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A Survey of Elliptic Cohomology - A-equivariant cohomology

This is a sub-entry of

• A Survey of Elliptic Cohomology

see there for background and context.

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

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

• A Survey of Elliptic Cohomology - equivariant cohomology

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

Next:

• A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves

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

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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 $E_{\mathrm{\infty }}$-ring. Let $E(A)$ denote the $\mathrm{\infty }$-groupoid of oriented elliptic curves over $\mathrm{Spec}A$. Note that $E(A)$ 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 $A\mapsto E(A)$ is representable by a derived Deligne-Mumford stack ${ℳ}^{\mathrm{Der}}=(ℳ,O_{ℳ})$. Further, $ℳ$ is equivalent to the topos underlying ${ℳ}_{\mathrm{1,1}}$ and ${\pi }_0{O}_{ℳ}=O_{{ℳ}_{\mathrm{1,1}}}$. Also, restricting to discrete rings, $O_{ℳ}$ provides a lift in sense of Hopkins and Miller.

$G$-Equivariant $A$-cohomology

The Strategy

1. Define $A_{S^1}(*)$;

2. Extend to $A_{S^1}(X)$ where $X$ is a trivial $S^1$-space;

3. Define $A_T(X)$ where $T$ is a compact abelian Lie group where $X$ is again a trivial $T$-space;

4. Extend to $A_T(X)$ for any (finite enough) $T$-space;

5. Define $A_G(X)$ for $G$ any compact Lie group.

$S^1$-equivariance

To accomplish (1) we need a map

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

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

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

which we may interpret as a preorientation $\sigma \in \mathrm{Map}({\mathrm{BS}}^1,G(A))$. 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 $\{{\mathrm{BS}}^1\to G(A)\}\leftrightarrow \{\mathrm{Spf}{A}^{{\mathrm{BS}}^1}\to G\}$;

2. Orientations of the multiplicative group $G_m$ associated to $A$ are in bijection with maps of $E_{\mathrm{\infty }}$-rings $\{K\to A\}$, 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, $\hat{T}$ of $T$ by $\hat{T}:={\mathrm{Hom}}_{\mathrm{Lie}}(T,S^1)$.

Examples

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

   $$\hat{T}\ni ({\theta }_1,\dots ,{\theta }_n)\mapsto (k_1{\theta }_1,\dots ,k_n{\theta }_n).$$\hat T \ni ( \theta_1 , \dots , \theta_n ) \mapsto (k_1 \theta_1 , \dots , k_n \theta_n ).

2. If $T=\{e\}$, then $\hat{T}=T$.

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

Definition Let $B$ be an $A$-algebra. Define $M_T$ by

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

Further, $M_T$ is representable.

Examples

1. $M_{S^1}(B)=\mathrm{Hom}(\mathbb{Z},G(B))=G(B).$

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

3. $M_{\mathbb{Z}/n}=\mathrm{hker}(\times n:G\to G).$

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

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

$$\mathrm{Bev}:\mathrm{BT}\to \mathrm{Hom}(\hat{T},{\mathrm{BS}}^1),p\mapsto (f\mapsto \mathrm{Bf}(p))$$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 $\hat{M}$ such that the assignment $T\mapsto M_T$ factors as $T\mapsto \hat{M}(\mathrm{BT})$. That is the functor $M$ factors through the category of classifying spaces of compact Abelian Lie groups $B(\mathrm{CALG})$ (considered as orbifolds). Further, such factorizations are in bijection with the preorientations of $G$.

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

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

as spaces. Now we need a map

$$\mathrm{BT}\prime \to \mathrm{Hom}(M_T,M_{T\prime }).$$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

$$\mathrm{BT}\prime \to \mathrm{Hom}(M_{\{e\}},M_{T\prime })=M_{T\prime }(A)$$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 $A_T(X)$ 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 $\{F_T\}$ 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. $F_T(*)=O(M_G)$;

4. If $T\subset T\prime$ and $X\prime =(X\times T\prime )/T$ then $F_{T\prime }(X\prime )\simeq f_{*}(F_T(X))$, 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. $X=T/T_0\times D^k$ for some subgroup $T_0\subset T$. Use (1) to reduce to the case where $X=T/T_0$. Use (3) to conclude that $F_T(T/T_0)=f_{*}{F}_{T_0}(*)$. Finally, (4) implies that $F_{T_0}(*)=\hat{M}(*/T_0)$, where $\hat{M}$ is specified by the preorientation.

For trivial actions there is no dependence on the preorientation.

Remark

1. $F_T(X)$ is actually a sheaf of algebras.

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

   $$F_T(X)\to F_T(X\times Y)\leftarrow F_T(Y)$$F_T (X) \to F_T (X \times Y) \leftarrow F_T (Y)

   and

   $$F_T(X)\otimes F_T(Y)\to F_T(X\times Y).$$F_T (X) \otimes F_T (Y) \to F_T (X \times Y).

3. Define relative version for $X_0\subset X$ by

   $$F_T(X,X_0)=\mathrm{hker}(F_T(X)\to F_T(X_0))$$F_T (X, X_0 ) = hker (F_T (X) \to F_T (X_0 ))

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

   $$F_T(X,X_0){\otimes }_A{F}_T(Y)\to F_T(X\times Y,X_0\times Y).$$F_T (X, X_0 ) \otimes_{A} F_T (Y) \to F_T (X \times Y , X_0 \times Y).

Definition $A_T(X)=\Gamma (F_T(X))$ as an $E_{\mathrm{\infty }}$-ring (algebra).

We now verify loop maps on $A_T$.

Recall that in the classical setting $A^n(X)$ is represented by a space $Z_n$ and we have suspension maps $Z_0\to (S^n\to Z_n)$. 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 $L_V=F_T(\mathrm{BV},\mathrm{SV})$. Then

1. $L_V$ is a line bundle on $M_T$, i.e. invertible;

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

$$L_V\otimes F_T(X)\to F_T(X\times \mathrm{BV},X\times \mathrm{SV})$$L_V \otimes F_T (X) \to F_T (X \times BV, X \times SV)

is an isomorphism.

Proof for $T=S^1=U(1)$ and $V=ℂ$. Then

$$L_V=\mathrm{hker}(F_T(\mathrm{BV})\to F_T(\mathrm{SV})).$$L_V = hker ( F_T (BV) \to F_T (SV)) .

As $\mathrm{BV}$ is contractible $F_T(\mathrm{BV})=O(G)$ and by property (3) above $F_T(\mathrm{SV})=f_{*}(O(\mathrm{Spec}A))$ 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 $L_W=L_U\otimes (L_{U\prime }{)}^{-1}.$

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

$$A_T^V(X)={\pi }_0\Gamma (F_T(X)\otimes L_V^{-1}).$$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 $E_{\mathrm{\infty }}$-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 $T_0\subset T$, $A_{T_0}(X)=A_T((X\times G)/T_0)$;

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

$$A_T(X)\to A_T(X\times E^{\mathrm{ab}}T)$$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 $T_0\subset T$, $(E^{\mathrm{ab}}T{)}^{T_0}$ is contractible and empty for $T_0$ not Abelian. Further, for Borel equivariant cohomology we require

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

2. $A_T(X)\to A_T(X\times \mathrm{ET})$ 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 $D^k\times T/T_0$ for $T_0$ Abelian. Via homotopy equivalence (1) we reduce to $X=T/T_0$. Using property (2) we see $A_T(X)=A_{T_0}(*)$, so (4) yields $A_{T_0}(*)=\hat{M}(*/T_0)$