALG}\right)$ (considered as orbifolds). Further, such factorizations are in bijection with the preorientations of $G$.

Proof. That such a factorization exists defines $\stackrel{^}{M}$ on objects. Now by choosing a base point in $\mathrm{BT}\prime$ we have

$\mathrm{Hom}\left(\mathrm{BT},\mathrm{BT}\prime \right)\simeq \mathrm{BT}\prime ×\mathrm{Hom}\left(T,T\prime \right)$\mathrm{Hom} (BT , BT' ) \simeq BT' \times \mathrm{Hom} (T, T')

as spaces. Now we need a map

$\mathrm{BT}\prime \to \mathrm{Hom}\left({M}_{T},{M}_{T\prime }\right).$BT' \to \mathrm{Hom} (M_T , M_{T'}) .

Because this map must be functorial in $T$ and $T\prime$ we can restrict to the universal case where $T$ is trivial and then

$\mathrm{BT}\prime \to \mathrm{Hom}\left({M}_{\left\{e\right\}},{M}_{T\prime }\right)={M}_{T\prime }\left(A\right)$BT' \to \mathrm{Hom} ( M_{\{e\}} , M_{T'} ) = M_{T'} (A)

is just a preorientation ${\sigma }_{T\prime }$.

The Abelian Case for General Spaces

We will see that ${A}_{T}\left(X\right)$ is the global sections of a quasi-coherent sheaf on ${M}_{T}$.

Theorem Let $G$ be preoriented and $X$ a finite $T$-CW complex. There exist a unique family of functors $\left\{{F}_{T}\right\}$ from finite $T$-spaces to the category of quasi-coherent sheaves on ${M}_{T}$ such that

1. ${F}_{T}$ maps $T$-equivariant (weak) homotopy equivalences to equivalence of quasi-coherent sheaves;

2. ${F}_{T}$ maps finite homotopy colimits to finite homotopy limits of quasi-coherent sheaves;

3. ${F}_{T}\left(*\right)=O\left({M}_{G}\right)$;

4. If $T\subset T\prime$ and $X\prime =\left(X×T\prime \right)/T$ then ${F}_{T\prime }\left(X\prime \right)\simeq {f}_{*}\left({F}_{T}\left(X\right)\right)$, where $f:{M}_{T}\to {M}_{T\prime }$ is the induced map;

5. The ${F}_{T}$ are compatible under finite chains of inclusions of subgroups $T\subset T\prime \subset T″\dots$.

Proof. Use (2) to reduce to the case where $X$ is a $T$-equivariant cell, i.e. $X=T/{T}_{0}×{D}^{k}$ for some subgroup ${T}_{0}\subset T$. Use (1) to reduce to the case where $X=T/{T}_{0}$. Use (3) to conclude that ${F}_{T}\left(T/{T}_{0}\right)={f}_{*}{F}_{{T}_{0}}\left(*\right)$. Finally, (4) implies that ${F}_{{T}_{0}}\left(*\right)=\stackrel{^}{M}\left(*/{T}_{0}\right)$, where $\stackrel{^}{M}$ is specified by the preorientation.

For trivial actions there is no dependence on the preorientation.

Remark

1. ${F}_{T}\left(X\right)$ is actually a sheaf of algebras.

2. If $X,Y$ are $T$-spaces then we have maps

${F}_{T}\left(X\right)\to {F}_{T}\left(X×Y\right)←{F}_{T}\left(Y\right)$F_T (X) \to F_T (X \times Y) \leftarrow F_T (Y)

and

${F}_{T}\left(X\right)\otimes {F}_{T}\left(Y\right)\to {F}_{T}\left(X×Y\right).$F_T (X) \otimes F_T (Y) \to F_T (X \times Y).
3. Define relative version for ${X}_{0}\subset X$ by

${F}_{T}\left(X,{X}_{0}\right)=\mathrm{hker}\left({F}_{T}\left(X\right)\to {F}_{T}\left({X}_{0}\right)\right)$F_T (X, X_0 ) = hker (F_T (X) \to F_T (X_0 ))

and for all $T$-spaces $Y$ we have a map

${F}_{T}\left(X,{X}_{0}\right){\otimes }_{A}{F}_{T}\left(Y\right)\to {F}_{T}\left(X×Y,{X}_{0}×Y\right).$F_T (X, X_0 ) \otimes_{A} F_T (Y) \to F_T (X \times Y , X_0 \times Y).

Definition ${A}_{T}\left(X\right)=\Gamma \left({F}_{T}\left(X\right)\right)$ as an ${E}_{\infty }$-ring (algebra).

We now verify loop maps on ${A}_{T}$.

Recall that in the classical setting ${A}^{n}\left(X\right)$ is represented by a space ${Z}_{n}$ and we have suspension maps ${Z}_{0}\to \left({S}^{n}\to {Z}_{n}\right)$. Now we need to consider all possible $T$-equivariant deloopings, that is $T$-maps from ${S}^{n}\to {Z}_{n}$.

