# Morava $K$-theory and filtrations by powers

Tobias Barthel  Max Planck Institute for Mathematics, Vivatsgasse 7, 53111
Bonn, Germany<EMAIL_ADDRESS>and  Piotr Pstrągowski  Harvard
University, Department of Mathematics, 1 Oxford Street, Cambridge MA 02139,
USA<EMAIL_ADDRESS>
###### Abstract.

We prove the convergence of the Adams spectral sequence based on Morava
$K$-theory and relate it to the filtration by powers of the maximal ideal in
the Lubin–Tate ring through a Miller square. We use the filtration by powers
to construct a spectral sequence relating the homology of the $K$-local sphere
to derived functors of completion and express the latter as cohomology of the
Morava stabilizer group. As an application, we compute the zeroth limit at all
primes and heights.

###### Contents

  1. 1 Introduction
  2. I Morava $K$-theories and Milnor modules
    1. 2 Morava $K$-theories and their Hopf algebroids
    2. 3 Digression: Minimal Morava $K$-theory
    3. 4 Milnor modules
    4. 5 Cohomology
    5. 6 Milnor filtration
  3. II The $K$-based Adams spectral sequence
    1. 7 Convergence of the $K$-based Adams spectral sequence
    2. 8 The finite height Miller square
    3. 9 The $K$-based Adams spectral sequence at large primes
    4. 10 Height one $K$-based Adams spectral sequence at an odd prime
  4. III Homology of inverse limits and the algebraic chromatic splitting conjecture
    1. 11 The homology of inverse limits
    2. 12 The Morava $E$-homology of inverse limits
    3. 13 Digression: Cohomology of $\mathbb{G}_{n}$ as $\operatorname{Ext}$ in comodules
    4. 14 Algebraic chromatic splitting conjecture
    5. 15 Cooperations at height one

###  1\. Introduction

In this paper, which consists of three closely related parts, we investigate
the relationship between the filtration by powers of the maximal ideal in the
Lubin–Tate ring and chromatic homotopy theory from several perspectives. In
the first part, we focus on Morava $K$-theories and their associated Hopf
algebroids. In particular, we give an invariant description of their
cohomology and prove its finiteness.

We then establish convergence of the $K$-based Adams spectral sequence and
relate it to the $K$-local Adams–Novikov and filtration by powers spectral
sequences through a Miller square, and describe it completely at large enough
primes. In the last part, we construct Hopkins’ spectral sequence relating
homology of the $K$-local sphere with derived functors of completion, and
identify the latter with cohomology of the Morava stabilizer group. As an
application, we compute the zeroth limit at all primes and heights.

#### Main results

A classical approach to the study of an arithmetic problem is to first
consider its reduction to residue fields and to then reassemble the local
solutions. Over the $p$-local integers, there are just two reside fields,
namely the field $\mathbb{Q}$ of rational numbers and the finite field
$\mathbb{F}_{p}$ corresponding to the closed point. Informally, these detect,
respectively, the torsion-free and torsion phenomena.

The approach of reducing to residue fields also works very well in stable
homotopy theory, where one is interested in classifying stable homotopy
classes of maps between finite complexes. Identifying a classical field $k$
with the corresponding Eilenberg–MacLane spectrum, we have a descent spectral
sequence of signature

$E_{2}^{s,t}\cong\operatorname{Ext}_{\pi_{*}(k\otimes_{S^{0}}k)}^{s,t}(k,k)\implies\pi_{t-s}(S^{0}_{k}),$
(1.1)

where $S^{0}_{k}$ is the appropriate Bousfield localization. When
$k=\mathbb{Q}$, this collapses immediately, recovering Serre’s calculation
that $\pi_{*}S^{0}\otimes_{\mathbb{Z}}\mathbb{Q}\simeq\mathbb{Q}$.

The situation is much more complicated when $k=\mathbb{F}_{p}$, in which case
$\operatorname{\mathcal{A}}_{*}=\pi_{*}(\mathbb{F}_{p}\otimes_{S^{0}}\mathbb{F}_{p})$
is the dual Steenrod algebra and the above recovers the classical Adams
spectral sequence. In this case, the spectral sequence converges completely,
but it does not collapse at any finite page and there is no known algorithmic
way of understanding its structure111A common saying about the classical Adams
spectral sequence is “ _Every differential is a theorem._ ”. It is arguably
the most important tool for computing the stable homotopy groups of spheres.

A new feature in homotopy theory which is not visible in the algebraic
contexts is that, even $p$-locally, $\mathbb{Q}$ and $\mathbb{F}_{p}$ are not
the only residue fields of spectra. Instead, we have an infinite family of
prime fields of “intermediate characteristic”, given by the Morava
$K$-theories $K(n)$ for $0<n<\infty$. These are homotopy ring spectra with the
properties that

  1. (1)

$\pi_{*}K(n)\simeq k[u^{\pm 1}]$, where $k$ is a perfect field of
characteristic $p$ and $u$ is an invertible variable of degree $|u|=2$ and

  2. (2)

the Quillen formal group
$\textbf{G}_{0}:=\textnormal{Spf}(K(n)^{0}(\mathbf{CP}^{\infty}))$ over $k$ is
of finite height $n$.

One can show that such a homotopy ring spectrum exists for any
$\textbf{G}_{0}$ of finite height over a perfect field. By a celebrated result
of Devinatz, Hopkins, and Smith, together with $\mathbb{Q}$ and
$\mathbb{F}_{p}$ these are the only residue fields of $p$-local spectra.

Morava $K$-theories are very calculable for at least two distinct reasons:

  1. (1)

the coefficients $K(n)_{*}$ form a graded field, so that we have an
unrestricted Künneth isomorphism $K(n)_{*}(X\otimes Y)\simeq
K(n)_{*}(X)\otimes_{K(n)_{*}}K(n)_{*}(Y)$ for any $X,Y$;

  2. (2)

there’s a canonical Poincare duality isomorphism $K(n)_{*}(X)\simeq
K(n)^{*}(X)$ whenever $X$ is a $\pi$-finite space, in particular a classifying
space of a finite group.

The second property, known as ambidexterity, is one way in which $K(n)$ behave
as if they were of characteristic zero, despite $K(n)_{*}$ being of positive
characteristic. This surprising feature makes Morava $K$-theories a powerful
tool also in more geometric contexts, such as in the breakthrough work of
Abouzaid and Blumberg on the Arnold conjecture in symplectic topology [AB21].

In the present paper, we set up and analyse the descent spectral sequence
associated to the Morava $K$-theories. As a first result, we verify that we
indeed do have convergence.

######  Theorem A (7.3, 7.4).

The $K(n)$-based Adams spectral sequence of signature

$\operatorname{Ext}^{s,t}_{K(n)_{*}K(n)}(K(n)_{*},K(n)_{*}(X))\implies\pi_{t-s}L_{K(n)}X$

is conditionally convergent for any spectrum $X$ and converges completely
whenever $X$ is $K(n)$-locally dualizable.

At first glance, this result is quite surprising, as each group on the second
page of this spectral sequence for the sphere is torsion, while the abutment
$\pi_{*}L_{K(n)}S^{0}$ is known to contain torsion-free summands. In
particular, this implies that there cannot exist a horizontal vanishing line
on any finite page of the spectral sequence. In order to establish conditional
convergence, we instead reduce to the case of a finite $n$ complex, where such
vanishing lines exist. Complete convergence is the consequence of the
degreewise finiteness of the $E_{2}$-term, which we prove for dualizable
spectra using the Cartan–Eilenberg spectral sequence

Going back to the case of non-localized spectra, the utility of the classical
Adams spectral sequence based on $\mathbb{F}_{p}$ is further amplified when
coupled with the Adams–Novikov spectral sequence, which is the descent
spectral sequence based on the Brown–Peterson spectrum $BP$. As one
spectacular application, Miller constructs a square relating the two spectral
sequences and compares the differentials to compute the $v_{1}$-periodic
homotopy of Moore spaces, thereby proving the telescope conjecture at height 1
and for all odd primes [Mil81].

The situation is similar in the case of the $K(n)$-based Adams, which has a
cousin which might be more familiar to the working chromatic homotopy
theorist, namely the $K(n)$-local $E_{n}$-based Adams spectral sequence. Here,
$E_{n}$ denotes the Lubin–Tate spectrum associated to the formal group
$\mathbf{G}_{0}$, parametrizing derived deformations of the latter. We show
that the relation between the two spectral sequences is again governed by a
square of spectral sequences of the following explicit form.

######  Theorem B (8.8, 8.10).

The $K(n)$-local Miller square associated to $(E,K)=(E_{n},K(n))$ is of the
form

${{\operatorname{Ext}_{E_{*}^{\vee}E}(E_{*},\bigoplus_{k}\mathfrak{m}^{k}/\mathfrak{m}^{k+1}E_{*}^{\vee}X)}}$${{\operatorname{Ext}_{E_{*}^{\vee}E}(E_{*},E^{\vee}_{*}X)}}$${{\operatorname{Ext}_{K_{*}K}(K_{*},K_{*}X)}}$${{\pi_{*}L_{K}X},}$$\scriptstyle{E-\textnormal{Adams}}$$\scriptstyle{K-\textnormal{Adams}}$MayMahowald

where the May spectral sequence is obtained from the $\mathfrak{m}$-adic
filtration of $E_{*}^{\vee}X$.

We also identify the Mahowald spectral sequence in the square above with a
Cartan–Eilenberg spectral sequence of a certain extension of Hopf algebroids
canonically attached to each Morava $K$-theory, see Section 8 for details.

A theorem of Miller relates the $d_{2}$-differentials in the $K$-Adams and May
spectral sequences, but at large primes we can say much more. If
$2p-2>n^{2}+n$, then it is well-known that the $K$-local $E$-based Adams
spectral sequence for the sphere collapses at the second page. Using a
variation on the classical argument of Milnor, we equip $K_{*}K$ at odd primes
with an additional grading which also forces the collapse of the Mahowald
spectral sequence.

If follows that at sufficiently large primes, two of the spectral sequences in
Miller’s square collapse, and it is thus natural to expect that the other two
can be identified. This is indeed the case, as we show the following.

######  Theorem C (9.22).

If $2p-2>n^{2}+n+1$, then the $K$-based Adams spectral sequence for
$S^{0}_{K}$ can be given an additional grading so that it becomes isomorphic
to the May spectral sequence

$\operatorname{Ext}_{E_{*}^{\vee}E}(E_{*},\bigoplus_{k}\mathfrak{m}^{k}/\mathfrak{m}^{k+1}E_{*})\implies\operatorname{Ext}_{E_{*}^{\vee}E}(E_{*},E_{*})$
(1.2)

induced by the filtration of $E_{*}$ by the powers of the maximal ideal.

Note that one new major phenomenon in intermediate characteristic is that
Morava $K$-theories do depend on choices; in particular, on the choice of a
formal group. By a result of Lazard, any two such formal groups of the same
height are isomorphic over the separable closure; it follows that for our
purposes all $K(n)$ of the same height are essentially interchangeable, in
particular, they determine the same Adams spectral sequence.

To be more precise, while the Hopf algebroid $K(n)_{*}K(n)$ depends on the
choice of a Morava $K$-theory, its category of comodules does not, by a result
of Hovey and Strickland [HS05a]. In the body of the text, we describe a
different derivation of this result, identifying the category of
$K(n)_{*}K(n)$-comodules with the category $\operatorname{Mil}_{abs}$ of
_absolute Milnor modules_ , which are certain sheaves on finite spectra, see
Section 4. This gives an approach to the Adams spectral sequence based on
Morava $K$-theory which only depends on the prime and the height, but no other
choices.

On a different note, the completion tower of comodules of the form

$\ldots\rightarrow E_{*}/\mathfrak{m}^{3}\rightarrow
E_{*}/\mathfrak{m}^{2}\rightarrow E_{*}/\mathfrak{m}$

appearing in Theorem C also leads to a _another_ spectral sequence, first
studied by Hopkins and Sadofsky222The construction of Hopkins and Sadofsky is
unpublished.. The latter spectral sequence approaches the transchromatic
information contained in $L_{K(n)}S^{0}$ through its _uncompleted_
$E$-homology $E_{*}(L_{K(n)}S^{0})$, as opposed to the homotopy groups, as we
now describe in more detail.

To provide some context, recall that it is known that at large primes there
exists an equivalence $h\mathcal{S}p_{E}\simeq h\mathcal{D}^{per}(E_{*}E)$
between the homotopy categories of $E$-local spectra and differential
comodules. Under this equivalence, the limit $\varprojlim
E_{*}/\mathfrak{m}^{k}$ in the derived $\infty$-category corresponds to the
$K$-local sphere $S^{0}_{K}$ [Pst18a]; thus, the derived completion of $E_{*}$
can be thought of as the algebraic analogue of the $K$-local sphere.

At smaller primes, we do not have algebraic models for the homotopy category,
but we instead have a spectral sequence relating the homology groups of the
derived completion to $E_{*}(S^{0}_{K})$. This spectral sequence, originally
due to Hopkins and Sadofsky, was subsequently worked out by Peterson [Pet20a],
adapting the classical construction of the Adams spectral sequence. We follow
a slightly different approach, based on the notion of an adapted homology
theory and Devinatz–Hopkins’ modified Adams spectral sequence [Dev97].

Our construction is closely related to work of Hovey on derived products of
comodules [Hov07], and in fact works in much greater generality, see Section
11. In the case of $E$-homology, we are able to express these derived limits
as cohomology of the Morava stabilizer group, leading to the following
statement.

######  Theorem D (11.13, 12.13, 12.11).

Let $E_{*}E:=\pi_{*}(E\otimes E)$ be the uncompleted $E$-homology of itself,
given the unique topology which restricts to the $\mathfrak{m}$-adic topology
on each finitely generated $E_{*}$-submodule. Then, there exists a canonical
isomorphism

$\mathrm{H}^{s}_{\operatorname{cts}}(\mathbb{G}_{n},E_{t}E)\cong(\textstyle\varprojlim_{E_{*}E}^{s}E_{*}/\mathfrak{m}^{k})_{t}$

between the continuous cohomology of the Morava stabilizer group and the
derived functors of the limit in $E_{*}E$-comodules. Moreover, there exists a
spectral sequence of $E_{*}E$-comodules of the form

$E_{2}^{s,t}\cong\mathrm{H}^{s}_{\operatorname{cts}}(\mathbb{G}_{n},E_{t}E)\implies
E_{t-s}(L_{K}S^{0})$ (1.3)

with differentials $d_{r}^{s,t}\colon E_{r}^{s,t}\to E_{r}^{s+r,t+r-1}$. This
spectral sequences converges completely and collapses at a finite page with a
horizontal vanishing line.

The subtle point in this result is the convergence of the spectral sequence,
which relies crucially on the finite virtual cohomological dimension of the
Morava stabilizer group. Note that here it is important to consider the
derived sequential limits in the category of comodules as opposed to ordinary
modules over $E_{*}$; indeed, in positive degrees the latter all vanish in
this case and there would be no hope of having a convergent spectral sequence.

The above spectral sequence underlies one approach towards Hopkins’ chromatic
splitting conjecture [Pet], [Hov95]. In a nutshell, the latter provides a
precise description of the transchromatic behaviour of the $K(n)$-local
sphere; that is, its behaviour under applying chromatic localizations $L_{h}$
for intermediate heights $0\leq h<n$. As such it has been verified for all
heights $n\leq 2$ and all primes [SY95, GHM12, Bea17], with a minor
modification to the original prediction at $n=p=2$ ([BGH17]). Beyond height 2,
the conjecture remains wide open.

The standard way of approaching the splitting conjecture is through an
explicit computation of the homotopy groups of $L_{K(n)}S^{0}$. However, as
mentioned, its transchromatic information is also encoded in the _uncompleted_
$E$-homology, and this is what the spectral sequence (11.2) abuts to.

Using the identification of derived functors of the limit with continuous
cohomology of the Morava stabilizer group, we are able to compute the zeroth
limit at all primes and heights.

######  Theorem E (14.13).

There is a canonical isomorphism

$\textstyle\varprojlim_{E_{*}E}E_{*}/\mathfrak{m}^{k}E_{*}\simeq
E_{*}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$

where the limit on the left is taken in the category of $E_{*}E$-comodules.

Note that $E_{*}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}\simeq E_{*}(S^{0}_{p})$,
the homology of the $p$-complete sphere. Thus, the above result can be
interpreted as detecting the conjectured copy of $L_{n-1}S^{0}_{p}$ inside
$L_{n-1}S^{0}_{K}$ through $E$-homology.

At height one, the Morava stabilizer group is particularly simple, and we are
able to compute all of the derived limits.

######  Theorem F (15.12).

At height $n=1$ and any prime, we have

$\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{1},E_{*}E)\simeq\begin{cases}E_{*}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}&\mbox{when
}$s=0$,\\\ E_{*}\otimes_{\mathbb{Z}}\mathbb{Q}_{p}&\mbox{when }$s=1$,\\\
0&\mbox{otherwise}.\end{cases}$

Note that the above in particular completely recovers the height one case of
the chromatic splitting conjecture, which states that
$L_{0}L_{K(1)}S^{0}\simeq L_{0}S^{0}_{p}\oplus L_{0}S^{-1}_{p}$. The latter
can be equivalently obtained by explicitly computing $\pi_{*}L_{K}S^{0}$,
which if we ignore torsion contains exactly two $\mathbb{Z}_{p}$ summands, one
for each sphere. However, the computation of homology gives more precise
information, as the two copies of $\mathbb{Z}_{p}$ are isomorphic as abelian
groups, but the comodules $E_{*}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$ and
$E_{*}\otimes_{\mathbb{Z}}\mathbb{Q}$ are not.

