# On $P$-algebras and their duals

Andrew Baker  School of Mathematics & Statistics, University of Glasgow,
Glasgow G12 8QQ, Scotland<EMAIL_ADDRESS>http://www.maths.gla.ac.uk/$\sim$ajb https://orcid.org/0000-0002-9369-7702

(Date: last updated 05/12/2022 – version 2 arXiv:2205.09541 )

###### Abstract.

The notion of $P$-algebra due to Margolis, building on work of Moore &
Peterson, was motivated by the case of the Steenrod algebra at a prime and its
modules. We develop aspects of this theory further, focusing especially on
coherent modules and finite dimensional modules. We also discuss the dual Hopf
algebra of $P$-algebra and its comodules. One of our aims is provide a
collection of techniques for calculating cohomology groups over $P$-algebras
and their duals, in particular giving vanishing results. Much of our work is
implicit in that of Margolis and others but we are unaware of systematic
discussions in the literature. We give some examples illustrating topological
applications which follow easily from our results.

###### Key words and phrases:

Steenrod algebra, Hopf algebras, Homotopy theory

###### 2020 Mathematics Subject Classification:

Primary 55S10; Secondary 55P42, 57T05

I would like to thank the following: The Max-Planck-Institut für Mathematik in
Bonn for supporting my visit in January 2020 and April-May 2022; Tobias
Barthel, Ken Brown, Bob Bruner, Doug Ravenel, Nige Ray, John Rognes and Chuck
Weibel for numerous helpful conversations on algebra and topology.

###### Contents

  1. 1 $P$-algebras and their modules
  2. 2 The dual of a $P$-algebra and its comodules
  3. 3 Some homological algebra
  4. 4 Finite comodule filtrations
  5. 5 Recollections on the Steenrod algebra and its dual
  6. 6 Some comodules and their cohomology
  7. 7 Some topological applications

## Introduction

This paper is a rewrite of the algebraic part of [AB:MSpBousfieldclass] and we
intend writing a companion paper on the topological applications. Shortly
after posting the earlier version, we discovered some significant gaps in the
calculations and this led to some substantial reworking of the algebra which
we feel is of enough interest to present it independently. Indeed, as far as
we are aware, although there has been significant use of properties of
$P$-algebras in work involving the Steenrod algebra, there does not appear to
be much discussion of the dual notion so it seems worthwhile recording some
basic results and in particular the existence of certain spectral sequences
for their cohomology.

The theory of $P$-algebras of Margolis [HRM:Book] which builds on work of
Moore & Peterson [JCM-FPP:NearlyFrobAlgs], provides an important theoretical
framework for understanding modules over the Steenrod algebra at a prime with
many applications in Algebraic Topology. In this paper we review some basic
results on $P$-algebras and their modules, particularly emphasising properties
of coherent modules and their relationship with finite dimensional modules; we
also mention some properties of pseudo-coherent modules. Next we consider the
dual Hopf algebra of a $P$-algebra and its comodules. In order to set up some
Cartan-Eilenberg spectral sequences we discuss the homological algebra
required to make a bivariant derived functor of the homomorphism for comodules
with the aim of obtaining three such spectral sequences, one of which involves
dualising to modules over the $P$-algebra itself because comodules over Hopf
algebras do not always have resolutions by projective objects (although recent
work of Salch related to [AS:ProjComods] sheds light on this for connective
graded comodules over a connective graded Hopf algebra). After reviewing some
properties of the mod $2$ Steenrod algebra and its dual, we discuss doubling
and then describe some families of subHopf algebras and their duals. Finally
we give some applications of our results on coherent modules to show vanishing
of homotopy groups of maps between some spectra, rederiving and extending
results of Lin, Margolis and Ravenel.

An important motivation for our work is to rework some the algebraic machinery
used in Ravenel’s seminal paper [DCR:Localn] and we will apply this in the
planned sequel. In [AB:LocFrobAlg] we have also developed an ungraded analogue
of the theory of $P$-algebras which sheds light on some aspects of it.

##  1\. $P$-algebras and their modules

We will make use material on $P$-algebras and their modules contained in
[HRM:Book]*chapter 13. Here is a summary of our assumptions, conventions and
notations, some of which differ slightly from those of Margolis.

  * •

We will often suppress explicit mention of internal grading in cohomology and
just write $M$ for $M^{*}$ when discussing a module over $A=A^{*}$. When
working in homology we will usually write $M_{*}$ or $A_{*}$.

  * •

We will work with graded vector spaces over a field $\Bbbk$ and in particular,
$\Bbbk$-algebras and their (co)modules. In our topological applications,
$\Bbbk=\mathbb{F}_{2}$ although similar results for odd primes are easily
found.

For a finite type graded vector space $V_{*}$ we think of $V_{n}$ as dual to
$V^{n}=\operatorname{Hom}_{\Bbbk}(V_{n},\Bbbk)$, so $V^{*}$ is the
cohomologically graded degree-wise dual, and bounded below means that $V^{*}$
is bounded below. We can also start with a finite type graded vector space
$V^{*}$ and form $V_{*}$ where
$V_{n}=\operatorname{Hom}_{\Bbbk}(V^{n},\Bbbk)$; the double dual of $V_{*}$ is
canonically isomorphic to $V_{*}$, and vice versa. We will denote the positive
degree part of a graded vector space by $V^{+}$; of course for a graded
$\Bbbk$-algebra $R$, $R^{+}$ is an ideal which is maximal if $R$ is also
connected, making $R$ local.

A graded vector space which is finite dimensional will be referred to as a
_finite_ ; when $\Bbbk$ is finite this terminology agrees with the use of
finite for a module over a $P$-algebra by Margolis.

  * •

In this paper, a _$P$ -algebra_ $A$ will always be a _$P$ -Hopf algebra_,
i.e., a strictly increasing union of connected finite dimensional
cocommutative Hopf algebras $A(n)\subset A(n+1)\subset A$. Thus each $A(n)$ is
a Poincaré duality algebra and we will denote its highest non-trivial degree
by $\operatorname{pd}(n)$ as it is also the Poincaré duality degree; this
number satisfies the inequality $\operatorname{pd}(n)<\operatorname{pd}(n+1)$.
We also stress that each $A(n)$ is a local graded ring and
$A(n)^{\operatorname{pd}(n)}=A(n)^{0}=\Bbbk$. Other important properties are
that $A$ is free as a left or right $A(n)$-module and $A$ is a coherent
$\Bbbk$-algebra. Although in general Margolis does not require $P$-algebras to
be of finite type, all the examples we consider have that property and it is
required in some of our homological results so we will assume it holds.

It is important to note that $P$-algebras are _coherent_ (but not Noetherian)
and their coherent modules play an important rôle in this paper. For general
properties of coherence and pseudo-coherence we refer the reader to Bourbaki
[Bourbaki:HomAlg]*ex. §3.10, and also Cohen [JMC:Coherent] for an account
aimed at an algebraic topology audience.

In Algebraic Topology the important examples of $P$-algebras are the Steenrod
algebra $\mathcal{A}$ at a prime $p$ as well as infinite dimensional sub and
quotient Hopf algebras.

We will use the following basic result stated on page 195 of [HRM:Book] but
left as an exercise.

######  Proposition 1.1.

Suppose that $A$ is a P-algebra. Let $M$ be a finite $A$-module and $F$ a
bounded below free $A$-module. Then

$\operatorname{Ext}_{A}^{*}(M,F)=0.$

###### Proof.

By [HRM:Book]*theorem 13.12, bounded below projective $A$-modules are also
injective, so $\operatorname{Ext}_{A}^{s}(M,F)=0$ when $s>0$, therefore we
only have to show that $\operatorname{Hom}_{A}(M,F)=0$. It suffices to prove
this for the case $F=A$.

Suppose that $0\neq\theta\in\operatorname{Hom}_{A}(M,A)$. The image of
$\theta$ contains a simple submodule in its top degree, so let $\theta(x)\neq
0$ be in this submodule; then $a\theta(x)=0$ for every positive degree element
$a\in A$. Now for some $n$, $\theta(x)\in A(n)^{k}$ where
$k<\operatorname{pd}(n)$, and by Poincaré duality for $A(n)$ there exists
$z\in A(n)$ for which $0\neq z\theta(x)\in A(n)^{\operatorname{pd}(n)}$. This
gives a contradiction, hence no such $\theta$ can exist.

For a different proof, see Lorentz [ML:TourRepThy]*proposition 10.6, which
shows that an infinite dimensional Hopf algebra has no non-trivial finite
dimensional left or right ideals. ∎

A particular concern for us will be the situation where we have a pair of
$P$-algebras $B\subseteq A$ with $B$ a subHopf algebra of $A$.

######  Proposition 1.2.

For a pair of P-algebras $B\subseteq A$,

$\operatorname{Ext}^{*}_{A}(A\otimes_{B}\Bbbk,A)\cong\operatorname{Ext}^{*}_{B}(\Bbbk,A)=\operatorname{Hom}_{B}(\Bbbk,A)=0.$

###### Proof.

First we recall a classic result of Milnor & Moore [M&M:HopfAlg]*proposition
4.9: $A$ is free as a left or right $B$-module. This guarantees the change of
rings isomorphism (which is valid for any subalgebra where $A$ is a flat
$B$-module) and the second isomorphism follows from Proposition 1.1. ∎

Since a $P$-algebra $A$ is coherent, its finitely presented modules are its
coherent modules, and they form a full abelian subcategory
$\mathbf{Mod}^{\mathrm{coh}}_{A}$ of $\mathbf{Mod}_{A}$ with finite limits and
colimits (in particular it has kernels, cokernels and images). A coherent
$A$-module $M$ admits a finite presentation

$\textstyle{A^{\oplus
k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{A^{\oplus\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

which can be defined over some $A(n)$, i.e., there is a finite presentation

$\textstyle{A(n)^{\oplus
k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\textstyle{A(n)^{\oplus\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

of $A(n)$-modules and a commutative diagram of $A$-modules

$\textstyle{A\otimes_{A(n)}A(n)^{\oplus
k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}\otimes\pi^{\prime}}$$\scriptstyle{\cong}$$\textstyle{A\otimes_{A(n)}A(n)^{\oplus\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A\otimes_{A(n)}M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{A^{\oplus
k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{A^{\oplus\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

with exact rows. It is standard that every coherent $A$-module admits a
resolution by finitely generated free modules. It is also true that a
homomorphism between $M\to N$ coherent modules is induced from a homomorphism
between finitely generated modules over some $A(m)$.

For a $P$-algebra we also have injective resolutions by finitely generated
free modules.

######  Proposition 1.3.

Let $M$ be a coherent module over a $P$-algebra $A$. Then $M$ admits an
injective resolution by finitely generated free modules.

###### Proof.

By [HRM:Book]*theorem 13.12, bounded below projective $A$-modules are
injective.

