group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
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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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 - derived group schemes and (pre-)orientations
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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.
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 $E(A)$ denote the $\infty$-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 $\mathcal{M}^{Der} = (\mathcal{M} , O_\mathcal{M} )$. Further, $\mathcal{M}$ is equivalent to the topos underlying $\mathcal{M}_{1,1}$ and $\pi_0 O_\mathcal{M} = O_{\mathcal{M}_{1,1}}$. Also, restricting to discrete rings, $O_\mathcal{M}$ provides a lift in sense of Hopkins and Miller.
Define $A_{S^1 } (*)$;
Extend to $A_{S^1} (X)$ where $X$ is a trivial $S^1$-space;
Define $A_T (X)$ where $T$ is a compact abelian Lie group where $X$ is again a trivial $T$-space;
Extend to $A_T (X)$ for any (finite enough) $T$-space;
Define $A_G (X)$ for $G$ any compact Lie group.
To accomplish (1) we need a map
over $\mathrm{Spec} A$. Then we can define $A_{S^1} (*) = O (\mathbf{G})$. Such a map arises from a completion map
which we may interpret as a preorientation $\sigma \in \mathrm{Map} (BS^1 , \mathbf{G} (A))$. Recall that such a map $\sigma$ is an orientation if the induced map to the formal completion of $\mathbf{G}$ is an isomorphism.
Recall two facts:
There is a bijection $\{ BS^1 \to \mathbf{G} (A) \} \leftrightarrow \{ \mathrm{Spf} A^{BS^1} \to \mathbf{G} \}$;
Orientations of the multiplicative group $\mathbf{G}_m$ associated to $A$ are in bijection with maps of $E_\infty$-rings $\{ K \to A\}$, where $K$ is the K-theory spectrum.
Theorem We can define equivariant $A$-cohomology using $\mathbf{G}_m$ if and only if $A$ is a $K$-algebra.
Fix $\mathbf{G}/A$ 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
$T=T^n$, the $n$-fold torus. Then $\hat T = \mathbb{Z}^n$ as
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
Further, $M_T$ is representable.
Examples
$M_{S^1} (B) = \mathrm{Hom} ( \mathbb{Z} , \mathbf{G} (B)) = \mathbf{G} (B).$
$M_{T^n} = \mathbf{G} \times_{\mathrm{Spec} A} \dots \times_{\mathrm{Spec} A} \mathbf{G}.$
$M_{\mathbb{Z}/n} = \mathrm{hker} (\times n: \mathbf{G}\to \mathbf{G}).$
$M_{\{e\}} (B) = \mathrm{Hom} ( \{e \} , \mathbf{G} (B) = \{e\}$, so $M_{\{e\}}$ is final over $\mathrm{Spec} A$, hence it is isomorphic to $\mathrm{Spec} A$.
How do we get a completion map $\sigma_T : BT \to M_T (A)$ for all $T$ given an orientation $\sigma_{S^1} : BS^1 \to \mathbf{G} (A)$? By a composition: define
then define
Proposition There exists a map $\hat M$ such that the assignment $T \mapsto M_T$ factors as $T \mapsto \hat M (BT)$. That is the functor $M$ factors through the category of classifying spaces of compact Abelian Lie groups $B(CALG)$ (considered as orbifolds). Further, such factorizations are in bijection with the preorientations of $\mathbf{G}$.
Proof. That such a factorization exists defines $\hat M$ on objects. Now by choosing a base point in $BT'$ we have
as spaces. Now we need a map
Because this map must be functorial in $T$ and $T'$ we can restrict to the universal case where $T$ is trivial and then
is just a preorientation $\sigma_{T'}$.
We will see that $A_T (X)$ is the global sections of a quasi-coherent sheaf on $M_T$.
Theorem Let $\mathbf{G}$ 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
$F_T$ maps $T$-equivariant (weak) homotopy equivalences to equivalence of quasi-coherent sheaves;
$F_T$ maps finite homotopy colimits to finite homotopy limits of quasi-coherent sheaves;
$F_T (*) = O (M_\mathbf{G})$;
If $T \subset T'$ and $X' = (X \times T' )/T$ then $F_{T'} (X') \simeq f_* (F_T (X))$, where $f: M_T \to M_{T'}$ is the induced map;
The $F_T$ are compatible under finite chains of inclusions of subgroups $T \subset T' \subset T'' \dots$.
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
$F_T (X)$ is actually a sheaf of algebras.
If $X,Y$ are $T$-spaces then we have maps
and
Define relative version for $X_0 \subset X$ by
and for all $T$-spaces $Y$ we have a map
Definition $A_T (X) = \Gamma (F_T (X))$ as an $E_\infty$-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 $\mathbf{G}$ be oriented, $V$ a finite dimensional unitary representation of $T$. Denote by $SV \subset BV$ the unit sphere inside of the unit ball. Define $L_V = F_T (BV, SV)$. Then
$L_V$ is a line bundle on $M_T$, i.e. invertible;
For all (finite) $T$-spaces $X$ the map
is an isomorphism.
Proof for $T = S^1 = U(1)$ and $V = \mathbb{C}$. Then
As $BV$ is contractible $F_T (BV) = O (\mathbf{G})$ and by property (3) above $F_T (SV) = f_* (O (\mathrm{Spec} A))$ for $f: \mathrm{Spec} A \to \mathbf{G}$ is the identity section. As $\mathbf{G}$ is oriented, $\pi_0 \mathbf{G} / \pi_0 \mathrm{Spec} A$ is smooth of relative dimension 1, so $L_V$ can be though of as the invertible sheaf of ideals defining the identity section of $\mathbf{G}$.
Suppose $V$ and $V'$ are representations of $T$ then $L_V \otimes L_{V'} \to L_{V \oplus V'}$ is an equivalence. So if $W$ is a virtual representation (i.e. $W = U - U'$) then $L_W = L_U \otimes (L_{U'} )^{-1} .$
Definition Let $V$ be a virtual representation of $T$ and define
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 $\mathbf{G}$.
Let $A$ be an $E_\infty$-ring, $\mathbf{G}$ 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.
$A_T$ preserves equivalence;
For $T_0 \subset T$, $A_{T_0} (X) = A_T ((X \times G) / T_0 )$;
$A_T$ maps homotopy colimits to homotopy limits;
If $T$ is Abelian, then $A_T$ is defined as above;
For all spaces $X$ the map
where $E^{ab} T$ is a $T$-space characterized by the requirement that for all Abelian subgroups $T_0 \subset T$, $(E^{ab} T )^{T_0}$ is contractible and empty for $T_0$ not Abelian. Further, for Borel equivariant cohomology we require
If $T = \{e \}$, then $A_T(X) = A(X) = A^X$;
$A_T (X) \to A_T (X \times ET)$ is an isomorphism.
Proof. In the case of ordinary equivariant cohomology we can use property (5) to reduce to the case where $X$ 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 $A_{T_0} (*) = \hat M (*/T_0 )$