#### Outline of document

We hope that the structure of the document becomes apparent from the choice of
section titles. In brief, the first part of the paper, comprising Section 2
through Section 6 deals with Morava $K$-theories, their Hopf algebroids,
categories of comodules, and their cohomology, through the perspective of
Milnor modules. We pay special attention to the (in)dependence of our
constructions on the choice of Morava $K$-theories, and revisit the relation
to continuous cohomology.

The $K$-based Adams spectral sequence and the finite height Miller square are
the topic of Part 2. We establish convergence properties in Section 7, then
construct the finite height Miller square, and use it to relate it to the
filtration by powers spectral sequence. In Section 10, the final section of
this part, we illustrate our results by going through explicit computations at
height 1.

The third part of the paper consists of Section 11 to Section 15. Here, the
focus lies on the derived category of $E_{*}E$-comodules and the construction
and study of the inverse limit spectral sequence. We then give our application
to the algebraic chromatic splitting conjecture based on the connection to the
continuous cohomology of the Morava stabilizer group. Each of the two parts
ends with some explicit computations at height one, which we hope elucidate
our methods and results.

#### Acknowledgements

We would like to thank Agnès Beaudry, Robert Burklund, Mike Hopkins, Eric
Peterson, and Hal Sadofsky for useful conversations related to this work. We
acknowledge the hospitality of the Max Planck Institute for Mathematics in
Bonn.

##  Part I Morava $K$-theories and Milnor modules

In this part of the paper, we study Morava $K$-theories, which play the role
of residue fields of Lubin–Tate spectra. We describe their Hopf algebroids,
categories of comodules, and cohomology.

###  2\. Morava $K$-theories and their Hopf algebroids

Let $\kappa$ be a perfect field equipped with a choice of a height $n$ formal
group $\mathbf{G}_{0}$. Associated to this data is the Lubin–Tate ring $E_{0}$
classifying deformations [LT66]. This is a complete local $W(\kappa)$-algebra
such that there exists a non-canonical choice of regular generators inducing
an isomorphism of rings $E_{0}\simeq W(\kappa)\llbracket
u_{1},\ldots,u_{n-1}\rrbracket$. We write
$\mathfrak{m}=(p,u_{1}\ldots,u_{n-1})$ for the maximal ideal.

The ring $E_{0}$ equipped with the universal deformation is Landweber exact
and so can be lifted to a $2$-periodic ring spectrum $E$ with $E_{*}\simeq
E_{0}[u^{\pm 1}]$ for some unit $u\in E_{2}$, and the property that the formal
group $\mathbf{G}:=\textnormal{Spf}(E^{0}(BS^{1}))$ is the universal
deformation of $\mathbf{G}_{0}$. It is well-known that $E$ admits a unique
$\mathbf{E}_{\infty}$-ring structure, functorial in the choice of $\kappa$ and
the formal group, see [GH05]. More recently, it was proven by Lurie that $E$
arises as a solution to a moduli problem involving formal groups over
$\mathbf{E}_{\infty}$-rings, see [Lur18].

The canonical nature of the Lubin–Tate spectrum makes it a good starting point
for the study of chromatic phenomena. Since $E_{0}$ is local, one would like
to construct a spectrum which plays the role of the residue field of $E$.
Following Hopkins and Lurie [HL17], we make the following definition.

######  Definition 2.1.

An $\mathbf{E}_{1}$-$E$-algebra $K$ is a _Morava $K$-theory_ if the unit map
$E\rightarrow K$ induces an isomorphism $K_{*}\simeq E_{*}/\mathfrak{m}$.

One can show that an algebra satisfying the above conditions always exists,
for any Lubin–Tate spectrum $E$; in fact:

  1. (1)

there are uncountably many Morava $K$-theories which are not equivalent as
$E$-algebras,

  2. (2)

none of which is preferred and

  3. (3)

none of which can be promoted to an $\mathbf{E}_{2}$-$E$-algebra,

see [HL17] for more information.

######  Warning 2.2.

Large parts of the literature are written in terms of the
$(2p^{n}-2)$-periodic spectrum $K(n)$ with
$\pi_{*}K(n)\simeq\mathbb{F}_{p}[v_{n}^{\pm 1}]$; this cannot be a made into a
Morava $K$-theory according to the above definition as it not $2$-periodic. We
will not use $K(n)$ in the current work, but in the interest of completness we
collect some results about it in §3 below.

Throughout the rest of this section, the letter $K$ will denote a choice of a
Morava $K$-theory in the sense of Definition 2.1.

######  Remark 2.3.

While two different Morava $K$-theories need not be equivalent as
$E$-algebras, one can show that they are always equivalent as $E$-modules
[HL17, Corollary 3.6.6]. In particular, they are always equivalent as spectra.

Observe that we have $K_{*}\simeq E_{*}/\mathfrak{m}E_{*}$ by definition, so
$K_{*}\simeq\kappa[u^{\pm 1}]$, which is a graded field. Note that the first
isomorphism is completely canonical, while the second is not, as it depends on
the choice of a unit $u\in K_{2}$. In any case, it follows that the ring
spectrum $K$ is even periodic, and so complex-orientable. The reduction map
$E^{0}(BS^{1})\rightarrow K^{0}(BS^{1})$ induces a canonical isomorphism
$\textnormal{Spf}(K^{0}(BS^{1}))\simeq\mathbf{G}_{0}$.

######  Remark 2.4.

It is not strictly necessary to start relative to a chosen Lubin–Tate
spectrum. Instead, one could say that an $\mathbf{E}_{1}$-algebra $K$ in
spectra is a _Morava $K$-theory_ if there exists an isomorphism
$K_{*}\simeq\kappa[u^{\pm 1}]$ with $\kappa$ a perfect field and such that
$\mathbf{G}_{0}:=\textnormal{Spf}(K^{0}(BS^{1}))$ is of finite height. One can
show that in this case $K$ can be canonically lifted to an algebra over the
corresponding Lubin–Tate ring spectrum. Thus, we lose no generality by working
relative to $E$.

In this paper, we will be interested in the $K$-based Adams spectral sequence.
By standard arguments, the $E_{2}$-page of this spectral sequence has a
description in terms of homological algebra of comodules over the Hopf
algebroid $K_{*}K$. Thus, we begin by giving a partial description of the
latter.

Our strategy is to exploit the $E$-algebra structure on $K$ to divide $K_{*}K$
into two parts, the first of which admits a convenient interpretation in terms
of formal groups and the second of which is more mysterious, but manageably
small. We start with the latter, for which we will need to work relative to
$E$.

######  Notation 2.5.

If $M,N$ are $E$-modules, we will write

$M_{*}^{E}N:=\pi_{*}(M\otimes_{E}N)$

for their homology relative to $E$ and

$M^{*}_{E}N:=[N,M]^{*}_{E}\simeq\pi_{-*}F_{E}(N,M)$

for their relative cohomology. Here, $F_{E}$ is the internal mapping
$E$-module; that is, the right adjoint to the tensor product of $E$-modules.

By usual arguments involving flatness of $K_{*}^{E}K$ over $K_{*}$, the map

$K\otimes_{E}K\rightarrow K\otimes_{E}K\otimes_{E}K$

induces a $K_{*}$-coalgebra structure on $K_{*}^{E}K$. Note that this is an
honest coalgebra structure; that is, the left and right units coincide, as
they are necessarily maps of $E_{*}$-algebras, of which $K_{*}$ is a quotient.
Together with multiplication, this makes $K_{*}^{E}K$ into a Hopf algebra,
which a priori need not be either commutative nor cocommutative.

######  Lemma 2.6.

The coalgebra structure on $K_{*}^{E}K$ is dual to an exterior algebra on an
$n$-dimensional vector space $V$ in degree $-1$. In particular, $K_{*}^{E}K$
is cocommutative.

###### Proof.

Since the coefficients of $K$ form a field, we have an isomorphism of algebras

$\operatorname{Hom}_{K_{*}}(K_{*}^{E}K,K_{*})\simeq[K,K]_{E},$

where on the right hand side we have maps of $E$-modules. One can show that
the latter is always isomorphic to an exterior algebra over a vector space of
the needed dimension, see [HL17, Proposition 6.5.1]. ∎

######  Remark 2.7.

Hopkins and Lurie show that $K_{*}^{E}K$ is actually isomorphic to an exterior
algebra _as a Hopf algebra_ , though there is in general no canonical such
isomorphism [HL17, Proposition 5.2.4]. In particular, it is also always
commutative.

######  Remark 2.8.

As both sides are $K_{*}$-vector spaces of dimension $2^{n}$, the latter by
Lemma 2.6, the Künneth spectral sequence

$\operatorname{Tor}^{t,s}_{E_{*}}(K_{*},K_{*})\implies K_{s+t}^{E}K$

collapses. Indeed, since $E_{0}$ is regular ring of dimension $n$, the
relevang $\operatorname{Tor}$-groups are canonically isomorphic to an exterior
algebra on $n$ generators.

The second, “understandable” part of $K_{*}K$ is the image of the canonical
map from $E_{*}E$, which can be identified with $K_{*}E$. The analogue of this
Hopf algebroid for the minimal Morava $K$-theory of §3 is denoted in Ravenel’s
book by $\Sigma(n)$, see [Rav03, 6.2].

Notice that since $E_{*}E$ is flat over $E_{*}$, we have canonical
isomorphisms

$K_{*}E\simeq K_{*}\otimes_{E_{*}}E_{*}E\simeq E_{*}E/\mathfrak{m},$

which in turn has the following consequence.

######  Proposition 2.9.

The category of $K_{*}E$-comodules is independent of $K$; more precisely, for
any two choices of Morava $K$-theories, possibly over different Lubin–Tate
spectra, these categories of comodules are canonically equivalent as symmetric
monoidal categories.

###### Proof.

By a result of Hovey–Strickland, see [HS05a, Theorem C], the categories of
$E_{*}E$-comodules for varying Lubin–Tate spectra $E$ are canonically
equivalent as locally graded symmetric monoidal categories.

The category of $K_{*}E$-comodules can be identified with $K_{*}$-modules in
the symmetric moonidal category $\operatorname{Comod}_{E_{*}E}$, and
$K_{*}\simeq E_{*}/\mathfrak{m}$ itself is uniquely determined as the minimal
quotient of $E_{*}$ as a comodule by [HS05a, Theorem D]. Thus, the equivalence
of Hovey and Strickland induces one between categories of $K_{*}E$-comodules.
∎

In fact, we can give a direct algebro-geometric description of this category
of comodules. By standard results about Landweber exact homology theories we
have

$E_{*}E\simeq E_{*}\otimes_{BP_{*}}BP_{*}BP\otimes_{BP_{*}}E_{*}.$

and by again invoking flatness we see that

$K_{*}E\simeq K_{*}\otimes_{BP_{*}}BP_{*}BP\otimes_{BP_{*}}E_{*}$

Since the ideal $I_{n}$ is invariant, we can instead write

$K_{*}E\simeq K_{*}\otimes_{BP_{*}}BP_{*}BP\otimes_{BP_{*}}K_{*}.$

This is a familiar Hopf algebroid, commutative ring homomorphisms out of which
classify pairs of homomorphisms $f_{1},f_{2}:K_{*}\rightarrow A$ together with
a strict isomorphism of the resulting formal groups
$(f_{1})^{*}\mathbf{G}_{0}=(f_{2})^{*}\mathbf{G}_{0}$. In other words, we have
a pullback of algebraic stacks

${{\mathrm{Spec}(K_{*}E)}}$${{\mathrm{Spec}(K_{*})}}$${{\mathrm{Spec}(K_{*})}}$${{\mathcal{M}_{fg}^{\omega=\mathrm{triv}}},}$

where $\mathcal{M}_{fg}^{\omega=\mathrm{1}}$ is the moduli of formal groups
with a trivialized Lie algebra. The maps from $\operatorname{Spec}(K_{*})$ are
faithfully flat surjections onto the height $n$ point in this moduli stack,
and we deduce that the groupoid
$(\operatorname{Spec}(K_{*}),\operatorname{Spec}(K_{*}E))$ is its
presentation.

######  Proposition 2.10.

The category of even graded $K_{*}E$-comodules is equivalent to the category
of quasi-coherent sheaves over the moduli stack $\mathcal{M}_{fg}^{n}$ of
formal groups of height exactly $n$.

###### Proof.

By the discussion above, the category of ungraded $K_{*}E$-comodules can be
identified with the category of quasi-coherent sheaves over the moduli stack
$\mathcal{M}_{fg}^{n,\omega=1}$ of formal groups of height $n$ with
trivialized Lie algebra.

The even grading of $K_{*}E$ corresponds under this equivalence to the
$\mathbf{G}_{m}$-action on the chosen trivialization of the Lie algebra, so
that even graded comodules can be identified with the quasi-coherent sheaves
over the quotient stack

$\faktor{\mathcal{M}_{fg}^{n,\omega=1}}{\mathbf{G}_{m}}\simeq\mathcal{M}_{fg}^{n},$

which is exactly the claim. ∎

The above discussion identifies $K_{*}E$ with familiar objects from the theory
of formal groups, which is why we referred to it above as the “understandable”
part of $K_{*}K$. The following gives some control over how these two parts
are related.

######  Lemma 2.11.

Consider $E\otimes K$ as an $E$-module with the module structure inherited
only from the left factor. Then, $E\otimes K$ is equivalent as an $E$-module
to a direct sum of $K$.

###### Proof.

Since $E_{*}E$ is flat over $E_{*}$, it follows from Corollary 8.5 below that
$L_{K}(E\otimes E)$ is equivalent as an $E$-modules to a $K$-local direct sum
of $E$. This is a corollary to the work of Hovey and it is independent of the
arguments given here.

Thus, it follows that there is a map $\bigoplus E\rightarrow L_{K}(E\otimes
E)$ which is an equivalence after applying $-\otimes K$. As the relative
tensor product $-\otimes_{E}K$ can be computed using the bar construction, it
also takes $K$-local equivalences of $E$-modules to equivalences. We deduce
that

$\bigoplus K\simeq(\bigoplus E)\otimes_{E}K\simeq L_{K}(E\otimes
E)\otimes_{E}K\simeq E\otimes E\otimes_{E}K\simeq E\otimes K,$

which is what we wanted to show. ∎

######  Remark 2.12.

Note that one can alternatively consider $E\otimes K$ as an $E$-module with
the module structure inherited form the right factor. In this case, it is
clear that it is a direct sum of $K$ as an $E$-module, as it is even a module
over the $E$-algebra $K$.

######  Corollary 2.13.

Consider $K\otimes E$ as an $E$-module with the module structure inherited
only from the right. Then, the Künneth spectral sequence

$\operatorname{Tor}_{E_{*}}(K_{*}E,K_{*})\implies\pi_{*}((K\otimes
E)\otimes_{E}K)\simeq K_{*}K$

in $E$-modules collapses.

###### Proof.

This is immediate from Lemma 2.11 and Remark 2.8, as the Künneth spectral
sequence depends only on the $E$-module structure. ∎

######  Proposition 2.14.

The map $K_{*}E\rightarrow K_{*}K$ is injective, central, and presents the
target as a free $K_{*}E$-module of rank $2^{n}$.

###### Proof.

To see that the map is central, notice that $K\otimes K$ is canonically an
$E\otimes E$-algebra. It follows that the map $E_{*}E\rightarrow K_{*}K$ is
central, and thus so must be its image $K_{*}E\simeq
K_{*}\otimes_{E_{*}}E_{*}E$.

As a consequence of Corollary 2.13, the Künneth spectral sequence

$K_{*}E\otimes\operatorname{Tor}_{E_{*}}(K_{*},K_{*})\simeq\operatorname{Tor}_{E_{*}}(K_{*}E,K_{*})\implies
K_{*}K$

in $E$-modules collapses. Since the $\operatorname{Tor}$-groups on the left
form a $K_{*}$-vector space of dimension $2^{n}$, it follows from the above
collapse that $K_{*}K$ has a finite filtration as a $K_{*}E$-module such that
the associated graded is free of rank $2^{n}$. Thus, $K_{*}K$ itself must be
free of this rank as well. ∎

######  Proposition 2.15.

Passing to homotopy groups in the diagram

${{K\otimes E}}$${{K\otimes K}}$${{K\otimes_{E}E}}$${{K\otimes_{E}K}}$

induces an an isomorphism

$K_{*}K\otimes_{K_{*}E}K_{*}\simeq K_{*}^{E}K.$

###### Proof.

Both of the Künneth spectral sequences for $K\otimes K\simeq(K\otimes
E)\otimes_{E}K$ and $K\otimes_{E}K$ collapse, as a consequence of Corollary
2.13 and Remark 2.8. Thus, the map $K_{*}K\rightarrow K_{*}^{E}K$ is
surjective after passing to associated graded, and so surjective by an
inductive argument. The conclusion follows by observing that
$K_{*}\otimes_{K_{*}E}K_{*}K\rightarrow K_{*}^{E}K$ is a surjective map of
$K_{*}$-vector spaces of the same dimension. ∎

We are now ready to assemble the above information. By standard arguments, the
pair $(K_{*},K_{*}K)$ acquires the structure of a Hopf algebroid. More
precisely, the two maps $K\rightarrow K\otimes K$ induce left and right units,
both of which are central as a consequence of Proposition 2.14, as they factor
through $K_{*}E$. These make $K_{*}K$ into a $K_{*}$-bimodule, and similarly
we get a suitable comultiplication and antipode.

