$H_{*}(K_{0}^{\otimes_{E_{0}}n})\simeq\operatorname{Tor}_{*}^{E_{0}}(K_{0},K_{0})^{\otimes_{K_{0}}n-1},$
(9.1)

where on the left we have the derived tensor product of $E_{0}$-modules and on
the right the ordinary tensor product of graded $K_{0}$-modules.

Using that $E_{0}$ is a regular ring with residue field $K_{0}$, the canonical
isomorphism

$\operatorname{Tor}^{1}_{E_{0}}(K_{0},K_{0})\simeq(\mathfrak{m}/\mathfrak{m}^{2})^{\vee}$

with the tangent space extends to a grading-preserving isomorphism

$\operatorname{Tor}_{*}^{E_{0}}(K_{0},K_{0})\simeq\Lambda_{K_{0}}((\mathfrak{m}/\mathfrak{m}^{2})^{\vee})$

with the exterior algebra. This is flat over $K_{0}$, which is a field, so by
standard arguments the cosimplicial object $K_{0}^{\otimes\bullet}$ on the
right of (9.1) encodes the cobar complex computing

$\textnormal{Cotor}_{\Lambda_{K_{0}}((\mathfrak{m}/\mathfrak{m}^{2})^{\vee})}(K_{0},K_{0})\simeq\operatorname{Sym}_{K_{0}}(\mathfrak{m}/\mathfrak{m}^{2})\simeq\bigoplus_{n\geq
0}\mathfrak{m}^{n}/\mathfrak{m}^{n+1}.$

Note that this is bigraded using the internal grading of the exterior algebra
and the Cotor-grading, with the summand $\mathfrak{m}^{n}/\mathfrak{m}^{n+1}$
being concentrated in degrees $(n,n)$.

Using the isomorphism in Equation 9.1 and passing to cohomology, we deduce
that

$H^{s}(H_{n}(K_{0}^{\otimes_{E_{0}}\bullet}))\simeq\begin{cases}\mathfrak{m}^{n}/\mathfrak{m}^{n+1}&\mbox{when
}$s=n$,\\\ 0&\mbox{otherwise,}\end{cases}$

where $H^{s}$ is the cohomology of the given cosimplicial abelian group. Thus,
the totalization spectral sequence for
$H_{n}(K_{0}^{\otimes_{E_{0}}\bullet})$, which we consider as a cosimplicial
object of the heart of the derived $\infty$-category of $E_{0}$-modules,
collapses and we deduce that

$\operatorname{Tot}(H_{n}(K_{0}^{\otimes\bullet}))\simeq\Sigma^{-n}\mathfrak{m}^{n}/\mathfrak{m}^{n+1},$

It follows that the graded pieces of the homological décalage filtration of
$E_{*}$ are given by

$\mathrm{fib}(\operatorname{Dec}_{n}\rightarrow\operatorname{Dec}_{n-1})\simeq\operatorname{Tot}(\Sigma^{n}H_{n}(K_{*}^{\otimes\bullet}))\simeq\mathfrak{m}^{n}/\mathfrak{m}^{n+1}\otimes_{E_{0}}E_{*}$

and so by induction the tower itself must be of the claimed form. ∎

######  Theorem 9.22.

If $2p-2>n^{2}+n+1$, the $K$-based Adams spectral sequence can be given an
additional grading so that it becomes isomorphic to the spectral sequence
induced by applying $\operatorname{Ext}_{E_{*}E}(E_{*},-)$ to the filtration
of $E_{*}$ given by powers $\mathfrak{m}^{n}E_{*}$ of the maximal ideal.

###### Proof.

This is a combination of Proposition 9.14 and Proposition 9.21. ∎

######  Remark 9.23.

If we pick $E=E_{n}$ to be the Morava $E$-theory of the Honda formal group law
over $\mathbb{F}_{p^{n}}$, then since
$\mathfrak{m}^{n}/\mathfrak{m}^{n+1}E_{*}$ is $\mathfrak{m}$-torsion, we have

$\operatorname{Ext}_{E_{*}E}^{*}(E_{*},\mathfrak{m}^{n}/\mathfrak{m}^{n+1}E_{*})\simeq\mathrm{H}_{\operatorname{cts}}^{*}(\mathbb{G}_{n},\mathfrak{m}^{n}/\mathfrak{m}^{n+1}E_{*}).$

Thus, the spectral sequence of Theorem 9.22 is isomorphic to the conditionally
convergent spectral sequence

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

obtained by the same filtration of $E_{*}$, but in continuous
$\mathbb{G}_{n}$-modules.

###  10\. Height one $K$-based Adams spectral sequence at an odd prime

Let us describe how using our methods one obtains an explicit description of
the $K$-based Adams spectral sequence at $n=1$ and $p$ odd. Since in this case

$2p-2\geq 4>3=n^{2}+n+1,$

Theorem 9.22 and Remark 9.23 imply that the relevant spectral sequence is
isomorphic to the one computing cohomology
$\mathrm{H}_{\operatorname{cts}}^{*}(\mathbb{G}_{n},E_{*})$ using the
filtration by powers of the maximal ideal, where $E_{0}$ is the Lubin–Tate
ring of the Honda formal group law.

######  Notation 10.1.

We have $E_{*}\simeq\mathbb{Z}_{p}[u^{\pm 1}]$, $\mathfrak{m}=(p)$ and
$\mathbb{G}_{1}\simeq\mathbb{Z}_{p}^{\times}$ with the action determined by
$\lambda_{*}(u)=\lambda u$ for $\lambda\in\mathbb{Z}_{p}$. We will denote the
associated graded of $E_{*}$ by

$A_{*}\simeq\mathbb{F}_{p}[b][u^{\pm 1}],$

with $|b|=(0,1)$ the equivalence class of $p\in E_{0}$, and $|u|=(2,0)$, where
the first degree is internal and the second one is coming from the filtration.

The $p$-adic group $\mathbb{Z}_{p}^{\times}$ is topologically cyclic at odd
primes, generated by $\psi=\sigma(1+p)$ for $\sigma$ a primitive $(p-1)$-th
root of unity. It follows that for a profinite
$\mathbb{Z}_{p}^{\times}$-module $M$, its cohomology can be computed using the
$2$-term complex

$\partial\colon M\rightarrow M,$ (10.1)

where $\partial=id_{M}-\psi_{*}$.

The complex of (10.1) is a quasi-isomorphic quotient of the standard group
cohomology cochain complex

$M\rightarrow\operatorname{Map}_{\operatorname{cts}}(\mathbb{Z}_{p}^{\times},M)\rightarrow\operatorname{Map}_{\operatorname{cts}}(\mathbb{Z}^{\times}_{p}\times\mathbb{Z}_{p}^{\times},M)\rightarrow\ldots,$
(10.2)

the quotient map being given by the identity in degree zero and evaluation at
the generator $\psi\in\mathbb{Z}_{p}^{\times}$ in degree one.

From this explicit description we see that if we equip $M$ with a filtration,
the above quotient map is a quasi-isomorphism of filtered complexes, where we
consider both (10.1) and (10.2) with the filtration induced from that of $M$.
In particular, both filtrations induced isomorphic spectral sequences.

######  Remark 10.2.

The 2-term complex of (10.1) can be used to directly compute the cohomology
groups
$\mathrm{H}_{\operatorname{cts}}^{*}(\mathbb{Z}_{p}^{\times},\mathbb{Z}_{p}[u^{\pm
1}])$, recovering the classical image of $J$ pattern [Lur10, Lecture 35]. Our
goal is not to compute the above groups, which are well-known, but rather
analyze the structure of the spectral sequence associated to filtration by
powers of $p$.

We first determine the cohomology groups of the associated graded $A_{*}$. We
have $\psi_{*}b=b$, because the latter can be represented by $p\in E_{0}$
which is necessarily acted on trivially, and since $\psi_{*}u\equiv\sigma
u\pmod{p}$, we have $\psi_{*}u=[\sigma]u$ in the associated graded, where
$[\sigma]\in\mathbb{F}_{p}$ is the image of $\sigma$. Thus, in the associated
graded we have that $\partial(b)=0$ and $\partial(u^{k})$ is a unit multiple
of $u^{k}$ for $(p-1)\nmid k$ and zero otherwise. We deduce the following.

######  Proposition 10.3.

The $E_{2}$-term of the $K$-based Adams for $S^{0}$ at $n=1$ and $p>2$ is
given by

$\operatorname{Ext}_{K_{*}K}^{*}(K_{*},K_{*})\cong\mathrm{H}_{\operatorname{cts}}^{*}(\mathbb{G}_{1},A_{*})\cong\mathrm{H}_{\operatorname{cts}}^{*}(\mathbb{Z}_{p}^{\times},\mathbb{F}_{p}[b][u^{\pm
1}])\cong\mathbb{F}_{p}[b][v_{1}^{\pm 1}]\otimes\Lambda(\zeta),$

where $v_{1}=u^{p-1}$ is of degree $(2p-2,0,0)$ and $\zeta$ is the class of
$b$ of degree $(0,1,1)$, with the last degree the cohomological one.

By Remark 9.23, if the $K$-based Adams is isomorphic to the spectral sequence
computing
$\mathrm{H}_{\operatorname{cts}}^{*}(\mathbb{Z}_{p}^{\times},\mathbb{Z}_{p}[u^{\pm
1}])$ induced by the filtration of (10.1) by powers of $p$, with $E_{2}$-term
as above and with differentials $d_{r}\colon E_{r}\rightarrow E_{r}$ of degree
$(0,r,1)$. We will now analyze this spectral sequence.