For some $n$, $M\cong A\otimes_{A(n)}M^{\prime}$ where $M^{\prime}$ is a
finitely generated $A(n)$-module. Since $A(n)$ is a Poincaré duality algebra,
it is standard that $M^{\prime}$ admits a monomorphism $M^{\prime}\to
J^{\prime}$ into a finitely generated free $A(n)$-module which is also
injective (this is obvious for a simple module and can be proved by induction
on dimension). By flatness the composition

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A\otimes_{A(n)}M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{A(n)}J^{\prime}}$

is a monomorphism of $A$-modules into a finitely generated free module with
coherent cokernel. Iterating this we can build a resolution of the stated form
and since $A$ is self-injective this is an injective resolution. ∎

Our next result summarises the properties of coherent modules.

######  Proposition 1.4.

The coherent modules over a $P$-algebra $A$ form a full subcategory
$\mathbf{Mod}^{\mathrm{coh}}_{A}$ of $\mathbf{Mod}_{A}$ with enough
projectives and injectives, and all finite limits and colimits.

We can generalise Proposition 1.1.

######  Proposition 1.5.

Let $M$ be a finite $A$-module and $N$ a coherent $A$-module. Then

$\operatorname{Ext}_{A}^{*}(M,N)=0.$

###### Proof.

The left exact functor $\operatorname{Hom}_{A}(M,-)$ has the right derived
functors $\operatorname{Ext}_{A}^{*}(M,-)$. By Proposition 1.1, for each
finitely generated free module $F$, $\operatorname{Ext}_{A}^{*}(M,F)=0$. This
means that $F$ is $\operatorname{Hom}_{A}(M,-)$-acyclic, and it is well-known
that these derived functors can be computed using resolutions by such modules;
see [CAW:HomAlg] on $F$-acyclic objects and dimension shifting.

Now every coherent $A$-module $N$ admits a resolution $0\to N\to J^{*}$ where
each $J^{s}$ is a finitely generated free module and these are
$\operatorname{Hom}_{A}(M,-)$-acyclic. Therefore
$\operatorname{Ext}_{A}^{*}(M,N)=0$. ∎

For example, every left $A$-module of the form

$A/\\!/A(n)=A\otimes_{A(n)}\Bbbk\cong A/AA(n)^{+}$

is coherent so it admits such an injective resolution and then
$\operatorname{Ext}_{A}^{s}(\Bbbk,A/\\!/A(n))=0$.

Here is a more general statement.

######  Proposition 1.6.

Let $A$ be a $P$-algebra and suppose that $B\subseteq A$ is a subalgebra which
is also a $P$-algebra and that $A$ is a free left $B$-module. Let $L$ be a
finite $B$-module and $N$ a coherent $A$-module. Then

$\operatorname{Ext}^{*}_{A}(A\otimes_{B}L,N)=0.$

###### Proof.

We have

$\operatorname{Ext}^{*}_{A}(A\otimes_{B}L,N)\cong\operatorname{Ext}^{*}_{B}(L,N).$

By Proposition 1.3, $N$ admits an injection resolution

$0\to N\to J^{*}$

by finitely generated free $A$-modules. For each $s\geqslant 0$,

$\operatorname{Hom}_{A}(A\otimes_{B}L,J^{s})\cong\operatorname{Hom}_{B}(L,J^{s})=0$

since a $B$-module homomorphism $L\to J^{s}$ must factor through finitely many
$B$-summands and therefore it is trivial. Therefore the cohomology
$\operatorname{Ext}^{*}_{B}(L,N)$ is trivial. A similar argument also applies
to the case where $N$ is a coproduct of coherent $A$-modules. ∎

### Pseudo-coherent modules over a coherent ring

In Bourbaki [Bourbaki:HomAlg]*ex. §3.10, as well as coherent modules, pseudo-
coherent modules are considered, where a module is _pseudo-coherent_ if every
finitely generated submodule is finitely presented (so over a coherent ring it
is coherent). The reader is warned that pseudo-coherent is sometimes used in a
different sense; for example, over a coherent ring the definition of Weibel
[CAW:Ktheory]*example II.7.1.4 corresponds to our coherent.

Examples of pseudo-coherent modules over a coherent ring $A$ include
coproducts of coherent modules such as the finitely related modules defined by
Lam [TYL:LectModules&Rings]*chapter 2§4. However pseudo-coherent modules do
not form a full abelian subcategory of $\mathbf{Mod}_{A}$ since cokernels of
homomorphisms between pseudo-coherent modules need not be pseudo-coherent.
Nevertheless, pseudo-coherent modules do occur quite commonly when working
with coherent modules over $P$-algebras, for example as tensor products over a
Hopf algebra.

######  Lemma 1.7.

Let $A$ be a ring and $M$ a pseudo-coherent $A$-module. Then $M$ is the union
of its coherent submodules and therefore their colimit.

###### Proof.

Every element $m\in M$ generates a cyclic submodule which is coherent. ∎

######  Lemma 1.8.

Let $A$ be a coherent ring and $B$ an Artinian subring where $A$ is flat as a
right $B$-module. Then every extended $A$-module $A\otimes_{B}N$ is pseudo-
coherent.

###### Proof.

Let $U\subseteq A\otimes_{B}N$ be a finitely generated submodule. Taking a
finite generating set and expressing each element as a sum of basic tensors we
find that $U\subseteq A\otimes_{B}U^{\prime}$ where $U^{\prime}\subseteq N$ is
a finitely generated submodule. As $B$ is Artinian and so Noetherian,
$U^{\prime}$ is finitely presented, so by flatness of $A$,
$A\otimes_{B}U^{\prime}$ is a finitely presented $A$-module. Since $A$ is
coherent, $U$ is also finitely presented. ∎

An important special case of this occurs when $A$ is a coherent algebra over a
field $\Bbbk$ and $B$ is a finite dimensional subalgebra. If $N$ is a
$B$-module then it is locally finite, i.e., every element is contained in a
finite dimensional submodule.

In general, the tensor product of two coherent modules over a Hopf algebra
that is a $P$-algebra is not a coherent module. However, as we will soon see,
it turns out that it is pseudo-coherent.

######  Assumption 1.9.

We will assume from now on that $A$ is a cocommutative Hopf algebra over a
field $\Bbbk$, and $B\subseteq A$ a subHopf algebra where $A$ is $B$-flat as a
left and right $B$-module.

Now given two left $A$-modules $L$ and $M$, their tensor product $L\otimes M$
is a left $A$-module with the diagonal action given by

$a\cdot(\ell\otimes m)=\sum_{i}a^{\prime}_{i}\ell\otimes
a^{\prime\prime}_{i}m,$

where the coproduct on $a$ is

$\psi a=\sum_{i}a^{\prime}_{i}\otimes a^{\prime\prime}_{i}.$

In particular, given a left $A$-module $M$ and a left $B$-module $N$, the
tensor product of $M$ and $A\otimes_{B}N$ is a left $A$-module
$M\otimes(A\otimes_{B}N)$. There is an isomorphism of $A$-modules

(1.1) $M\otimes(A\otimes_{B}N)\xrightarrow[\cong]{\Theta}A\otimes_{B}(M\otimes
N);\quad m\otimes(a\otimes
n)\mapsto\sum_{i}a^{\prime}_{i}\otimes(\overline{a^{\prime\prime}_{i}}m\otimes
n)$

where $\overline{x}=\chi(x)$ and $M\otimes N$ is a left $B$-module with the
diagonal action.

A particular instance of this is

(1.2) $(A/\\!/B)\otimes(A/\\!/B)\cong A\otimes_{B}(A/\\!/B)$

where

$A/\\!/B=A/AB^{+}\cong A\otimes_{B}\Bbbk.$

We will be especially interested in the case where the $B$-module $A/\\!/B$ is
a coproduct of finitely generated modules.

######  Lemma 1.10.

Suppose that $A$ is coherent and $B$ is finite dimensional. Let $M$ be a left
$A$-module and $N$ a left $B$-module. Then the $A$-module
$M\otimes(A\otimes_{B}N)$ is pseudo-coherent.

###### Proof.

Using (1.1),

$M\otimes(A\otimes_{B}N)\cong A\otimes_{B}(M\otimes N)$

which is pseudo-coherent by Lemma 1.8. ∎

######  Corollary 1.11.

Suppose that $A$ is a $P$-algebra and that $M_{1}$ and $M_{2}$ are two
coherent $A$-modules. Then $M_{1}\otimes M_{2}$ is pseudo-coherent.

###### Proof.

Every coherent $A$-module is induced from a finitely generated $A(n)$-module
for some $n$. By choosing a large enough $n$ we can assume that $M_{1}\cong
A\otimes_{A(n)}M_{1}^{\prime}$ for some finitely generated $A(n)$-module
$M_{1}^{\prime}$. Then

$M_{1}\otimes M_{2}\cong A\otimes(M_{1}^{\prime}\otimes M_{2}),$

so it is a pseudo-coherent $A$-module. ∎

Pseudo-coherence itself is probably of limited practical use, but stronger
versions are perhaps more likely to be of use for computational purposes.

######  Definition 1.12.

  * •

A module over a coherent ring is _strongly pseudo-coherent_ if it is a
coproduct of coherent modules.

  * •

A module $M$ over a $P$-algebra $A$ is _$n$ -strongly pseudo-coherent_ if

$M\cong A\otimes_{A(n)}M^{\prime}$

where the $A(n)$-module $M^{\prime}$ is a coproduct of finitely generated
$A(n)$-modules.

Over a coherent ring every free module is strongly pseudo-coherent as is every
finitely related module since it is the sum of a coherent module and a free
module.

Of course an $n$-strongly pseudo-coherent module over a $P$-algebra $A$ is
strongly pseudo-coherent. For a given $n$, the cyclic $A$-module
$A/\\!/A(n)=A/AA(n)^{+}\cong A\otimes_{A(n)}\Bbbk$ can be viewed as an
$A(n)$-module, and by (1.2),

$A/\\!/A(n)\otimes A/\\!/A(n)\cong A\otimes_{A(n)}A/\\!/A(n).$

If $A/\\!/A(n)$ is a coproduct of finitely generated (i.e., finite
dimensional) $A(n)$-modules then $A/\\!/A(n)\otimes A/\\!/A(n)$ is
$n$-strongly pseudo-coherent. This occurs when $A$ is the Steenrod algebra for
a prime $p$.

######  Example 1.13.

When $A=\mathcal{A}$ is the mod $2$ Steenrod algebra it is known that there is
a coproduct decomposition of the $\mathcal{A}(n)$-module
$\mathcal{A}/\\!/\mathcal{A}(n)$ into finite dimensional modules which are
generalised Brown-Gitler modules. For details see Behrens et al [MB-MAH-MJH-
MM:v2to32]*corollary 5.6.

