* characteristic class
* universal characteristic class
* secondary characteristic class
* differential characteristic class
* fiber sequence
/long exact sequence in cohomology
* fiber ∞-bundle
, principal ∞-bundle
, associated ∞-bundle
* ∞-group extension
### Special and general types ###
* cochain cohomology
* ordinary cohomology
, singular cohomology
* group cohomology
, nonabelian group cohomology
, Lie group cohomology
* Galois cohomology
* groupoid cohomology
, nonabelian groupoid cohomology
* generalized (Eilenberg-Steenrod) cohomology
* cobordism cohomology theory
* integral cohomology
* elliptic cohomology
* abelian sheaf cohomology
* Deligne cohomology
* de Rham cohomology
* Dolbeault cohomology
* etale cohomology
* group of units
, Picard group
, Brauer group
* crystalline cohomology
* syntomic cohomology
* motivic cohomology
* cohomology of operads
* Hochschild cohomology
, cyclic cohomology
* string topology
* nonabelian cohomology
* principal ∞-bundle
* universal principal ∞-bundle
, groupal model for universal principal ∞-bundles
* principal bundle
, Atiyah Lie groupoid
* principal 2-bundle
* covering ∞-bundle
* (∞,1)-vector bundle
/ (∞,n)-vector bundle
* quantum anomaly
, Spin structure
, Spin^c structure
, String structure
, Fivebrane structure
* cohomology with constant coefficients
/ with a local system of coefficients
* ∞-Lie algebra cohomology
* Lie algebra cohomology
, nonabelian Lie algebra cohomology
, Lie algebra extensions
, Gelfand-Fuks cohomology
* bialgebra cohomology
### Special notions
* Čech cohomology
### Variants ###
* equivariant cohomology
* equivariant homotopy theory
* Bredon cohomology
* twisted cohomology
* twisted bundle
* twisted K-theory
, twisted spin structure
, twisted spin^c structure
* twisted differential c-structures
* twisted differential string structure
, twisted differential fivebrane structure
* differential cohomology
* differential generalized (Eilenberg-Steenrod) cohomology
* differential cobordism cohomology
* Deligne cohomology
* differential K-theory
* differential elliptic cohomology
* differential cohomology in a cohesive topos
* Chern-Weil theory
* ∞-Chern-Weil theory
* relative cohomology
### Extra structure
* Hodge structure
, in generalized cohomology
### Operations ###
* cohomology operations
* cup product
* connecting homomorphism
, Bockstein homomorphism
* fiber integration
* cohomology localization
* universal coefficient theorem
* Künneth theorem
* de Rham theorem
, Poincare lemma
, Stokes theorem
* Hodge theory
, Hodge theorem
nonabelian Hodge theory
, noncommutative Hodge theory
* Brown representability theorem
* hypercovering theorem
* Eckmann-Hilton-Fuks duality
## Algebraic theories
* algebraic theory
/ 2-algebraic theory
/ (∞,1)-algebraic theory
## Algebras and modules
* algebra over a monad
∞-algebra over an (∞,1)-monad
* algebra over an algebraic theory
∞-algebra over an (∞,1)-algebraic theory
* algebra over an operad
∞-algebra over an (∞,1)-operad
* associated bundle
, associated ∞-bundle
## Higher algebras
* monoidal (∞,1)-category
* symmetric monoidal (∞,1)-category
* monoid in an (∞,1)-category
* commutative monoid in an (∞,1)-category
* symmetric monoidal (∞,1)-category of spectra
* smash product of spectra
* symmetric monoidal smash product of spectra
* ring spectrum
, module spectrum
, algebra spectrum
* A-∞ algebra
* A-∞ ring
, A-∞ space
* C-∞ algebra
* E-∞ ring
, E-∞ algebra
, (∞,1)-module bundle
* multiplicative cohomology theory
* L-∞ algebra
* deformation theory
## Model category presentations
* model structure on simplicial T-algebras
/ homotopy T-algebra
* model structure on operads
model structure on algebras over an operad
## Geometry on formal duals of algebras
* Isbell duality
* derived geometry
* Deligne conjecture
* delooping hypothesis
* monoidal Dold-Kan correspondence
This is a sub-entry of
see there for background and context.
This entry contains a basic introduction to derived group schemes and their orientations.
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
Recall from last time that given an algebraic group such that the formal spectrum is the completion , we could define then passing to germs gave a completion map
The problem we (begin) to address here is how to extend this equivariant cohomology to other spaces besides the point. This requires derived algebraic geometry.
Derived group schemes
Recall that a commutative group scheme over a scheme is a functor
such that composition with the forgetful functor is representable.
We would like extend this definition to the world of derived schemes. There are two problems
Because of the higher categorical nature of derived schemes Hom sets are spaces.
Everything should in the -setting, that is defined only up to homotopy.
We will not worry about the second concern and address the first by replacing the category with and with .
The following definition is somewhat restrictive and really should incorporate more of the -structure.