We know that $\partial(p)=0$ in $E_{0}$, so that $b$ is a permanent cycle. The
same is true for $\zeta$, for degree reasons, and we deduce that all of the
differentials are both $b$ and $\zeta$-linear, and so the structure of the
spectral sequence is completely determined by what it does on $v_{1}^{k}$. One
computes with no difficulty that if we write $k=p^{n}m$, where $p\nmid m$,
then

$\partial(u^{(p-1)k})\equiv\text{unit}\cdot p^{n+1}u^{(p-1)k}\pmod{p^{n+2}}$

from which we deduce that

$d_{i}(v_{1}^{k})=\begin{cases}0&\text{for }i<n+1\\\ \zeta
b^{n+1}v_{1}^{k}&\text{for }i={n+1}.\end{cases}$ (10.3)

Note that the first computation is also implied by the Leibniz rule for $p$-th
powers. The extension problems are resolved using multiplication by $b$,
recovering the classical answer that

$\mathrm{H}_{\operatorname{cts}}^{s,t}(\mathbb{Z}_{p}^{\times},\mathbb{Z}_{p}[u^{\pm
1}])=\begin{cases}\mathbb{Z}_{p}&\mbox{if }(s,t)=(0,0),\\\
\mathbb{Z}_{p}\\{\zeta\\}&\mbox{if }(s,t)=(1,0),\\\ \mathbb{Z}/p^{n+1}\\{\zeta
u^{(p-1)p^{n}m}\\}&\mbox{if }(s,t)=(1,(2p-2)p^{n}m)\mbox{, where }p\nmid m,\\\
0&\mbox{otherwise,}\end{cases}$

where $\\{-\\}$ denotes a generator of the given group. Thus, the spectral
sequence is as in the following familiar “image of J” pattern

${\bullet}$${\ldots}$${\bullet}$${\ldots}$${\bullet}$${\ldots}$${\bullet}$${\ldots}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${{\bullet_{\zeta
b}}}$${{\bullet_{b^{2}}}}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${{\bullet_{\zeta}}}$${{\bullet_{b}}}$${{\bullet_{v_{1}\zeta}}}$${{\bullet_{v_{1}b}}}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${{\bullet_{1}}}$${{\bullet_{v_{1}}}}$${{\bullet_{v_{1}^{2}}}}$${{\bullet_{v_{1}^{p}}}}$${{-1}}$${0}$${{2p-2}}$${\ldots}$${\ldots}$$\scriptstyle{t-s}$$\scriptstyle{s}$

with even longer differentials supported on $v_{1}^{p^{k}}$ for $k\geq 1$.

It is interesting to observe that this shows that already at height one, the
$K$-based Adams spectral sequence has non-zero differentials of arbitrary
length.

######  Example 10.4 (Non-completely convergent $K$-Adams).

Using the above calculation, we can give an example of a $K$-local spectrum
for which its $K$-based Adams spectral sequence is not completely convergent.
Note that we know that it is always conditionally convergent by Proposition
7.2.

To this end, we will construct a spectrum $X$ for which
$\operatorname{lim}_{r}^{1}E_{r}^{0,0}(X)$ does not vanish at $n=1$ and $p>2$.
Set $\lambda_{k}=2(p-1)p^{k}$ and consider the spectrum

$X=L_{K(1)}\bigoplus_{k\geq 0}S^{-\lambda_{k}}.$

In bidegree $(s,t)=(0,0)$, the $E_{2}$-page is given by

$E_{2}^{0,0}(X)\cong\bigoplus_{k\geq 0}\mathbb{F}_{p}\\{v_{1}^{p^{k}}\\},$

where the generators correspond to the maps $v_{1}^{p^{k}}\colon K(n)_{*}\to
K(n)_{*}S^{\lambda_{k}}$. The formulas for the differentials of Equation 10.3
show that (up to reindexing) the filtration on $E_{2}^{0,0}(X)$ is given by

$\ldots\subset\bigoplus_{k\geq
2}\mathbb{F}_{p}\\{v_{1}^{p^{k}}\\}\subset\bigoplus_{k\geq
1}\mathbb{F}_{p}\\{v_{1}^{p^{k}}\\}\subset\bigoplus_{k\geq
0}\mathbb{F}_{p}\\{v_{1}^{p^{k}}\\}=E_{2}^{0,0}(X).$

Therefore, we obtain an exact sequence

$0\to\operatorname{lim}_{r}E_{r}^{s,t}(X)\to E_{2}^{0,0}(X)=\bigoplus_{k\geq
0}\mathbb{F}_{p}\\{v_{1}^{p^{k}}\\}\to\prod_{k\geq
0}\mathbb{F}_{p}\\{v_{1}^{p^{k}}\\}\to\operatorname{lim}_{r}^{1}E_{r}^{s,t}(X)\to
0.$

In particular, we see that $\operatorname{lim}_{r}E_{r}^{s,t}(X)=0$, while
$\operatorname{lim}_{r}^{1}E_{r}^{s,t}(X)\neq 0$.

##  Part III Homology of inverse limits and the algebraic chromatic splitting
conjecture

In the third part of the work, we largely move away from Morava $K$-theory,
and we focus on homology of $K$-local spectra. From the perspective of
$E$-local category, $K$-localization is a form of completion, and so we first
focus on homology of inverse limits, construction a spectral sequence
computing these in the generality of an arbitrary adapted homology theory.

In the case of $K$-localization, the $E_{2}$-page of this spectral sequence is
given by derived functors of limits in $E_{*}E$-comodules. We describe the
latter in terms of cohomology of the Morava stabilizer group, and use it to
compute the zeroth one at all heights and primes. At height one, we compute
these derived functors completely.

###  11\. The homology of inverse limits

We begin with a general review of the context for the construction of the
modified Adams spectral sequence, following the approach of Devinatz and
Hopkins [Dev97]. This approach will then be employed to construct a spectral
sequence that computes the homology of inverse limits in a presentable stable
$\infty$-category.

In particular, this gives rise to a spectral sequence for computing the
$E$-homology of the inverse limit of a tower of spectra $(X_{\alpha})$ from
the derived functors of inverse limits of the tower of $E_{*}E$-comodules
$(E_{*}X_{\alpha})$, for suitable ring spectra $E$. The question of when this
spectral sequence converges is subtle, and will be studied in detail in the
case of Morava $E$-theory. The material in this section is based on
unpublished work of Mike Hopkins and Hal Sadofsky.

####  11.1. Adapted homology theories

In this short section we will recall basic facts about adapted homology
theories. Informally, these are exactly those homology theories which admit an
Adams spectral sequence based on injectives. Everything here is classical,
although our presentation will be most close to [PP21, §2].

A locally graded $\infty$-category is an $\infty$-category equipped with a
distinguished autoequivalence
$[1]_{\mathcal{C}}\colon\mathcal{C}\rightarrow\mathcal{C}$. A locally graded
functor $f\colon\mathcal{C}\rightarrow\mathcal{D}$ is functor equipped with a
natural isomorphism $f\circ[1]_{\mathcal{C}}\simeq[1]_{\mathcal{D}}\circ f$.

######  Definition 11.1.

Let $\mathcal{C}$ be a stable $\infty$-category, considered as a locally
graded $\infty$-category using the suspension functor, and $\mathcal{A}$ an
arbitrary locally graded abelian category. We say a locally graded functor
$H\colon\mathcal{C}\rightarrow\mathcal{A}$ is a _homology theory_ if for any
cofibre sequence

$X\rightarrow Y\rightarrow Z$

in $\mathcal{C}$, the induced diagram

$H(X)\rightarrow H(Y)\rightarrow H(Z)$

in $\mathcal{A}$ is exact in the middle.

######  Remark 11.2.

The requirement that $H$ is locally graded amounts to specifying an
isomorphism

$H(\Sigma X)\simeq H(X)[1]$

natural in $X\in\mathcal{C}$.

In cases of interest to us, $H$ will be _Grothendieck_ in the sense that

  1. (1)

$\mathcal{C}$ is presentable,

  2. (2)

$\mathcal{A}$ is Grothendieck abelian and

  3. (3)

$H$ preserves arbitrary direct sums.

In particular, this means that $\mathcal{A}$ has enough injectives. Any
injective object $I$ of $\mathcal{A}$ gives rise to a cohomological functor

$\operatorname{Hom}_{\mathcal{A}}(H(-),I)\colon\mathcal{C}^{\operatorname{op}}\to\mathcal{A}.$

Since $\mathcal{C}$ is presentable and stable, Brown’s representability
theorem holds [PP21, Proposition 2.15], so that there exists some
$D(I)\in\operatorname{Ho}\mathcal{C}$ representing
$\operatorname{Hom}_{\mathcal{A}}(H(-),I)$. It follows that, for any object
$X\in\mathcal{C}$, there is a natural equivalence

$\operatorname{Hom}_{\mathcal{A}}(H(X),I)\simeq\pi_{0}\operatorname{Hom}_{\mathcal{C}}(X,D(I)).$

Note that the object $D(I)$ is $H$-local, in the sense that for any object
$Y\in\mathcal{C}$ with $H(Y)=0$ we have $\operatorname{Hom}(Y,D(I))=0$.
Moreover, the identity map on $D(I)$ corresponds under this isomorphism to a
natural counit map $H(D(I))\to I$.

######  Definition 11.3.

We say $H\colon\mathcal{C}\to\mathcal{A}$ is _adapted_ if the counit map
$H(D(I))\to I$ is an equivalence for any injective $I\in\mathcal{A}$.

######  Example 11.4.

Rational homology, viewed as a functor from spectra to graded rational vector
spaces, is an adapted homology theory. In this case, the lift $D(V)$ of a
(graded) $\mathbb{Q}$-vector space $V$ is the (generalized) Eilenberg–MacLane
spectrum $HV$.

######  Example 11.5.

In contrast to the previous example, mod $p$ homology is not adapted for any
$p$; indeed, the lift of $\mathbb{F}_{p}$ is $H\mathbb{F}_{p}$, but its mod
$p$-cohomology is the mod $p$ Steenrod algebra
$\operatorname{\mathcal{A}}_{p}$ which is larger than $\mathbb{F}_{p}$.

However, $H_{*}(-,\mathbb{F}_{p})$ has more structure: it takes values in the
category $\operatorname{Comod}_{\operatorname{\mathcal{A}}_{p}}$ of graded
$\operatorname{\mathcal{A}}_{p}$-comodules. It turns out that the homology
theory

$H_{*}(-;\mathbb{F}_{p})\colon\operatorname{Sp}\to\operatorname{Comod}_{\operatorname{\mathcal{A}}_{p}}$

is adapted.

######  Remark 11.6.

The phenomena visible in Example 11.5 is typical in the sense that asking for
$H$ to be adapted is to ask for $\mathcal{A}$ to encode all available
homological information. This generalizes considerably, one can show that any
homology theory factors uniquely through an adapted one followed by an exact
comonadic functor of abelian categories [PP21, §3.3]

In fact, the previous example generalizes considerably. Let $R$ be a homotopy
commutative ring spectrum which satisfies the Adams condition, that is, it can
be written as a filtered colimit of finite spectra $X_{\alpha}$ with the
properties that

  1. (1)

$R_{*}X_{\alpha}$ is a finitely generated projective $R_{*}$-module and

  2. (2)

the Künneth map
$E^{*}X_{\alpha}\rightarrow\operatorname{Hom}_{E_{*}}(E_{*}X_{\alpha},E_{*})$
is an isomorphism.

Note that the second condition follows from the first whenever $R$ can be made
$\mathbf{E}_{1}$.