Note that it is common in algebraic topology literature to assume in the
definition of a Hopf algebroid that multiplication is commutative. However,
this is not strictly necessary; in our case, as the units are central,
$K_{*}K$ is a Hopf algebroid in the more general sense of Maltsiniotis
[Mal92]. The category of comodules can be defined in the usual way, it will be
monoidal with the tensor product lifting that of $K_{*}$-modules. It need not
in general be symmetric monoidal.

######  Remark 2.16.

We believe it is plausible that $K_{*}K$ is in fact always commutative; it is
a finite algebra over the commutative $K_{*}E$ with the relative tensor
product $K_{*}\otimes_{K_{*}E}K_{*}K\simeq K_{*}^{E}K$ also commutative, as a
consequence of Remark 2.7. We were, however, not able to resolve this
question. Note that $K_{*}K$ is automatically commutative if we choose a
homotopy commutative Morava $K$-theory, as can always be done when $p>2$
[HL17, Proposition 3.5.2].

######  Theorem 2.17.

The natural maps

$\displaystyle(K_{*},K_{*}E)\rightarrow(K_{*},K_{*}K)\rightarrow(K_{*},K_{*}^{E}K)$
(2.1)

form an extension of Hopf algebroids in the sense of Ravenel.

###### Proof.

Since $K_{*}^{E}K$ is a cocommutative, consulting Ravenel’s definition of an
extension [Rav03, §A.1.1], we see that we have to verify that

$\operatorname{Ext}^{0}_{K_{*}^{E}K}(K_{*},K_{*})\simeq K_{*}$

and

$\operatorname{Ext}^{0}_{K_{*}^{E}K}(K_{*},K_{*}K)\simeq K_{*}E.$

The first part is clear, as $K_{*}^{E}K$ is a Hopf algebra, while the second
part requires a little bit of work. If we write

$K_{*}K\simeq\pi_{*}(K\otimes K)\simeq\pi_{*}(K\otimes_{E}E\otimes K)\simeq
K_{*}^{E}(E\otimes K),$

then the left $K_{*}^{E}K$-comodule structure on $K_{*}K$ given by the
quotient map $K_{*}K\rightarrow K_{*}^{E}K$ corresponds to the standard
comodule structure

$K_{*}^{E}(E\otimes K)\rightarrow K_{*}^{E}K\otimes_{K_{*}}K_{*}^{E}(E\otimes
K)$

on the homology $K_{*}^{E}(E\otimes K)$ relative to the $\infty$-category of
$E$-modules.

The comodule structure map is determined by the structure of $E\otimes K$ as a
left $E$-module with module structure coming only from the left factor. By
Lemma 2.11, $E\otimes K$ is equivalent as an $E$-module to a direct sum of
$K$, and so admits a compatible structure of a $K$-module. Choosing such a
$K$-module structure determines an isomorphism

$K_{*}^{E}(E\otimes K)\simeq K_{*}^{E}K\otimes\pi_{*}(E\otimes K)\simeq
K_{*}^{E}K\otimes E_{*}K$

so that

$\operatorname{Ext}^{0}_{K_{*}^{E}K}(K_{*},K_{*}K)\simeq\operatorname{Ext}^{0}_{K_{*}^{E}K}(K_{*},K_{*}^{E}(E\otimes
K))\simeq\operatorname{Ext}^{0}_{K_{*}^{E}K}(K_{*},K_{*}^{E}K\otimes
E_{*}K)\simeq E_{*}K$

as needed. ∎

###  3\. Digression: Minimal Morava $K$-theory

Large parts of the literature on Morava $K$-theories are written in terms of
the _minimal Morava $K$-theory spectrum_ $K(n)$, the unique up to equivalence
$BP$-module with $K(n)_{*}\simeq\mathbb{F}_{p}[v_{n}^{\pm 1}]$. Note that it
is not a Morava $K$-theory in the sense of Definition 2.1 as it is not
$2$-periodic, but as a consequence of nilpotence theorem any Morava $K$-theory
is equivalent as a spectrum to a direct sum of $K(n)$ [HS98, Lemma 1.8].

Our focus on $2$-periodic Morava $K$-theories stems from our preference for
Lubin–Tate spectra as they admit canonical $\mathbf{E}_{\infty}$-structures,
unlike the Brown–Peterson spectrum. In this short section, we collect some
results about $K(n)$ to highlight similarities and differences with the
$E$-algebra version. We do this for completness; these results will not be
used elsewhere in the current work.

It is a result of Robinson that $K(n)$ can be made into an
$\mathbf{E}_{1}$-ring spectrum, but there is no canonical way to do so
[Rob89]. A surprising result of Angelveit shows that all choices of
$\mathbf{E}_{1}$-multiplication yield equivalent $\mathbf{E}_{1}$-algebras in
spectra [Ang11].

One can also ask about less structured multiplication. By a theorem of Würgler
[Wür91], in the homotopy category $h\mathcal{S}p$ of spectra, the spectrum
$K(n)$ admits

  1. (1)

if $p>2$, a unique product $K(n)\otimes K(n)\rightarrow K(n)$;

  2. (2)

if $p=2$, exactly two products $K(n)\otimes K(n)\rightarrow K(n)$ differing by
the symmetry on $K(n)\otimes K(n)$,

which make it into an associative $BP$-algebra in a way compatible with its
$BP$-module structure [Wür91, Theorem 1.5].

The ring of cooperations of $K(n)$ has been computed completely [Wür91,
Theorem 2.4]. The map $K(n)_{*}BP\rightarrow K(n)_{*}K(n)$ is injective and we
have an isomorphism

$K(n)_{*}BP\simeq K(n)_{*}[t_{i}]/(v_{n}t_{i}^{p^{n}}-v_{n}^{p^{i}}t_{i}),$

where $|t_{i}|=2p^{i}-2$ for $i\geq 1$. At odd primes, this extends to an
isomorphism

$K(n)_{*}K(n)\simeq
K(n)_{*}BP\otimes_{K(n)_{*}}\Lambda_{K(n)_{*}}(\tau_{0},\ldots,\tau_{n-1}),$

where the latter factor is an exterior algebra on $|\tau_{i}|=2p^{i}-1$. When
$p=2$, we instead have that

$K(n)_{*}K(n)\simeq
K(n)_{*}BP\otimes_{K(n)_{*}}K(n)_{*}[\tau_{0},\ldots,\tau_{n-1}]/(\tau_{i}^{2}=t_{i+1}).$

This is analogous to the variation in the dual Steenrod algebra depending on
whether $p$ is even or odd. Note that the latter computation implies that
$K(n)_{*}K(n)$ is commutative also at $p=2$, despite $K(n)$ not being homotopy
commutative. The corresponding question in the $2$-periodic case is open, see
Remark 2.16.

###  4\. Milnor modules

As we have seen in the previous section, there are many different Morava
$K$-theory spectra, which makes any discussion of this subject dependent on
these choices. However, as in the case of Proposition 2.9, one show that many
cohomological invariants are canonical and do not depend on any choices. In
this section, we will make this canonical nature transparent by introducing an
absolute variant of Hopkins and Lurie’s categories of _Milnor modules_ , see
[HL17].

Before we work in the absolute case, let us first describe the case relative
to a fixed Lubin–Tate spectrum $E$. Note that by the uniqueness part of
Goerss–Hopkins–Miller, this is the same as choosing a perfect field equipped
with a finite height formal group.

######  Definition 4.1.

We say that an $E$-module is _molecular_ if it is equivalent to a finite sum
of shifts of Morava $K$-theories of $E$, and denote the full subcategory of
$E$-modules on the molecular objects by $\operatorname{Mod}^{mol}_{E}$.

######  Remark 4.2.

Note that since any two Morava $K$-theories of $E$ are equivalent as
$E$-modules, the notion of being molecular does not depend on any choices.

######  Definition 4.3.

The category of $E$-based _Milnor modules_ is the category

$\operatorname{Mil}_{E}:=P_{\Sigma}(\operatorname{Mod}^{mol}_{E},\operatorname{Ab})$

of additive presheaves of abelian groups on molecular $E$-modules.

By construction, the category of Milnor modules is a compactly generated
Grothendieck abelian category. Since molecular $E$-modules are stable under
the $E$-tensor product, it acquires a non-unital symmetric monoidal structure
via left Kan extension. One can show this extends uniquely to a unital
symmetric monoidal structure [HL17, Propositions 4.4.1 and 4.4.10].

The main property of Milnor modules is that it canonically captures cohomology
of _any_ Morava $K$-theory relative to $E$, in the following sense.

######  Proposition 4.4.

Let $K$ be a Morava $K$-theory of $E$. Then, the left Kan extension of the
functor

$K_{*}^{E}(-):\operatorname{Mod}_{E}^{mol}\rightarrow\operatorname{Comod}_{K_{*}^{E}K}$

which associates to any molecular $E$-module its relative $K$-homology induces
a monoidal equivalence

$\operatorname{Mil}_{E}\simeq\operatorname{Comod}_{K_{*}^{E}K}$

between Milnor modules and $K_{*}^{E}K$-comodules.

###### Proof.

This is [HL17, Corollary 6.4.13], where we replaced the category of modules
over $K^{*}_{E}K$ with comodules over its linear dual. ∎

######  Corollary 4.5.

As a monoidal category, $\operatorname{Comod}_{K_{*}^{E}K}$ does not depend on
the choice of $K$, but only on its corresponding formal group.

######  Remark 4.6.

Note that in the context of Corollary 4.5, the situation is somewhat better
than just saying that the categories $\operatorname{Comod}_{K_{*}^{E}K}$ for
fixed $E$ and varying $K$ are all equivalent to each other. In fact, they are
all canonically equivalent to a single category which doesn’t depend on any
choices, namely the category of Milnor modules of $E$.

######  Remark 4.7.

The category $\operatorname{Mil}_{E}$ is not only canonical, but in a certain
sense gives the right answer even if $K_{*}^{E}K$-comodules do not. As an
example, the restricted Yoneda embedding

$y\colon\operatorname{Mod}_{E}\rightarrow\operatorname{Mil}_{E},$

which can be thought of as a way of taking relative $K$-homology without
choosing $K$, is always a symmetric monoidal functor [HL17, Variant 4.4.11].

On the other hand, even if $K$ is not homotopy commutative, $K_{*}^{E}K$ is
both commutative and cocommutative so that the category
$\operatorname{Comod}_{K_{*}^{E}K}$ acquires a symmetric monoidal structure
from $K_{*}$-vector spaces. However, in this case the relative $K$-homology
functor

$K_{*}^{E}(-):\operatorname{Mod}_{E}\rightarrow\operatorname{Comod}_{K_{*}^{E}K}$

need not be symmetric monoidal. In fact, one can show that if $p=2$ and $n>0$,
this functor is _never_ symmetric monoidal, even when restricted to molecular
modules. Indeed, any such functor would induce a symmetric monoidal
equivalence $\operatorname{Comod}_{K_{*}^{E}K}\simeq\operatorname{Mil}_{E}$,
which one can show is not possible at the even prime333Personal communication
with Jacob Lurie..

If $K$ is homotopy commutative, then the above homology theory is canonically
symmetric monoidal so that we have a symmetric monoidal equivalence
$\operatorname{Comod}_{K_{*}^{E}K}\simeq\operatorname{Mil}_{E}$. Informally,
any algebraic structure which one can put on
$\operatorname{Comod}_{K_{*}^{E}K}$ in a way compatible with the homology
functor from $E$-modules must already be present in the category of Milnor
modules.

Let us now describe the global analogue of this situation, which describes the
Adams spectral sequence in spectra rather than $E$-modules; alternatively,
which describes comodules over $K_{*}K$ rather than the relative variant
$K_{*}^{E}K$.

In Miller’s approach to the Adams spectral sequence, the latter is determined
by the class of maps which are $K$-split; that is, which are split
epimorphisms after applying $K\otimes-$. This only depends on the structure of
$K$ as a spectrum and does not require any coherent multiplication.

As a consequence of the nilpotence theorem, all Morava $K$-theories at a fixed
prime and height are, as spectra, direct sums of the minimal Morava $K$-theory
spectrum of Section 3. It follows that they all determine the same class of
epimorphisms, and hence isomorphic Adams spectral sequences. In fact, the
class of epimorphisms is enough to describe canonically the abelian category
whose $\operatorname{Ext}$-groups form the $E_{2}$-page. To do so, we will
consider sheaves with respect to a certain topology, as in the construction of
synthetic spectra [Pst18b].

######  Definition 4.8.

The _$K_{*}$ -epimorphism_ topology on the $\infty$-category
$\mathcal{S}p^{fin}$ of finite spectra is the Grothendieck pretopology in
which a covering families $\\{X\rightarrow Y\\}$ consist of a single map such
that $K_{*}X\rightarrow K_{*}Y$ is surjective.

######  Remark 4.9.

A reader familiar with [Pst18b] might recall that in the construction of
synthetic spectra, one works with the site of those finite spectra $X$ such
that $K_{*}X$ is projective as a $K_{*}$-module [Pst18b, Definition 3.12].
Since Morava $K$-theories are fields, this yields the class of all finite
spectra, as above.

######  Definition 4.10.

The category of _absolute Milnor modules_ is the category

$\operatorname{Mil}_{abs}:=\operatorname{Shv}_{\Sigma}(\operatorname{Sp}^{fin},\operatorname{Ab})$

of additive sheaves of abelian groups on finite spectra with respect to the
$K_{*}$-epimorphism topology.

The category of absolute Milnor modules is a compactly-generated Grothendieck
abelian category with a symmetric monoidal structure induced by that of the
tensor product of finite spectra.

######  Remark 4.11.

A reader familiar with synthethic spectra will immediately recognize that the
category of Milnor modules can be canonically identified as
$\operatorname{Mil}_{abs}\simeq\mathcal{S}yn_{K}^{\heartsuit}$, the heart of
the stable $\infty$-category of $K$-based synthetic spectra of [Pst18b].

The following is the absolute analogue of Proposition 4.4.

######  Proposition 4.12.

If $K$ is a Morava $K$-theory, then the left Kan extension of the functor

$K_{*}:\operatorname{Sp}^{fin}\rightarrow\operatorname{Comod}_{K_{*}K}$

induces a monoidal equivalence

$\operatorname{Mil}_{abs}\simeq\operatorname{Comod}_{K_{*}K}$

between absolute Milnor modules and $K_{*}K$-comodules.

###### Proof.

This is an instance of [Pst18b, Remark 3.26]. ∎

As a consequence, we deduce the following result proven earlier by Hovey and
Strickland using slightly different methods [HS05a, §8].

######  Corollary 4.13.

The monoidal category $\operatorname{Comod}_{K_{*}K}$ of $K_{*}K$-comodules
does not depend on the choice of Morava $K$-theory, but only on the prime and
height.

######  Remark 4.14.

By the same argument as in the relative case of Remark 4.7, if $K$ is homotopy
commutative, then we have a canonical symmetric monoidal equivalence
$\operatorname{Mil}_{abs}\simeq\operatorname{Comod}_{K_{*}K}$.

######  Remark 4.15 (Homological residue fields).

In the terminology of Balmer, the category of Milnor modules is the
_homological residue field_ of the tensor-triangulated category of finite
spectra at the prime ideal of spectra of type at least $n$, see [Bal20],
[BC21].

######  Remark 4.16 (Milnor modules as sheaves).

The reader can observe that there is a certain asymmetry in our definitions of
Milnor modules and absolute Milnor modules; namely, in the former case we use
as our indexing category only molecular $E$-modules and in the latter case we
use _all_ finite spectra. Moreover, in the latter case we use sheaves while in
the former presheaves are already enough.

In fact, this difference is but a trick of light, as there is an alternative
description of Milnor modules which is more in line with the absolute case.
Namely, one can show that $\operatorname{Mil}_{E}$ is equivalent to the
category of sheaves on the $\infty$-category $\operatorname{Mod}^{perf}_{E}$
of perfect $E$-modules with respect to the $K_{*}^{E}(-)$-epimorphism
topology. Under this topology, the functor

$E\otimes-\colon\mathcal{S}p^{fin}\rightarrow\operatorname{Mod}_{E}^{perf}$

becomes a morphism of sites, so that left Kan extension yields a cocontinuous,
symmetric monoidal functor

$\operatorname{Mil}_{abs}\rightarrow\operatorname{Mil}_{E}.$

In this sense absolute Milnor modules determine a relative Milnor module, for
any choice of $E$, justifying the terminology.

Under the equivalences of Proposition 4.4 and Proposition 4.12 the above left
Kan extension corresponds to the forgetful functor

$\operatorname{Comod}_{K_{*}K}\rightarrow\operatorname{Comod}_{K_{*}^{E}K}.$

The image of the monoidal unit under the corresponding right adjoint can be
identified with the Hopf algebroid $K_{*}E$ studied previously. As the Adams
spectral sequence based on $K_{*}E$ in the appropriate derived
$\infty$-categories can be identified with the Cartan–Eilenberg spectral
sequence associated to this extension of Hopf algebroids [Bel20, 1.1], we
deduce that the Cartan–Eilenberg spectral sequence itself can be constructed
using only the categories of Milnor modules, so it only depends on $E$.

