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A Survey of Elliptic Cohomology - A-equivariant cohomology

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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.

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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 See also at equivariant elliptic cohomology.


Contents

Derived Elliptic Curves

Definition A derived elliptic curve over an (affine) derived scheme SpecA\mathrm{Spec} A is a commutative derived group scheme (CDGS) E/AE /A such that E¯Specπ 0A\overline{E} \to \mathrm{Spec} \pi_0 A is an elliptic curve.

Let AA be an E E_\infty-ring. Let E(A)E(A) denote the \infty-groupoid of oriented elliptic curves over SpecA\mathrm{Spec} A. Note that E(A)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 AE(A)A \mapsto E(A) is representable by a derived Deligne-Mumford stack Der=(,O )\mathcal{M}^{Der} = (\mathcal{M} , O_\mathcal{M} ). Further, \mathcal{M} is equivalent to the topos underlying 1,1\mathcal{M}_{1,1} and π 0O =O 1,1\pi_0 O_\mathcal{M} = O_{\mathcal{M}_{1,1}}. Also, restricting to discrete rings, O O_\mathcal{M} provides a lift in sense of Hopkins and Miller.

G\mathbf{G}-Equivariant AA-cohomology

The Strategy

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

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

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

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

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

S 1S^1-equivariance

To accomplish (1) we need a map

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

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

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

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

Recall two facts:

  1. There is a bijection {BS 1G(A)}{SpfA BS 1G}\{ BS^1 \to \mathbf{G} (A) \} \leftrightarrow \{ \mathrm{Spf} A^{BS^1} \to \mathbf{G} \};

  2. Orientations of the multiplicative group G m\mathbf{G}_m associated to AA are in bijection with maps of E E_\infty-rings {KA}\{ K \to A\}, where KK is the K-theory spectrum.

Theorem We can define equivariant AA-cohomology using G m\mathbf{G}_m if and only if AA is a KK-algebra.

The Abelian Lie Group Case for a Point

Fix G/A\mathbf{G}/A oriented. Now let TT be a compact abelian Lie group. We construct a commutative derived group scheme M TM_T over AA whose global sections give A TA_T which is equipped with an appropriate completion map.

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

Examples

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

    T^(θ 1,,θ n)(k 1θ 1,,k nθ n).\hat T \ni ( \theta_1 , \dots , \theta_n ) \mapsto (k_1 \theta_1 , \dots , k_n \theta_n ).
  2. If T={e}T = \{ e \}, then T^=T\hat T = T.

Pontryagin Duality If TT is an abelian, locally compact topological group then T^^T\hat \hat T \simeq T.

Definition Let BB be an AA-algebra. Define M TM_T by

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

Further, M TM_T is representable.

Examples

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

  2. M T n=G× SpecA× SpecAG.M_{T^n} = \mathbf{G} \times_{\mathrm{Spec} A} \dots \times_{\mathrm{Spec} A} \mathbf{G}.

  3. M /n=hker(×n:GG).M_{\mathbb{Z}/n} = \mathrm{hker} (\times n: \mathbf{G}\to \mathbf{G}).

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

How do we get a completion map σ T:BTM T(A)\sigma_T : BT \to M_T (A) for all TT given an orientation σ S 1:BS 1G(A)\sigma_{S^1} : BS^1 \to \mathbf{G} (A)? By a composition: define

Bev:BTHom(T^,BS 1),p(fBf(p)) Bev: BT \to \mathrm{Hom}(\hat T , BS^1), \; p \mapsto (f \mapsto Bf(p))

then define

σ T:=σ S 1Bev.\sigma_T := \sigma_{S^1} \circ Bev.

Proposition There exists a map M^\hat M such that the assignment TM TT \mapsto M_T factors as TM^(BT)T \mapsto \hat M (BT). That is the functor MM factors through the category of classifying spaces of compact Abelian Lie groups B(CALG)B(CALG) (considered as orbifolds). Further, such factorizations are in bijection with the preorientations of G\mathbf{G}.

Proof. That such a factorization exists defines M^\hat M on objects. Now by choosing a base point in BTBT' we have

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

as spaces. Now we need a map

BTHom(M T,M T).BT' \to \mathrm{Hom} (M_T , M_{T'}) .

Because this map must be functorial in TT and TT' we can restrict to the universal case where TT is trivial and then

BTHom(M {e},M T)=M T(A)BT' \to \mathrm{Hom} ( M_{\{e\}} , M_{T'} ) = M_{T'} (A)

is just a preorientation σ T\sigma_{T'}.

The Abelian Case for General Spaces

We will see that A T(X)A_T (X) is the global sections of a quasi-coherent sheaf on M TM_T.

Theorem Let G\mathbf{G} be preoriented and XX a finite TT-CW complex. There exist a unique family of functors {F T}\{ F_T \} from finite TT-spaces to the category of quasi-coherent sheaves on M TM_T such that

  1. F TF_T maps TT-equivariant (weak) homotopy equivalences to equivalence of quasi-coherent sheaves;

  2. F TF_T maps finite homotopy colimits to finite homotopy limits of quasi-coherent sheaves;

  3. F T(*)=O(M G)F_T (*) = O (M_\mathbf{G});

  4. If TTT \subset T' and X=(X×T)/TX' = (X \times T' )/T then F T(X)f *(F T(X))F_{T'} (X') \simeq f_* (F_T (X)), where f:M TM Tf: M_T \to M_{T'} is the induced map;

  5. The F TF_T are compatible under finite chains of inclusions of subgroups TTTT \subset T' \subset T'' \dots.

Proof. Use (2) to reduce to the case where XX is a TT-equivariant cell, i.e. X=T/T 0×D kX = T/ T_0 \times D^k for some subgroup T 0TT_0 \subset T. Use (1) to reduce to the case where X=T/T 0X = T/T_0. Use (3) to conclude that F T(T/T 0)=f *F T 0(*)F_T (T/T_0) = f_* F_{T_0} (*). Finally, (4) implies that F T 0(*)=M^(*/T 0)F_{T_0} (*) = \hat M (*/T_0 ), where M^\hat M is specified by the preorientation.