For example, $H\mathbb{F}_{p}$ or any Landweber exact homology theory satisfy
the Adams condition [Hov04a, Theorems 1.4.7-9], but $H\mathbb{Z}$ does not
[GH, Page 17]. This condition guarantees that the associated Hopf algebroid
$(R_{*},R_{*}R)$ is Adams and thus has a well-behaved category of comodules
$\operatorname{Comod}_{R_{*}R}$; see [Hov04a, Section 1.4] for details.

One way in which the Adams-type condition implies that the category
$\operatorname{Comod}_{R_{*}R}$ is well-behaved also in the topological sense
is the following result of Devinatz.

######  Lemma 11.7 ([Dev97, Theorem 1.5 and Section 2]).

Let $R$ be a topologically flat commutative ring spectrum. For any injective
$R_{*}R$-comodule $I$, the identity map on $D(I)$ induces an isomorphism
$R_{*}D(I)\cong I$. In other words, the functor

$R_{0}\colon\mathcal{S}p\to\operatorname{Comod}_{R_{*}R}$

is an adapted homology theory. ∎

####  11.2. Construction of the spectral sequence

The starting point of our construction is the existence of modified Adams
towers for any homology theory $H\colon\mathcal{C}\to\mathcal{A}$. We collect
its properties in the next result:

######  Lemma 11.8.

Consider an adapted homology theory $H\colon\mathcal{C}\to\mathcal{A}$. Let
$X\in\mathcal{C}$ and suppose

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H(X)=C_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{0}}$$\textstyle{I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{0}}$$\textstyle{I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{1}}$$\textstyle{\ldots}$$\textstyle{C_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{0}}$$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{1}}$$\textstyle{\ldots}$

is an injective resolution of $H(X)$ in $\mathcal{A}$, where
$C_{i}=\ker(\tau_{i})$ for all $i$. There exists a tower of objects in
$\mathcal{C}$ over $X$ of the form

$\textstyle{X=X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{0}}$$\scriptstyle{f_{1}}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\scriptstyle{f_{2}}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\textstyle{\Sigma^{0}D(I_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{0}}$$\textstyle{\Sigma^{-1}D(I_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{1}}$$\textstyle{\Sigma^{-2}D(I_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{2}}$$\textstyle{\ldots,}$

where the dotted maps are the boundary maps shifting degree by $1$, and such
that the following properties are satisfied for all $i$:

  1. (1)

$X_{i+1}\simeq\operatorname{fib}(f_{i})$;

  2. (2)

$H(X_{i})\cong C_{i}[{-i}]$;

  3. (3)

$H(f_{i})=\eta_{i}[{-i}]$;

  4. (4)

$H(g_{i})=0$;

  5. (5)

$H(\delta_{i})$ is the surjection $I_{i}[{-i}]\to C_{i+1}[{-i}]$.

If $X$ is $H$-local, then $X_{i}$ and $D(I_{i})$ are $H$-local for all $i$.
Moreover, this tower is weakly functorial; that is, a map $X\to Y$ can be
extended to a (non-canonical) map of towers $(X_{i})\to(Y_{i})$.

###### Proof.

This lemma is an axiomatization of the results proven in [Dev97, Section 1].
The tower so constructed consists of $H$-local spectra as noticed in Section
11.1, while naturality follows from the naturality of injective resolutions. ∎

######  Notation 11.9.

We refer to the tower $(X_{i})$ constructed in Lemma 11.8 as the modified
Adams tower of $X$ with respect to the adapted homology theory
$H\colon\mathcal{C}\to\mathcal{A}$.

Since the limit $\varprojlim I_{\alpha}$ of an injective tower of injectives
in $\mathcal{A}$ is injective, the construction of the previous subsection
yields an object $D(\varprojlim I_{\alpha})$ of $\mathcal{C}$. In order to
proceed, we need an auxiliary characterization of injective towers of objects
in a Grothendieck abelian category $\mathcal{A}$.

######  Lemma 11.10.

Let $\mathcal{A}$ be a Grothendieck abelian category. An object
$I=(I_{\alpha})\in\mathcal{A}^{\mathbb{N}}$ is injective if and only if
$I_{\alpha}$ is injective for all $\alpha\in J$ and all structure maps in $I$
are split epimorphisms.

###### Proof.

This is proven in [Jan88, Proposition 1.1]. We sketch the argument of the
implication we need. Let $j\colon\mathbb{N}^{\delta}\to\mathbb{N}$ be the
inclusion of the discrete set of natural numbers into the poset of natural
numbers. This induces an adjunction $(j^{*},j_{*})$ between diagram
categories. If $I$ is injective, then the natural monomorphism $I\to
j_{*}j^{*}I$ is split. By construction, the structure maps of $j_{*}j^{*}I$
are projections, so $I$ also has split surjective structure maps, and the
claim follows. ∎

######  Proposition 11.11.

If
$I=(I_{\alpha})\in\operatorname{Inj}(\operatorname{\mathcal{A}}^{\mathbb{N}})$
is an injective tower in $\mathcal{A}$, then there is a preferred equivalence

$\textstyle{D(\varprojlim_{\mathcal{A}}I_{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\varprojlim_{\mathcal{C}}D(I_{\alpha}).}$

of objects in $\mathcal{C}$, well-defined up to homotopy.

###### Proof.

For brevity, let us write $I_{\infty}:=\varprojlim_{\mathcal{A}}I_{\alpha}$;
note that this is an injective object of $\mathcal{A}$. The canonical
structure maps $\varprojlim_{\mathcal{A})}I_{\alpha}\to I_{\alpha}$ give
arrows

$D(I_{\infty})\rightarrow D(I_{\alpha}),$

well-defined and compatible up to homotopy. By the Milnor exact sequence,
these can be lifted to a homotopy class of maps

$\textstyle{\phi\colon
D(I_{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varprojlim
D(I_{\alpha}),}$

in $\mathcal{C}$. We claim this is an equivalence, which we will check by
verifying that both sides represent the same functor in the homotopy category.

Indeed, for any $X\in\mathcal{C}$, there are isomorphisms

$\displaystyle\textstyle[X,D(I_{\infty})]_{*}$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{A}}^{*}(H(X),I_{\infty})$
$\displaystyle\cong\varprojlim\
\operatorname{Hom}_{\mathcal{A}}^{*}(H(X),I_{\alpha})$
$\displaystyle\cong\varprojlim\ [X,D(I_{\alpha})]_{*}.$

The Milnor sequence associated to the limit $\varprojlim D(I_{n})$ takes the
form

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varprojlim^{1}[X,D(I_{n})]_{*+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[X,\varprojlim
D(I_{\alpha})]_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varprojlim[X,D(I_{\alpha})]_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
(11.1)

It follows from Lemma 11.10 that the structure maps in the tower

$([X,D(I_{\alpha})]_{*+1})\cong(\operatorname{Hom}_{\mathcal{A}}^{*+1}(H(X),I_{\alpha}))$

are split epimorphisms, so the $\varprojlim^{1}$-term in (11.1) vanishes.
Thus, the Yoneda lemma implies that $\phi$ is an equivalence, as claimed. ∎

The spectral sequence we construct will involve derived functors of the limit,
and so we first establish a consistent notation for these.

######  Notation 11.12.

If $\mathcal{A}$ is a Grothendieck abelian category and $F\colon
J\rightarrow\mathcal{A}$ is a diagram, then we write
$\textstyle\varprojlim_{\mathcal{A}}F$ for the limit taken in $\mathcal{A}$
itself. Formation of limits defines a functor

$\textstyle\varprojlim_{\mathcal{A}}\colon\operatorname{Fun}(J,\mathcal{A})\rightarrow\mathcal{A}$

and we denote its right derived functors by

$\textstyle\varprojlim^{s}_{\mathcal{A}}F.$

Alternatively, $\mathcal{A}$ has an associated derived $\infty$-category, and
we write

$\textstyle\varprojlim_{\mathcal{D}(\mathcal{A})}F$

for the limit of the composite of $F$ with the inclusion with the heart. These
two different notions of limits, one in $\mathcal{A}$ and the other in the
derived $\infty$-category, are related by the formula

$\textstyle\varprojlim^{s}_{\mathcal{A}}F\simeq
H_{-s}(\textstyle\varprojlim_{\mathcal{D}(\mathcal{A})}F),$

where on the right hand side we have the homology of an object of the derived
$\infty$-category. Thus, $\textstyle\varprojlim_{\mathcal{D}(\mathcal{A})}F$
can be thought of as the total derived functor of the limit.

######  Theorem 11.13.

Let $H\colon\mathcal{C}\to\mathcal{A}$ be an adapted homology theory. If
$(X^{\alpha})\in\mathcal{C}^{\mathbb{N}}$ is a tower in $\mathcal{C}$, then
there exists a natural spectral sequence in $\mathcal{A}$ of the form

$\textstyle
E_{2}^{s,t}\cong\varprojlim_{\mathcal{A}}^{s}H(X^{\alpha})[-t]_{\mathcal{A}}\implies
H(\varprojlim X^{\alpha})[s-t]_{\mathcal{A}}$ (11.2)

with differentials $d_{r}^{s,t}\colon E_{r}^{s,t}\to E_{r}^{s+r,t+r-1}$.

###### Proof.

If $H$ is adapted, then so is the induced homology theory

$H\colon\mathcal{C}^{\mathbb{N}}\rightarrow\mathcal{A}^{\mathbb{N}}$

between the $\infty$-categories of towers [PP21, Example 8.24]. Thus, we have
Adams resolutions of towers and we let $(X_{i}^{\alpha})_{i\geq 0}$ be a
modified Adams tower of $(X^{\alpha})$ as in Lemma 11.8.