##  2\. The dual of a $P$-algebra and its comodules

The theory of $P$-algebras can be dualised: given a finite type $P$-algebra
$A$, we define its (graded) dual $A_{*}$ by setting
$A_{n}=\operatorname{Hom}_{\Bbbk}(A^{n},\Bbbk)$ and making this a commutative
Hopf algebra by dualising the structure maps of $A$ (doing this the finite
type condition is essential). We will refer to the dual Hopf algebra of a
(finite type) $P$-algebra as a _$P_{*}$ -algebra_; although this is not
standard terminology it seems appropriate and convenient.

When working with comodules over a $P_{*}$-algebra $A_{*}$ we will use
homological grading. For left $A_{*}$-comodules which are bounded below and of
finite type there is no significant difference between working with them or
their (degree-wise) duals as $A$-modules. In particular,

$\operatorname{Cohom}_{A_{*}}(M_{*},N_{*})\cong\operatorname{Hom}_{A}(N^{*},M^{*}),$

where $M^{n}=\operatorname{Hom}_{\mathbb{F}_{2}}(M_{n},\mathbb{F}_{2})$ and
$M^{*}$ is made into a left $A$-module using the antipode. More generally,

$\operatorname{Coext}^{s,*}_{A_{*}}(M_{*},N_{*})\cong\operatorname{Ext}^{s,*}_{A}(N^{*},M^{*}),$

where $\operatorname{Coext}^{s,*}_{A_{*}}(M_{*},-)$ denotes the right derived
functor of $\operatorname{Cohom}_{A_{*}}(M_{*},-)$, which can be computed
using extended comodules which are injective comodules here since we are
working over a field.

Here are the dual versions of Propositions 1.1 and 1.2. Recall that a _cofree_
or _extended_ $A_{*}$-comodule is one of the form $A_{*}\otimes W_{*}$ where
$W_{*}$ is a graded vector space.

######  Proposition 2.1.

Suppose that $A_{*}$ is a $P_{*}$-algebra. Let $L_{*}$ a bounded below cofree
$A_{*}$-comodule and let $M_{*}$ be a finite $A_{*}$-comodule. Then

$\operatorname{Coext}_{A_{*}}^{*}(L_{*},M_{*})=0.$

######  Proposition 2.2.

For a surjective morphism of $P_{*}$-algebras $A_{*}\to B_{*}$,

$\operatorname{Coext}^{*,*}_{A_{*}}(A_{*},A_{*}\square_{B_{*}}\Bbbk)\cong\operatorname{Coext}^{*,*}_{B_{*}}(A_{*},\Bbbk)=\operatorname{Cohom}_{B_{*}}(A_{*},\Bbbk)=0.$

######  Definition 2.3.

If $A_{*}$ is a $P_{*}$-algebra then an $A_{*}$-comodule $M_{*}$ is _coherent_
if its dual $M^{*}$ is a coherent $A$-module; this is equivalent to the
existence of an exact sequence of $A_{*}$-comodules

$0\to M_{*}\to A_{*}\otimes U_{*}\to A_{*}\otimes V_{*}$

where $U_{*},V_{*}$ are finite $\Bbbk$-vector spaces.

Here is a dual version of Proposition 1.3.

######  Proposition 2.4.

Let $M_{*}$ be a coherent comodule over a $P_{*}$-algebra $A_{*}$. Then
$M_{*}$ admits a projective resolution by finitely generated cofree comodules.

###### Proof.

Take an injective resolution of $M^{*}$ as in Proposition 1.3 and then take
duals to obtain a projective resolution. ∎

Notice that since $A_{*}$ is an injective comodule we have

$\operatorname{Coext}_{A_{*}}^{*}(A_{*},A_{*})\cong\operatorname{Hom}_{\Bbbk}(A_{*},\Bbbk)\cong
A.$

######  Proposition 2.5.

Let $M_{*}$ be a coherent comodule over a $P_{*}$-algebra $A_{*}$ and let
$N_{*}$ be a finite $P_{*}$-comodule. Then

$\operatorname{Coext}_{A_{*}}^{*}(M_{*},N_{*})=0.$

###### Proof.

Let $P_{\bullet,*}\to M_{*}\to 0$ be a resolution of $M_{*}$ by cofree
comodules. Then by Proposition 2.1, for each $s\geqslant 0$ we have

$\operatorname{Cohom}_{A_{*}}(P_{s,*},N_{*})=0,$

and the result follows. ∎

######  Remark 2.6.

For bounded below finite type comodules over a $P_{*}$-algebra $A_{*}$ dual to
a $P$-algebra $A$, taking degree-wise duals defines an equivalence of
categories

$\textstyle{\mathbf{Comod}_{A_{*}}^{\flat,\,\mathrm{f.t.}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)^{*}}$$\textstyle{(\mathbf{Mod}_{A}^{\flat,\,\mathrm{f.t.}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces)^{\mathrm{op}}}$$\scriptstyle{(-)_{*}}$

between the $A_{*}$-comodule and the $A$-module categories. Moreover, these
functors are exact, so this equivalence identifies injective comodules (which
are retracts of extended comodules) with bounded below projective modules. By
[HRM:Book]*theorem 13.12, bounded below projective $A$-modules are injective
so it also identifies bounded below injective comodules as projective objects
(this is not true in general of course). In fact this equivalence fits into a
bigger diagram

(2.1)
$\textstyle{\mathbf{Mod}_{A}^{\sharp,\,\mathrm{f.t.}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)^{*}}$$\textstyle{\mathbf{Comod}_{A_{*}}^{\flat,\,\mathrm{f.t.}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)^{*}}$$\textstyle{(\mathbf{Mod}_{A}^{\flat,\,\mathrm{f.t.}})^{\mathrm{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)_{*}}$$\scriptstyle{(-)_{*}}$$\textstyle{\mathbf{Mod}_{A}^{\mathrm{op}}}$

where $\mathbf{Mod}_{A}^{\natural,\,\mathrm{f.t.}}$ denotes the category of
finite type bounded below homologically graded $A$-modules (with $A$ acting by
decreasing degree), $\mathbf{Mod}_{A}^{\flat,\,\mathrm{f.t.}}$ denotes the
category of finite type bounded below cohomologically graded $A$-modules and
$\mathbf{Mod}_{A}$ denoting the category of all $A$-modules. All of the
functors here are exact.

######  Remark 2.7.

Each object $M_{*}$ of $\mathbf{Mod}_{A}^{\natural,\,\mathrm{f.t.}}$ is a
locally finite $A$-module, i.e., it is a union of finite modules. It is
convenient to regrade $M_{*}$ so that $M_{n}$ is given cohomological degree
$-n$ and then multiplication by a positive degree element of $A$ increases
this cohomological degree. Then by Proposition 1.5 we have for any coherent
$A$-module $N$, $\operatorname{Ext}_{A}^{*}(M_{*},N)=0$.

For a fixed $A_{*}$-comodule $M_{*}$, the functor

$\operatorname{Cohom}_{A_{*}}(M_{*},-)=\mathbf{Comod}_{A_{*}}^{\flat,\,\mathrm{f.t.}}(M_{*},-)\to\mathbf{Mod}_{\Bbbk}^{\flat,\,\mathrm{f.t.}}$

is left exact and has right derived functors
$\operatorname{Coext}^{*}_{A_{*}}(M_{*},-)$. Since

$\operatorname{Cohom}_{A_{*}}(M_{*},-)\cong\operatorname{Hom}_{A}((-)^{*},M^{*})=\mathbf{Mod}_{A}^{\flat,\,\mathrm{f.t.}}((-)^{*},M^{*})=\mathbf{Mod}_{A}((-)^{*},M^{*})$

and injective comodules are sent to projective modules, we also have

(2.2)
$\operatorname{Coext}^{*}_{A_{*}}(M_{*},-)\cong\operatorname{Ext}^{*}_{A}((-)^{*},M^{*}).$

The contravariant functor
$\mathbf{Comod}_{A_{*}}^{\flat,\,\mathrm{f.t.}}\to\mathbf{Mod}_{A}^{\mathrm{op}}$
allows us to define cohomological invariants of comodules using injective
resolutions in $\mathbf{Mod}_{A}$ as a substitute for projective resolutions
in $\mathbf{Comod}_{A_{*}}^{\flat,\,\mathrm{f.t.}}$. In effect for a comodule
$N_{*}$ we define

$\operatorname{Coext}^{*}_{A_{*}}(-,N_{*})=\operatorname{Ext}_{A}(N^{*},(-)^{*}).$

Of course this is calculated using injective resolutions of $A$-modules; since
$\operatorname{Ext}_{A}(-,-)$ is a balanced functor, (2.2) implies that
$\operatorname{Coext}^{*}_{A_{*}}(-,-)$ is too, whenever we can use projective
comodule resolutions in the first variable. For example, if we restrict to the
subcategory of coherent comodules we obtain balanced bifunctors

$\operatorname{Coext}^{s}_{A_{*}}(-,-)\colon(\mathbf{Comod}_{A_{*}}^{\mathrm{coh}}(-,-))^{{\mathrm{op}}}\otimes\mathbf{Comod}_{A_{*}}^{\mathrm{coh}}(-,-)\to\mathbf{Mod}_{\Bbbk}^{\flat,\,\mathrm{f.t.}}.$

Given a surjection of $P_{*}$-algebras $A_{*}\to B_{*}$ there are adjunction
isomorphisms of the form

(2.3) $\displaystyle\operatorname{Cohom}_{A_{*}}(-,-)$
$\displaystyle\cong\operatorname{Cohom}_{B_{*}\backslash\\!\backslash
A_{*}}((B_{*}\backslash\\!\backslash A_{*})\square_{A_{*}}(-),-),$ (2.4)
$\displaystyle\operatorname{Cohom}_{B_{*}}(-,-)$
$\displaystyle\cong\operatorname{Cohom}_{A_{*}}(-,A_{*}\square_{B_{*}}(-)),$

where $B_{*}\backslash\\!\backslash A_{*}=\Bbbk\square_{B_{*}}A$. Later we
will use these adjunctions to construct composite functor spectral sequences.

######  Remark 2.8.

Since writing this we became aware of the revised version of Salch
[AS:ProjComods], where it is shown that the category of graded connected
comodules over a graded connected Hopf algebra over a field has enough
projectives. As our results are stronger but more restricted, we feel it
worthwhile presenting them despite the greater generality of Salch’s result.

##  3\. Some homological algebra

In this section we describe some Cartan-Eilenberg spectral sequences for
comodules over a commutative Hopf algebra over a field. Some of these are
similar to other examples in the literature such as that for computing
$\operatorname{Cotor}$ for Hopf algebroids in [DCR:GreenBook].

To ease notation, in this section we suppress internal gradings and assume
that all our objects are connective and of finite type over a field $\Bbbk$.
We refer to the classic [M&M:HopfAlg] as well as the more recent
[JPM&KP:MoreConcise] for notation and basic ideas about graded Hopf algebras.

