The goal of the next few posts is to compute :
Theorem 1 (Milnor) The complex cobordism ring is isomorphic to a polynomial ring where each is in degree .
We are also going to work out what the image of the Hurewicz map is on indecomposables. The strategy will be to apply the Adams spectral sequence to , at each prime individually.
1. Change-of-rings theorem
In order to apply the ASS, we’re going to need the groups because the spectral sequence runs
The groups are computed in the category of (graded) comodules over .
In the previous post, we computed
as a comodule over . In order to compute the groups, we need a general machine. The idea is that is almost a coinduced comodule—if it were, the groups would be trivial. It’s not, but the general “change-of-rings” machine will enable us to reduce the calculation of these groups to the calculation of (much simpler) groups over an exterior algebra.
Let be a coalgebra (over a field ). Let be another coalgebra, and consider a map
of -coalgebras. Then, there is a functor
which sends an -comodule to a -comodule ; the comodule structure on on comes from
In ordinary algebra, given a morphism of algebras , one gets a functor given by restriction; the left adjoint to restriction is given by . In the coalgebra case, everything is dualized.
We will see that that the restriction of scalars has a right adjoint in the dual case. To do this, we will need to define a dual construction to the tensor product in coalgebra.
Definition 2 Let be a right -comodule, and a left -comodule. Define the cotensor product via the equalizer
The two maps come from the two comodule structure maps. The unadorned tensor products are over .
This is completely dual to the definition of the tensor product of a right -module and a left -module , as the coequalizer . Then the cotensor product is a -vector space, and in general has no additional structure.
However, if is equipped with a dual left -comodule structure “commuting” with the right -comodule structure, then gets a left comodule structure by the left action on .
Example 1 If is a coalgebra, then is a left and a right comodule. Given a left -comodule , the vector space acquires the structure of a left -comodule and is, in fact, isomorphic to .
All this framework is completely dual to the ordinary case of algebra, and in fact, one can formulate the notion of an “algebra object” or a “module” internal to any monoidal category. Then, “coalgebra” just becomes “algebra” when one works in the opposite category to the category of -vector spaces.
This duality makes evident the following:
Proposition 3 The restriction functor has a right adjoint .
One of the reasons we care about the cotensor product is the following. Suppose we’re working with a Hopf algebra here, not just a coalgebra. Then the one-dimensional vector space becomes a comodule via the unit map . We have the identity
Both represent the “primitive” elements of . For the purposes of the ASS, we are interested in the derived functors of , and those will be equivalently (by this observation) provided by the derived functors of the cotensor product.
3. Quotienting by a normal subHopf algebra
We’ll be interested in a special case of this operation of “coextending” scalars. Let be a connected, graded Hopf algebra over the field . “Connected” means that in degree zero, is just .
Let be a subHopf algebra (graded). We say that is normal if , where consists of the augmentation ideal. If this case, we define:
Definition 4 We define the Hopf algebra as a vector space. The algebra and coalgebra structures are induced from that of .
In the case we are interested in, we will be taking the dual Steenrod algebra as , and will be the subalgebra defined earlier; normality is automatic since the Hopf algebras are commutative. Observe that is a quotient Hopf algebra of , and we have a (non-exact) sequence of Hopf algebra maps
The main result we will need is the following, which states that we can recover from :
Lemma 5 (Change of rings) We have .
Here we treat as a comodule via the unit map : that is, the comodule structure is somewhat uninteresting.
Proof: We need to compute the equalizer of the two maps
where the first map is (this comes from the -comodule structure) and the second map is where
is the comultiplication, reduced mod .
So, in other words, the claim is that if is an element such that , then . Let’s prove this, by induction on the degree. When the degree is one, then we have
and if this is equal to in , then we find that . Since has degree one, we must have itself.
Suppose is a homogeneous element and actually lies in . We can assume the are linearly independent in , and that one of the is and one is : in fact, we have
where the indicate terms where both degrees are positive.
Then by coassociativity,
This means, that if we consider the sum , then all the except for the one have the property that . Using induction on the dimension, we find that each of the must belong to itself, so . Since occurs as a term as well, we find that actually .
4. The change of rings theorem
The main computational tool for handling the ASS for is going to be a derived version of the cotensor adjunction together with with lemma of the previous section.
Proposition 6 (Change-of-rings theorem) Let be a graded, connected Hopf algebra, a connected subHopf algebra. Then we have, for an -comodule and a vector space ,
where is considered as an -comodule in the trivial way.
This is now a consequence of the cotensor-restriction adjunction: for , we have
by the analysis of the previous section. We can now derive this equality to get the desired statement.