###  5\. Cohomology

In this section, we establish finiteness properties of the cohomology of
$K_{*}K$-comodules using the Cartan–Eilenberg spectral sequence.

We have seen in Theorem 2.17 that we have an extension of Hopf algebroids

$(K_{*},K_{*}E)\rightarrow(K_{*},K_{*}K)\rightarrow(K_{*},K_{*}^{E}K)$

which induces a Cartan–Eilenberg spectral sequence of signature

$\operatorname{Ext}^{s}_{K_{*}E}(K_{*},\operatorname{Ext}^{s^{\prime}}_{K_{*}^{E}K}(K_{*},K_{*}))\implies\operatorname{Ext}^{s+s^{\prime}}_{K_{*}K}(K_{*},K_{*}),$

see [Rav03, A.1.3.14]. This can be thought as approximating the cohomology of
$K_{*}K$ using simplier pieces, namely the cohomology of $K_{*}E$ and
$K_{*}^{E}K$. To make best use of this, we need to begin by understanding the
latter two.

We have seen in Section 2 that the Hopf algebroid $(K_{*},K_{*}E)$ is an even
graded presentation of the moduli of formal groups of height exactly $n$, see
Proposition 2.10. To identify its cohomology, it is convenient to choose a
particularly small presentation of the latter moduli stack, which is what we
will do.

######  Notation 5.1.

Let $\mathbf{G}_{0}$ be the Honda formal group law over $\mathbb{F}_{q}$,
where $q=p^{n}$, and let $E_{n}$ be the associated Lubin–Tate spectrum and
$K_{n}$ a Morava $K$-theory associated to $E_{n}$. Note that we have a non-
canonical isomorphism

$\pi_{0}E_{n}\simeq W(\mathbb{F}_{q})[[u_{1},\ldots,u_{n-1}]].$

We write
$\mathbb{G}_{n}\simeq\mathbb{S}_{n}\rtimes\textnormal{Gal}(\mathbb{F}_{q}/\mathbb{F}_{p})$
for the _extended Morava stabilizer group_ , the automorphism group of the
pair $\mathbb{F}_{q}$ together with $\mathbf{G}_{0}$. By the uniquenses part
of Goerss–Hopkins–Miller, the action of $\mathbb{G}_{n}$ extends to an action
on the Lubin–Tate spectrum $E_{n}$.

######  Lemma 5.2.

The association $\mathbb{G}_{n}\times(K_{n})_{*}E_{n}\rightarrow(K_{n})_{*}$
sending an element $g\in\mathbb{G}_{n}$ and a homotopy class $S^{k}\rightarrow
K_{n}\otimes E_{n}$ to the homotopy class of the composite

${{S^{k}}}$${{K_{n}\otimes E_{n}}}$${{K_{n}\otimes
E_{n}}}$${K_{n},}$$\scriptstyle{id\otimes g}$$\scriptstyle{m}$

induces an isomorphism
$\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{n},(K_{n})_{*})\simeq(K_{n})_{*}E_{n}$
of algebras between the $K_{n}$-homology of $E_{n}$ and continuous functions
on the Morava stabilizer group.

###### Proof.

This is a result of Strickland, reduced modulo $\mathfrak{m}$ [Str00, Theorem
12]. ∎

One can check that the above isomorphism is compatible with the Hopf algebroid
structure, which is induced from the group structure on $\mathbb{G}_{n}$ and
its action on $(K_{n})_{*}\simeq\mathbb{F}_{q}[u^{\pm 1}]$, which yields the
following.

######  Corollary 5.3.

For any Morava $K$-theory $K$, there is a canonical equivalence of categories
between $K_{*}E$-comodules and graded $\mathbb{F}_{q}[u^{\pm 1}]$-vector
spaces equipped with a continuous action of the extended Morava stabilizer
group $\mathbb{G}_{n}$.

###### Proof.

We have proven previously in Proposition 2.9 that the category of
$K_{*}E$-comodules is indendent of the choice of $K$. For the particular
choice of $K=K_{n}$ as above, this isomorphism is induced by Lemma 5.2. ∎

######  Remark 5.4.

In terms of the equivalence
$\operatorname{Comod}_{K_{*}E}^{ev}\simeq\operatorname{QCoh}(\mathcal{M}_{fg}^{n})$
of Proposition 2.10, Corollary 5.3 is a consequence of the fact that the map
$\mathbb{F}_{q}\rightarrow\mathcal{M}_{fg}^{n}$ classifying the Honda formal
group law is a Galois covering with automorphism group $\mathbb{G}_{n}$; we
refer to [Goe08] for further details.

######  Corollary 5.5.

For any Morava $K$-theory, there is a canonical isomorphism

$\operatorname{Ext}_{K_{*}E}^{s,t}(K_{*},K_{*})\simeq\mathrm{H}^{*}_{\operatorname{cts}}(\mathbb{G}_{n},\mathbb{F}_{q}[u^{\pm
1}])$

between the cohomology of the Hopf algebraid $(K_{*},K_{*}E)$ and the
continuous cohomology of the Morava stabilizer group.

The group
$\mathbb{G}_{n}\simeq\mathbb{S}_{n}\rtimes\textnormal{Gal}(\mathbb{F}_{q}/\mathbb{F}_{p})$
is a semidirect product of the automorphism group $\mathbb{S}_{n}$ of
$\mathbf{G}_{0}$ and the Galois group. The former is a $p$-adic Lie group, in
fact a group of units in integers in a central division algebra over
$\mathbb{Q}_{p}$ of Hasse invariant $\frac{1}{n}$, which gives its excellent
cohomological properties.

The action of the Morava stabilizer group on $\mathbb{F}_{q}[u^{\pm 1}]$ is
easy to describe. The Galois group acts by its standard action on
$\mathbb{F}_{q}$. The action of $\mathbb{S}_{n}$ factors through the leading
coefficient homomorphism
$\rho\colon\mathbb{S}_{n}\rightarrow\mathbb{F}_{q}^{\times}$ and is determined
by the formula $s\cdot u=p(s)u$ for $s\in\mathbb{S}_{n}$. The kernel of $\rho$
is the strict Morava stabilizer group $S_{n}$; it is a pro-$p$-group acting
trivially. Morava [Mor85] essentially proved:

######  Theorem 5.6 (Morava).

We have isomorphisms of continuous cohomology groups

$\mathrm{H}_{\operatorname{cts}}^{*}(\mathbb{G}_{n},\mathbb{F}_{q}[u^{\pm
1}])\simeq\mathrm{H}^{*}_{\operatorname{cts}}(\mathbb{S}_{n},\mathbb{F}_{q}[u^{\pm
1}])^{\mathrm{Gal}}\simeq\mathrm{H}^{*}_{\operatorname{cts}}(S_{n},\mathbb{F}_{q}[u^{\pm
1}])^{\mathbb{F}_{q}^{\times}\rtimes\mathrm{Gal}}.$

Moreover, these groups are finite in each bidegree. If $(p-1)\nmid n$, this is
a Poincaré duality algebra over
$\mathrm{H}_{\operatorname{cts}}^{0}\simeq\mathbb{F}_{p}[v_{n}^{\pm 1}]$ of
dimension $n^{2}$, where $v_{n}:=u^{p^{n}-1}$, in particular their total
dimension is finite.

###### Proof.

These isomorphism are induced by the collapse of the respective
Lyndon–Hochschild–Serre spectral sequences, as neither $\mathrm{Gal}$ nor
$\mathbb{F}_{q}^{\times}$ have any higher cohomology with these coefficients,
the first one by Galois theory and the second one as it is of order coprime to
$p$.

The finiteness statements are due to Lazard, as $S_{n}$ is a $p$-adic analytic
Lie group of dimension $n^{2}$ and so has a finite index subgroup whose
cohomology is isomorphic to an exterior algebra [SW00, Theorem 5.1.5] of
dimension $n^{2}$. If $(p-1)\nmid n$, then $S_{n}$ has no $p$-torsion, see
[Hen98, Theorem 3.2.1], and so it has the same cohomological dimension as all
of its finite index subgroups [Ser65]. ∎

######  Corollary 5.7.

For any finite-dimensional $K_{*}E$-comodules $M,N$, the groups
$\operatorname{Ext}^{t,s}_{K_{*}E}(M,N)$ are finite in each bidegree. If
$(p-1)\nmid n$, then they vanish for $s>n^{2}$.

###### Proof.

We first claim that any finite-dimensional $K_{*}E$-comodule $M$ has an
invariant element; that is, there exists a non-zero map $K_{*}\rightarrow M$
of comodules. Observe that $M$ can be considered as an $E_{*}E$-comodule using
restriction along the maps $E_{*}E\rightarrow K_{*}E$ and $E_{*}\rightarrow
K_{*}$. As $M$ is finitely generated over $E_{*}$, it admits a non-zero map
$E_{*}\rightarrow M$ of comodules by a result of Hovey and Strickland [HS05a,
Theorem D]. As the target is $\mathfrak{m}$-torsion, the map will factor
through a map $K_{*}\rightarrow M$ of $K_{*}E$-comodules, which is what we
were looking for.

Observe that the result holds when $M=N=K_{*}$ as a combination of Corollary
5.5 and Theorem 5.6. Since $K_{*}$ is a field, it follows that from the above
that any finite-dimensional $K_{*}E$ comodule can be obtained as an iterated
extension of $K_{*}$. Thus, the general case is proven by presenting $M,N$ as
such iterated extensions and using the corresponding long exact sequences of
$\operatorname{Ext}$-groups. ∎

To get hold of the Cartan–Eilenberg spectral sequence, we also need to have
some control over the cohomology of $K_{*}^{E}K$. By a result of Hopkins and
Lurie, see Remark 2.7, the latter is isomorphic to an exterior algebras as a
Hopf algebra and so its cohomology is necessarily a polynomial ring. In fact,
the same authors identify the generators of this polynomial ring, as we now
recall.

Suppose that $x\in\mathfrak{m}$ is an element of the maximal ideal. We then
have a fibre sequence

$\Sigma^{-1}E\rightarrow E/x\rightarrow E$

of $E$-modules which induces a short exact sequence

$0\rightarrow K_{*}[-1]\rightarrow K_{*}^{E}(E/x)\rightarrow K_{*}\rightarrow
0$

of $K_{*}^{E}K$-comodules. Let us denote by $\psi(x)$ the correspoding element
of $\operatorname{Ext}^{1,-1}_{K_{*}^{E}K}(K_{*},K_{*})$.

######  Proposition 5.8 (Hopkins–Lurie).

The association $x\mapsto\psi(x)$ induces a map

$\mathfrak{m}/\mathfrak{m}^{2}\rightarrow\operatorname{Ext}_{K_{*}^{E}K}^{1,-1}(K_{*},K_{*})$

which extends to an isomorphism of bigraded rings

$K_{*}\otimes_{\kappa}\operatorname{Sym}_{\kappa}^{*}(\mathfrak{m}/\mathfrak{m}^{2})\simeq\operatorname{Ext}_{K_{*}^{E}K}(K_{*},K_{*}).$

###### Proof.

As we show in Proposition 4.4, the category of $K_{*}^{E}K$-comodules can be
identified, as a monoidal category, with the category of Milnor modules
associated to $E$. It follows that we have

$\operatorname{Ext}^{*}_{K_{*}^{E}K}(K_{*},K_{*})\simeq\operatorname{Ext}^{*}_{\operatorname{Mil}_{E}}(y(E),y(E)),$

the cohomology in Milnor modules, where $y(E)$ is the Milnor module associated
to $E$. For the right hand side, this isomorphism is [HL17, Proposition
7.2.7]. ∎

######  Remark 5.9.

If we again let $K_{n}$ be any Morava $K$-theory associated to the Honda
formal group over $\mathbb{F}_{q}$, it follows that the Cartan–Eilenberg
spectral sequence for cohomology of the unit has signature

$\mathrm{H}_{\operatorname{cts}}^{s_{0}}(\mathbb{G}_{n},\mathbb{F}_{q}[u^{\pm
1}]\otimes_{\mathbb{F}_{q}}\operatorname{Sym}^{s_{1}}_{\mathbb{F}_{q}}(\mathfrak{m}/\mathfrak{m}^{2}))\implies\operatorname{Ext}^{s_{0}+s_{1}}_{K_{*}K}(K_{*},K_{*})$

To determine the action on the symmetric algebra, it is convenient to work in
the context of Milnor modules, as in the proof of Proposition 5.8. The
category of Milnor modules $\operatorname{Mil}_{E}$ is by construction
functorial in automorphisms of $E$ and it it is clear that the association
$x\mapsto\psi(x)$ is natural with respect to the action of the Morava
stabilizer group. It follows that the action of $\mathbb{G}_{n}$ on this
symmetric algebra is the one induced by its action on the cotangent space
$\mathfrak{m}/\mathfrak{m}^{2}$ of the Lubin–Tate ring.

Using the Cartan–Eilenberg spectral sequence, we can prove the following
finiteness statement.

######  Theorem 5.10.

Let $M,N$ be finite-dimensional $K_{*}K$-comodules. Then, the group
$\operatorname{Ext}^{s,t}_{K_{*}K}(M,N)$ is finite in each bidegree.

###### Proof.

By replacing $N$ by $N\otimes M^{\vee}$, where
$M^{\vee}:=\operatorname{Hom}_{K_{*}}(M,K_{*})$ is the linear dual, we can
assume that $M=K_{*}$. We have a Cartan–Eilenberg spectral sequence of
signature

$\operatorname{Ext}^{s_{0}}_{K_{*}E}(K_{*},\operatorname{Ext}^{s_{1}}_{K_{*}^{E}K}(K_{*},N))\implies\operatorname{Ext}^{s_{0}+s_{1}}_{K_{*}K}(K_{*},N)$

The groups contributing to $\operatorname{Ext}^{s}_{K_{*}K}(K_{*},N)$ are
those indexed by $s_{0},s_{1}$ with $s_{0}+s_{1}=s$, of which there are
finitely many. For a fixed $s_{1}$, the group
$\operatorname{Ext}^{s_{1}}_{K_{*}^{E}K}(K_{*},N)$ is finite-dimensional over
$K_{*}$ since both $M$ and $K_{*}^{E}K$ are, and we can compute this
$\operatorname{Ext}$-group using the cobar complex. The finiteness of
$\operatorname{Ext}^{s_{0}}_{K_{*}E}(K_{*},-)$ applied to these groups follows
from Corollary 5.7. ∎

###  6\. Milnor filtration

In this section we describe a filtration on $K_{*}K$ which arises from its
module structure over $E$, which we call the _Milnor filtration_. We show that
at odd primes this filtration can be canonically refined to a grading which
forces the Cartan–Eilenberg spectral sequence associated to the extension
$K_{*}E\rightarrow K_{*}K\rightarrow K_{*}^{E}K$ to collapse.

######  Remark 6.1.

This existence of an additional grading at $p>2$ is a finite height analogue
of the classical argument involving the Cartan–Eilenberg spectral sequence
associated to $H\mathbb{F}_{p}$, see [Rav03, Theorem 4.3.3].

The Milnor filtration will be induced by a suitable filtration of the spectrum
$K\otimes K$. Before we work with such objects, we first recall some standard
definitions and establish terminology.

A non-negatively _filtered spectrum_ is a functor
$P_{\bullet}\colon\mathbb{N}\rightarrow\mathcal{S}p$, where we consider the
natural numbers as a partially ordered set. If $X$ is a spectrum, then a
_filtration_ on $X$ is a filtered spectrum $P_{\bullet}$ together with an
equivalence $\varinjlim P_{\bullet}\simeq X$.

######  Remark 6.2.

Note that according to our terminology, any filtered spectrum is canonically a
filtration of its colimit.

To any filtration $P_{\bullet}$ we can attach an associated non-negatively
graded spectrum

$\mathrm{gr}(P_{\bullet})=\bigoplus_{s}\mathrm{gr}_{s}(P_{\bullet})$

with the degree $s$ part defined by

$\mathrm{gr}_{s}(P_{\bullet}):=\mathrm{cofib}(P_{s-1}\rightarrow P_{s}),$

where $P_{-1}=0$ by convention. The boundary maps

$\Sigma^{-1}\mathrm{gr}_{s}(P_{\bullet})\rightarrow
P_{s-1}\rightarrow\mathrm{gr}_{s-1}(P_{\bullet})$

make $\Sigma^{-s}\mathrm{gr}_{s}(P_{\bullet})$ into a chain complex in the
homotopy category of spectra. Any filtration has a corresponding spectral
sequence of signature

$E^{1}_{s,t}:=\pi_{t}\mathrm{gr}_{s}(\Sigma^{-s}P_{\bullet})\implies\pi_{s+t}X,$

where the first differential is induced by the chain complex structure on the
associated graded [Lur16, §1.2.2]. The increasing filtration on $\pi_{*}X$
associated to this spectral sequence is given by the images of maps
$\pi_{*}P_{s}\rightarrow\pi_{*}X$.