Before discussing cohomology for comodules, we will recall the dual theory to
modules over algebras, where there are classical Cartan-Eilenberg spectral
sequences of [HC&SE:HomAlg] for a normal sequence of Hopf algebras over a
field $\Bbbk$,

(3.1) $R\to S\to S/\\!/R$

where $S$ is a free $R$-module. Then for a left $S/\\!/R$-module $L$ and a
left $S$-module $M$ there is a spectral sequence of the form

(3.2)
$\mathrm{E}_{2}^{s,t}=\operatorname{Ext}^{s}_{S/\\!/R}(L,\operatorname{Ext}^{t}_{R}(\Bbbk,M))\Longrightarrow\operatorname{Ext}^{s+t}_{S}(L,M).$

There is another similar spectral sequence for a left $S$-module $M$ and a
left $S/\\!/R$-module $N$ which has the form

(3.3)
$\mathrm{E}_{2}^{s,t}=\operatorname{Ext}^{s}_{S/\\!/R}(\operatorname{Tor}^{t}_{R}(\Bbbk,M),N)\Longrightarrow\operatorname{Ext}^{s+t}_{S}(M,N).$

Since $\operatorname{Ext}$ and $\operatorname{Tor}$ are balanced functors, one
approach to setting these spectral sequences is by resolving both variables
and using double complex spectral sequences. However, they can instead be
viewed as composite functor spectral sequences obtained using injective or
projective resolutions of the $S$-module $M$. When the algebras and modules
are graded $\Bbbk$-vector spaces, these spectral sequences are tri-graded;
also, in topological applications, (3.1) is often a sequence of cocommutative
Hopf algebras.

Now suppose we have a sequence of homomorphisms of connected commutative
graded Hopf algebras over $\Bbbk$,

$K\backslash\\!\backslash H\rightarrowtail H\twoheadrightarrow K,$

where in the notation of [M&M:HopfAlg]*definition 3.5,

$K\backslash\\!\backslash H=\Bbbk\square_{K}H=H\square_{K}\Bbbk\subseteq H.$

We also assume given a left $K\backslash\\!\backslash H$-comodule $M$ and a
left $H$-comodule $N$. Of course $M$ and $N$ inherit structures of
$H$-comodule and $K$-comodule respectively, where $M$ is trivial as a
$K$-comodule. Our aim is to calculate $\operatorname{Coext}^{*}_{H}(M,N)$, the
right derived functor of

$\operatorname{Cohom}_{H}(M,-)\colon\mathbf{Comod}_{K\backslash\\!\backslash
H}\to\mathbf{Mod}_{\Bbbk};\quad N\mapsto\operatorname{Cohom}_{H}(M,N).$

Following Hovey [MH:HtpyThyComods], we will write $U\overset{H}{\wedge}V$ to
indicate the tensor product of two $H$-comodules $U\otimes
V=U\otimes_{\Bbbk}V$ with the diagonal coaction given by the composition

$\textstyle{U\otimes
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu\otimes\mu}$$\textstyle{(H\otimes
U)\otimes(H\otimes
V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{(H\otimes
H)\otimes(U\otimes
V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\varphi}\otimes\operatorname{Id}}$$\textstyle{H\otimes(U\otimes
V).}$

For a vector space $W$, the notation $U\otimes W$ will be used to denote
$H$-comodule with coaction

$\textstyle{U\otimes
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu\otimes\operatorname{Id}}$$\textstyle{(H\otimes
U)\otimes
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{H\otimes(U\otimes
W)}$

carried on the first factor alone.

If $L$ is a left $H$-comodule, then there is a well-known isomorphism of left
$H$-comodules

(3.4) $K\backslash\\!\backslash
H\overset{H}{\wedge}L=(H\square_{K}\Bbbk)\overset{H}{\wedge}L\cong
H\square_{K}L,$

We can also regard $K\backslash\\!\backslash H=\Bbbk\square_{K}H$ as a right
$H$-comodule to form the left $K\backslash\\!\backslash H$-comodule

(3.5) $K\backslash\\!\backslash
H\square_{H}L=(\Bbbk\square_{K}H)\square_{H}L\cong\Bbbk\square_{K}L;$

in particular, if $L$ is a trivial $K$-comodule then as left
$K\backslash\\!\backslash H$-comodules,

(3.6) $K\backslash\\!\backslash H\square_{H}L\cong L.$

We will use two more functors

$\mathbf{Comod}_{H}\to\mathbf{Comod}_{K\backslash\\!\backslash H};\quad
N\mapsto K\backslash\\!\backslash
H\square_{H}N=(\Bbbk\square_{K}H)\square_{H}N\cong\Bbbk\square_{K}N$

and

$\mathbf{Comod}_{K\backslash\\!\backslash H}\to\mathbf{Mod}_{\Bbbk};\quad
N\mapsto\operatorname{Cohom}_{K\backslash\\!\backslash H}(M,N).$

Notice that there is a natural isomorphism

$\operatorname{Cohom}_{K\backslash\\!\backslash H}(M,K\backslash\\!\backslash
H\square_{H}(-))\cong\operatorname{Cohom}_{H}(M,-)$

and for an injective $H$-comodule $J$, $K\backslash\\!\backslash
H\square_{H}J$ is an injective $K\backslash\\!\backslash H$-comodule. This
means we are in a situation where we have a Grothendieck composite functor
spectral sequence which in this case is a form of Cartan-Eilenberg spectral
sequence; for details see [CAW:HomAlg]*section 5.8 for example.

######  Proposition 3.1.

Let $M$ be a left $K\backslash\\!\backslash H$-comodule and $N$ a left
$H$-comodule. Then there is a first quadrant cohomologically indexed spectral
sequence with

$\mathrm{E}_{2}^{s,t}=\operatorname{Coext}_{K\backslash\\!\backslash
H}^{s}(M,\operatorname{Cotor}_{K}^{t}(\Bbbk,N))\Longrightarrow\operatorname{Coext}_{H}^{s+t}(M,N).$

If $N$ is a trivial $K$-comodule then

$\mathrm{E}_{2}^{s,t}\cong\operatorname{Coext}_{K\backslash\\!\backslash
H}^{s}(M,\operatorname{Cotor}_{K}^{t}(\Bbbk,\Bbbk)\overset{K\backslash\\!\backslash
H}{\wedge}N).$

There is another spectral sequence that we will use whose construction
requires that one of the Hopf algebras involved is a $P_{*}$-algebra. The
reason for this is discussed in Remark 2.6: in the category of finite type
connected comodules, extended comodules are projective objects.

######  Proposition 3.2.

Assume that $H$ and $K\backslash\\!\backslash H$ are $P_{*}$-algebras. Let $M$
be a left $H$-comodule which admits a projective resolution and let $N$ be a
bounded below left $K\backslash\\!\backslash H$-comodule. Then there is a
first quadrant cohomologically indexed spectral sequence with

$\mathrm{E}_{2}^{s,t}=\operatorname{Coext}_{K\backslash\\!\backslash
H}^{s}(\operatorname{Cotor}_{K}^{t}(\Bbbk,M),N)\Longrightarrow\operatorname{Coext}_{H}^{s+t}(M,N).$

If $M$ is a trivial $K$-comodule then

$\mathrm{E}_{2}^{s,t}\cong\operatorname{Coext}_{K\backslash\\!\backslash
H}^{s}(\operatorname{Cotor}_{K}^{t}(\Bbbk,\Bbbk)\overset{K\backslash\\!\backslash
H}{\wedge}M,N).$

###### Proof.

The construction is similar to the other one, and involves expressing
$\operatorname{Cohom}_{H}(-,N)$ as a composition

$\operatorname{Cohom}_{K\backslash\\!\backslash
H}(-,N)\circ(K\backslash\\!\backslash
H\square_{H}(-))=\operatorname{Cohom}_{K\backslash\\!\backslash
H}(K\backslash\\!\backslash
H\square_{H}(-),N)\cong\operatorname{Cohom}_{H}(-,N).$

The functor $K\backslash\\!\backslash
H\square_{H}(-)\colon\mathbf{Comod}_{H}\to\mathbf{Comod}_{K\backslash\\!\backslash
H}$ sends projective objects to projective objects (see Remark 2.6), so the
standard construction can be used to give a spectral sequence. ∎

Of course the condition that $M$ admits a projective resolution is crucial; in
the case of $P_{*}$-algebras this is satisfied if $M$ is a coherent comodule.

##  4\. Finite comodule filtrations

In this section we will give some results based on a particular kind of
comodule filtration.

We will make use of the adjoint coaction dual to the adjoint action which is
given a thorough treatment in Singer [WMS:SteenrodSqsSS]*chapter 4; the
dualisation to the comodule setting is straightforward and details are left to
the reader. For our purposes we need to know the following: If $B\subseteq A$
is a conormal subHopf algebra of a commutative Hopf algebra over a field
$\Bbbk$, then there is a (left) _adjoint coaction_ $A\to A\otimes A$ which is
the composition of the solid arrows in the following commutative diagram,
where $\mathrm{T}$ is the switch map.

(4.1)
$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi\otimes\operatorname{Id}=\operatorname{Id}\otimes\psi}$$\textstyle{B\otimes
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes
A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes
A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}\otimes\mathrm{T}}$$\scriptstyle{\cong}$$\textstyle{A\otimes
A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}\otimes\chi\otimes\operatorname{Id}}$$\scriptstyle{\cong}$$\textstyle{A\otimes
A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\varphi}\otimes\operatorname{Id}}$$\textstyle{A\otimes
A}$

By [WMS:SteenrodSqsSS]*proposition 4.24, the adjoint coaction makes $A$ and
any conormal subHopf algebra $B$ into an $A$-comodule Hopf algebra;
furthermore, for any left $A$-comodule $M$ (which also becomes a left
$A/\\!/B$-comodule through the projection $A\to A/\\!/B$), there is an induced
left $A$-coaction on $\operatorname{Cotor}_{A/\\!/B}(\Bbbk,M)$ that factors
through a left $B$-coaction.