Taking limits in the Adams resolution of $X^{\alpha}$ yields a tower in
$\mathcal{C}$ of the form

$\textstyle{\varprojlim
X_{0}^{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\textstyle{\varprojlim
X_{1}^{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{0}}$$\scriptstyle{f_{1}}$$\textstyle{\varprojlim
X_{2}^{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\scriptstyle{f_{2}}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\textstyle{\varprojlim\Sigma^{0}D(I_{0}^{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{0}}$$\textstyle{\varprojlim\Sigma^{-1}D(I_{1}^{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{1}}$$\textstyle{\varprojlim\Sigma^{-2}D(I_{2}^{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{2}}$$\textstyle{\ldots.}$

For brevity, let us write
$[k]:=[k]_{\mathcal{A}}:\mathcal{A}\rightarrow\mathcal{A}$ for the internal
degree shift. Setting $D_{1}^{s,t}=H(\varprojlim X_{s}^{\alpha})[s-t]$ and
$E_{1}^{s,t}=H(\varprojlim\Sigma^{-s}D(I_{s}^{\alpha}))[s-t]$, this induces an
exact couple of bigraded objects in $\mathcal{A}$ of the form

---
$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}=H(g_{i})[s-t]}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}=H(f_{i})[s-t]}$$\textstyle{E,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k_{1}=H(\delta_{i})[s-t]}$

where the maps have bidegree $|i_{1}|=(-1,-1)$, $|j_{1}|=(0,0)$, and
$|k_{1}|=(1,0)$. The $E_{1}$-term can be identified using Proposition 11.11
and Lemma 11.7 as

$\textstyle E_{1}^{s,t}=\varprojlim\Sigma^{-s}D(I_{s}^{\alpha})[s-t]\cong
D(\varprojlim_{\mathcal{A}}I_{s}^{\alpha})[-t]=(\varprojlim_{\mathcal{A}}I_{s}^{\alpha})[-t]$

and $d_{1}=\varprojlim_{\mathcal{A}}\tau_{s}^{\alpha}$. This gives the
$E_{2}$-term

$\textstyle
E_{2}^{s,t}\cong\varprojlim_{\mathcal{A}}^{s}(H_{*}X^{\alpha})[-t],$

as needed. It is clear from this construction and Lemma 11.8 that the
resulting spectral sequence is natural in the tower $X^{\alpha}$ and that all
pages and differentials are in $\mathcal{A}$. ∎

######  Warning 11.14.

Note that since the local grading
$[1]_{\mathcal{A}}\colon\mathcal{A}\rightarrow\mathcal{A}$ is an equivalence,
it commutes with derived functors of the limit, and in the spectral sequence
of Theorem 11.13 we have

$E_{2}^{s,t}:=\textstyle\varprojlim_{\mathcal{A}}^{s}H(X^{\alpha})[-t]_{\mathcal{A}}\simeq(\textstyle\varprojlim_{\mathcal{A}}^{s}H(x^{\alpha}))[-t]_{\mathcal{A}}\simeq
E_{2}^{s}[-t]_{\mathcal{A}}$

Thus, the $t=0$ line in the spectral sequence already determines the whole
$E_{2}$-page. This is similar to the case of the Bockstein spectral sequence,
which also has a periodic $E_{2}$-page.

######  Remark 11.15.

Suppose that $(X^{\alpha})$ is a tower under $X$. Then, the image

$\mathrm{im}(H(X))\subseteq\textstyle\varprojlim^{0}_{\mathcal{A}}H(X^{\alpha})$

consists of permanent cycles; that is, it is in the kernel of all the
differentials in the spectral sequence of Theorem 11.13.

To see this, note that the assumption gives a map from the constant tower on
$X$ to $(X^{\alpha})$, inducing a map of spectral sequences

$\textstyle{\varprojlim_{\mathcal{A}}^{s}H(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varprojlim^{s}_{\mathcal{A}}H(X^{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H(\varprojlim
X^{\alpha})}$

The $E_{2}$-page of the top spectral sequence is concentrated in

$E_{2}^{0,t}\cong\textstyle\varprojlim_{\mathcal{A}}^{0}H(X)[-t]_{\mathcal{A}}\cong
H(X)[-t]_{\mathcal{A}},$

so that the spectral sequence collapses. The comparison map is readily
identified with the induced morphism
$H(X)\to\varprojlim_{\mathcal{A}}H(X^{\alpha})$, so the claim follows.

####  11.3. Derived limits of comodules

We now specialize the above to the case of a homology theory on spectra
corresponding to a ring spectrum of Adams type. In order to study the
$E_{2}$-term of the spectral sequence of Theorem 11.13, we begin with a
general discussion of inverse limits of comodules over suitable flat Hopf
algebroids.

Let $(A,\Psi)$ be an Adams Hopf algebroid in the sense of [Hov04a][1.4.3], so
that $\Psi$ is a filtered colimit of dualizable comodules. We will describe an
approach to computing limits in the derived $\infty$-category
$\mathcal{D}(\Psi)=\mathcal{D}(\operatorname{Comod}_{\Psi})$ of comodules by
reducing to the case of modules. The forgetful functor
$\epsilon_{*}\colon\operatorname{Comod}_{\Psi}\to\operatorname{Mod}(A)$ is
_left_ adjoint to the extended (or cofree) comodule functor $\epsilon^{*}$
which sends an $A$-module $M$ to $\Psi\otimes_{A}M$. These functors are exact
and thus give rise to an adjunction:

$\epsilon_{*}\dashv\epsilon^{*}\colon\mathcal{D}(\Psi)\leftrightarrows\mathcal{D}(A).$

In terms of chain complexes, both can be computed levelwise.

The corresponding monad $\epsilon^{*}\epsilon_{*}$ can be used to construct a
resolution of any comodule. Concretely, if $M$ is a $\Psi$-comodule, then we
have the associated cobar complex

$0\rightarrow
M\rightarrow\Psi\otimes_{A}M\rightarrow\Psi\otimes_{A}\Psi\otimes_{A}M\rightarrow\ldots,$

which is a resolution of $M$ by extended comodules, that is, those of the form
$\epsilon^{*}N=\Psi\otimes_{A}N$ for an $A$-module $N$.

One way to phrase that the cobar complex is exact is to say that the augmented
cosimplicial object from which it arises, namely

$M\rightarrow\Psi\otimes_{A}M\rightrightarrows\Psi\otimes_{A}\Psi\otimes_{A}M\Rrightarrow\ldots,$
(11.3)

is a limit diagram in the derived $\infty$-category $\mathcal{D}(\Psi)$. Now,
since limit diagrams are stable under levelwise limits, it follows that if
$(M_{i})$ is a diagram of comodules, then the diagram

$\textstyle\varprojlim_{\mathcal{D}(\Psi)}M_{i}\rightarrow\varprojlim_{\mathcal{D}(\Psi)}\Psi\otimes_{A}M_{i}\rightrightarrows\varprojlim_{\mathcal{D}(\Psi)}\Psi\otimes_{A}\Psi\otimes_{A}M_{i}\Rrightarrow\ldots,$

is also limit, giving an approach to computing
$\varprojlim_{\mathcal{D}(\Psi)}M_{i}$. To see this, notice that since
$\epsilon^{*}$ is a right adjoint and thus preserves limits, we have

$\textstyle\varprojlim_{\mathcal{D}(\Psi)}\Psi\otimes_{A}N_{i}\simeq\varprojlim_{\mathcal{D}(\Psi)}\epsilon^{*}N_{i}\simeq\epsilon^{*}(\varprojlim_{\mathcal{D}(A)}N_{i})\simeq\Psi\otimes_{A}(\varprojlim_{\mathcal{D}(A)}N_{i}).$

Applying this to the cosimplicial diagram (11.3), we see that we get a limit
diagram of the form

$\textstyle\varprojlim_{\mathcal{D}(\Psi)}M_{i}\rightarrow\Psi\otimes_{A}(\varprojlim_{\mathcal{D}(A)}M_{i})\rightrightarrows\Psi\otimes_{A}(\varprojlim_{\mathcal{D}(A)}\Psi\otimes_{A}M_{i})\Rrightarrow\ldots$

which expresses the limit in the derived category of comodules using only
limits taken in the category of modules. Passing to homology, which encodes
the derived functors of the limit in the categories of $A$-modules and
$\Psi$-comodules, we deduce the following.

######  Proposition 11.16.

If $M_{i}$ is a diagram of comodules, then there exists a spectral sequence of
comodules of signature

$E_{1}^{s,t}:=\textstyle\Psi\otimes_{A}\varprojlim_{A}^{s}(\Psi^{\otimes_{A}t}\otimes_{A}M_{i})\implies\varprojlim_{\Psi}^{s+t}M_{i}.$

computing the derived functors of the limit in comodules.

######  Remark 11.17.

Note that in the particular case of sequential inverse limits, that is,
diagrams of the form

$\ldots\rightarrow M_{2}\rightarrow M_{1}\rightarrow M_{0},$

this spectral sequence will collapse after at most two pages, since in this
case we have $\textstyle\varprojlim^{s}_{A}N_{i}=0$ for $s\neq 0,-1$ and any
diagram $N_{i}$ of $A$-modules.

######  Remark 11.18.

In an even more specific case, let us assume that we work with sequential
inverse limits and that the maps $M_{n+1}\rightarrow M_{n}$ are all
epimorphisms. If that is the case, the same is true for
$\Psi\otimes_{A}\ldots\otimes_{A}M_{n+1}\rightarrow\Psi\otimes_{A}\ldots\otimes_{A}M_{n}$
and we deduce using the Mittag-Leffler condition that all of the
$\textstyle\varprojlim^{1}_{A}$-terms vanish. In this case the spectral
sequence of Proposition 11.16 has only one potentially non-zero differential.

Phrasing it without the use of spectral sequences, we see that in this the
diagram of comodules

$\textstyle\Psi\otimes_{A}(\varprojlim_{A}M_{i})\rightarrow\Psi\otimes_{A}(\varprojlim_{A}\Psi\otimes_{A}M_{i})\rightarrow\ldots,$

is an explicit presentation of the derived limit
$\varprojlim_{\mathcal{D}(\Psi)}M_{i}$ as a chain complex.

######  Example 11.19.

Suppose that $R$ is a homotopy commutative ring spectrum of Adams-type. As
$R_{*}:\mathcal{S}p\rightarrow\operatorname{Comod}_{R_{*}R}$ is an adapted
homology theory by Lemma 11.7, Theorem 11.13 specializes to give a spectral
sequence

$\textstyle
E_{2}^{s,t}\cong\varprojlim_{R_{*}R}^{s}(R_{*+t}X^{\alpha})\implies
R_{*+s+t}(\operatorname{lim}_{\alpha}X^{\alpha})$

for a tower of spectra $(X^{\alpha})$, where the derived functors of the limit
are taken in $R_{*}R$-comodules.

This is a spectral sequence in $R_{*}R$-comodules, so that in particular the
$E_{2}$-page is _trigraded_. Passing to internal degree zero, we obtain a
more-pleasant looking bigraded spectral sequence

$\textstyle
E_{2}^{s,t}\cong(\varprojlim_{R_{*}R}^{s}R_{*}X^{\alpha})_{t}\implies
R_{s+t}(\operatorname{lim}_{\alpha}X^{\alpha})$

Note that in general we cannot rewrite the $E_{2}$-page as
“$\textstyle\varprojlim^{s}_{R_{*}R}(R_{t}X^{\alpha})$”, as the derived
functors of the limit are computed in $R_{*}R$-comodules, a structure which
cannot be restricted to a single degree.