There is a natural tensor product on the $\infty$-category of filtered
spectra, given by Day convolution. This has the properties that

  1. (1)

if $P_{\bullet}$ and $S_{\bullet}$ are filtrations of $X,Y$, then
$P_{\bullet}\otimes S_{\bullet}$ is a filtration of $X\otimes Y$ and

  2. (2)

$\mathrm{gr}(P_{\bullet}\otimes
S_{\bullet})\simeq\mathrm{gr}(P_{\bullet})\otimes\mathrm{gr}(S_{\bullet})$,
where the latter is the tensor product of graded objects.

In fact, the latter equivalence can be upgraded to an isomorphism in the
category of chain complexes in the homotopy category of spectra, where we
consider the latter as symmetric monoidal with the differential on the tensor
product given by the usual Leibniz rule [Rak20, 3.2.15].

######  Example 6.3.

We will need certain special filtrations of $E$-modules, the main reference
here being Lurie [Lur16, §7.2.1], who works with simplicial objects. By the
stable Dold–Kan equivalence, this is equivalent to working with filtered
objects as we do.

If $P_{\bullet}$ is an $E$-module filtration of some $M$, then we say
$P_{\bullet}$ is _$E$ -free_ if the associated simplicial object is $S$-free
with respect to the set $S=\\{\ \Sigma^{n}E\ |\ n\in\mathbb{Z}\ \\}$. We say
it is an $E$-_resolution_ if it is $E$-free and an $S$-hypercover. For precise
definitions of these terms, see [Lur16, Definition 7.2.1.2.]. The two main
facts we will need are that

  1. (1)

if $P_{\bullet}$ is $E$-free, then the associated graded $E$-module is
levelwise free and

  2. (2)

if it is a resolution, then $\pi_{*}\Sigma^{-n}\mathrm{gr}_{n}(P_{\bullet})$
is a projective resolution of $M_{*}$ in $E$-modules.

Lurie shows that any $E$-module $M$ admits a resolution, which is unique (as
an augmented simplicial object) up to simplicial homotopy.

If $P_{\bullet}$ is an $E$-module resolution of $M$ and $X$ is an arbitrary
spectrum, then $P_{\bullet}\otimes X$ is a filtration of $M\otimes X$ and so
we get a spectral sequence whose signature is readily shown to be

$E^{2}_{t,s}:=\operatorname{Tor}^{s,t}_{E_{*}}(M_{*},E_{*}X)\implies M_{*}X$

This is the Künneth spectral sequence which we previously mentioned in
Corollary 2.13.

We are now ready to make the key definition of this section.

######  Definition 6.4.

The _Milnor filtration_ on $K_{*}K\simeq\pi_{*}(K\otimes K)$ is the filtration
induced by the filtration $P_{\bullet}\otimes P_{\bullet}$ of $K\otimes K$,
where $P_{\bullet}$ is an $E$-free resolution of $K$ as an $E$-module.

In fact, the above filtration is readily identified with a familiar one.

######  Lemma 6.5.

The Milnor filtration coincides with the filtration induced by the Künneth
spectral sequence of signature

$\operatorname{Tor}_{E_{*}}^{s,t}(K_{*},E_{*}K)\implies K_{t+s}K.$

###### Proof.

The Künneth spectral sequence is induced by the filtration $P_{\bullet}\otimes
K$, and the obvious map $P_{\bullet}\rightarrow K$ into the constant
filtration of the latter induces a comparison map $P_{\bullet}\otimes
P_{\bullet}\rightarrow P_{\bullet}\otimes K$ of filtrations of $K\otimes K$.
We claim this induces an isomorphism of spectral sequences starting from the
second page onwards, and hence an isomorphism of filtrations of $K_{*}K$.

On the first pages of the corresponding spectral sequences, we are looking at
the map

$\pi_{*}(\mathrm{gr}(P_{\bullet})\otimes\mathrm{gr}(P_{\bullet}))\rightarrow\pi_{*}(\mathrm{gr}(P_{\bullet}\otimes
K)),$

where we have suppressed the suspensions needed to make the above into chain
complexes of graded abelian groups.

Note that if $M$ is an arbitrary $E$-module, then $E_{*}M\simeq
E_{*}E\otimes_{E_{*}}M$ as $E_{*}E$ is flat over $E_{*}$ so that the
corresponding Künneth spectral sequence collapses. It follows that since each
$\mathrm{gr}(P_{\bullet})$ is levelwise free as an $E$-module, we can rewrite
the above map on first pages of the corresponding spectral sequences as

$\pi_{*}\mathrm{gr}(P_{\bullet})\otimes_{E_{*}}E_{*}E\otimes_{E_{*}}\pi_{*}\mathrm{gr}(P_{\bullet})\rightarrow\pi_{*}\mathrm{gr}(P_{\bullet})\otimes_{E_{*}}E_{*}E\otimes_{E_{*}}K_{*}.$

The tensor product of the first two factors is a non-negatively graded complex
of flat $E_{*}$-modules, so that tensoring with it preserves quasi-
isomorphism. As $\mathrm{gr}_{*}(P_{\bullet})\rightarrow K_{*}$ is a quasi-
isomorphism, we deduce that so is the map on first pages, ending the argument.
∎

######  Corollary 6.6.

The Milnor filtration does not depend on the choice of the $E$-free
resolution.

###### Proof.

This is immediate from Lemma 6.5, as any two $E$-free filtrations
$P_{\bullet}^{1}$, $P_{\bullet}^{2}$ induce the same Künneth filtration, as
$P_{\bullet}^{1}\otimes K$ and $P_{\bullet}^{2}\otimes K$ are simplicially
homotopic. ∎

######  Remark 6.7.

By symmetry, Lemma 6.5 implies that the Milnor filtration also coincides with
the filtration induced by the other Künneth spectral sequence, namely that of
signature

$\operatorname{Tor}_{E_{*}}^{s,t}(K_{*}E,K_{*})\implies K_{t+s}K.$

The main advantage of using the Milnor filtration is that its definition is
manifestly symmetric, so that it is easier to work with.

Note that since the Milnor filtration coincides with the Künneth one, we
immediately deduce that the associated graded object is given by

$\operatorname{Tor}_{E_{*}}^{*,*}(K_{*},E_{*}K)\simeq\operatorname{Tor}_{E_{*}}^{*,*}(K_{*},K_{*})\otimes
E_{*}K,$

where the additional grading comes from the external
$\operatorname{Tor}$-grading. In particular, the elements of filtration zero
are given by

$F_{0}^{Mil}(K_{*}K)=E_{*}K,$

the image of the canonical map $E_{*}E\rightarrow K_{*}K$. As the latter is a
sub-Hopf algebroid, this suggests that the Milnor filtration is compatible
with the Hopf algebroid structure. This is indeed the case, as we will now
show.

######  Lemma 6.8.

The Milnor filtration on $K_{*}K$ is multiplicative.

###### Proof.

We first claim that the multiplication map $K\otimes K\rightarrow K$ lifts to
a morphism $P_{\bullet}\otimes P_{\bullet}\rightarrow P_{\bullet}$ of
filtrations.

Observe that since $K$ is an $E$-algebra, the multiplication factors
canonically through a map $K\otimes_{E}K\rightarrow K$. Observe that
$P_{\bullet}\otimes_{E}P_{\bullet}$ is an $E$-free filtration of
$K\otimes_{E}K$ (although not a resolution, but this is not needed). Then,
since $P_{\bullet}$ is an $E$-module resolution of $K$, it follows from
[Lur16, Proposition 7.2.1.5] (where $C=K$) that the map
$K\otimes_{E}K\rightarrow K$ of $E$-modules lifts to a map
$P_{\bullet}\otimes_{E}P_{\bullet}\rightarrow P_{\bullet}$ of filtrations. The
needed lift of $K\otimes K\rightarrow K$ is then given as composition

$P_{\bullet}\otimes P_{\bullet}\rightarrow
P_{\bullet}\otimes_{E}P_{\bullet}\rightarrow P_{\bullet},$

where the first map is the canonical one to the relative tensor product.

We now claim that the multiplication on $K\otimes K$ also lifts to a map of
corresponding filtrations of spectra, showing that it respects filtrations on
homotopy groups. This product map is given up to homotopy by the composite

$\leavevmode\hbox to308.08pt{\vbox
to15.09pt{\pgfpicture\makeatletter\hbox{\hskip
154.04123pt\lower-7.59286pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-154.04123pt}{-7.49303pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
41.0554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.74986pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{K\otimes K\otimes
K\otimes K}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
41.0554pt\hfil&\hfil\hskip 28.30551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
4.30554pt\hfil&\hfil\hskip 65.05537pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.74986pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{K\otimes K\otimes
K\otimes K}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
41.0554pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 43.62491pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.3194pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{K\otimes K}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
19.62494pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-71.73042pt}{-4.99303pt}\pgfsys@lineto{-15.91936pt}{-4.99303pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.71938pt}{-4.99303pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-57.51514pt}{-2.05695pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{K\otimes\sigma\otimes K}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{66.99141pt}{-4.99303pt}\pgfsys@lineto{114.19139pt}{-4.99303pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{114.39137pt}{-4.99303pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{81.92307pt}{-2.05693pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{m\otimes
m}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where $\sigma$ is the twist in the symmetric monoidal structure. This is
lifted to a map of filtrations by

${{(P_{\bullet}\otimes P_{\bullet})\otimes(P_{\bullet}\otimes
P_{\bullet})}}$${{P_{\bullet}\otimes P_{\bullet}\otimes P_{\bullet}\otimes
P_{\bullet}}}$${{P_{\bullet}\otimes
P_{\bullet}},}$$\scriptstyle{P_{\bullet}\otimes\sigma\otimes
P_{\bullet}}$$\scriptstyle{\widetilde{m}\otimes\widetilde{m}}$

where $\widetilde{m}$ is the lift of the multiplication map constructed above.
This ends the argument. ∎

######  Theorem 6.9.

The Milnor filtration makes $K_{*}K$ into a filtered Hopf algebroid over
$K_{*}$.

###### Proof.

We have already checked that the Milnor filtration is preserved by
multiplication on $K_{*}K$ in Lemma 6.8. Thus, we are left with
comultiplication and antipode. The latter is induced by the twist map
$K\otimes K$, which can be lifted to a map of filtration by the twist on
$P_{\bullet}\otimes P_{\bullet}$, hence it preserves the filtration on
homotopy groups.

We move on to comultiplication $\Delta\colon K_{*}K\rightarrow
K_{*}K\otimes_{K_{*}}K_{*}K$. This is induced by the map

${{K\otimes K\simeq K\otimes S^{0}\otimes K}}$${{K\otimes K\otimes
K\simeq(K\otimes K)\otimes_{K}(K\otimes K)},}$$\scriptstyle{K\otimes u\otimes
K}$

where $u\colon S^{0}\rightarrow K$ is the unit and we implicitly use the
Künneth isomorphism

$\pi_{*}((K\otimes K)\otimes_{K}(K\otimes K))\simeq
K_{*}K\otimes_{K_{*}}K_{*}K,$

which holds in full generality since $K_{*}$ is a field.

As $P_{\bullet}$ is an $E$-free filtration of $K_{*}$, it induces a trivial
filtration on $K_{*}$, so that the unit factors through some map
$S^{0}\rightarrow P_{0}$. The latter induces a map $\widetilde{u}\colon
S^{0}\rightarrow P_{\bullet}$ of filtrations, where we equip the sphere with
the trivial filtration, which makes it into the monoidal unit of filtered
spectra.

Consider the composite

${{P_{\bullet}\otimes P_{\bullet}}}$${{P_{\bullet}\otimes P_{\bullet}\otimes
P_{\bullet}}}$${{P_{\bullet}\otimes K\otimes
P_{\bullet}\simeq(P_{\bullet}\otimes K)\otimes_{K}(K\otimes
P_{\bullet})}}$$\scriptstyle{P_{\bullet}\otimes\widetilde{u}\otimes
P_{\bullet}}$

which by construction lifts the map inducing comultiplication. As the
filtrations of $K_{*}K$ induced by $P_{\bullet}\otimes K$ and $K\otimes
P_{\bullet}$ coincide with the Milnor filtration by Lemma 6.5 and Remark 6.7,
we deduce that comultiplication preserves the Milnor filtration as needed. ∎

We will show that if $p>2$ and $K_{*}K$ is commutative—for example, if $K$ is
homotopy commutative—then the Milnor filtration can be canonically promoted to
a grading. We will prove that the existence of this grading implies that the
Cartan–Eilenberg spectral sequence associated to the extension
$K_{*}E\rightarrow K_{*}K\rightarrow K_{*}^{E}K$ collapses.

Assume that $p>2$ and $K_{*}K$ is commutative, we will first use the Milnor
filtration to describe $K_{*}K$ completely as an algebra. By Lemma 6.5, the
associated graded to the Milnor filtration is given by

$\operatorname{Tor}^{*,*}_{E_{*}}(K_{*},E_{*}K)\simeq\operatorname{Tor}^{*,*}_{E_{*}}(K_{*},K_{*})\otimes
E_{*}K.$

Observe that since $F_{0}^{Mil}(K_{*}K)\simeq E_{*}K$ is in even degree, the
external degree one part in the associated graded object, which is

$\operatorname{Tor}^{1}_{E_{*}}(K_{*},K_{*})\otimes E_{*}K,$

can be identified with elements of Milnor filtration at most one in $K_{*}K$
of odd internal degree. Thus, the inclusion

$\operatorname{Tor}^{1}_{E_{*}}(K_{*},K_{*})\hookrightarrow\operatorname{Tor}^{1}_{E_{*}}(K_{*},K_{*})\otimes
E_{*}K,$

can be canonically lifted to a unique map

$\operatorname{Tor}^{1}_{E_{*}}(K_{*},K_{*})\rightarrow K_{*}K.$

which reduces to the canonical inclusion relative to elements of Milnor
filtration zero. Since $\operatorname{Tor}^{*,*}_{E_{*}}(K_{*},K_{*})$ is an
exterior algebra on elements of external degree one, when $p>2$ this
$\operatorname{Tor}$-algebra is a free commutative $K_{*}$-algebra on the
vector space $\operatorname{Tor}^{1,*}$. If $K_{*}K$ is commutative, it
follows that the above lift uniquely extends to a map of $K_{*}$-algebras

$\operatorname{Tor}^{*}_{E_{*}}(K_{*},K_{*})\rightarrow K_{*}K.$

with the following property.

######  Proposition 6.10.

Let $p>2$ and assume that $K_{*}K$ is commutative. Then, the induced map of
$E_{*}K$-algebras

$\displaystyle\operatorname{Tor}_{E_{*}}^{*}(K_{*},K_{*})\otimes_{K_{*}}E_{*}K\rightarrow
K_{*}K$ (6.1)

is an isomorphism.

###### Proof.

By construction, the above map is surjective on the associated graded of
$K_{*}K$ with respect to the Milnor filtration. Thus, it is surjective by an
inductive argument, and as a surjective map of free $E_{*}K$-modules of the
same rank it must be an isomorphism. ∎

As a consequence of Proposition 6.10, $K_{*}K$ inherits an additional external
grading from the grading of $\operatorname{Tor}$-groups. This grading can be
made explicit, as in the following definition.

######  Definition 6.11.

Let $p>2$ and assume that $K_{*}K$ is commutative. Then, the _Milnor grading_
on $K_{*}K$ is the unique algebra grading such that

  1. (1)

elements of Milnor filtration zero are of Milnor degree zero and

  2. (2)

odd internal degree elements of Milnor filtration at most one are of Milnor
degree one.

######  Lemma 6.12.

The Milnor grading is well-defined.

###### Proof.

This is immediate from Proposition 6.10, as the external grading on the
algebra tensor product is
$\operatorname{Tor}^{*}_{E_{*}}(K_{*},K_{*})\otimes_{K_{*}}E_{*}K$ is uniquely
determined by which elements are of degrees zero and one. ∎

######  Remark 6.13.

The increasing filtration associated to the Milnor grading is exactly the
Milnor filtration. Thus, this grading can be considered as a refinement of the
latter canonical filtration, which exists at all primes and with no
assumptions on commutativity.

######  Theorem 6.14.

Let $p>2$ and assume that $K_{*}K$ is commutative. Then, the Milnor grading on
$K_{*}K$ is compatible with comultiplication and antipode, making it into a
bigraded Hopf algebroid.

###### Proof.

As $K_{*}K$ is generated as an algebra by elements in Milnor degrees zero and
one, it is enough to check that these two degrees are preserved. The subspace
of elements of Milnor degree most one coincides with elements of filtration at
most one, which get preserved by comultiplication and antipode by Theorem 6.9.

Since the ones in Milnor degree zero are precisely those of even internal
degree, and the ones in Milnor degree one are exactly the ones of odd internal
degree, we deduce that each degree gets preserved separately by both
comultiplication and antipode. ∎

######  Remark 6.15.

To see what goes wrong if $p=2$, observe that even if $K_{*}K$ were
commutative, odd degree elements do not have to square to zero and so there
need not be a multiplicative extension of the canonical map
$\operatorname{Tor}^{1}_{E_{*}}(K_{*},K_{*})\rightarrow K_{*}K$ to a map on
the whole $\operatorname{Tor}$-algebra.

One can try to define the Milnor grading at $p=2$ directly similarly to
Definition 6.11, by declaring that elements in Milnor degree $s$ are the
$s$-fold products of those in Milnor degree one. This is bound to fail; the
calculations of Würgler in the case of the minimal Morava $K$-theory given in
Section 3 show that the elements $\tau_{i}$ of Milnor degree one square to
non-zero elements of Milnor degree zero.