(4.2)
$\textstyle{\operatorname{Cotor}_{A/\\!/B}(\Bbbk,M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes\operatorname{Cotor}_{A/\\!/B}(\Bbbk,M)}$$\textstyle{B\otimes\operatorname{Cotor}_{A/\\!/B}(\Bbbk,M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Now let $C$ be a cocommutative coalgebra over a field $\Bbbk$. A $C$-comodule
$M$ is _unipotent_ if it has a finite length descending filtration by
subcomodules

$M=M^{\ell}\supset M^{\ell-1}\supset\cdots\supset M^{1}\supset M^{0}=0$

where each quotient comodule $M^{i+1}/M^{i}$ has trivial coaction. We will
refer to such a filtration as a _unipotent filtration of length_ $\ell$. Every
comodule $M$ contains a _primitive sequence_ of subcomodules

$M\supseteq\cdots\supseteq M^{[i+1]}\supseteq M^{[i]}\supseteq\cdots\supseteq
M^{[2]}\supseteq M^{[1]}\supseteq M^{[0]}=0$

defined recursively by

$M^{[i]}=\pi_{i-1}^{-1}\operatorname{Prim}_{C}(M/M^{[i-1]})$

where $\pi_{i-1}\colon M\to M/M^{[i-1]}$ is the quotient homomorphism, so
$\pi_{i-1}$ induces an isomorphism

$M^{[i]}/M^{[i-1]}\xrightarrow{\cong}\operatorname{Prim}_{C}(M/M^{[i-1]}).$

In general this need not be exhaustive or become stable, but if it has both
these properties then it is a unipotent filtration of $M$ and we will say that
$M$ has a _finite primitive filtration_.

######  Lemma 4.1.

Suppose that the comodule $M$ is unipotent. Then the primitive sequence of $M$
is finite.

###### Proof.

Suppose that

$M=M^{\ell}\supset M^{\ell-1}\supset\cdots\supset M^{1}\supset M^{0}=0$

is a unipotent filtration. Then we clearly have $M^{1}\subseteq M^{[1]}$. Now
suppose that for some $k\geqslant 1$, $M^{k}\subseteq M^{[k]}$. There is a
commutative diagram of comodules

$\textstyle{M^{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{k+1}/M^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M/M^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M/M^{[k]}}$$\textstyle{\operatorname{Prim}_{C}(M/M^{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{[k+1]}/M^{[k]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

from which it follows that $M^{k+1}\subseteq M^{[k+1]}$. By Induction we find
that $M^{n}\subseteq M^{[n]}$ for all $n\geqslant 1$ and in particular
$M=M^{\ell}\subseteq M^{[\ell]}$, so $M^{[\ell]}=M$. ∎

######  Lemma 4.2.

Let $L,M,N$ be $C$-comodules fitting into a short exact sequence

$0\to L\to M\to N\to 0.$

Then $M$ is unipotent if and only if $L$ and $N$ are unipotent.

###### Proof.

Suppose that $M$ and $N$ are unipotent. A unipotent filtration of $N$ pulls
back to a comodule filtration of $M$ where each stage contains the image of
$L$, and this can be extended to a unipotent filtration of $M$ using a
unipotent filtration of $L$.

Suppose that $M$ has a finite primitive filtration

$M=M^{[\ell]}\supset M^{[\ell-1]}\supset\cdots\supset M^{[1]}\supset
M^{[0]}=0.$

Set $L^{1}=\operatorname{Prim}_{C}(L)=L\cap M^{[1]}$. Then

$\operatorname{Prim}_{C}(L/L^{1})\cong\operatorname{Prim}_{C}(L+M^{[1]}/M^{[1]})=\bigl{(}(L+M^{[1]})\cap
M^{[2]}\bigr{)}/M^{[1]}.$

Now for each $1\leqslant k\leqslant\ell$ define $L^{k}=L\cap M^{[k]}$. Notice
that $L^{k}=L\cap M^{[k]}\subseteq=L^{k+1}$ and $L^{\ell}=L$. Also,

$L^{k+1}/L^{k}\cong(L^{k+1}+M^{[k]})/M^{[k]}\subseteq\bigl{(}(L+M^{[k]})\cap
M^{[k+1]}\bigr{)}/M^{[k]}\subseteq M^{[k+1]}/M^{[k]}$

so the coaction of $L^{k+1}/L^{k}$ is trivial. Now taking $N=M/L$ define

$N^{k}=(L+M^{[k]})/L\cong M^{[k]}/L^{k}\cap M^{[k]}$

so that

$N^{k+1}/N^{k}\cong(L+M^{[k+1]})/(L+M^{[k]})$

which is easily seen to have trivial coaction. ∎

######  Remark 4.3.

Suppose that $M$ is a unipotent $C$-comodule. Then $M$ is also unipotent as a
$D$-comodule where $C\to D$ is a surjective morphism of coalgebras. Also, if
the coaction factors through a $C^{\prime}$-coaction $M\to C_{0}\otimes M$
where $C^{\prime}\subseteq C$ is a subcoalgebra, then $M$ is a unipotent
$C_{0}$-comodule.

Our main use of such unipotent filtrations is to situations described in the
next result.

######  Proposition 4.4.

Let $B\subseteq A$ be a conormal subHopf algebra of a commutative Hopf algebra
over a field $\Bbbk$. Suppose that $M$ is a unipotent left $A$-comodule and
that for every $k\geqslant 0$,
$\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,\Bbbk)$ is a unipotent left
$A$-comodule. Then each $\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,M)$ is a
unipotent $A$-comodule.

###### Proof.

We remark that the long exact sequence for
$\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,-)$ used below is one of
$A$-comodules; the dual module case follows from the details given in Singer
[WMS:SteenrodSqsSS]*chapter 4.

We prove this by induction on the length $\ell$ of the unipotent filtration on
$M$. When $\ell=1$,

$\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,M)\cong\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,\Bbbk)\otimes
M$

and the result holds. Now suppose that it holds for $\ell<n$ and let $\ell=n$.
Consider the short exact sequence

(4.3) $0\to M^{[n-1]}\to M\to M/M^{[n-1]}\to 0$

where we know that for all $k\geqslant 0$,

$\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,M^{[n-1]}),\quad\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,M^{[n]}/M^{[n-1]})$

are unipotent $A$-comodules. On applying
$\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,-)$ to (4.3) we obtain a long exact
sequence where in degree $k$ we have

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,M^{[n-1]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,M^{[n-1]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

so by Lemma 4.2 there is a short exact sequence

$0\to N^{\prime}_{k}\to\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,M)\to
N^{\prime\prime}_{k}\to 0$

for unipotent comodules $N^{\prime}_{k}$ and $N^{\prime\prime}_{k}$. Again
using Lemma 4.2 we find that $\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,M)$ is
unipotent. ∎

######  Remark 4.5.

Although we have stated the last result in terms the $A$-comodule structure,
in our applications we will use it by passing to a conormal quotient Hopf
algebra of $A$.

######  Proposition 4.6.

Let $B\subseteq A$ be a conormal subHopf algebra of a commutative Hopf algebra
over a field $\Bbbk$ where $B$ is cocommutative. Let $M$ be a left
$A$-comodule such that for every $k\geqslant 0$,
$\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,M)$ is a unipotent left
$B$-comodule. Then each $\operatorname{Cotor}^{k}_{A}(\Bbbk,M)$ is a unipotent
$B$-comodule.

###### Proof.

There is a Cartan-Eilenberg spectral sequence with

$\mathrm{E}_{2}^{s,t}=\operatorname{Cotor}_{B}^{s}(\Bbbk,\operatorname{Cotor}_{A/\\!/B}^{t}(\Bbbk,M))\Longrightarrow\operatorname{Cotor}_{A}^{s+t}(\Bbbk,M).$

When $B$ is cocommutative this is a first quadrant cohomological spectral
sequence of left $B$-comodules, and the differentials are homomorphisms of
unipotent $B$-comodules; this is verified by noting that the functor
$\operatorname{Cotor}_{A/\\!/B}^{t}(\Bbbk,-)\cong\operatorname{Cotor}_{A}^{t}(B,-)$
takes values in the category of left $B$-comodules, and when $B$ is
cocommutative and $U$ is finite dimensional, $\operatorname{Cohom}_{B}(U,V)$
is naturally a left comodule. By Lemma 4.2, each $\mathrm{E}_{\infty}^{s,t}$
is a unipotent $B$-comodule and the same is true of each
$\operatorname{Cotor}_{A}^{n}(\Bbbk,M)$. ∎

In order to state a special case which will be used later, we recall a
standard result which can be found in [JPM&KP:MoreConcise]*corollary 21.24.
Let $B\subseteq A$ and $C\subseteq A$ be conormal subHopf algebras of a
commutative Hopf algebra over a field $\Bbbk$ with $C\subseteq B$. Then there
is an induced normal inclusion $B/\\!/C\to A/\\!/B$ and an isomorphism of Hopf
algebras

(4.4) $(A/\\!/C)/\\!/(B/\\!/C)\cong A/\\!/B.$

######  Corollary 4.7.

Suppose that $B/\\!/C$ is cocommutative and $M$ is a left $A/\\!/C$-comodule
such that for every $k\geqslant 0$,
$\operatorname{Cotor}^{k}_{A/\\!/B}(\Bbbk,M)$ is a unipotent left
$A/\\!/C$-comodule. Then each cohomology group
$\operatorname{Cotor}^{k}_{A/\\!/C}(\Bbbk,M)$ is a unipotent
$B/\\!/C$-comodule.

###### Proof.

Use the Cartan-Eilenberg spectral sequence in the proof of Proposition 4.6,

$\mathrm{E}_{2}^{s,t}=\operatorname{Cotor}_{B/\\!/C}^{s}(\Bbbk,\operatorname{Cotor}_{(A/\\!/C)/\\!/(B/\\!/C)}^{t}(\Bbbk,M))\Longrightarrow\operatorname{Cotor}_{A/\\!/C}^{s+t}(\Bbbk,M).$

where by (4.4),

$\operatorname{Cotor}_{(A/\\!/C)/\\!/(B/\\!/C)}^{t}(\Bbbk,M)\cong\operatorname{Cotor}_{A/\\!/B}^{t}(\Bbbk,M).\qed$

Our motivation for developing these ideas is the following result.

######  Proposition 4.8.

Suppose that $A_{*}$ is a $P_{*}$-algebra and $M_{*}$ a unipotent left
$A_{*}$-comodule. Then

$\operatorname{Cohom}_{A_{*}}(A_{*},M_{*})=0.$

###### Proof.

A non-trivial comodule homomorphism $f\colon A_{*}\to M_{*}$ factors through
some subcomodule $M_{*}^{k}\subseteq M_{*}$ in a unipotent filtration where
and $k$ is minimal. Now choose a non-zero element $x\in
M_{*}^{k}\cap\operatorname{im}f$ and choose a $\Bbbk$-linear map
$M_{*}^{k}/M_{*}^{k-1}\to\Bbbk$ so that $x\mapsto 1$ under the composition.
The resulting composition

$\textstyle{A_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{M_{*}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{*}^{k}/M_{*}^{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Bbbk}$

is a non-trivial comodule homomorphism which cannot exist by Proposition 2.1.
∎

######  Corollary 4.9.

For $s\geqslant 0$,

$\operatorname{Coext}^{s}_{A_{*}}(A_{*},M_{*})=0.$

###### Proof.

This can be proved by induction of the length of a unipotent filtration for
$M_{*}$. ∎

We will use this repeatedly in what follows to show that certain
$\operatorname{Coext}$ groups vanish.

