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.
2. Coalgebra
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.
May 25, 2012 at 12:26 pm
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