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.
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.
Let be an -ring. Let denote the -groupoid of oriented elliptic curves over . Note that 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 is representable by a derived Deligne-Mumford stack . Further, is equivalent to the topos underlying and . Also, restricting to discrete rings, provides a lift in sense of Hopkins and Miller.
Extend to where is a trivial -space;
Define where is a compact abelian Lie group where is again a trivial -space;
Extend to for any (finite enough) -space;
Define for any compact Lie group.
To accomplish (1) we need a map
over . Then we can define . Such a map arises from a completion map
which we may interpret as a preorientation . Recall that such a map is an orientation if the induced map to the formal completion of is an isomorphism.
Recall two facts:
There is a bijection ;
Orientations of the multiplicative group associated to are in bijection with maps of -rings , where is the K-theory spectrum.
Theorem We can define equivariant -cohomology using if and only if is a -algebra.
Fix oriented. Now let be a compact abelian Lie group. We construct a commutative derived group scheme over whose global sections give which is equipped with an appropriate completion map.
Definition Define the Pontryagin dual, of by .
, the -fold torus. Then as
If , then .
Pontryagin Duality If is an abelian, locally compact topological group then .
Definition Let be an -algebra. Define by
Further, is representable.
, so is final over , hence it is isomorphic to .
How do we get a completion map for all given an orientation ? By a composition: define
Proposition There exists a map such that the assignment factors as . That is the functor factors through the category of classifying spaces of compact Abelian Lie groups (considered as orbifolds). Further, such factorizations are in bijection with the preorientations of .
Proof. That such a factorization exists defines on objects. Now by choosing a base point in we have
as spaces. Now we need a map
Because this map must be functorial in and we can restrict to the universal case where is trivial and then
is just a preorientation .
We will see that is the global sections of a quasi-coherent sheaf on .
Theorem Let be preoriented and a finite -CW complex. There exist a unique family of functors from finite -spaces to the category of quasi-coherent sheaves on such that
maps -equivariant (weak) homotopy equivalences to equivalence of quasi-coherent sheaves;
maps finite homotopy colimits to finite homotopy limits of quasi-coherent sheaves;
If and then , where is the induced map;
The are compatible under finite chains of inclusions of subgroups .
Proof. Use (2) to reduce to the case where is a -equivariant cell, i.e. for some subgroup . Use (1) to reduce to the case where . Use (3) to conclude that . Finally, (4) implies that , where is specified by the preorientation.
For trivial actions there is no dependence on the preorientation.
is actually a sheaf of algebras.
If are -spaces then we have maps
Define relative version for by
and for all -spaces we have a map
Definition as an -ring (algebra).
We now verify loop maps on .
Recall that in the classical setting is represented by a space and we have suspension maps . Now we need to consider all possible -equivariant deloopings, that is -maps from .
Theorem Let be oriented, a finite dimensional unitary representation of . Denote by the unit sphere inside of the unit ball. Define . Then
is a line bundle on , i.e. invertible;
For all (finite) -spaces the map
is an isomorphism.
Proof for and . Then
As is contractible and by property (3) above for is the identity section. As is oriented, is smooth of relative dimension 1, so can be though of as the invertible sheaf of ideals defining the identity section of .
Suppose and are representations of then is an equivalence. So if is a virtual representation (i.e. ) then
Definition Let be a virtual representation of and define
The point is that in order to define equivariant cohomology requires functors for all representations of , not just the trivial ones. In the derived setting we obtain this once we have an orientation of .
Let be an -ring, an orientated commutative derived group scheme over , and a (not necessarily Abelian) compact Lie group.
Theorem There exists a functor from (finite?) -spaces to Spectra which is uniquely characterized by the following.
For , ;
maps homotopy colimits to homotopy limits;
If is Abelian, then is defined as above;
For all spaces the map
where is a -space characterized by the requirement that for all Abelian subgroups , is contractible and empty for not Abelian. Further, for Borel equivariant cohomology we require
If , then ;
is an isomorphism.
Proof. In the case of ordinary equivariant cohomology we can use property (5) to reduce to the case where has only Abelian stabilizer groups. Then via (3) we reduce to being a colimit of -equivariant cells for Abelian. Via homotopy equivalence (1) we reduce to . Using property (2) we see , so (4) yields