Theorem Let $G$ be oriented, $V$ a finite dimensional unitary representation of $T$. Denote by $\mathrm{SV}\subset \mathrm{BV}$ the unit sphere inside of the unit ball. Define ${L}_{V}={F}_{T}\left(\mathrm{BV},\mathrm{SV}\right)$. Then

1. ${L}_{V}$ is a line bundle on ${M}_{T}$, i.e. invertible;

2. For all (finite) $T$-spaces $X$ the map

${L}_{V}\otimes {F}_{T}\left(X\right)\to {F}_{T}\left(X×\mathrm{BV},X×\mathrm{SV}\right)$L_V \otimes F_T (X) \to F_T (X \times BV, X \times SV)

is an isomorphism.

Proof for $T={S}^{1}=U\left(1\right)$ and $V=ℂ$. Then

${L}_{V}=\mathrm{hker}\left({F}_{T}\left(\mathrm{BV}\right)\to {F}_{T}\left(\mathrm{SV}\right)\right).$L_V = hker ( F_T (BV) \to F_T (SV)) .

As $\mathrm{BV}$ is contractible ${F}_{T}\left(\mathrm{BV}\right)=O\left(G\right)$ and by property (3) above ${F}_{T}\left(\mathrm{SV}\right)={f}_{*}\left(O\left(\mathrm{Spec}A\right)\right)$ for $f:\mathrm{Spec}A\to G$ is the identity section. As $G$ is oriented, ${\pi }_{0}G/{\pi }_{0}\mathrm{Spec}A$ is smooth of relative dimension 1, so ${L}_{V}$ can be though of as the invertible sheaf of ideals defining the identity section of $G$.

Suppose $V$ and $V\prime$ are representations of $T$ then ${L}_{V}\otimes {L}_{V\prime }\to {L}_{V\oplus V\prime }$ is an equivalence. So if $W$ is a virtual representation (i.e. $W=U-U\prime$) then ${L}_{W}={L}_{U}\otimes \left({L}_{U\prime }{\right)}^{-1}.$

Definition Let $V$ be a virtual representation of $T$ and define

${A}_{T}^{V}\left(X\right)={\pi }_{0}\Gamma \left({F}_{T}\left(X\right)\otimes {L}_{V}^{-1}\right).$A_T^V (X) = \pi_0 \Gamma (F_T (X) \otimes L_V^{-1}) .

The point is that in order to define equivariant cohomology requires functors ${A}_{G}^{W}$ for all representations of $G$, not just the trivial ones. In the derived setting we obtain this once we have an orientation of $G$.

The Non-Abelian Lie Group Case

Let $A$ be an ${E}_{\infty }$-ring, $G$ an orientated commutative derived group scheme over $\mathrm{Spec}A$, and $T$ a (not necessarily Abelian) compact Lie group.

Theorem There exists a functor ${A}_{T}$ from (finite?) $T$-spaces to Spectra which is uniquely characterized by the following.

1. ${A}_{T}$ preserves equivalence;

2. For ${T}_{0}\subset T$, ${A}_{{T}_{0}}\left(X\right)={A}_{T}\left(\left(X×G\right)/{T}_{0}\right)$;

3. ${A}_{T}$ maps homotopy colimits to homotopy limits;

4. If $T$ is Abelian, then ${A}_{T}$ is defined as above;

5. For all spaces $X$ the map

${A}_{T}\left(X\right)\to {A}_{T}\left(X×{E}^{\mathrm{ab}}T\right)$A_T (X) \to A_T (X \times E^{ab} T)

where ${E}^{\mathrm{ab}}T$ is a $T$-space characterized by the requirement that for all Abelian subgroups ${T}_{0}\subset T$, $\left({E}^{\mathrm{ab}}T{\right)}^{{T}_{0}}$ is contractible and empty for ${T}_{0}$ not Abelian. Further, for Borel equivariant cohomology we require

1. If $T=\left\{e\right\}$, then ${A}_{T}\left(X\right)=A\left(X\right)={A}^{X}$;

2. ${A}_{T}\left(X\right)\to {A}_{T}\left(X×\mathrm{ET}\right)$ is an isomorphism.

Proof. In the case of ordinary equivariant cohomology we can use property (5) to reduce to the case where $X$ has only Abelian stabilizer groups. Then via (3) we reduce to $X$ being a colimit of $T$-equivariant cells ${D}^{k}×T/{T}_{0}$ for ${T}_{0}$ Abelian. Via homotopy equivalence (1) we reduce to $X=T/{T}_{0}$. Using property (2) we see ${A}_{T}\left(X\right)={A}_{{T}_{0}}\left(*\right)$, so (4) yields ${A}_{{T}_{0}}\left(*\right)=\stackrel{^}{M}\left(*/{T}_{0}\right)$

Revised on July 7, 2010 19:57:56 by Anonymous Coward (75.64.209.193)