######  Theorem 6.16.

Suppose that $p>2$ and that $K_{*}K$ is commutative. Then, the
Cartan–Eilenberg spectral sequence associated to the extension

$K_{*}E\rightarrow K_{*}K\rightarrow K_{*}^{E}K$

collapses on the second page, inducing a canonical isomorphism

$\operatorname{Ext}_{K_{*}K}(K_{*},K_{*})\simeq\operatorname{Ext}_{K_{*}E}(K_{*},\operatorname{Ext}_{K_{*}^{E}K}(K_{*},K_{*})).$

###### Proof.

Under the given assumptions, $K_{*}K$ is bigraded using its internal grading
and the Milnor grading of Definition 6.11. This grading is compatible with the
above extension, with all of $K_{*}E$ of Milnor degree zero.

As $K_{*}^{E}K\simeq\Lambda_{K_{*}}(V)$ as Hopf algebras, where $V$ is the
$K_{*}$-vector subspace of primitive elements, we have

$\operatorname{Ext}^{*,*,*}_{K_{*}^{E}K}(K_{*},K_{*})\simeq\operatorname{Sym}_{K_{*}}(V^{\vee}),$

an isomorphism with the symmetric algebra. Here, $V^{\vee}$ is of homological
degree one and Milnor degree one, and so the whole above
$\operatorname{Ext}$-algebra is concentrated on the plane of elements for
which Milnor degree is equal to the homological one.

Since the Cartan–Eilenberg differentials lower the homological degree and
preserve the Milnor degree, we deduce that they are all zero. This gives an
isomorphism as above up to passing to associated graded to the homological
filtration, as the extension problems all get trivialized by the additional
grading. ∎

##  Part II The $K$-based Adams spectral sequence

In this part, we study the Adams spectral sequence based on a Morava
$K$-theory $K$ at arbitrary height $n$ and prime $p$, constructed as part of a
finite height analogue of Miller’s square of spectral sequences. In
particular, we establish its (perhaps surprisingly) good convergence
properties and identify it with the filtration by powers spectral sequence, at
least for sufficiently large primes $p$. We then conclude this part with an
illustration of the spectral sequence at heights 1 and odd primes, already
exhibiting the existence of arbitrary long differentials.

######  Remark 6.17 (Adams spectral sequence based on Milnor modules).

We will not proceed in this way, but one can phrase the $K$-based Adams
spectral sequence entirely in terms of absolute Milnor modules, following
Miller’s observation that the Adams spectral sequence is determined by the
class of $K_{*}$-injective maps.

To see this, notice that the restricted Yoneda embedding, followed by a
sheafification, gives a symmetric monoidal functor

$y\colon\operatorname{Sp}\rightarrow\operatorname{Mil}_{abs}$

which can be considered a Milnor modules-valued homology theory. That is, the
functor $y$ takes cofibre sequences to exact sequences, preserves arbitrary
direct sums, and takes the suspension in spectra to the distinguished auto-
equivalence of Milnor modules induced by suspension functor on
$\operatorname{Sp}^{fin}$.

By construction, any injective object in Milnor modules can be lifted to a
representing object in spectra, and this allows one to construct a “$y$-based”
Adams spectral sequence based on injectives, where the $E_{2}$-terms will be
canonically given by $\operatorname{Ext}$-groups in absolute Milnor modules
[PP21]. Through the equivalence of Proposition 4.12, this spectral sequence
will be canonically isomorphic to the $K$-based Adams spectral sequence for
any choice of Morava $K$-theory.

###  7\. Convergence of the $K$-based Adams spectral sequence

In this short section, we prove that the $K$-based Adams spectral sequence is
conditionally convergent for any $K$-local spectrum, and it converges
completely in the dualizable case.

######  Remark 7.1.

As observed in Remark 6.17, the $K$-based Adams spectral sequence depends only
on the choice of the prime and the height. However, this independence from
choices will not be needed in this section, as the arguments we use apply
equally well to any Morava $K$-theory.

By standard arguments, the $K$-based Adams spectral sequence is isomorphic to
the totalization spectral sequence obtained by mapping into the _canonical
$K$-Adams resolution_

$X\rightarrow K\otimes X\rightrightarrows K\otimes K\otimes
X\Rrightarrow\ldots$

Thus, it follows that the $K$-based Adams spectral sequence converges to the
homotopy of the _$K$ -nilpotent completion of $X$_:

$\operatorname{Tot}(K^{\otimes\bullet+1}\otimes
X):=\textstyle\varprojlim_{s\in\Delta}K^{\otimes\bullet+1}\otimes X.$

Since the totalization is $K$-local as a limit of $K$-local spectra, there is
a canonical comparison map from the $K$-localization of $X$. Our first task is
to determine when this map is an equivalence.

######  Proposition 7.2.

For any spectrum $X$, the $K$-nilpotent completion of $X$ is equivalent to the
$K$-localization of $X$; that is, the canonical map

$L_{K}X\rightarrow\operatorname{Tot}(K^{\otimes\bullet+1}\otimes X)$ (7.1)

is an equivalence.

###### Proof.

Since both sides are $K$-local, it is enough to show that this map is a
$K_{*}$-isomorphism. The latter can be checked after tensoring with any
non-$K$-acyclic spectrum as $K_{*}$ is a field, so that we have the Künneth
isomorphism, and tensoring with a non-zero $K_{*}$-vector space reflects
isomorphisms.

After tensoring with a finite spectrum $F(n)$ of type $n$, the above map
becomes

$L_{K}(X\otimes F(n))\rightarrow\operatorname{Tot}(K^{\otimes\bullet+1}\otimes
X\otimes F(n)),$

as both sides of (7.1) define exact functors in $X$ and so commute with
tensoring with a finite spectrum. Thus, it is enough to check the result holds
for $X\otimes F(n)$.

The proof of the smash product theorem in [Rav92, Chapter 8] shows that
$S^{0}$ is $E$-prenilpotent; that is,
$L_{E}S^{0}\in\operatorname{Thick}^{\otimes}(E)$, where
$\operatorname{Thick}^{\otimes}(Y)$ denotes the smallest full subcategory of
$\mathcal{S}p$ which contains $Y$, is thick, and is closed under smashing with
arbitrary objects in $\mathcal{S}p$. It follows that

$L_{K}F(n)\simeq L_{E}F(n)\in\operatorname{Thick}^{\otimes}(E\otimes
F(n))\subseteq\operatorname{Thick}^{\otimes}(K).$

In other words, $F(n)$ is $K$-prenilpotent. By definition, $X\otimes F(n)$ is
thus also $K$-prenilpotent for all $X\in\mathcal{S}p$. Therefore, Bousfield’s
theorem [Bou79, Theorem 6.10] applies to provide an equivalence
$L_{K}(X\otimes F(n))\simeq\operatorname{Tot}(K^{\otimes\bullet+1}\otimes
X\otimes F(n))$, which is what we wanted. ∎

If $X,Y$ are spectra, then by applying $[Y,-]$ to the canonical $K$-Adams
resolution of $X$ we obtain the Adams spectral sequence as the totalization
spectral sequence. Since $K_{*}$ is a field, $K_{*}Y$ is projective over the
base ring, so that by standard arguments the resulting spectral sequence has
signature

$E_{2}^{s,t}(Y,X)\cong\operatorname{Ext}_{K_{*}K}^{s,t}(K_{*}Y,K_{*}X)\implies[Y,L_{K}X]^{t-s}.$

Indeed, if we use the canonical Adams resolution, then the first page of the
Adams spectral sequence will be given by the cobar complex, which in the
projective case computes the $\operatorname{Ext}$-groups in comodules [Rav03,
Corollary A.1.2.12].

######  Corollary 7.3.

For any $K$-local spectrum $X$ and any $Y$, the $K$-based Adams spectral
sequence converges conditionally in the sense of Boardman to $[Y,L_{K}X]$.

###### Proof.

The totalization spectral sequence is the spectral sequence induced by the
tower of partial totalizations. Conditional convergence to $[-,L_{K}X]$ in the
sense of Boardman is equivalent to the natural map from $L_{K}X$ into the
limit of this tower being an equivalence, so that the statement follows from
Proposition 7.2. ∎

Let us write $E_{r}^{*,*}(Y,X)$ for the $r$-th page of the $K$-based Adams
spectral sequence converging conditionally to $[Y,L_{K}X]$, so that we have
$E_{2}^{s,t}(Y,X)\simeq\operatorname{Ext}_{K_{*}K}^{s,t}(K_{*}Y,K_{*}X)$. The
differentials in this spectral sequence are of the form

$d_{r}\colon E_{r}^{s,t}(Y,X)\to E_{r}^{s+r,t+r-1}(Y,X).$

In particular, for $r>s$ there are inclusions $E_{r+1}^{s,t}(Y,X)\subseteq
E_{r}^{s,t}(Y,X)$, and we write

$E_{\infty}^{s,t}(Y,X)=\operatorname{lim}_{r}E_{r}^{s,t}(Y,X).$

Let $F^{*}[Y,L_{K}X]_{*}$ denote the filtration on $[Y,L_{K}X]_{*}$ induced by
the spectral sequence. Following Bousfield [Bou79, §6], we say that the
$K$-based Adams spectral sequence for $X$ _converges completely_ if the
canonical maps

$[Y,L_{K}X]_{*}\twoheadrightarrow\operatorname{lim}_{s}[Y,L_{K}X]_{*}/F^{s}[Y,L_{K}X]_{*}$

and

$F^{s}[Y,L_{K}X]_{*}/F^{s+1}[Y,L_{K}X]_{*}\hookrightarrow
E_{\infty}^{s,s+*}(Y,X)$

are isomorphisms. By the work of Bousfield, see [Bou79, Proposition 6.3], this
is equivalent to the condition that
$\operatorname{lim}_{r>s}^{1}E_{r}^{s,t}(Y,X)$ vanishes.

######  Proposition 7.4.

The $K$-based Adams spectral sequence of signature

$\operatorname{Ext}_{K_{*}K}^{s,t}(K_{*}Y,K_{*}X)\implies[Y,L_{K}X]^{t-s}.$

converges completely for any spectra $X$ and $Y$ with $K_{*}X$ and $K_{*}Y$
finite-dimensional.

###### Proof.

By Theorem 5.10, the groups in the $E_{2}$-term are all finite. Therefore, the
system

$(E_{r}^{s,t}(Y,X))_{r>s}$

is Mittag-Leffler, so that Bousfield’s convergence criterion [Bou79,
Proposition 6.3] applies. ∎

######  Remark 7.5.

For a spectrum $X$, the homology $K_{*}X$ being finite-dimensional is
equivalent to $X$ being $K$-locally dualizable, see [HS99, Theorem 8.6].

######  Remark 7.6.

In general, the $K$-based Adams spectral sequence may fail to converge
completely in the sense of Bousfield, see Example 10.4.

###  8\. The finite height Miller square

In his seminal paper [Mil81] on relations between Adams spectral sequences,
Miller uses the relationship between the Adams–Novikov and classical Adams
spectral sequences to prove the telescope conjecture at height one and odd
primes. In this section, we apply his axiomatic framework to the case of
Lubin–Tate and Morava $K$-theory spectra.

In Miller’s framework, one starts with a map $A\rightarrow B$ of homotopy ring
spectra; that is, an arrow in the category
$\operatorname{Alg}(\operatorname{ho}(\operatorname{Sp}))$. In this context,
it is not hard to show that a map $X\rightarrow Y$ of spectra which is _$A$
-monic_, in the sense that $A\otimes X\rightarrow A\otimes Y$ is a split
inclusion, is also $B$-monic. Thus, any $B$-injective spectrum is also
$A$-injective, see [Mil81, Lemma 2.1].

######  Definition 8.1 (Miller).

A spectrum $X$ is _$(A,B)$ -primary_ if there exists an $A$-Adams resolution
$X\rightarrow I^{\bullet}$ such that for each ${m}$, the $B$-Adams spectral
sequence computing $\pi_{*}I^{m}$ converges and collapses on the second page.

When $X$ is $(A,B)$-primary, Miller constructs a square of spectral sequences

${{E^{2}_{A,B}(X)}}$${{E^{2}_{A}(X)}}$${{E^{2}_{B}(X)}}$${{\pi_{*}X}}$$\scriptstyle{A-\textnormal{Adams}}$$\scriptstyle{B-\textnormal{Adams}}$MayMahowald

which we will now explain. The terms $E_{A}^{2}(X),E^{2}_{B}(X)$ form the
second pages of the relevant Adams spectral sequences, and the bottom two
arrows are the corresponding Adams spectral sequences.

The construction of the other two spectral sequences is more involved, but let
us briefly recall the arguments of Miller. Observe that $E^{2}_{A}(X)$ arises
as homology of the cochain complex $\pi_{*}I^{\bullet}$, where $I^{\bullet}$
is an $A$-Adams resolution as in Definition 8.1. As this cochain complex is
given by homotopy groups, it has an additional filtration given by the
$B$-Adams filtration. By definition, the term $E^{2}_{A,B}(X)$ is the
cohomology of the associated graded of $\pi_{*}I^{\bullet}$ with respect to
this filtration, and the top left arrow is the associated cohomology of
filtered complex spectral sequence, which Miller calls the _May spectral
sequence_.

If $X$ is primary in the sense of Definition 8.1, the cohomology
$E^{2}_{A,B}(X)$ of the associated graded of $\pi_{*}I^{\bullet}$ is the same
as the cohomology of the complex $E^{2}_{B}(I^{\bullet})$ consisting of the
$B$-Adams $E_{2}$-pages of the spectra $I^{\bullet}$. As all $A$-split maps
are $B$-split, these $E_{2}$-pages are related by long exact sequences and can
be collected into an $E_{1}$ of an exact couple using that $I^{\bullet}$ is a
resolution of $X$, yielding a spectral sequence computing $E^{2}_{B}(X)$. This
is the spectral sequence in the top right corner, which Miller calls the
_Mahowald spectral sequence_.

######  Remark 8.2.

Miller proves that the above diagram of spectral sequences is commutative in
the sense that, after making appropriate choices, the $d_{2}$-differentials in
the May spectral sequence can be identified with $d_{2}$-differentials in the
$B$-Adams spectral sequence [Mil81, Theorem 4.2]. In the case of Morava
$K$-theory and under the assumption that the prime is large enough compared to
the height, we will generalize this result to higher differentials in
Proposition 9.13.

Let us give an explicit description of these spectral sequences for Lubin–Tate
spectra and Morava $K$-theories. Note that in this case, the $E$-based Adams
spectral sequence computes the $E$-local homotopy groups, which are quite
different from the $K$-local ones. Thus, to get a commutative square, we
should work internally to $K$-local spectra.

Our first goal is to show that the $K$-based Adams spectral sequence has a
particularly simple form for $E$-modules with flat homotopy groups. To do so,
we will reduce to the pro-free case using the work of Hovey.

Before we proceed, let us recall that the completion functor
$M_{*}\mapsto(M_{*})_{\mathfrak{m}}^{\vee}:=\varprojlim
M_{*}/\mathfrak{m}^{n}M_{*}$ is neither left or right exact, but it has left
derived functors
$L_{s}\colon\operatorname{Mod}_{E_{*}}\rightarrow\operatorname{Mod}_{E_{*}}$.
For more properties of these functors, see [HS99, Appendix B] or [BF15,
Appendix A].

######  Lemma 8.3.

Let $M_{*}$ be an $E_{*}$-module such that
$\operatorname{Tor}^{s}_{E_{*}}(M_{*},K_{*})=0$ for $s>0$; for example,
$M_{*}$ can be flat. Then, the completion $(M_{*})^{\vee}_{\mathfrak{m}}$ is
pro-free.

###### Proof.

This is proven by Hovey in [Hov04c, 1.2]. ∎

######  Lemma 8.4.

Let $M$ be an $E$-module such that $M_{*}$ is flat over $E_{*}$. Then, the
$K$-localization map $M\rightarrow L_{K}M$ induces an isomorphism
$\pi_{*}L_{K}M\simeq(M_{*})^{\vee}_{\mathfrak{m}}$ with the completion.

###### Proof.

Hovey constructs for any $E$-module $M$ a strongly convergent spectral
sequence of signature

$L_{s}M_{t}\rightarrow\pi_{t+s}L_{K}M.$

In the present case, the spectral is concentrated on the zero line by Lemma
8.3, hence the spectral sequence collapses inducing the needed isomorphism. ∎

######  Corollary 8.5.

Let $M$ be an $E$-module such that $M_{*}$ is flat. Then, $L_{K}M$ is
equivalent as an $E$-module to the $K$-localization of a direct sum of copies
of $E$.

###### Proof.

We have $\pi_{*}L_{K}M\simeq(M_{*})^{\vee}_{\mathfrak{m}}$ by Lemma 8.4 and
$(M_{*})^{\vee}_{\mathfrak{m}}$ is pro-free by Lemma 8.3. Choosing pro-
generators of the latter induces a map $L_{K}(\bigoplus E)\rightarrow M$ from
a direct sum of shifts of $E$ which is an equivalence by inspection of
homotopy groups. ∎

######  Proposition 8.6.

If $M$ is an $E$-module such that $M_{*}$ is $E_{*}$-flat, then the $K$-Adams
filtration on $M_{*}$ is the $\mathfrak{m}$-adic one and the $K$-based Adams
spectral sequence for $M$ converges completely to
$\pi_{*}L_{K}M:=(M_{*})^{\vee}_{\mathfrak{m}}$ and collapses on the second
page.