######  Warning 11.20.

Even for quite reasonable $R$, such as $R=H\mathbb{Q}$, convergence of the
spectral sequence of Example 11.19 is a subtle problem. For example, let
$(M_{p^{i}})$ be the tower of mod $p^{i}$ Moore spectra with the canonical
structure maps, and suppose $R=H\mathbb{Q}$. Since

$\operatorname{lim}(\ldots\longrightarrow M_{p^{2}}\longrightarrow
M_{p})\simeq S_{p}^{0},$

the $p$-complete sphere, the abutment of the spectral sequence is non-trivial,
while the $E_{2}$-term is zero. We will study this question in more detail in
the next section in the case of Morava $E$-theory.

######  Remark 11.21.

Same methods as those leading to Example 11.19 have been employed by Hovey to
set up a spectral sequence computing the $R$-homology of a product of spectra,
see [Hov07]. In [Pet20b, Appendix A], a different construction of the spectral
sequence of Example 11.19 is given, based on Adams resolutions rather than
modified Adams resolutions.

###  12\. The Morava $E$-homology of inverse limits

The goal of this section is to study the spectral sequence constructed in
Theorem 11.13 in the special case of (uncompleted) Morava $E$-homology. In
particular, we will identify the $E_{2}$-page for the sphere spectrum in terms
of the continuous cohomology of the Morava stablizer group and exhibit
conditions on the tower that ensure convergence.

We will work with a particular form of Morava $E$-theory, which we used
previously in §5. For convenience of the reader, let us fix our notation for
the remainder of the current work.

We fix a prime $p$ and a height $n>0$ and we let $\mathbf{G}_{0}$ be the Honda
formal group law over $\mathbb{F}_{q}:=\mathbb{F}_{p^{n}}$. We write $E$ for
the associated Lubin-Tate spectrum, so that we have a non-canonical
isomorphism

$E_{*}\simeq W(\mathbb{F}_{p^{n}})\llbracket
u_{1},\ldots,u_{n-1}\rrbracket[u^{\pm 1}].$

We write $\mathfrak{m}=(p,u_{1},\ldots,u_{n-1})$ for the maximal ideal of
$E_{0}$. By Goerss-Hopkins-Miller, the spectrum $E$ is acted on by the
extended Morava stabilizer group
$\mathbb{G}_{n}:=\mathrm{Aut}(\mathbf{G}_{0}/\mathbb{F}_{p})\rtimes\mathrm{Gal}(\mathbb{F}_{q}/\mathbb{F}_{p})$.

######  Warning 12.1.

We remind the reader that, unless otherwise noted, we work with uncompleted
$E$-homology, the uncompleted homology cooperations $E_{*}E:=\pi_{*}(E\otimes
E)$, and uncompleted $E_{*}E$-comodules.

####  12.1. Continuous cohomology of filtered colimits

In this subsection, we prove that continuous cohomology groups of
$\mathbb{G}_{n}$ commute with certain filtered colimits. To this end, it will
be useful for us to consider the following variant on the $\mathfrak{m}$-adic
topology on an $E_{0}$-module.

######  Definition 12.2.

Let $M$ be an $E_{0}$-module. The _local $\mathfrak{m}$-adic_ topology on $M$
is the linear topology in which a submodule $U\subseteq M$ is open if for
every finitely generated $E_{0}$-submodule $M^{\prime}\subseteq M$,

$U\cap M^{\prime}\subseteq M^{\prime}$

is open in the usual $\mathfrak{m}$-adic topology on $M^{\prime}$; that is, we
have $\mathfrak{m}^{n}M^{\prime}\subseteq U\cap M^{\prime}$ for some $n$
depending on $M^{\prime}$.

######  Example 12.3.

Suppose that $M$ is finitely generated over $E_{0}$. Then, the local
$\mathfrak{m}$-adic topology on $M$ coincides with the usual
$\mathfrak{m}$-adic topology.

######  Example 12.4.

Suppose that $M$ is an arbitrary $E_{0}$-module. Then, we can write
$M\simeq\varinjlim M_{\alpha}$ as a filtered colimit of finitely generated
$E_{0}$-modules. The local $\mathfrak{m}$-adic topology coincides with the
colimit topology if we equip each $M_{\alpha}$ with its usual
$\mathfrak{m}$-adic topology.

The importance of the local $\mathfrak{m}$-adic topology in our context comes
down to the following observation.

######  Remark 12.5.

Let $X$ be a spectrum. Then, $E_{0}X$ is a continuous $\mathbb{G}_{n}$-module
with respect to its local $\mathfrak{m}$-adic topology. To see this, observe
that we can write $X\simeq\varinjlim X_{\alpha}$ as a filtered colimit of
finite spectra, so that

$E_{0}X\simeq\varinjlim E_{0}X_{\alpha}.$ (12.1)

Each of $E_{0}X_{\alpha}$ is a continuous $\mathbb{G}_{n}$-module when
considered with its $\mathfrak{m}$-adic topology, and $E_{0}X$ becomes a
topological $\mathbb{G}_{n}$-module since it has the colimit topology with
respect to (12.1)

We will be interested in the continuous cohomology of $\mathbb{G}_{n}$ with
coefficients in an $E_{0}$-module equipped with its local $\mathfrak{m}$-adic
topology. As we will see, this has an elegant description in terms of
cohomologies of its finitely generated submodules. The key is the following
lemma.

######  Lemma 12.6.

Let $K$ be a compact Hausdorff topological space and $M$ be an $E_{0}$-module
equipped with its local $\mathfrak{m}$-adic topology. Then, any continuous map
$f\colon K\rightarrow M$ factors through a finitely generated submodule. In
other words, we have

$\operatorname{map}_{\operatorname{cts}}(K,M)\simeq\varinjlim\operatorname{map}_{\operatorname{cts}}(K,M_{\alpha}),$

where $M_{\alpha}$ is the poset of finitely generated submodules of $M$.

###### Proof.

Suppose by contradiction that for every finitely generated submodule
$M^{\prime}$, there exists a $k\in K$ such that $f(k)\not\in M^{\prime}$.
Proceeding inductively, we produce an increasing sequence

$M_{0}\subseteq M_{1}\subseteq M_{2}\subseteq\ldots$

of finitely generated submodules and points $k_{n}\in K$ such that
$f(k_{n})\in M_{n+1}$, but $f(k_{n})\notin M_{n}$.

We claim that an arbitrary finitely generated submodule $N$ contains at most
finitely many of the $f(k_{n})$. To see this, let us write $M_{\infty}=\cup
M_{n}$ and consider $N\cap M_{\infty}$; the latter is again finitely
generated, as $E_{0}$ is noetherian, and its intersection with
$\\{f(k_{n})\\}$ is the same as that of $N$. By finite generation, we have
$N\cap M_{\infty}=N\cap M_{n}$ for sufficiently large $n$, proving the claim,
as the latter contains at most $n$ of the $f(k_{n})$.

Since each finitely generated submodule is Hausdorff, its finite subspaces are
closed and discrete and we deduce that $T=\\{f(k_{n})\\}$ is closed and
discrete as a subspace of $M$, as this is true for its intersection with each
finitely generated submodule. This is a contradiction, as $T=f(f^{-1}(T))$
must be compact Hausdorff as an image of a closed subspace of $K$, so it
cannot be both infinite and discrete. ∎

######  Proposition 12.7.

Let $M\simeq\varinjlim M_{\alpha}$ be a filtered colimit diagram of
$E_{0}$-modules equipped with their local $\mathfrak{m}$-adic topologies and
let $K$ be compact Hausdorff. Then, the induced map

$\theta\colon\varinjlim\operatorname{map}_{\operatorname{cts}}(K,M_{\alpha})\rightarrow\operatorname{map}_{\operatorname{cts}}(K,M)$

is an isomorphism of abelian groups.

###### Proof.

We first show surjectivity of $\theta$. Suppose that $f\colon K\rightarrow M$
is a continuous map, by Lemma 12.6 it factors through a finitely generated
submodule $N\subseteq M$. If we write $N_{\alpha}=N\times_{M}M_{\alpha}$, then
since filtered colimits are exact we have

$N\simeq\varinjlim N_{\alpha}$

As $N$ is finitely generated and $E_{0}$ is noetherian, it is finitely
presented and we deduce that there exists a section $N\rightarrow N_{\alpha}$
for some $\alpha$. Then, the composite

$K\rightarrow N\rightarrow N_{\alpha}\rightarrow M_{\alpha}$

determines the needed element of the filtered colimit.

For injectivity, suppose that we have a continuous map $f\colon K\rightarrow
M_{\alpha}$ such that the composite $s_{\alpha}\circ f\colon K\rightarrow M$
where $s_{\alpha}\colon M_{\alpha}\rightarrow M$ is the canonical map, is
zero. By another application of Lemma 12.6, $K$ factors through a finitely
generated submodule $N_{\alpha}\subseteq M_{\alpha}$ and we necessarily have
$N_{\alpha}\subseteq\mathrm{ker}(s_{\alpha})$. As $N_{\alpha}$ is finitely
generated, the second conidtion implies that we can find a larger index
$\beta$ such that the composite

$N_{\alpha}\rightarrow M_{\alpha}\rightarrow M_{\beta}$

is zero. It follows that $f$ determines the zero element of the filtered
colimit, as needed. ∎

######  Corollary 12.8.

Let $M\simeq\varinjlim M_{\alpha}$ be a filtered colimit of $E_{0}$-modules
equipped with their local $\mathfrak{m}$-adic topology and compatible
continuous $\mathbb{G}_{n}$-actions. Then,

$\mathrm{H}^{*}_{\operatorname{cts}}(\mathbb{G}_{n},M)\simeq\varinjlim\mathrm{H}^{*}_{\operatorname{cts}}(\mathbb{G}_{n},M_{\alpha}).$

In particular, for any $E_{0}$-module $N$, we have

$\mathrm{H}^{*}_{\operatorname{cts}}(\mathbb{G}_{n},N)\simeq\varinjlim\mathrm{H}^{*}_{\operatorname{cts}}(\mathbb{G}_{n},N_{\alpha}).$

where the colimit is taken over the poset $N_{\alpha}$ of finitely generated
submodules.