Definition A commutative derived group scheme over a derived scheme is a topological functor
such that composition with the forgetful functor is representable (up to weak equivalence) by an object which is flat? over .
Let be a scheme, then we have an associated derived scheme . The structure sheaf of is obtained by viewing the structure sheaf of as a presheaf of -rings and then sheafifying. We then have an equivalence between commutative derived group schemes over and commutative group schemes which are flat over .
For a derived scheme we have a map from commutative derived group schemes over to commutative group schemes which are flat over .
Throughout will be an -ring, the affine derived scheme , a commutative derived group scheme over , the -ring given by , and the -ring given by
Definition(Preliminary) A preorientation of is a morphism of commutative derived group schemes over
where is the completion wrt the ideal . A preorientation is an orientation if the induced map
is an isomorphism.
Suppose that is affine, then a map
corresponds to a map which is the same as a map
Hence we are led to the following definition.
Definition Let be a derived scheme and a commutative derived group scheme over . A preorientation of is a morphism of topological commutative monoids
Notice that is nearly freely generated. Indeed it follows from the fundamental theorem of algebra that as a topological monoid is generated by subject to the single relation that is the monoidal unit.
Proposition A preorientation up to homotopy is a map
that is an element of .
Hence, we always have at least one preorientation: the trivial one which corresponds to .
As motivation recall that a map of 1-dim formal groups is an isomorphism if and only if is invertible. We would like to encode this in our derived language (without defining ).
Definition Let be an -ring, a commutative derived group scheme over and a preorientation. Then is an orientation if
is smooth of relative dimension 1, and
The map induced by
is an isomorphism for each .
Note that (2) implies that is weakly periodic. Conversely, if is weakly periodic then (2) is equivalent to being an isomorphism.
Before defining and we extend the above definition to derived group schemes over an arbitrary derived scheme.
Definition Let be a derived scheme, a commutative derived group scheme over and a preorientation. Then is an orientation if is an orientation for all affine.
We now define the module and the map . Let be an -ring, a commutative derived group scheme over and let a scheme over . Let denote the sheaf of differentials on . Then define
is the identity section.
Now let be an open affine subscheme containing the identity section, so for some -ring . Then induces a map which is the same as a map
It is a fact that the map is a derivation over and hence has a classifying map which yields a map
The Multiplicative Derived Group Scheme
The naive guess for is , where . It is true that is a derived scheme over , however it is not flat, nor is an Abelian group as is an -ring and not an honestly commutative ring.
If is rational, that is there is a map , then can be given an Abelian group structure. Hence, is a perfectly good group scheme defined on the category of rational -rings, however this category is too small; there are too few rational -rings.
Recall that for a ring , . Further, recall that for a group we can form the group algebra which is really a Hopf algebra. Then is a group scheme over . Further, is characterized by
where is the category of monoids and the ring is thought of as a monoid wrt to multiplication. Motivated by these observations we make the following definitions.
Definition Let be an -ring and a topological Abelian monoid, then we can define which is characterized by
Recall that because of the higher categorical nature of things, the hom-sets above are spaces and the symbol indicates weak equivalence of spaces.
Definition Let be an -ring. We define the multiplicative group corresponding to as
is a derived commutative group scheme over .
Note that . Also, the map is smooth of relative dimension 1.
Proposition For any -ring , we have a bijection (of sets) between preorientations of and maps .
The proof follows from the fact that is initial in the category of -rings and the mapping property of .
Corollary is the moduli space of preorientations of . That is, if is defined over , then a preorientation of is the same as a map .
We consider the map where
and is the identity section. Note that , hence it follows that is canonically trivial, so an orientation is just an element such that is invertible in .
Let denote the (universal) orientation of . Then we have the following.
Theorem is the moduli space of orientations of .
It is a theorem of Snaith, that this moduli space has the homotopy type of the spectrum of complex K-theory. Note that by considering the homtopy fixed points of a certain action there is a way to recover as well.
Connection to complex orientation
Let be an -ring, so in particular defines a cohomology theory. An orientation of over is a map . A complex orientation of is a map . Recalling that is complex oriented, we see that an orientation of gives a complex orientation by precomposing with the map .
The Additive Derived Group Scheme
The naive definition of is , where is the additive group of . It is true that is a derived scheme over , however it is not flat as for an -ring
where as if it were flat we would have
Also, is not commutative. is an infinite loop space, but not an Abelian monoid. Again is a derived group scheme when restricted to rational -rings.
We no restrict to the category of integral -rings, i.e. those equipped with a map . Note that in this category is initial.
Definition For an integral -ring define
It can be shown that is flat and has the correct amount of commutativity.
Why can’t we just use ?
Proposition For all integral -rings, preorientations of are in bijective correspondence with maps . Consequently, is the moduli space of preorientations of .
Now, . The right side is a free divided power series on a generator where .
Proposition is the moduli space of orientations of .
Proposition . Hence the Chern character yields an isomorphism with rational periodic cohomology.