###### Proof.

By flatness we have $\pi_{*}L_{K}M\simeq(M_{*})^{\vee}_{\mathfrak{m}}$. Since
the $\mathfrak{m}$-adic filtration on $M_{*}$ is induced by that of
completion, we can replace $M$ by its $K$-localization. In this case, $M$ is
$K$-locally equivalent to a direct sum of $E$ by Corollary 8.5. As the Adams
spectral sequence of a direct sum is a direct sum of spectral sequences, we
can assume that $M=E$.

We have a base-change isomorphism

$\operatorname{Ext}_{K_{*}K}(K_{*}E,K_{*})\simeq\operatorname{Ext}_{K_{*}^{E}K}(K_{*},K_{*})$

and by Proposition 5.8 a further isomorphism

$\operatorname{Ext}_{K_{*}^{E}K}(K_{*},K_{*})\simeq
K_{*}\otimes_{\kappa}\operatorname{Sym}_{\kappa}^{*}(\mathfrak{m}/\mathfrak{m}^{2}).$

Since $E_{0}$ is regular,
$\operatorname{Sym}_{\kappa}^{*}(\mathfrak{m}/\mathfrak{m}^{2})$ can be
identified with the associated graded to the $\mathfrak{m}$-adic filtration on
$E_{0}$.

We deduce from the above that the $E_{2}$-term of the $K$-based Adams for $E$
is abstractly isomorphic to the associated graded of the $\mathfrak{m}$-adic
filtration on $E_{*}$. Moreover, since the $E_{2}$-term is concentrated in
even Adams degree, we deduce that the Adams spectral sequence collapses.

We are left with showing that the $K$-based Adams filtration on $E_{*}$
coincides with the $\mathfrak{m}$-adic one. Observe that since $E\rightarrow
K$ is a $K_{*}$-monomorphism by Proposition 2.14, the elements of positive
Adams filtration are precisely given by
$\mathfrak{m}E_{*}=\mathrm{ker}(E_{*}\rightarrow K_{*})$.

Using compatibility of the Adams filtration with products, we deduce that any
element of $\mathfrak{m}^{n}E_{*}$ is of Adams filtration at least $n$. Thus,
one filtration is contained in the other. As their associated gradeds have the
same dimension over $K_{*}$ by what was said above, we deduce that the two
filtrations must coincide, as needed. ∎

######  Proposition 8.7.

Let $X$ be a spectrum such that $E_{*}^{\vee}(X):=\pi_{*}L_{K}(E\otimes X)$ is
pro-free; for example, such that $E_{*}X$ is flat. Then, $X$ is
$(E,K)$-primary in the $K$-local category.

###### Proof.

The implication that $E_{*}X$ flat implies $E_{*}^{\vee}X$ pro-free is a
combination of Lemma 8.3 and Proposition 8.6.

Since $E_{*}E$ is flat over $E_{*}$, this in particular implies that
$E_{*}^{\vee}E$ is pro-free, which can be also proven directly using evenness,
see [HS99, THeorem 8.6]. It follows (as in [BH16, Corollary 1.24], for
example) that

$\pi_{*}L_{K}(E^{\bullet}\otimes
X)\simeq(E_{*}^{\vee}E)^{\otimes^{\vee}_{E_{*}}\bullet-1}\otimes^{\vee}_{E_{*}}E_{*}^{\vee}X,$

where on the right hand side we have the completed tensor products, which are
again pro-free. This together with Proposition 8.6 implies that the standard
$K$-local $E$-based Adams resolution

${L_{K}X}$${{L_{K}(E\otimes X)}}$${{L_{K}(E\otimes\overline{E}\otimes
X)}}$${\ldots}$${{L_{K}(\overline{E}\otimes
M)}}$${{L_{K}(\overline{E}\otimes\overline{E}\otimes X)}}$

satisfies the primarity condition. ∎

If $E_{*}^{\vee}X$ is pro-free, then the $E_{1}$-page of the $K$-local
$E$-based Adams spectral sequence is given by the cobar complex

$\pi_{*}L_{K}(E^{\bullet}\otimes
X)\simeq(E_{*}^{\vee}E)^{\otimes^{\vee}_{E_{*}}\bullet-1}\otimes_{E_{*}}E_{*}^{\vee}X.$

It follows from Proposition 8.6 that the $K$-Adams filtration on these
homotopy groups coincides with the filtration by powers of the maximal ideal
$\mathfrak{m}$; this is the “filtration by powers” appearing in the title.
This has the following consequence.

######  Corollary 8.8.

If $E_{*}^{\vee}X$ is pro-free, then Miller’s May spectral sequence associated
to $(E,K)$ coincides with the spectral sequence of signature

$\operatorname{Ext}_{E_{*}^{\vee}E}^{s,t}(E_{*},\bigoplus\mathfrak{m}^{n}/\mathfrak{m}^{n+1}E_{*}^{\vee}X)\implies\operatorname{Ext}_{E_{*}^{\vee}E}^{t-s}(E_{*},E^{\vee}_{*}X)$

obtained by filtering $E_{*}^{\vee}X$ by powers of the maximal ideal; that is,
by $\mathfrak{m}^{n}E_{*}^{\vee}X$.

###### Proof.

Since Miller’s May spectral sequence is induced by the $K$-Adams filtration on
the homotopy groups of the $K$-local $E$-Adams resolution, this is a
consequence of Proposition 8.6, as under the above assumption all of the
spectra in the resolution are pro-free. ∎

######  Remark 8.9.

The identification of the $E_{2}$-page of the $K$-local $E$-based Adams
spectral sequence is somewhat subtle. However, in the case where
$E_{*}^{\vee}X$ is pro-free, it follows from [BH16] that it acquires a
structure of a suitably complete comodule over $E_{*}^{\vee}E$ and that the
$E_{2}$-page is given by the $\operatorname{Ext}$-groups in comodules as
above.

We deduce that the $K$-local Miller square associated to $(E,K)$ is of the
form

${{\operatorname{Ext}_{E_{*}^{\vee}E}(E_{*},\bigoplus\mathfrak{m}^{n}/\mathfrak{m}^{n+1}E_{*}^{\vee}X)}}$${{\operatorname{Ext}_{E_{*}^{\vee}E}(E_{*},E^{\vee}_{*}X)}}$${{\operatorname{Ext}_{K_{*}K}(K_{*},K_{*}X)}}$${{\pi_{*}L_{K}X},}$$\scriptstyle{E-\textnormal{Adams}}$$\scriptstyle{K-\textnormal{Adams}}$MayMahowald
(8.1)

whenever $E^{\vee}_{*}X$ is profree. This leaves the question of identifying
the Mahowald spectral sequence, which is not too difficult.

######  Proposition 8.10.

If $E_{*}^{\vee}X$ is pro-free, then Miller’s Mahowald spectral sequence based
on $(E,K)$ can be identified with the Cartan–Eilenberg spectral sequence
associated to the extension of Hopf algebroids

$K_{*}E\rightarrow K_{*}K\rightarrow K_{*}^{E}K.$

###### Proof.

Taking the standard $K$-local $E$-Adams resolution

$X\rightarrow L_{K}(E\otimes X)\rightarrow L_{K}(E\otimes\overline{E}\otimes
X)\rightarrow\ldots$

and applying $K$-homology gives the long exact sequence of comodules of the
form

$K_{*}X\rightarrow K_{*}E\otimes K_{*}X\rightarrow
K_{*}E\otimes_{K_{*}}\overline{K_{*}E}\otimes_{K_{*}}K_{*}X\rightarrow\ldots.$

Applying $\operatorname{Ext}_{K_{*}K}(K_{*},-)$ to the above leads to an exact
couple which yields the Mahowald spectral sequence.

Note that from the description of the $E_{1}$-term we see that this is just
the $K_{*}E$-based Adams spectral sequence in the derived category of
$K_{*}K$-comodules, sometimes called the Margolis–Palmieri spectral sequence.
This is known to be isomorphic to the Cartan–Eilenberg spectral sequence
associated to the extension, see [Bel20, Theorem 4.2]. ∎

######  Remark 8.11.

We have shown in Theorem 6.16 that if $p>2$, then the Cartan–Eilenberg
spectral sequence computing $\operatorname{Ext}_{K_{*}K}(K_{*},K_{*})$
collapses on the second page. It follows from the isomorphism of Proposition
8.10 that the same is true for Miller’s $(E,K)$-Mahowald spectral sequence
associated to $X=S^{0}$.

######  Remark 8.12.

Note that applying Proposition 8.10 to the starting terms of the respective
spectral sequences implies in particular that

$\operatorname{Ext}_{E_{*}^{\vee}E}(E_{*},\bigoplus\mathfrak{m}^{n}/\mathfrak{m}^{n+1}E_{*})\simeq\operatorname{Ext}_{E_{*}K}(K_{*},\operatorname{Ext}_{K_{*}^{E}K}(K_{*},K_{*})).$

This can be proven directly, without resorting to Miller’s work, using Hopkins
and Lurie’s calculation of the cohomology of $K_{*}^{E}K$, which we stated as
Proposition 5.8. The key is that since $E_{0}$ is regular, we have
$\operatorname{Sym}_{\kappa}(\mathfrak{m}/\mathfrak{m}^{2})\simeq\bigoplus\mathfrak{m}^{n}/\mathfrak{m}^{n+1}$.

######  Remark 8.13.

If we again pick $E=E_{n}$, the Lubin–Tate spectrum associated to the Honda
formal group law of height $n$ over $\mathbb{F}_{p^{n}}$, then the relevant
$\operatorname{Ext}$-groups appearing in the Miller square can be described in
terms of the cohomology of the Morava stabilizer group, see [BH16, Theorem
4.3]. Keeping this choice of $E$ in mind, we can rewrite the Miller square as

${{\mathrm{H}^{*}(\mathbb{G}_{n},\bigoplus\mathfrak{m}^{n}/\mathfrak{m}^{n+1}E_{*}^{\vee}X)}}$${{\mathrm{H}^{*}(\mathbb{G}_{n},E^{\vee}_{*}X)}}$${{\operatorname{Ext}_{K_{*}K}(K_{*},K_{*}X)}}$${{\pi_{*}L_{K}X}.}$$\scriptstyle{E-\textnormal{Adams}}$$\scriptstyle{K-\textnormal{Adams}}$MayMahowald

######  Remark 8.14.

In the important case of $(MU,H\mathbb{F}_{p})$, the Miller square can also be
described explicitly, see for example [GWX21]. In this context, the Miller
square foreshadows the later development of the $\mathrm{C}\tau$-philosophy of
Gheorghe, Isaksen, Wang and Xu which applied motivic methods to breakthrough
results in stable homotopy groups of spheres [IWX20].

###  9\. The $K$-based Adams spectral sequence at large primes

It is well-known that if $p$ is large compared to the height, then the
$E$-based Adams–Novikov spectral sequence for the $K$-local sphere collapses.
As the same is true for the Mahowald spectral sequence, as we have shown in
Remark 8.11, we deduce that under these assumptions, two of the sides of the
Miller square of (8.1) represent collapsing spectral sequences.

It is then only natural to expect that the other two sides can be identified
up to regrading; in this section, we show that this is indeed the case. More
precisely, we will show that if $2p-2>n^{2}+n+1$, then the $K$-based Adams
spectral sequence for the sphere is isomorphic to the filtration by powers
spectral sequence

$\mathrm{H}^{*}(\mathbb{G}_{n},\bigoplus\mathfrak{m}^{n}/\mathfrak{m}^{n+1})\implies\mathrm{H}^{*}(\mathbb{G}_{n},E_{*}),$

which we can identify with Miller’s May spectral sequence by Corollary 8.8.

######  Remark 9.1.

Note that a result of Miller [Mil81, Theorem 4.2] relates the $d_{2}$-s in the
May and $K$-based Adams spectral sequence, at all primes and heights. Our
first approach to this problem was to try to push Miller’s methods further
into higher differentials, hoping that the collapses of the other two spectral
sequences in the square would simplify the situation. This approach was
unsuccessful, and below we use quite different methods.

######  Remark 9.2.

While we will show in this section that at large primes the $K$-based Adams
spectral sequence admits an algebraic description, it _never_ collapses for
$n>0$, unlike the Adams–Novikov spectral sequence. Indeed, the element
$v_{n}=u^{p^{n}-1}\in\operatorname{Ext}^{0,2p^{n}-2}_{K_{*}K}(K_{*},K_{*})$ is
never a permanent cycle, as the $K$-local sphere is of type $0$ and so cannot
admit a $v_{n}$-periodic map.

To prove our comparison result, we will use an algebraicity result for
homotopy categories of $E$-local spectra at large primes due to the second
author, strengthened in recent work with Patchkoria, which compares it the
following algebraic construction.

######  Notation 9.3.

Let us write $\mathcal{D}(E_{*}E)$ for the derived $\infty$-category of the
abelian category of comodules. This is a symmetric monoidal stable
$\infty$-category, with monoidal unit given by $E_{*}$ itself, considered as a
chain complex concentrated in degree zero. The unit defines for any
$X\in\mathcal{D}(E_{*}E)$ its _homotopy groups_

$\operatorname{Ext}_{\mathcal{D}(E_{*}E)}^{s,t}(X):=[\Sigma^{s}E_{*}[t],X]_{\mathcal{D}(E_{*}E)}.$

Notice that the homotopy groups are bigraded, since the abelian category of
$E_{*}E$-comodules has an internal grading. This endows the derived
$\infty$-category with _two_ gradings,

  1. (1)

the _homological_ one given by the suspension and corresponding above to $s$
and

  2. (2)

the _internal one_ induced from the grading shift functor on
$\operatorname{Comod}_{E_{*}E}$ and corresponding above to $t$.

In the special case of $X$ being a comodule, considered as an object of the
heart, these homotopy groups encode the $E_{2}$-term of the $E$-based Adams
spectral sequence in the $\infty$-category of spectra.

######  Definition 9.4.

The _periodicity algebra_ $P(E_{*})$ is the $\mathbf{E}_{\infty}$-algebra in
$\mathcal{D}(E_{*}E)$ determined by the commutative algebra

$E_{*}[\tau^{\pm 1}],\textnormal{where }|\tau|=(1,-1)$

in chain complexes of comodules, equipped with the zero differential.

######  Notation 9.5.

For brevity, let us write

$\operatorname{Mod}_{P(E_{*})}:=\operatorname{Mod}_{P(E_{*})}(\mathcal{D}(E_{*}E)).$

The following result shows that modules over the periodicity algebra give an
algebraic model for the homotopy category of $E$-local spectra.

######  Theorem 9.6.

If $2p-2>n^{2}+n$, there exists an equivalence

$\phi\colon h\mathcal{S}p_{E}\simeq
h\operatorname{Mod}_{P(E_{*})}\mathcal{D}(E_{*}E)$

of homotopy categories of $E$-local spectra and $P(E_{*})$-modules which is
moreover compatible with homology functors in the sense that the following
diagram commutes

${{h\mathcal{S}p_{E}}}$${{h\operatorname{Mod}_{P(E_{*})}\mathcal{D}(E_{*}E)}}$${\operatorname{Comod}_{E_{*}E}.}$$\scriptstyle{E_{*}(-)}$$\scriptstyle{\mathrm{H}_{0}}$$\scriptstyle{\phi}$

###### Proof.

This is proven under the assumption $2p-2>2(n^{2}+n)$ in the second author’s
thesis [Pst18a]. The bound was improved to the one given above in joint work
with Patchkoria [PP21]. ∎

######  Remark 9.7.

Note that as an $\mathbf{E}_{1}$-algebra in $\mathcal{D}(E_{*}E)$, the
connective cover of the periodicity algebra $P(E_{*})$ can be identified with
the free $\mathbf{E}_{1}$-algebra on the object $\Sigma E_{*}[-1]$, and so is
uniquely determined by its homology groups.

The category of modules over $P(E_{*})$ (considered only as a stable
$\infty$-category) can be identified with the subcategory of modules over the
connective cover spanned by those modules on which $\tau$ acts invertibly.
Thus, to define the target of the equivalence of Theorem 9.6, it is not
strictly necessary to work with algebras in chain complexes as in Definition
9.4, but we do so out of convenience.

######  Remark 9.8.

For any $M,N\in\operatorname{Mod}_{P(E_{*})}$ we have an Adams spectral
sequence of signature

$\operatorname{Ext}^{s,t}(\mathrm{H}_{0}(M),\mathrm{H}_{0}(N))\implies[M,N]_{t-s},$

where on the right hand side we have homotopy classes of maps of
$P(E_{*})$-modules.

This observation, due to Franke [Fra], is the starting point for the
comparison of $\operatorname{Mod}_{P(E_{*})}$ to the $E$-local category. The
general form of Franke’s conjecture implies that all stable
$\infty$-categories admitting a suitably convergent Adams-type spectral
sequence of this signature must have equivalent homotopy categories [PP21].

One of the advantage of $P(E_{*})$-modules over $E$-local spectra is the
existence of the free module functor

$P(E_{*})\otimes-\colon\mathcal{D}(E_{*}E)\rightarrow\operatorname{Mod}_{P(E_{*})}$

(also called the _periodicization_) which is cocontinuous and symmetric
monoidal. The homotopy groups of free modules can be understood explicitly, as
the following shows.