###### Proof.

The continuous cohomology is computed by the standard cochain complex

$M\rightarrow\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{n},M)\rightarrow\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{n}\times\mathbb{G}_{n},M)\rightarrow\ldots$

of continuous cochains. The statement follows from an application of
Proposition 12.7 to each term separately, as
$\mathbb{G}_{n}\times\ldots\times\mathbb{G}_{n}$ is compact Hausdorff and
taking cohomology of cochain complexes commutes with filtered colimits. ∎

####  12.2. Inverse limits of $E_{*}E$-comodules and continuous cohomology

In this section, we will give a description of derived functors of the limit

$\textstyle\varprojlim_{E_{*}E}^{s}M/\mathfrak{m}^{k}M\simeq\mathrm{H}_{-s}(\
\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M/\mathfrak{m}^{k}M)$

in $E_{*}E$-comodules as cohomology of the Morava stabilizer group. The
importance of these derived functors stems from the fact that they form the
$E_{2}$-page of the spectral sequence constructed in §11.2.

######  Lemma 12.9.

If $M$ is a dualizable $E_{*}E$-comodule, then we have a canonical isomorphism

$\operatorname{Ext}_{\mathcal{D}(E_{*}E)}^{s}(E_{*},\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M/\mathfrak{m}^{k}M):=\pi_{-s}\operatorname{Hom}_{\mathcal{D}(E_{*}E)}(E_{*},\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M/\mathfrak{m}^{k}M)\simeq\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{n},M)$

for any $s\geq 0$, where $E_{*}/\mathfrak{m}^{k}$ is the $\mathfrak{m}$-adic
tower of $E_{*}$.

###### Proof.

For brevity, if $X\in\mathcal{D}(E_{*}E)$, let us write

$\operatorname{Ext}^{s}(X):=\operatorname{Ext}^{s}_{\mathcal{D}(E_{*}E)}(E_{*},X)\simeq\pi_{-s}\operatorname{Hom}_{\mathcal{D}(E_{*}E)}(E_{*},X)$

Since we have

$\operatorname{Hom}_{\mathcal{D}(E_{*}E)}(E,\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M/\mathfrak{m}^{k}M)\simeq\varprojlim\operatorname{Hom}_{\mathcal{D}(E_{*}E)}(E_{*},M/\mathfrak{m}^{k}M),$

there is a Milnor sequence

$0\to\textstyle\varprojlim^{1}\operatorname{Ext}^{s-1}(M/\mathfrak{m}^{k}M)\to\operatorname{Ext}^{s}(\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M/\mathfrak{m}^{k}M)\to\varprojlim\operatorname{Ext}^{s}(M/\mathfrak{m}^{k}M)\to
0,$

where we have omitted subscripts for simplicity and the derived functors of
the limit are taken in abelian groups.

We first claim that the $\varprojlim^{1}$-term above vanishes. Note that since
$M/\mathfrak{m}^{k}M$ is finitely generated and $\mathfrak{m}$-torsion for any
$k\geq 1$, we have a canonical isomorphism

$\operatorname{Ext}(M/\mathfrak{m}^{k}M)\simeq\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{n},M/\mathfrak{m}^{k}M)$

with the cohomology of the Morava stabilizer group. For $k=1$, we have a
further isomorphism

$\operatorname{Ext}(M/\mathfrak{m}M)\simeq\operatorname{Ext}_{E_{*}E}(E_{*},K_{*}\otimes_{E_{*}}M)\simeq\operatorname{Ext}_{E_{*}K}(K_{*},K_{*}\otimes_{E_{*}}M)$

and the group on the right is degreewise finite by Corollary 5.7. It follows
by induction that $\operatorname{Ext}(M_{*}/\mathfrak{m}^{k}M)$ is degreewise
finite for each $k\geq 1$. Therefore, the groups
$\operatorname{Ext}(M/\mathfrak{m}^{k}M)$ satisfy the Mittag-Leffler condition
as $k$ varies, so that the corresponding $\varprojlim^{1}$-term vanishes as
claimed.

Consequently, we get a string of isomorphisms

$\displaystyle\operatorname{Ext}^{s}(\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M/\mathfrak{m}^{k}M)$
$\displaystyle\cong\varprojlim\operatorname{Ext}^{s}(E_{*},M/\mathfrak{m}^{k}M)$
$\displaystyle\cong\textstyle\varprojlim\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{n},M/\mathfrak{m}^{k}M)$
$\displaystyle\cong\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{n},M).$

Here, the last isomorphism uses that $M$ is $\mathfrak{m}$-adically complete
as a finitely generated $E_{*}$-module, and that taking continuous cohomology
of a finitely generated profinite group commutes with taking countable inverse
limits of finite modules [NSW13, §7]. ∎

In order to pass from Lemma 12.9 to a description of the derived functors
$\textstyle\varprojlim^{s}_{E_{*}E}E_{*}/\mathfrak{m}^{k}$, we have to
understand the homology groups of
$\varprojlim_{\mathcal{D}(E_{*}E)}M/\mathfrak{m}^{k}M$, rather than its
homotopy. To do so, we will make use of the Adams condition, which involves
certain filtered colimits. To commute past the latter, we will need a
technical lemma stating that homotopy groups

$\operatorname{Ext}^{s}(E_{*},-):=\pi_{-s}\operatorname{Hom}_{\mathcal{D}(E_{*}E)}(E_{*},-)$

preserve certain filtered colimits of bounded complexes. If $p>n+1$, then the
monoidal unit of $\mathcal{D}(E_{*}E)$ is compact, so no boundedness
hypotheses are required, but this can fail for general $n$. In general, we
still have the following result.

######  Lemma 12.10.

Suppose $(L_{\alpha})_{\alpha}$ is a filtered system of complexes of
$E_{*}E$-comodules which are uniformly bounded above in the standard
$t$-structure on $\mathcal{D}(E_{*}E)$; that is, there exists an $N$ such that
for all $s>N$ and all $\alpha$ we have $H_{s}(L_{\alpha})=0$. Then the
canonical comparison map

$\phi\colon\varinjlim_{\alpha}\operatorname{Ext}^{s}(L_{\alpha})\longrightarrow\operatorname{Ext}^{s}(\varinjlim_{\alpha}L_{\alpha})$

is an isomorphism for any $s\in\mathbb{Z}$.

###### Proof.

By shifting the system $(L_{\alpha})_{\alpha}$ if necessary, we may reduce to
the case of $s=0$. In other words, we have to show that that the canonical map

$\varinjlim_{\alpha}\
[E_{*},L_{\alpha}]\longrightarrow[E_{*},\varinjlim_{\alpha}L_{\alpha}]$

between homotopy classes of maps in the derived $\infty$-category, is an
isomorphism.

Since $E_{*}$ is connective in the standard $t$-structure, we can then replace
$L_{\alpha}$ by its connective cover $\tau_{\geq 0}L_{\alpha}$; that is,
reduce to the case in which there exists a non-negative integer $N$ such that
$L_{\alpha}$ has homology concentrated in degrees $[0,N]$ for all $\alpha$. By
induction and the five-lemma, we can reduce further to the case that
$L_{\alpha}$ is concentrated in a single non-negative degree $[s,s]$ for all
$\alpha$.

Summarizing these reduction steps, it remains to prove that the canonical map

$\varinjlim_{\alpha}\operatorname{Ext}_{E_{*}E}^{s}(E_{*},L_{\alpha})\longrightarrow\operatorname{Ext}_{E_{*}E}^{s}(E_{*},\varinjlim_{\alpha}L_{\alpha})$

is an isomorphism for any filtered system of comodules
$L_{\alpha}\in\operatorname{Comod}_{E_{*}E}$. This is clear, these
$\operatorname{Ext}$-groups can be computed by the cobar complex, formation of
which commutes with filtered colimits. ∎

######  Proposition 12.11.

For any dualizable $E_{*}E$-comodule $M$, there exists a canonical isomorphism

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

between the derived functors of the limit in comodules and the continuous
cohomology of the Morava stabilizer group with coefficients in
$E_{*}E\otimes_{E_{*}}M$ equipped with its local $\mathfrak{m}$-adic topology.

###### Proof.

We have

$\textstyle{\varprojlim_{E_{*}E}^{s}}M_{*}/\mathfrak{m}^{k}M\simeq
H_{-s}(\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M_{*}/\mathfrak{m}^{k}M)$

which we can further rewrite as

$H_{-s}(\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M/\mathfrak{m}^{k}M)\simeq\operatorname{Ext}^{s}(E_{*}E\otimes_{E_{*}}\varprojlim_{\mathcal{D}(E_{*}E)}M_{*}/\mathfrak{m}^{k}M),$

where the tensor product is the derived one. Writing $E_{*}E\simeq\varinjlim
N_{\alpha}$ as a filtered colimit of dualizable comodules, we have

$\displaystyle\operatorname{Ext}^{s}(E_{*}E\otimes_{E_{*}}\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M/\mathfrak{m}^{k}M)$
$\displaystyle\cong\operatorname{Ext}^{s}(\varinjlim\
(N\otimes_{E_{*}}\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M/\mathfrak{m}^{k}M))$
$\displaystyle\cong\varinjlim\operatorname{Ext}^{s}(N\otimes_{E_{*}}\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M/\mathfrak{m}^{k}M)$
$\displaystyle\cong\varinjlim\operatorname{Ext}^{s}(\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}N\otimes_{E_{*}}M/\mathfrak{m}^{k}(N\otimes_{E_{*}}M))$
$\displaystyle\cong\varinjlim\
\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{n},N\otimes_{E_{*}}M)$
$\displaystyle\cong\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{n},\varinjlim\
N\otimes_{E_{*}}M)$
$\displaystyle\cong\mathrm{H}^{s}_{\operatorname{cts}}(\mathbb{G}_{n},E_{*}E\otimes_{E_{*}}M).$

Here, the first isomorphism uses that the derived tensor product commutes with
filtered colimits and the second one is Lemma 12.10. The third isomorphism
uses that tensoring with a dualizable object commutes with limit, the fourth
one is Lemma 12.9 while the fifth one is Corollary 12.8. ∎

######  Remark 12.12.

The relationship between derived functors of the limit and derived completion
in the context of comodules over a flat Hopf algebroid is studied further in
[BHV20].

####  12.3. The inverse limit spectral sequence and its convergence

In this subsection, we study the convergence of the inverse limit spectral
sequence based on Morava $E$-theory and combine the previous results in a
spectral sequence computing the (uncompleted) $E$-homology of the $K$-local
sphere spectrum from continuous group cohomology with coefficients in the
(uncompleted) cooperations $E_{*}E$.