##  5\. Recollections on the Steenrod algebra and its dual

The theory of $P$-algebras applies to many situations involving sub and
quotient Hopf algebras of the Steenrod algebra and its dual for a prime. We
will focus attention on the prime $2$ but the methods are applicable for all
primes.

To illustrate this, here is a simple application involving the mod $2$
Steenrod algebra; this result appears in [DCR:Localn]*corollary 4.10. We
denote the mod $2$ Eilenberg-Mac Lane spectrum by $H=H\mathbb{F}_{2}$ and
recall that

$\mathcal{A}_{*}=H_{*}(H)=\mathbb{F}_{2}[\xi_{1},\ldots,\xi_{n},\ldots]=\mathbb{F}_{2}[\zeta_{1},\ldots,\zeta_{n},\ldots]$

where $\zeta_{n}=\chi(\xi_{n})\in\mathcal{A}_{2^{n}-1}$ and the coproduct is
given by the equivalent formulae

$\psi\xi_{n}=\sum_{0\leqslant i\leqslant
n}\xi_{n-i}^{2^{i}}\otimes\xi_{i},\quad\psi\zeta_{n}=\sum_{0\leqslant
i\leqslant n}\zeta_{i}\otimes\zeta_{n-i}^{2^{i}}.$

### Doubling

The operation of _doubling_ has been used frequently in studying
$\mathcal{A}$-modules. The reader is referred to the account of Margolis
[HRM:Book]*section 15.3 which we will use as background.

Since the dual $\mathcal{A}_{*}$ is a commutative Hopf algebra, it admits a
Frobenius endomorphism $\mathcal{A}_{*}\to\mathcal{A}_{*}$ which doubles
degrees and has Hopf algebra cokernel

$\mathcal{E}_{*}=\mathcal{A}_{*}/\\!/\mathcal{A}_{*}^{(1)}=\Lambda_{\mathbb{F}_{2}}(\overline{\zeta}_{s}:s\geqslant
1),$

where $\mathcal{A}_{*}^{(1)}=\mathbb{F}_{2}[\zeta_{s}^{2}:s\geqslant 1]$.
Dually, there is a Verschiebung $\mathcal{A}\to\mathcal{A}$ which halves
degrees and satisfies

$\operatorname{Sq}^{r}\mapsto\begin{dcases*}\operatorname{Sq}^{r/2}&if $r$ is
even,\\\ \phantom{\;\;\;}0&if $r$ is odd.\end{dcases*}$

The kernel of this Verschiebung is the ideal generated by the Milnor
primitives $\mathrm{P}^{0}_{t}$ ($t\geqslant 1$), hence there is a degree-
halving isomorphism of Hopf algebras
$\mathcal{A}/\\!/\mathcal{E}\xrightarrow{\cong}\mathcal{A}$, where
$\mathcal{E}\subseteq\mathcal{A}$ is the subHopf algebra generated by the
primitives $\mathrm{P}^{0}_{t}$ and dual to the exterior quotient Hopf algebra
$\mathcal{E}_{*}$.

Given a left (graded) $\mathcal{A}$-module $M$, we can induce an
$\mathcal{A}/\\!/\mathcal{E}$-module $M_{(1)}$ where

$M_{(1)}^{n}=\begin{dcases*}M^{n/2}&if $n$ is even,\\\ 0&if $n$ is
odd,\end{dcases*}$

and we write $x_{(1)}$ to indicate the element $x\in M$ regarded as an element
of $M_{(1)}$; the module structure is given by

$\operatorname{Sq}^{r}(x_{(1)})=\begin{dcases*}(\operatorname{Sq}^{r/2}x)_{(1)}&if
$r$ is even,\\\ 0&if $r$ is odd,\end{dcases*}$

Using this construction, the category of left $\mathcal{A}$-modules
$\mathbf{Mod}_{\mathcal{A}}$ admits an additive functor to the category of
evenly graded $\mathcal{A}/\\!/\mathcal{E}$-modules,

$\Phi\colon\mathbf{Mod}_{\mathcal{A}}\to\mathbf{Mod}_{\mathcal{A}/\\!/\mathcal{E}}^{\mathrm{ev}};\quad
M\mapsto M_{(1)}$

which is an isomorphism of categories. The quotient homomorphism
$\rho\colon\mathcal{A}\to\mathcal{A}/\\!/\mathcal{E}$ also induces an additive
isomorphism of categories
$\rho^{*}\colon\mathbf{Mod}_{\mathcal{A}/\\!/\mathcal{E}}^{\mathrm{ev}}\to\mathbf{Mod}_{\mathcal{A}}^{\mathrm{ev}}$
and it is often useful to consider the composition
$\rho^{*}\circ\Phi\colon\mathbf{Mod}_{\mathcal{A}}\to\mathbf{Mod}_{\mathcal{A}}^{\mathrm{ev}}$.

By iterating $\Phi^{(1)}=\Phi$ we obtain isomorphisms

$\Phi^{(s)}=\Phi\circ\Phi^{(s-1)}\colon\mathbf{Mod}_{\mathcal{A}}\to\mathbf{Mod}_{\mathcal{A}/\\!/\mathcal{E}^{(s-1)}}^{(s)};\quad
M\mapsto M_{(s)}$

where the codomain is the category of
$\mathcal{A}/\\!/\mathcal{E}^{(s-1)}$-modules concentrated in degrees
divisible by $2^{s}$ and $\mathcal{E}^{(s-1)}\subseteq\mathcal{A}$ is the
subHopf algebra multiplicatively generated by the elements

(5.1) $\mathrm{P}^{a}_{b}\quad(s-1\geqslant a\geqslant 0,\;b\geqslant 1),$

and $\mathcal{E}^{(0)}=\mathcal{E}$.

By doubling all three of the variables involved the following homological
result is immediate for $e\geqslant 1$ and two $\mathcal{A}$-modules $M,N$:

(5.2)
$\displaystyle\operatorname{Ext}_{\mathcal{A}_{(e)}}^{s,2^{e}t}(M_{(e)},N_{(e)})\cong\operatorname{Ext}_{\mathcal{A}}^{s,t}(M,N).$

Because doubling is induced using a grade changing Hopf algebra endomorphism,
the double $\mathcal{A}_{(1)}$ is also a Hopf algebra isomorphic to the
quotient Hopf algebra $\mathcal{A}/\\!/\mathcal{E}$ and dual to the subHopf
algebra of squares $\mathcal{A}^{(1)}_{*}\subseteq\mathcal{A}_{*}$ which is
also given by

$\mathcal{A}^{(1)}_{*}=\mathcal{A}_{*}\square_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}\mathbb{F}_{2}=\mathbb{F}_{2}\square_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}\mathcal{A}_{*}=(\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*})\backslash\\!\backslash\mathcal{A}_{*}.$

More generally, for any $s\geqslant 1$, $\mathcal{A}_{(s)}$ is isomorphic to
the quotient Hopf algebra of $\mathcal{A}/\\!/\mathcal{E}^{(s)}$ dual to the
subalgebra of $2^{s}$-th powers

$\mathcal{A}^{(s)}_{*}=(\mathcal{A}_{*}/\\!/\mathcal{A}^{(s)}_{*})\backslash\\!\backslash\mathcal{A}_{*}\subseteq\mathcal{A}_{*}.$

In many ways, doubling is more transparent when viewed in terms of comodules.
For an $\mathcal{A}_{*}$-comodule $M_{*}$, we can define a
$\mathcal{A}_{*}^{(1)}$-coaction $\mu^{(1)}\colon
M_{*}^{(1)}\to\mathcal{A}_{*}^{(1)}\otimes M_{*}^{(1)}$ where $M_{*}^{(1)}$
denotes $M_{*}$ with its degrees doubled; this is given on elements by the
composition

$\textstyle{M_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{\mu^{(1)}}$$\textstyle{\mathcal{A}_{*}\otimes
M_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)^{2}\otimes\operatorname{Id}}$$\textstyle{\mathcal{A}_{*}^{(1)}\otimes
M_{*}.}$

By iterating we also obtain a $\mathcal{A}_{*}^{(s)}$-coaction
$\mu^{(s)}\colon M_{*}^{(s)}\to\mathcal{A}_{*}^{(s)}\otimes M_{*}^{(s)}$.

Then the comodule analogue of (5.2) is

(5.3)
$\displaystyle\operatorname{Coext}_{\mathcal{A}_{*}^{(e)}}^{s,2^{e}t}(M_{*}^{(e)},N_{*}^{(e)})\cong\operatorname{Coext}_{\mathcal{A}_{*}}^{s,t}(M_{*},N_{*}).$

We can use iterated doubling combined with Proposition 1.2 to show that for
any $d\geqslant 1$,

(5.4)
$\operatorname{Coext}_{\mathcal{A}_{*}}^{s,t}(\mathcal{A}_{*},\mathcal{A}_{*}^{(d)})\cong\operatorname{Ext}_{\mathcal{A}}^{s,t}(\mathcal{A}_{(d)},\mathcal{A})=0.$

By doubling all three of the variables involved here we can also prove that
for $e\geqslant 0$,

(5.5)
$\operatorname{Coext}_{\mathcal{A}_{*}^{(e)}}^{s,2^{e}t}(\mathcal{A}_{*}^{(e)},\mathcal{A}_{*}^{(d+e)})\cong\operatorname{Coext}_{\mathcal{A}_{*}}^{s,t}(\mathcal{A}_{*},\mathcal{A}_{*}^{(d)})=0.$

### Some families of quotient $P_{*}$-algebras of $\mathcal{A}_{*}$

We will begin by describing some quotients of the dual Steenrod algebra
$\mathcal{A}_{*}$. For any $n\geqslant 1$,
$(\zeta_{1},\ldots,\zeta_{n})\lhd\mathcal{A}_{*}$ is a Hopf ideal so there is
a quotient Hopf algebra $\mathcal{A}_{*}/(\zeta_{1},\ldots,\zeta_{n})$
together with the subHopf algebra

$\mathcal{P}(n)_{*}=\mathcal{A}_{*}\square_{\mathcal{A}_{*}/(\zeta_{1},\ldots,\zeta_{n})}\mathbb{F}_{2}=\mathbb{F}_{2}[\zeta_{1},\ldots,\zeta_{n}]\subseteq\mathcal{A}_{*}$

and in fact

$\mathcal{A}_{*}/\\!/\mathcal{P}(n)_{*}=\mathcal{A}_{*}/(\zeta_{1},\ldots,\zeta_{n}).$

Similarly, for any $s\geqslant 0$, the ideal
$(\zeta_{1}^{2^{s}},\ldots,\zeta_{n}^{2^{s}})\lhd\mathcal{A}_{*}$ is a Hopf
ideal and there is a quotient Hopf algebra