######  Remark 9.9.

Since the periodicity algebra is a sum of shifts of a unit as an object of the
derived $\infty$-category, the homotopy groups of the free module are simple
to understand. More precisely, for any $M\in\mathcal{D}(E_{*}E)$ we have

$[P(E_{*}),P(E_{*})\otimes_{E_{*}}M]_{P(E_{*})}^{t}\simeq[E_{*},P(E_{*})\otimes_{E_{*}}M)]_{\mathcal{D}(E_{*}E)}^{t}\simeq\bigoplus_{s}\operatorname{Ext}^{s,t-s}_{E_{*}E}(E_{*},M).$

As a consequence, whenever a $P(E_{*})$-module can be written as a
periodicization of some $M$ as above, its homotopy groups acquire an
additional grading. Note that any given module can be written as a
periodicization in many different ways and this additional grading depends on
that choice.

Our plan is to use Theorem 9.6 to compare the $K$-based Adams spectral
sequence in $E$-local spectra to the one in its algebraic model, and
subsequently give an explicit description of the latter in terms of filtration
by powers.

######  Lemma 9.10.

Let $X$ be either the $E$-local sphere or a Morava $K$-theory; that is,
$X=S^{0}_{E}$ or $X=K$. Then, there exists a canonical isomorphism

$\phi(X)\simeq P(E_{*})\otimes_{E_{*}}E_{*}X,$

in $h\operatorname{Mod}_{P(E_{*})}$, where $\phi$ is the algebraicity
equivalence of Theorem 9.6.

###### Proof.

It will be convenient to work here with the Johnson-Wilson homology, the
$p$-local Landweber exact homology theory with

$\pi_{*}E(n)\simeq\mathbb{Z}_{(p)}[v_{1},\ldots,v_{n-1},v_{n}^{\pm 1}]$

Both this ring and $E(n)_{*}E(n)$ are concentrated in degrees divisible by
$2p-2$, which is a key ingredient in the construction of the equivalence
$\phi$ [Pst18a, §2].

Choosing coordinates for the Quillen formal group over $E_{*}$ such that
$v_{i}=0$ for $i>n$ we obtain a classifying map $E(n)_{*}\rightarrow E_{*}$
which is faithfully flat and hence

$E_{*}X\simeq E_{*}\otimes_{E(n)_{*}}E(n)_{*}X$

The induced functor
$\operatorname{Comod}_{E(n)_{*}E(n)}\rightarrow\operatorname{Comod}_{E_{*}E}$
is an equivalence of categories by a result of Hovey and Strickland [HS05a].

Now suppose that $X$ is a spectrum with $E(n)_{*}X$ concentrated in degrees
divisible by $2p-2$. Since $\phi$ is compatible with taking homology, we have

$\mathrm{H}_{0}(\phi(X))\simeq E_{*}X\simeq E_{*}\otimes_{E(n)_{*}}E(n)_{*}X$

as comodules. By Remark 9.8, we have an Adams spectral sequence

$\operatorname{Ext}^{s,t}(E_{*}X,E_{*}X)\implies[\phi(X),P(E_{*})\otimes_{E_{*}}E_{*}X]^{t-s}$

Using the equivalence of categories of comodules, we can rewrite the
$E_{2}$-page as

$\operatorname{Ext}^{s,t}(E(n)_{*}X,E(n)_{*}X)\simeq\operatorname{Ext}^{s,t}(E_{*}X,E_{*}X)$

This vanishes above $n^{2}+n$ by [Pst18a, Remark 2.5] and is concentrated in
degrees $t$ divisible by $2p-2$. Since $2p-2>n^{2}+n$ by assumption, we deduce
that the spectral sequence collapses on the second page, so that the identity
of $E_{*}X$ descends to an equivalence $\phi(X)\simeq
P(E_{*})\otimes_{E_{*}}E_{*}X$.

This establishes the result for $X=S^{0}_{E}$, which has $E(n)_{*}X$ homology
concentrated in degrees divisible by $2p-2$. If $K$ is a Morava $K$-theory,
then as a consequence of the nilpotence theorem it is a module over the
minimal Morava $K$-theory of §3 and so equivalent as a spectrum to a direct
sum of $K(n)$. As $\phi$ preserves direct sums and

$E(n)_{*}K(n)\simeq K(n)_{*}\otimes_{E(n)_{*}}E(n)_{*}E(n)$

is concentrated in degrees divisible by $2p-2$, the result follows also for
$X=K$. ∎

######  Remark 9.11.

The detour into Johnson-Wilson homology Lemma 9.10 can be avoided, by directly
defining a splitting of order $(2p-2)$ on the category
$\operatorname{Comod}_{E_{*}E}$ by using the equivalence with quasi-coherent
sheaves on the moduli of formal groups of height at most $n$. We decided
against it, as the degree divisibility argument involving $E(n)$ is classical
and well-known.

Let us write $K_{alg}:=P(E_{*})\otimes_{E_{*}}E_{*}K$ for the algebraic model
for the Morava $K$-theory spectrum. Note that its homotopy groups satisfy

$\pi_{s}K_{alg}:=[P(E_{*}),K_{alg}]^{P(E_{*})}_{s}\simeq K_{s},$

so that $K_{alg}$ is a field object in $P(E_{*})$. Consequently, for a map
$M\rightarrow N$ of $P(E_{*})$-modules, the following two conditions are
equivalent:

  1. (1)

$K_{alg}\otimes_{P(E_{*})}M\rightarrow K_{alg}\otimes_{P(E_{*})}N$ is a split
monomorphism of $P(E_{*})$-modules or

  2. (2)

$[N,K_{alg}]_{*}\rightarrow[M,K_{alg}]_{*}$ is an epimorphism of graded
abelian groups.

This forms a class of monomorphisms and so determines an Adams spectral
sequence [PP21, §3.1].

######  Definition 9.12.

We call the Adams spectral sequence in $P(E_{*})$-modules determined by the
above class of monomorphisms the _$K_{alg}$ -based Adams spectral sequence_.

We are now ready to verify the correspondence between the topological and
algebraic Adams spectral sequences; we learned the following elegant argument
from Robert Burklund.

######  Proposition 9.13.

If $2p-2>n^{2}+n+1$, then the $K$-based Adams spectral sequence in
$\mathcal{S}p_{E}$ for $S^{0}_{E}$ is isomorphic to the $K_{alg}$-based Adams
spectral sequence for $P(E_{*})$ in $P(E_{*})$-modules.

###### Proof.

We claim that the equivalence $\phi$ of Theorem 9.6 induces an isomorphism of
exact couples leading to these spectral sequences. Both of these are obtained
by mapping into suitable towers, so we have to verify that $\phi$ takes a
$K$-Adams resolution of $S^{0}_{E}$ to a $K_{alg}$-Adams resolution of
$\phi(S^{0}_{E)})\simeq P(E_{*})$.

If $2p-2>n^{2}+n+1$, the equivalence $\phi$ is compatible with triangulated
structures [Pst18a, B.8], so it remains to show that it identifies
$K$-injectives with $K_{alg}$-injectives, as the corresponding class of
monomorphism is then uniquely determined. This follows immediately from Lemma
9.10, as on both sides each injective is a direct sum of shifts of,
respectively, $K$ and $K_{alg}$. ∎

In our case, in Proposition 9.13 we considered a $K_{alg}=P(E_{*})\otimes
E_{*}K$-based Adams spectral sequence for the monoidal unit $P(E_{*})$. Both
objects arise through periodicization, giving us additional information, which
we will now make explicit.

######  Proposition 9.14.

If $2p-2>n^{2}+n+1$, then the $K$-based Adams spectral sequence for
$S^{0}_{E}$ can be given an additional grading so that it becomes isomorphic
to the $E_{*}K$-based Adams spectral sequence in $\mathcal{D}(E_{*}E)$.

###### Proof.

By Proposition 9.13, the $K$-Adams spectral sequence in $\mathcal{S}p_{E}$ is
isomorphic to the $K_{alg}$-based Adams spectral sequence in
$P(E_{*})$-modules. The latter is isomorphic to the spectral sequence induced
by the Amitsur resolution

$P(E_{*})\rightarrow K_{alg}\rightrightarrows
K_{alg}\otimes_{P(E_{*})}K_{alg}\Rrightarrow\ldots$

which is image of the $E_{*}K$-Adams resolution of $E_{*}$ in
$\mathcal{D}(E_{*}E)$ under the periodicization functor. The statement follows
from the isomorphism of homotopy groups of Remark 9.9 applied to the exact
couple giving rise to the spectral sequence. ∎

The above result reduces the study of the $K$-based Adams spectral sequence at
large primes to the study of the algebrac $E_{*}K$-based in the derived
$\infty$-category of comodules. The latter can be described explicitly in
terms of the cohomology of the Morava stabilizer group, with no assumptions on
the prime, which is our next step.

We will need to recall some results on décalage, following Deligne and Levine
[Lev15].

######  Construction 9.15.

If $X^{\bullet}$ is a cosimplicial spectrum, we have an associated
conditionally convergent spectral sequence

$\prescript{\operatorname{Tot}}{}{E}^{2}_{s,t}:=\mathrm{H}^{s}\pi_{t}X^{\bullet}\implies\pi_{t-s}\operatorname{Tot}(X^{\bullet}).$

The décalage of this cosimplicial object is the tower

$\operatorname{Dec}_{n}:=\operatorname{Tot}(\tau_{\leq n}X^{\bullet})$

obtained by totalizing the Postnikov truncations of $X$. Since limits commute
with limits we have

$\varprojlim\operatorname{Dec}_{n}\simeq\varprojlim\operatorname{Tot}(\tau_{\leq
n}X^{\bullet})\simeq\operatorname{Tot}(\varprojlim\tau_{\leq
n}X^{\bullet})\simeq\operatorname{Tot}(X^{\bullet}).$

The above equivalence induces a spectral sequence associated to the tower on
the left, which is a conditionally convergent and of signature

$\prescript{\operatorname{Dec}}{}{E}^{1}_{s,t}:=H^{s}\pi_{t}X^{\bullet}\implies\pi_{t-s}\operatorname{Tot}(X^{\bullet}).$

Levine proves that the evident isomorphisms between the $E^{2}$-term of the
$\operatorname{Tot}$-spectral sequence and the $E_{1}$-term of the spectral
sequence associated to the décalage extends to an isomorphism of spectral
sequences [Lev15].

In our case, we will be working with cosimplicial objects in
$\mathcal{D}(E_{*}E)$, rather than in spectra. Thus, the role of the Postnikov
towers will be played by the truncations in the standard $t$-structure on the
derived category.

######  Definition 9.16.

Let $X^{\bullet}\colon\Delta\rightarrow\mathcal{D}(E_{*}E)$ be a cosimplicial
object in the derived $\infty$-category of comodules. The _homological
décalage_ of $X^{\bullet}$ is the tower

$h\operatorname{Dec}_{n}:=\operatorname{Tot}(\tau_{\leq n}X^{\bullet}),$

where $\tau_{\leq n}$ denotes the truncation in the standard $t$-structure on
the derived $\infty$-category.

######  Warning 9.17.

Note that if $X^{\bullet}$ is a cosimplicial object in $\mathcal{D}(E_{*}E)$,
then the homological décalage tower induces a spectral sequence obtained by
applying $\operatorname{Ext}(E_{*},-)$. Since Postnikov towers converge in
$\mathcal{D}(E_{*}E)$, this will be conditionally convergent to
$\operatorname{Ext}(E_{*},\operatorname{Tot}(X^{\bullet}))$.

The layers of the tower are given, up to a shift, by
$\operatorname{Tot}(H_{k}(X^{\bullet}))$ so that this spectral sequence has
signature

$\operatorname{Ext}(E_{*},\operatorname{Tot}(H_{k}(X^{\bullet}))\implies\operatorname{Ext}(E_{*},\operatorname{Tot}(X^{\bullet})).$

This spectral sequence is usually not the same as the one obtained by applying
$\operatorname{Ext}(E_{*},-)$ to $X^{\bullet}$ itself, because the homological
$t$-structure does not interact in an easy way with $E_{*}$-homotopy groups,
unlike in the case of the standard $t$-structure on the $\infty$-category of
spectra.

Despite the above warning, for a certain restricted class of cosimplicial
objets, we _do_ have an isomorphism between the spectral sequence of a
cosimplicial object and its homological décalage.

######  Lemma 9.18.

Let $X^{\bullet}\colon\Delta\rightarrow\mathcal{D}(E_{*}E)$ be a cosimplicial
object and assume that each $X^{m}$ is a direct sum of shifts of objects in
the heart, each of which is a cofree $E_{*}E$-comodule. Then

$F(E_{*}[t],h\operatorname{Dec}_{n}X^{\bullet})\simeq\operatorname{Dec}_{n}F(E_{*},X^{\bullet}),$

where $F(E_{*}[t],-)$ is the internal mapping spectrum in
$\mathcal{D}(E_{*}E)$. That is, for such cosimplicial objects, mapping out of
$E_{*}[t]$ takes homological décalage to the spectral one.

###### Proof.

Since $F(E_{*},-)$ preserves totalizations, it is enough to check that
$F(E_{*},-)$ takes homological truncations to Postnikov truncations. Since
$X^{\bullet}$ is levelwise a direct sum of shifts of cofree comodules, it is
enough to check this in the latter case, where it is clear since
$\operatorname{Ext}^{s}(E_{*}[t],C)=0$ for $s>0$ whenever $C$ is cofree. ∎

######  Corollary 9.19.

The $E_{*}K$-based Adams spectral sequence in $\mathcal{D}(E_{*}E)$ has a
second page isomorphic to the first page of the spectral sequence associated
to the homological décalage $h\operatorname{Dec}_{n}(E_{*}K)^{\otimes\bullet}$
of the Amitsur resolution.

###### Proof.

Both the Adams spectral sequence and the spectral sequence of the tower are
obtained by applying $F(E_{*}[t],-)$ (for all $t$ at once) and using the
relevant spectral sequence in spectra. Thus, the result then follows from
Levine’s work and Lemma 9.18, since

$E_{*}K^{\otimes n}\simeq(E_{*}E)^{\otimes
n}\otimes_{E_{*}}K_{*}^{\otimes_{E_{*}}n}$

and

$K_{*}^{\otimes_{E_{*}}n}\simeq\operatorname{Tor}_{E_{*}}(K_{*},K_{*})\otimes_{K_{*}}\ldots\otimes_{K_{*}}\operatorname{Tor}_{E_{*}}(K_{*},K_{*}),$

where the right hand side has $(n-1)$ factors and the unadorned tensor
products are the derived tensor products over $E_{*}$. This is a direct sum of
shifts of $K_{*}$, as needed. ∎

The idea is now to relate the homological décalage of $E_{*}K$ to that of
$K_{*}$.

######  Lemma 9.20.

The morphism $K_{*}\rightarrow E_{*}E\otimes K_{*}\simeq E_{*}K$ of algebras
in the derived category of $E_{*}E$-comodules induces an isomorphism of
spectral sequence associated to the homological décalage of Amitsur
resolutions.

###### Proof.

Since $E_{*}E$ is flat, for any $X\in\mathcal{D}(E_{*}E)$ we have
$\mathrm{H}_{n}(E_{*}E\otimes X)\simeq E_{*}E\otimes\mathrm{H}_{n}(X)$. Thus,
the map between the layers of the homological décalage towers is given by
applying totalization to the map

$\mathrm{H}_{n}(K_{*}^{\otimes\bullet})\rightarrow
E_{*}E^{\otimes\bullet}\otimes\mathrm{H}_{n}(K_{*}^{\otimes\bullet}).$

The map is obtained by tensoring the map

$E_{*}^{\otimes\bullet}\rightarrow E_{*}E^{\otimes\bullet},$

which is a quasi-isomorphism of levelwise flat cosimplicial comodules with
$\mathrm{H}_{n}(K_{*}^{\otimes\bullet})$, so it will be a quasi-isomorphism
again. It follows that it is an equivalence after taking totalizations. ∎

######  Proposition 9.21.

From the second page on, the Adams spectral sequence associated to $E_{*}K$ is
isomorphic to the spectral sequence obtained from the tower

$\ldots\rightarrow E_{*}/\mathfrak{m}^{2}\rightarrow E_{*}/\mathfrak{m},$

that is, it is the filtration by powers spectral sequence.

###### Proof.

We know from Corollary 9.19 that this Adams spectral sequence has second page
isomorphic to the spectral sequence of the homological décalage tower, which
is in turn equivalent to the homological décalage tower associated to the
$K_{*}$-Adams resolution by Lemma 9.20. Thus, it is enough to identify the
latter.

We’re interested in the cosimplicial object

$K_{*}\rightrightarrows K_{*}\otimes_{E_{*}}K_{*}\Rrightarrow\ldots,$

all tensor products being implicitly derived. Everything here is $2$-periodic
and concentrated in even degrees in the internal grading, so that we can
instead focus on the cosimplicial object

$K_{0}\rightrightarrows K_{0}\otimes_{E_{0}}K_{0}\Rrightarrow\ldots,$

working in the derived category of $E_{0}$-modules. As an object of the
latter, $K_{0}\otimes_{E_{0}}K_{0}$ is a direct sum of its homology groups,
and we have

$H_{*}(K_{0}\otimes_{E_{0}}K_{0})\simeq\operatorname{Tor}^{E_{0}}_{*}(K_{0},K_{0})$

and more generally