######  Proposition 12.13.

If $R=E$ is Morava $E$-theory and
$(X^{\alpha})\in\mathcal{S}p_{E}^{\mathbb{N}}$ is a tower of $E$-local
spectra, then the spectral sequence of Example 11.19 converges conditionally
and strongly, and has a horizontal vanishing line at the $E_{r}$-page for some
$r\geq 2$.

###### Proof.

By Lemma 11.8 and assumption on $X^{\alpha}$, the modified Adams tower
$(X_{i}^{\alpha})$ is $E$-local for all $i$. Therefore, the proof of the smash
product theorem due to Hopkins and Ravenel [Rav92, Theorem 7.5.6] yields a
constant $N$ independent of $\alpha$ such that

$\textstyle{g_{s}^{\alpha}\circ\ldots\circ g_{s+N-1}^{\alpha}\colon
X_{s+N}^{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{s}^{\alpha}}$

is null for all $\alpha$; see [HS05b, Theorem 5.10]. Consequently,

$\textstyle{g_{s}\circ\ldots\circ g_{s+N-1}\colon\varprojlim
X_{s+N}^{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varprojlim
X_{s}^{\alpha}}$ (12.2)

is null as well, so [Boa99] shows that the spectral sequence is conditionally
convergent. Similarly, we obtain the horizontal vanishing line from (12.2),
which in turn gives strong convergence. ∎

######  Theorem 12.14.

If $(X^{\alpha})\in\mathcal{S}p_{E}^{\mathbb{N}}$ is a tower of $E$-local
spectra, then there is a strongly convergent spectral sequence

$\textstyle{E_{2}^{s,t}\cong(\varprojlim_{E_{*}E}^{s}E_{*}X^{\alpha})_{t}\implies
E_{t-s}(\varprojlim X^{\alpha}),}$

with a horizontal vanishing line at the $E_{r}$-page for some $r\geq 2$. In
particular, this spectral sequence specializes to:

$\textstyle{\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{n},E_{t}E)\cong(\varprojlim_{E_{*}E}^{s}E_{*}/\mathfrak{m}^{k})_{t}\implies
E_{t-s}(L_{K}S^{0}).}$ (12.3)

###### Proof.

The spectral sequence was constructed in Theorem 11.13, see also Example
11.19, while the convergence properties were established in Proposition 12.13.
The final statement follows by applying this spectral sequence to a cofinal
tower of $E$-local generalized type $n$ Moore spectra as in [HS99, Section 4].
The identification of the $E_{2}$-page in terms of continuous cohomology was
proven in Proposition 12.11. ∎

######  Remark 12.15.

The spectral sequence (12.3) may be thought of as an $E_{*}$-homology version
of the $K$-local $E$-based Adams spectral sequence for the sphere, which has
signature

$\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{n},E_{*})_{t}\implies\pi_{*}L_{K}S^{0}$

and is also strongly convergent. In contrast, working with completed
$E_{*}$-homology would result in an isomorphism $E_{*}\cong
E_{*}^{\vee}L_{K}S^{0}\cong\mathrm{H}_{\operatorname{cts}}^{*}(\mathbb{G}_{n},E_{*}^{\vee}E_{*})$.
This amply highlights the extra information contained in the terms
$\mathrm{H}_{\operatorname{cts}}^{*}(\mathbb{G}_{n},E_{*}E)$.

###  13\. Digression: Cohomology of $\mathbb{G}_{n}$ as $\operatorname{Ext}$
in comodules

In Lemma 12.9, we have expressed continuous cohomology of a Morava stabilizer
group with coefficients in a dualizable $E_{*}E$-comodule $M$ as homotopy
groups of a the derived completion. In this short section, we will use a form
of local duality in the context of comodules to express it as an
$\operatorname{Ext}$-groups between actual comodules, rather than objects of
the derived $\infty$-category. Using this calculation we will define a
comparison map between $\operatorname{Ext}$-groups and continuous cohomology.

Let $I_{k}=(p,v_{1},\ldots,v_{k-1})$ be the $k$-th invariant chromatic ideal
in $E_{*}$; in particular, $I_{n}=\mathfrak{m}$. We define
$E_{*}=E_{*}/I_{0}^{\infty}$ and then iteratively construct $E_{*}E$-comodules
via the cofiber sequences

$E_{*}/I_{k}^{\infty}\rightarrow v_{k}^{-1}E_{*}/I_{k}^{\infty}\rightarrow
E_{*}/I_{k+1}^{\infty}$ (13.1)

in $\mathcal{D}(E_{*}E)$. Note that $E_{*}/I_{k}^{\infty}\simeq\varinjlim
E_{*}/v_{0}^{l_{0}},\ldots,v_{k-1}^{l_{k-1}}$, where the colimit is taken over
a set of indices with $l_{i}\rightarrow\infty$. For the constituent pieces, we
have the following self-duality result:

######  Lemma 13.1.

In the derived category of $E_{*}E$-comodules, the comodule
$E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k}^{i_{k}}$ is self-dual with a
shift in the sense that there exists an equivalence

$D(E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k}^{i_{k}})\simeq\Sigma^{-k}E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k}^{i_{k}},$

where $DX=F(X,E_{*})$ denotes the monoidal dual.

###### Proof.

We prove this by induction on $k$, where $k=-1$ is clear, since $E_{*}$ is the
monoidal unit.

Now assume that the statement is known for
$E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k-1}^{i_{k-1}}$. Multiplication
by $v_{k}^{i_{k}}$ induces a cofiber sequence

$E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k-1}^{i_{k-1}}\rightarrow
E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k-1}^{i_{k-1}}\rightarrow
E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k}^{i_{k}}$

which after taking duals yields a cofiber sequence

$D(E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k}^{i_{k}})\rightarrow
D(E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k-1}^{i_{k-1}})\rightarrow
D(E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k-1}^{i_{k-1}}).$

By the inductive assumption, this sequence can be rewritten as

$D(E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k}^{i_{k}})\rightarrow\Sigma^{k-1}E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k-1}^{i_{k-1}}\rightarrow\Sigma^{k-1}E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k-1}^{i_{k-1}}.$

The self-map of
$\Sigma^{k-1}E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k-1}^{i_{k-1}}\rightarrow\Sigma^{k-1}E_{*}/v_{0}^{i_{0}},v_{1}^{i_{1}},\ldots,v_{k-1}^{i_{k-1}}$
appearing on the right cannot be non-zero, since it is a dual to the non-zero
class $v_{k}^{i_{k}}$. It follows that it must be itself a unit times
$v_{k}^{i_{k}}$ since there is nothing else in this degree, see for example
[Rav86, Thm. 4.3.2(b)]. The assertion then follows. ∎

######  Lemma 13.2.

If $M$ is a finitely generated $E_{*}E$-comodule, then its derived comodule
completion of $M$ is given by

$\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}(M\otimes_{E_{*}}E_{*}/v_{0}^{k_{0}},\ldots,v_{n-1}^{k_{n-1}})\simeq\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M/v_{0}^{k_{0}},\ldots,v_{n-1}^{k_{n-1}},$

where on the left hand side we have used the derived tensor product of
comodules.

###### Proof.

This is proven in [BHV20, Proposition 2.25] ∎

The next result is an incarnation of local duality for comodules [BHV18, §5],
relating local cohomology (derived torsion) and local homology (derived
completion), combined with Lemma 12.9.

######  Theorem 13.3.

Let $M$ be a finitely generated $E_{*}E$-comodule. Then, there exists an
isomorphism

$\operatorname{Ext}^{n+s}_{E_{*}E}(E_{*}/I_{n}^{\infty},M)\simeq\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{n},M)$

between $\operatorname{Ext}$-groups in comodules and the cohomology of
$\mathbb{G}_{n}$, for any $s\geq 0$.

###### Proof.

As noted before, we have $E_{*}/I_{n}^{\infty}\simeq\varinjlim
E_{*}/v_{0}^{k_{0}},\ldots,v_{n-1}^{k_{n-1}}$, where the colimit is taken over
a set of indices with $k_{i}\rightarrow\infty$. By by the duality equivalence
of Lemma 13.1 we have

$\operatorname{Hom}_{\mathcal{D}(E_{*}E)}(\varinjlim\Sigma^{-n}E_{*}/v_{0}^{k_{0}},\ldots,v_{n-1}^{k_{n-1}},M)\simeq\operatorname{Hom}_{\mathcal{D}(E_{*}E)}(E_{*},\textstyle\varprojlim_{\mathcal{D}(E_{*}E)}M\otimes
E_{*}/v_{0}^{k_{0}},\ldots,v_{n-1}^{k_{n-1}}).$

In light of the Lemma 13.2, the latter term is equivalent to
$\operatorname{Hom}_{\mathcal{D}(E_{*}E)}(E_{*},\varprojlim
M/v_{0}^{k_{0}},\ldots,v_{n-1}^{k_{n-1}})$. It follows that

$\operatorname{Ext}^{n+s}_{E_{*}E}(E_{*}/I_{n}^{\infty},M)\simeq\varprojlim\operatorname{Ext}^{s}_{E_{*}E}(E_{*},M/v_{0}^{k_{0}},\ldots,v_{n-1}^{k_{n-1}}),$

up to possible $\varprojlim^{1}$-terms. However, all of the Ext groups on the
right are finite, since $M$ is finitely generated, so these
$\varprojlim^{1}$-terms vanish. We then deduce as in Lemma 12.9 that

$\operatorname{Ext}^{n+s}_{E_{*}E}(E_{*}/I_{n}^{\infty},M)\simeq\varprojlim\operatorname{Ext}_{E_{*}E}^{s}(E_{*},M/v_{0}^{k_{0}},\ldots,v_{n-1}^{k_{n-1}})\simeq\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{n},M),$

which is what we wanted to show. ∎

The boundary maps $\delta_{k}\colon E_{*}/I_{k+1}^{\infty}\to\Sigma
E_{*}/I_{k}^{\infty}$ associated with the cofiber sequences (13.1) compose to
give a ‘residue map’ in the derived category:

$\delta:=\delta_{n-1}\circ\ldots\circ\delta_{0}\colon
E_{*}/I_{n}^{\infty}\to\Sigma^{n}E_{*}.$

For any finitely generated $E_{*}E$-comodule $N$, the map $\delta$ induces a
natural map on Ext-groups that fits, for any $s$, into a commutative square

$\textstyle{\operatorname{Ext}^{s}_{E_{*}E}(E_{*},N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ext}(\delta,N)}$$\scriptstyle{\cong}$$\textstyle{\operatorname{Ext}^{s+n}_{E_{*}E}(E_{*}/I_{n}^{\infty},N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{[N^{\operatorname{dual}},E_{*}]^{-s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[N^{\operatorname{dual}},\varprojlim_{\mathcal{D}(E_{*}E)}E_{*}/I_{n}^{k}]^{-s}.}$
(13.2)

Here, the left vertical isomorphism is by duality, while the right vertical
isomorphism is established as in the proof of the previous theorem. Unwinding
the construction, the bottom horizontal map is then induced by the canonical
completion map $E_{*}\to\varprojlim_{\mathcal{D}(E_{*}E)}E_{*}/I_{n}^{k}$.