$\mathcal{A}_{*}/\\!/\mathcal{P}(n)^{(s)}_{*}=\mathcal{A}_{*}/(\zeta_{1}^{2^{s}},\ldots,\zeta_{n}^{2^{s}})$

with associated subHopf algebra

$\mathcal{P}(n)^{(s)}_{*}=\mathcal{A}_{*}\square_{\mathcal{A}_{*}/\\!/\mathcal{P}(n)^{(s)}_{*}}\mathbb{F}_{2}=\mathbb{F}_{2}[\zeta_{1}^{2^{s}},\ldots,\zeta_{n}^{2^{s}}]\subseteq\mathcal{A}_{*}.$

For each $t\geqslant 0$ there is a finite quotient Hopf algebra

$\mathcal{P}(n)^{(s)}_{*}/(\zeta_{1}^{2^{s+t}},\zeta_{2}^{2^{s+t-1}},\ldots,\zeta_{t}^{2^{s+1}},\zeta_{t+1}^{2^{s}},\ldots,\zeta_{n}^{2^{s}})$

and we have

$\mathcal{P}(n)^{(s)}_{*}=\lim_{t}\mathcal{P}(n)^{(s)}_{*}/(\zeta_{1}^{2^{s+t}},\zeta_{2}^{2^{s+t-1}},\ldots,\zeta_{t}^{2^{s+1}},\zeta_{t+1}^{2^{s}},\ldots,\zeta_{n}^{2^{s}})$

where the limit is computed degree-wise. The graded dual Hopf algebra

$\mathcal{P}(n)_{(s)}=(\mathcal{P}(n)_{*}^{(s)})^{*}=\operatorname{Hom}(\mathcal{P}(n)^{(s)}_{*},\mathbb{F}_{2})$

is the colimit of the finite dual Hopf algebras

$\operatorname{Hom}(\mathcal{P}(n)^{(s)}_{*}/(\zeta_{1}^{2^{s+t}},\zeta_{2}^{2^{s+t-1}},\ldots,\zeta_{t}^{2^{s+1}},\zeta_{t+1}^{2^{s}},\ldots,\zeta_{n}^{2^{s}}),\mathbb{F}_{2}),$

i.e.,

$\mathcal{P}(n)_{(s)}=\operatorname*{colim}_{t}\operatorname{Hom}(\mathcal{P}(n)^{(s)}_{*}/(\zeta_{1}^{2^{s+t}},\zeta_{2}^{2^{s+t-1}},\ldots,\zeta_{t}^{2^{s+1}},\zeta_{t+1}^{2^{s}},\ldots,\zeta_{n}^{2^{s}}),\mathbb{F}_{2}).$

Therefore $\mathcal{P}(n)_{(s)}$ is a $P$-algebra and
$\mathcal{P}(n)^{(s)}_{*}$ is a $P_{*}$-algebra.

##  6\. Some comodules and their cohomology

Later we will need to determine various cohomology groups such as

$\operatorname{Coext}^{*,*}_{\mathcal{A}_{*}}(\mathcal{A}^{(1)}_{*},\mathbb{F}_{2}).$

using the Cartan-Eilenberg spectral sequence of Proposition 3.1,

(6.1)
$\mathrm{E}^{s,t}_{2}=\operatorname{Coext}^{s}_{\mathcal{A}^{(1)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{t}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2}))\Longrightarrow\operatorname{Coext}^{s+t}_{\mathcal{A}_{*}}(\mathcal{A}^{(1)}_{*},\mathbb{F}_{2}),$

where we have suppressed the internal grading. In fact,
$\mathcal{A}^{(1)}_{*}$ is a projective $\mathcal{A}^{(1)}_{*}$-comodule, so
$\mathrm{E}^{s,t}_{2}=0$ when $s>0$, therefore we only need to consider

$\mathrm{E}^{0,t}_{2}=\operatorname{Cohom}_{\mathcal{A}^{(1)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{t}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2})).$

Here the $\operatorname{Cotor}$ term is bigraded with

$\operatorname{Cotor}^{*,*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2})=\mathbb{F}_{2}[q_{n}:n\geqslant
0]$

for $q_{n}=[\overline{\zeta_{n+1}}]\in\operatorname{Cotor}^{1,2^{n+1}-1}$
represented in the cobar construction by the residue class of
$\overline{\zeta_{n+1}}\in\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}$. We need
to understand the comodule structure on this and similar
$\operatorname{Cotor}$ groups.

Now we can consider the adjoint coaction for $\mathcal{A}_{*}$.

######  Lemma 6.1.

The left adjoint coaction of $\mathcal{A}_{*}$ is given by

$\mu\colon\mathcal{A}_{*}\to\mathcal{A}_{*}\otimes\mathcal{A}_{*};\quad\mu(\zeta_{n})=\sum_{i\geqslant
0}\sum_{j\geqslant 0}\zeta_{i}\xi_{n-i-j}^{2^{i+j}}\otimes\zeta_{j}^{2^{i}}.$

###### Proof.

This follows by iterating the coaction and using the formulae

$\psi\zeta_{n}=\sum_{0\leqslant k\leqslant
n}\zeta_{k}\otimes\zeta_{n-k}^{2^{k}},\quad\xi_{r}=\chi(\zeta_{r}).\qed$

The left coaction on $q_{n}$ can be deduced from that on $\zeta_{n+1}$ where
we can ignore all terms in the sum for $\mu(\zeta_{n+1})$ with $i>0$, thus
giving

(6.2) $\mu q_{n}=\sum_{0\leqslant j\leqslant n}\xi_{n-j}^{2^{j+1}}\otimes
q_{j}.$

This extends to polynomials in the $q_{n}$ using multiplicativity.

This coaction is visibly defined over
$\mathcal{A}^{(1)}_{*}\subseteq\mathcal{A}_{*}$, and in practise we will only
require the coaction over the quotient
$\mathcal{A}^{(1)}_{*}/\\!/\mathcal{A}^{(3)}_{*}$ so we will denote this
coaction by

$\overline{\mu}\colon\operatorname{Cotor}^{*,*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2})\to\mathcal{A}^{(1)}_{*}/\\!/\mathcal{A}^{(3)}_{*}\otimes\operatorname{Cotor}^{*,*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2}),$

where for $n\geqslant 2$,

$\overline{\mu}q_{n}=\overline{\xi_{n}^{2}}\otimes
q_{0}+\overline{\xi_{n-1}^{4}}\otimes q_{1}+1\otimes q_{n}$

and

$\overline{\mu}q_{1}=\overline{\xi_{1}^{2}}\otimes q_{0}+1\otimes q_{1}.$

In what follows we will make use of this
$\mathcal{A}^{(1)}_{*}/\\!/\mathcal{A}^{(3)}_{*}$-comodule structure and also
the induced $\mathcal{A}^{(1)}_{*}/\\!/\mathcal{A}^{(2)}_{*}$-comodule
structure.

######  Proposition 6.2.

For $k\geqslant 0$ there are no non-trivial
$\mathcal{A}^{(1)}_{*}/\\!/\mathcal{A}^{(2)}_{*}$-comodule homomorphisms

$\mathcal{A}^{(1)}_{*}\to\operatorname{Cotor}^{k,*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2}).$

Hence

$\operatorname{Cohom}_{\mathcal{A}^{(1)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{k,*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2}))=\operatorname{Cohom}_{\mathcal{A}^{(1)}_{*}/\\!/\mathcal{A}^{(2)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{k,*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2}))=0.$

Before giving the proof, we will define for each $k\geqslant 0$, an increasing
filtration of the $\mathcal{A}^{(1)}_{*}/\\!/\mathcal{A}^{(2)}_{*}$-comodule

$\operatorname{Cotor}^{k,*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2})=\mathbb{F}_{2}\\{q_{0}^{r_{0}}q_{1}^{r_{1}}\cdots
q_{\ell}^{r_{\ell}}:\sum_{0\leqslant i\leqslant\ell}r_{i}=k\\}$

by setting

$\mathrm{F}^{k,s}=\mathbb{F}_{2}\\{q_{0}^{r_{0}}q_{1}^{r_{1}}\cdots
q_{\ell}^{r_{\ell}}:r_{0}\geqslant k-s,\;\sum_{0\leqslant
i\leqslant\ell}r_{i}=k\\}.$

Each $\mathrm{F}^{k,s}$ is a subcomodule of
$\operatorname{Cotor}^{k,*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2})$
and

$\mathrm{F}^{k,0}=\mathbb{F}_{2}\\{q_{0}^{k}\\},\
\quad\mathrm{F}^{k,k}=\mathbb{F}_{2}\\{q_{1}^{r_{1}}\cdots
q_{\ell}^{r_{\ell}}:\sum_{0\leqslant i\leqslant\ell}r_{i}=k\\}.$

Now suppose that $f\colon
M_{*}\to\operatorname{Cotor}^{k,*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2})$
is a non-trivial comodule homomorphism. Choose $s_{0}$ to be minimal so that
$\operatorname{im}f\subseteq\mathrm{F}^{k,s_{0}}$. Then the composition

(6.3)
$\textstyle{M_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\overline{f}}$$\textstyle{\mathrm{F}^{k,s_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{F}^{k,s_{0}}/\mathrm{F}^{k,s_{0}-1}}$

is a non-trivial comodule homomorphism, where
$\mathrm{F}^{k,s_{0}}/\mathrm{F}^{k,s_{0}-1}$ has the trivial coaction. Now we
may choose any non-trivial linear map
$\mathrm{F}^{k,s_{0}}/\mathrm{F}^{k,s_{0}-1}\to\mathbb{F}_{2}[d]$ which is
non-trivial on $\operatorname{im}\overline{f}$ and so obtain a non-trivial
comodule homomorphism $M_{*}\to\mathbb{F}_{2}[d]$. This is the key observation
required for our proof.

###### Proof of _Proposition 6.2_.

Such a comodule homomorphism leads to a non-trivial
$\mathcal{A}_{*}^{(1)}/\\!/\mathcal{A}_{*}^{(2)}$-comodule homomorphism
$\mathcal{A}^{(1)}_{*}\to\mathbb{F}_{2}[d]$ for some $d$. By the Milnor-Moore
theorem, $\mathcal{A}^{(1)}_{*}$ is an extended
$\mathcal{A}_{*}^{(1)}/\\!/\mathcal{A}_{*}^{(2)}$-comodule, so there must be a
non-trivial $\mathcal{A}_{*}^{(1)}/\\!/\mathcal{A}_{*}^{(2)}$-comodule
homomorphism
$\mathcal{A}_{*}^{(1)}/\\!/\mathcal{A}_{*}^{(2)}\to\mathbb{F}_{2}[d^{\prime}]$
for some $d^{\prime}$. But since
$\mathcal{A}_{*}^{(1)}/\\!/\mathcal{A}_{*}^{(2)}$ is a $P_{*}$-algebra, this
contradicts Proposition 2.5.