For trivial actions there is no dependence on the preorientation.

Remark

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

  2. If X,YX,Y are TT-spaces then we have maps

    F T(X)F T(X×Y)F T(Y)F_T (X) \to F_T (X \times Y) \leftarrow F_T (Y)

    and

    F T(X)F T(Y)F T(X×Y).F_T (X) \otimes F_T (Y) \to F_T (X \times Y).
  3. Define relative version for X 0XX_0 \subset X by

    F T(X,X 0)=hker(F T(X)F T(X 0))F_T (X, X_0 ) = hker (F_T (X) \to F_T (X_0 ))

    and for all TT-spaces YY we have a map

    F T(X,X 0) AF T(Y)F T(X×Y,X 0×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)=Γ(F T(X))A_T (X) = \Gamma (F_T (X)) as an E E_\infty-ring (algebra).

We now verify loop maps on A TA_T.

Recall that in the classical setting A n(X)A^n (X) is represented by a space Z nZ_n and we have suspension maps Z 0(S nZ n)Z_0 \to (S^n \to Z_n). Now we need to consider all possible TT-equivariant deloopings, that is TT-maps from S nZ nS^n \to Z_n.

Theorem Let G\mathbf{G} be oriented, VV a finite dimensional unitary representation of TT. Denote by SVBVSV \subset BV the unit sphere inside of the unit ball. Define L V=F T(BV,SV)L_V = F_T (BV, SV). Then

  1. L VL_V is a line bundle on M TM_T, i.e. invertible;

  2. For all (finite) TT-spaces XX the map

L VF T(X)F T(X×BV,X×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)T = S^1 = U(1) and V=V = \mathbb{C}. Then

L V=hker(F T(BV)F T(SV)).L_V = hker ( F_T (BV) \to F_T (SV)) .

As BVBV is contractible F T(BV)=O(G)F_T (BV) = O (\mathbf{G}) and by property (3) above F T(SV)=f *(O(SpecA))F_T (SV) = f_* (O (\mathrm{Spec} A)) for f:SpecAGf: \mathrm{Spec} A \to \mathbf{G} is the identity section. As G\mathbf{G} is oriented, π 0G/π 0SpecA\pi_0 \mathbf{G} / \pi_0 \mathrm{Spec} A is smooth of relative dimension 1, so L VL_V can be though of as the invertible sheaf of ideals defining the identity section of G\mathbf{G}.

Suppose VV and VV' are representations of TT then L VL VL VVL_V \otimes L_{V'} \to L_{V \oplus V'} is an equivalence. So if WW is a virtual representation (i.e. W=UUW = U - U') then L W=L U(L U) 1.L_W = L_U \otimes (L_{U'} )^{-1} .

Definition Let VV be a virtual representation of TT and define

A T V(X)=π 0Γ(F T(X)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 WA_G^W for all representations of GG, not just the trivial ones. In the derived setting we obtain this once we have an orientation of G\mathbf{G}.

The Non-Abelian Lie Group Case

Let AA be an E E_\infty-ring, G\mathbf{G} an orientated commutative derived group scheme over SpecA\mathrm{Spec} A, and TT a (not necessarily Abelian) compact Lie group.

Theorem There exists a functor A TA_T from (finite?) TT-spaces to Spectra which is uniquely characterized by the following.

  1. A TA_T preserves equivalence;

  2. For T 0TT_0 \subset T, A T 0(X)=A T((X×G)/T 0)A_{T_0} (X) = A_T ((X \times G) / T_0 );

  3. A TA_T maps homotopy colimits to homotopy limits;

  4. If TT is Abelian, then A TA_T is defined as above;

  5. For all spaces XX the map

A T(X)A T(X×E abT) A_T (X) \to A_T (X \times E^{ab} T)

where E abTE^{ab} T is a TT-space characterized by the requirement that for all Abelian subgroups T 0TT_0 \subset T, (E abT) T 0(E^{ab} T )^{T_0} is contractible and empty for T 0T_0 not Abelian. Further, for Borel equivariant cohomology we require

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

  2. A T(X)A T(X×ET)A_T (X) \to A_T (X \times ET) is an isomorphism.

Proof. In the case of ordinary equivariant cohomology we can use property (5) to reduce to the case where XX has only Abelian stabilizer groups. Then via (3) we reduce to XX being a colimit of TT-equivariant cells D k×T/T 0D^k \times T/T_0 for T 0T_0 Abelian. Via homotopy equivalence (1) we reduce to X=T/T 0X = T / T_0. Using property (2) we see A T(X)=A T 0(*)A_T (X) = A_{T_0} (*), so (4) yields A T 0(*)=M^(*/T 0)A_{T_0} (*) = \hat M (*/T_0 )

Revised on April 15, 2014 06:15:54 by Urs Schreiber (88.128.80.29)