Since the natural homomorphism

$\operatorname{Cohom}_{\mathcal{A}^{(1)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{k,*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2}))\to\operatorname{Cohom}_{\mathcal{A}^{(1)}_{*}/\\!/\mathcal{A}^{(2)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{k,*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2}))$

is injective, the last statement follows. ∎

######  Corollary 6.3.

The $\mathrm{E}_{2}$-term of the spectral sequence (6.1) is trivial, hence

$\operatorname{Coext}^{*,*}_{\mathcal{A}_{*}}(\mathcal{A}^{(1)}_{*},\mathbb{F}_{2})=0.$

A generalisation of these results is

######  Proposition 6.4.

For any $n\geqslant 0$,

$\operatorname{Coext}^{*,*}_{\mathcal{A}_{*}}(\mathcal{A}^{(1)}_{*},\mathcal{P}(n)^{(1)}_{*})=0.$

###### Proof.

We need to deal with the case $n\geqslant 1$. By Proposition 3.1 there is a
Cartan-Eilenberg spectral sequence of form

$\mathrm{E}^{s,t}_{2}=\operatorname{Coext}^{s}_{\mathcal{A}^{(1)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{t}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathcal{P}(n)^{(1)}_{*}))\Longrightarrow\operatorname{Coext}^{s+t}_{\mathcal{A}_{*}}(\mathcal{A}^{(1)}_{*},\mathbb{F}_{2}).$

Since the $\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}$-coaction on
$\mathcal{P}(n)^{(1)}_{*}$ is trivial,

$\operatorname{Cotor}^{t}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathcal{P}(n)^{(1)}_{*})\cong\operatorname{Cotor}^{t}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2})\overset{\mathcal{A}^{(1)}_{*}}{\wedge}\mathcal{P}(n)^{(1)}_{*}.$

As left $\mathcal{A}^{(1)}_{*}$-comodules

$\mathcal{P}(n)^{(1)}_{*}\cong\mathcal{A}^{(1)}_{*}\square_{\mathcal{A}^{(1)}_{*}/\\!/\mathcal{P}(n)^{(1)}_{*}}\mathbb{F}_{2}$

so by a standard ‘change of rings’ isomorphism,

$\mathrm{E}^{s,t}_{2}\cong\operatorname{Coext}^{s}_{\mathcal{A}^{(1)}_{*}/\\!/\mathcal{P}(n)^{(1)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{t}_{\mathcal{A}_{*}/\\!/\mathcal{A}_{*}^{(1)}}(\mathbb{F}_{2},\mathbb{F}_{2})).$

Again the Milnor-Moore theorem shows that $\mathcal{A}^{(1)}_{*}$ is a cofree
$\mathcal{A}^{(1)}_{*}/\\!/\mathcal{P}(n)^{(1)}_{*}$-comodule and since this
is a $P_{*}$-algebra, $\mathrm{E}^{s,t}_{2}=0$ for $s>0$. Now a similar
argument to that in the proof of Proposition 6.2 shows that
$\mathrm{E}^{0,t}_{2}=0$. ∎

Before passing on to discuss other examples, we note that in $\mathcal{A}_{*}$
there is a subcomodule $\leftidx{{}^{\leqslant k}}{\mathcal{A}}{{}_{*}}$
spanned by the monomials in the $\xi_{i}$ of degree at most $k$. Under the
doubling isomorphism this corresponds to a subcomodule $\leftidx{{}^{\leqslant
k}}{\mathcal{A}}{{}_{*}^{(1)}}$ of $\mathcal{A}^{(1)}_{*}$ spanned by
monomials in the $\xi_{i}^{2}$ of degree at most $k$.

######  Proposition 6.5.

For $k\geqslant 0$ there is an isomorphism of
$\mathcal{A}^{(1)}_{*}$-comodules

$\operatorname{Cotor}^{k,*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2})\xrightarrow{\cong}\leftidx{{}^{\leqslant
k}}{\mathcal{A}}{{}_{*}^{(1)}};\quad q_{0}^{r_{0}}q_{1}^{r_{1}}\cdots
q_{\ell}^{r_{\ell}}\leftrightarrow\xi_{1}^{2r_{1}}\cdots\xi_{\ell}^{2r_{\ell}}.$

The following result is an analogue of Proposition 6.2 whose proof can be
adapted using the filtration of $\leftidx{{}^{\leqslant
k}}{\mathcal{A}}{{}_{*}}$ based on polynomial degree.

######  Proposition 6.6.

For $k\geqslant 0$,

$\operatorname{Coext}^{*}_{\mathcal{A}_{*}}(\mathcal{A}_{*},\leftidx{{}^{\leqslant
k}}{\mathcal{A}}{{}_{*}})=0.$

We will also require some other vanishing results.

######  Proposition 6.7.

For $n\geqslant 0$,

$\operatorname{Coext}^{*,*}_{\mathcal{A}_{*}}(\mathcal{A}^{(1)}_{*},\mathcal{P}(n)^{(2)}_{*})=0$

and

$\operatorname{Coext}^{*,*}_{\mathcal{A}_{*}}(\mathcal{A}^{(1)}_{*},\mathcal{A}^{(2)}_{*})=0.$

###### Proof.

By setting $\mathcal{P}(\infty)^{(2)}_{*}=\mathcal{A}^{(2)}_{*}$ we can
present the proofs of these in a uniform faahion.

There is a Cartan-Eilenberg spectral sequence

$\mathrm{E}^{s,t}_{2}=\operatorname{Coext}^{s}_{\mathcal{A}^{(1)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{t}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathcal{P}(n)^{(2)}_{*}))\Longrightarrow\operatorname{Coext}^{s+t}_{\mathcal{A}_{*}}(\mathcal{A}^{(1)}_{*},\mathcal{P}(n)^{(2)}_{*}).$

Here

$\operatorname{Cotor}^{t}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathcal{P}(n)^{(2)}_{*})\cong\operatorname{Cotor}^{t}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2})\overset{\mathcal{A}^{(1)}_{*}}{\wedge}\mathcal{P}(n)^{(2)}_{*}$

and

$\mathcal{P}(n)^{(2)}_{*}\cong\mathcal{A}^{(1)}_{*}\square_{\mathcal{A}^{(1)}_{*}/\\!/\mathcal{P}(n)^{(2)}_{*}}\mathbb{F}_{2}$

so

$\mathrm{E}^{s,t}_{2}\cong\operatorname{Coext}^{s}_{\mathcal{A}^{(1)}_{*}/\\!/\mathcal{P}(n)^{(2)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{t}_{\mathcal{A}_{*}/\\!/\mathcal{A}_{*}^{(1)}}(\mathbb{F}_{2},\mathbb{F}_{2})).$

Since $\mathcal{A}^{(1)}_{*}$ is a cofree
$\mathcal{A}^{(1)}_{*}/\\!/\mathcal{P}(n)^{(2)}_{*}$-comodule,
$\mathrm{E}^{s,t}_{2}=0$ when $s>0$. Also the change of coalgebra homomorphism

$\operatorname{Cohom}_{\mathcal{A}^{(1)}_{*}/\\!/\mathcal{P}(n)^{(2)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{t}_{\mathcal{A}_{*}/\\!/\mathcal{A}_{*}^{(1)}}(\mathbb{F}_{2},\mathbb{F}_{2}))\to\operatorname{Cohom}_{\mathcal{A}^{(1)}_{*}/\\!/\mathcal{A}^{(2)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{t}_{\mathcal{A}_{*}/\\!/\mathcal{A}_{*}^{(1)}}(\mathbb{F}_{2},\mathbb{F}_{2}))$

is injective. By Proposition 6.2, the codomain is trivial so
$\mathrm{E}^{0,t}_{2}=0$. ∎

##  7\. Some topological applications

We illustrate our theory with a few calculations of homotopy groups of
spectra. Some of the following examples where found in response to questions
of John Rognes. Results of this type were proved by Lin, Margolis, Ravenel and
others. The interested reader will be able to give others, especially at odd
primes.

For simplicity we assume that all spectra are $2$-completed. We are interested
in determining when $[X,Y]^{*}=0$ for two connective finite type spectra
$X,Y$. Here the Adams spectral sequence

$\mathrm{E}_{2}^{s,t}(X,Y)=\operatorname{Ext}_{\mathcal{A}}^{s,t}(H^{*}(Y),H^{*}(X))=\operatorname{Coext}_{\mathcal{A}}^{s,t}(H_{*}(X),H_{*}(Y))\Longrightarrow[X,Y]^{s-t}$

converges by work of Boardman [JMB:CCSS], so we are interested in examples
where $\mathrm{E}_{2}^{*,*}(X,Y)=0$. The general result of Proposition 1.5
applies to many interesting examples.

  * •

For each the spectra $X$ where $H=H\mathbb{F}_{2}$, $H\mathbb{Z}_{2^{s}}$
($s\geqslant 2$), $H\mathbb{Z}$, ${k\mathrm{O}}$, ${k\mathrm{U}}$,
$\mathrm{tmf}$, and $BP\langle n\rangle$ ($n\geqslant 1$), $[X,S]^{*}=0$ since
$H^{*}(X)$ is a coherent $\mathcal{A}$-module and
$\mathrm{E}_{2}^{*,*}(X,S)=0$.

  * •

For each of the above spectra $X$, $[X,BP]^{*}=0$. This follows since

$H^{*}(BP)\cong\mathcal{A}\otimes_{\mathcal{E}}\mathbb{F}_{2}$

where $\mathcal{E}\subseteq\mathcal{A}$ is the subHopf algebra generated by
the Milnor primitives and this is also a $P$-algebra. So by Proposition 1.6,

$\mathrm{E}_{2}^{*,*}(X,BP)\cong\operatorname{Ext}_{\mathcal{E}}^{*,*}(\mathbb{F}_{2},H^{*}(X))=0.$

  * •

The case of $[BP,S]^{*}$ involves more work. We will use the dual version of
the Adams spectral sequence,

$\mathrm{E}_{2}^{*,*}(BP,S)=\operatorname{Coext}_{\mathcal{A}_{*}}^{s,t}(\mathcal{A}^{(1)}_{*},\mathbb{F}_{2})\Longrightarrow[BP,S]^{s-t}.$

We can calculate the $\mathrm{E}_{2}$-term using the Cartan-Eilenberg spectral
sequence of (6.1), and by Corollary 6.3 this $\mathrm{E}_{2}$-term

$\operatorname{Coext}^{*}_{\mathcal{A}^{(1)}_{*}}(\mathcal{A}^{(1)}_{*},\operatorname{Cotor}^{*}_{\mathcal{A}_{*}/\\!/\mathcal{A}^{(1)}_{*}}(\mathbb{F}_{2},\mathbb{F}_{2}))$

is trivial. Therefore $\mathrm{E}_{2}^{*,*}(BP,S)=0$ and $[BP,S]^{*}=0$.

## References