######  Definition 13.4.

For a finitely generated $E_{*}E$-comodule $N$, we define the comparison map
$\alpha_{N}$ as the composite

$\alpha_{N}\colon\operatorname{Ext}^{s}_{E_{*}E}(E_{*},N)\to\operatorname{Ext}^{n+s}_{E_{*}E}(E_{*}/I_{n}^{\infty},N)\simeq\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{n},N)$

of the map in (13.2) and the isomorphism of Theorem 13.3.

Note that, by construction, the maps $\alpha_{N}$ are natural and compatible
with the long exact sequences of $\operatorname{Ext}$-groups and cohomology.

######  Remark 13.5.

Alternatively, it is possible to define the comparison map $\alpha_{M}$
directly using the cobar complex. More precisely, one can show that if $M$ is
a finitely generated $E_{*}E$-comodule, then the differentials in the cobar
complex

$M\rightarrow E_{*}E\otimes_{E_{*}}M\rightarrow
E_{*}E\otimes_{E_{*}}E_{*}E\otimes_{E_{*}}M\rightarrow\ldots$

respect the $\mathfrak{m}$-adic filtration (even though they are not
$E_{*}$-linear). The completion of the cobar complex is exactly the cochain
complex computing continuous $\mathbb{G}_{n}$-cohomology, and the induced map
on homology can be shown to be exactly $\alpha_{M}$.

###  14\. Algebraic chromatic splitting conjecture

In this penultimate section we use the comparison map of the previous section
to compute the zeroth derived limit as
$\varprojlim^{0}_{E_{*}E}E_{*}/\mathfrak{m}^{k}\simeq
E_{*}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$. This result should be viewed in
light of Hopkins’ algebraic chromatic splitting conjecture, see [Pet, §14],
which we briefly review for context.

######  Notation 14.1.

To simplify notation, if $N$ is a finitely generated $E_{*}E$-comodule, we
will sometimes use abbreviations
$\operatorname{Ext}^{s}(N):=\operatorname{Ext}^{s}_{E_{*}E}(E_{*},N)$ and
$\mathrm{H}^{s}(N):=\mathrm{H}_{\operatorname{cts}}^{s}(\mathbb{G}_{n},N)$.

####  14.1. The algebraic chromatic splitting conjecture

Motivated by computations in height 2, this conjecture was formulated by
Hopkins and recorded later in [Pet, Conjecture 134]444We note that the
statement given there contains a typo: the left hand side of the equation
should have $v_{n-1}$ inverted. as well as [BSS20, Conjecture 6.10], and
studied further in [BH18]. To state it, we first recall the construction of
the class $\zeta\in\pi_{-1}L_{K(n)}S^{0}$ due to Devinatz and Hopkins [DH04,
Section 8].

If we write $\mathbb{G}_{n}^{1}$ for the kernel of the determinant map
$\det\colon\mathbb{G}_{n}\to\mathbb{Z}_{p}$, there is a residual action of
$\mathbb{Z}_{p}$ on $E_{n}^{h\mathbb{G}_{n}^{1}}$. For $t$ is a topological
generator of $\mathbb{Z}_{p}$, we then obtain a fiber sequence of spectra

$\textstyle{L_{K(n)}S^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{n}^{h\mathbb{G}_{n}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}-t}$$\textstyle{E_{n}^{h\mathbb{G}_{n}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{L_{K(n)}S^{1}.}$

The composition $\delta\circ\eta\colon S^{0}\to L_{K(n)}S^{0}$ of the unit map
with the boundary map gives rise to an element
$\zeta\in\pi_{-1}L_{K(n)}S^{0}$. In [DH04, Proposition 8.2], Devinatz and
Hopkins prove that $\zeta$ is non-trivial as long as the height $n>0$.

######  Conjecture 14.2 (Algebraic chromatic splitting conjecture).

For all heights $n\geq 2$ and sufficiently large $p$, there are isomorphisms
of $E_{*}E$-comodules

$\textstyle\varprojlim_{E_{*}E}^{s}E_{*}/(p,\ldots,v_{n-2},v_{n-1}^{i})\cong\begin{cases}E_{*}/(p,\ldots,v_{n-2})&s=0\\\
v_{n-1}^{-1}E_{*}/(p,\ldots,v_{n-2})&s=1\\\ 0&\mathrm{otherwise.}\end{cases}$

Moreover, these isomorphisms are topologically realized by the unit map
$\iota\colon S^{0}\to L_{K(n)}S^{0}$ and the map $\zeta\colon S^{-1}\to
L_{K(n)}S^{0}$, respectively.

######  Remark 14.3.

As the name suggests, the algebraic chromatic splitting conjecture provides an
algebraic counterpart to Hopkins’ chromatic splitting conjecture [Hov95,
Conjecture 4.2] in the edge case: If the algebraic chromatic splitting
conjecture holds at height $n$ and prime $p$, then the maps $\iota$ and
$\zeta$ induce a splitting

$(\iota,\zeta)\colon L_{n-1}F\oplus\Sigma^{-1}L_{n-1}F\simeq L_{n-1}L_{K(n)}F$

for any finite spectrum $F$ of type $n-1$. Indeed, the conclusion of the
algebraic chromatic splitting conjecture forces the inverse limit sequence
(11.2) for $L_{K(n)}F$ to collapse at the $E_{2}$-page, thereby showing
$(\iota,\zeta)$ is an $E_{*}$-isomorphism and hence an $E$-local equivalence.

######  Example 14.4.

In Section 15, we will verify a height 1 variant of the algebraic chromatic
splitting conjecture at any prime, see in particular Theorem 15.12 and also
[Pet, §14]. The appearance of the additional tensor factors is the only reason
we excluded the case of height 1 from the statement of the algebraic chromatic
splitting conjecture above.

####  14.2. The comparison map in degree 0

We work with 14.1. Our goal is to study the comparison map
$\operatorname{Ext}^{s}(M)\rightarrow\mathrm{H}^{s}(M)$, constructed in
Definition 13.4. We will focus on the particular case when $s=0$; note that in
this case this map is always injective as both the source and target are
abelian subgroups of $M$.

Thus, the real question is how far this comparison map is from being
surjective. When $M$ is the monoidal unit, we have the following folklore
result of Hopkins.

######  Lemma 14.5 (Hopkins).

We have $\mathrm{H}^{0}(E_{*})\simeq\mathbb{Z}_{p}$ and the map
$\operatorname{Ext}^{0}(E_{*})\rightarrow\mathrm{H}^{0}(E_{*})$ is a
$p$-completion; that is, it induces an isomorphism
$\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\operatorname{Ext}^{0}(E_{*})\simeq\mathrm{H}^{0}(E_{*})$.

###### Proof.

The first part is [BG18, Lemma 1.33]. The second part follows from the fact
that $\operatorname{Ext}^{0}(E_{*})\simeq\mathbb{Z}_{(p)}$, as in both cases
the relevant invariants are generated by the unit of the Lubin–Tate ring. ∎

######  Lemma 14.6.

The map $\operatorname{Ext}^{1}(E_{*})\rightarrow\mathrm{H}^{1}(E_{*})$ is a
monomorphism.

###### Proof.

Since it vanishes rationally, the source of this map is $p$-torsion. For a
given $x\in\operatorname{Ext}^{1}(E_{*})$ we can thus find a $k$ and
$y\in\operatorname{Ext}^{0}(E_{*}/p^{k})$ so that $x$ can be written as
$x=\delta(y)$, where $\delta$ denotes the boundary homomorphism. We have a
commutative diagram

${{\operatorname{Ext}^{0}(E_{*})}}$${{\operatorname{Ext}^{0}(E_{*})}}$${{\operatorname{Ext}^{0}(E_{*}/p^{k})}}$${{\operatorname{Ext}^{1}(E_{*})}}$${{\mathrm{H}^{0}(E_{*})}}$${{\mathrm{H}^{0}(E_{*})}}$${{\mathrm{H}^{0}(E_{*}/p^{k})}}$${{\mathrm{H}^{1}(E_{*})}}$$\scriptstyle{p^{k}}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$

with exact rows. Thus, it is enough to show that if the image
$\widetilde{y}\in\mathrm{H}^{0}(E_{*}/p^{k})$ of $y$ can be lifted to
$\mathrm{H}^{0}(E_{*})$, then $y$ can be lifted to
$\operatorname{Ext}^{0}(E_{*})$. By Lemma 14.5, the latter two groups are
isomorphic to respectively $\mathbb{Z}_{p}$ and $\mathbb{Z}_{(p)}$.

By multiplying by an appropriate unit $\sigma$ in the $p$-adics congruent to
$1$ modulo $p^{k}$, and replacing the chosen lift
$\widetilde{y}\in\mathrm{H}^{0}(E_{*})\simeq\mathbb{Z}_{(p)}$ of
$\widetilde{x}$ by $\sigma\widetilde{y}$, we can assume that
$\widetilde{y}\in\mathbb{Z}_{(p)}$. It follows that $x$ can be lifted to
$\operatorname{Ext}^{0}(E_{*})$, which is what we wanted. ∎

Our goal is to use an inductive argument, building on the calculation of
invariants of $E_{*}$ due to Hopkins and extending it to a larger class of
comodules. The starting point will be the following class.

######  Definition 14.7.

We say a finitely generated $E_{*}E$-comodule $M$ is _pure_ if it belongs to
the smallest class of comodules closed under extensions and containing all
shifts of $E_{*}$.

######  Remark 14.8.

Any pure $M$ is finitely generated and projective over $E_{*}$, and so
dualizable. It follows that if

$0\rightarrow K\rightarrow M\rightarrow N\rightarrow 0$

is a short exact sequence of pure comodules, then it is split over $E_{*}$, so
that the induced sequence

$0\rightarrow N^{\operatorname{dual}}\rightarrow
M^{\operatorname{dual}}\rightarrow K^{\operatorname{dual}}\rightarrow 0$

of linear duals is again exact. We deduce that the class of pure comodules is
closed under taking duals.

######  Remark 14.9.

It is plausible that it follows from the arguments of Hovey and Strickland,
who establish an analogue of the Landweber filtration for $E_{*}E$-comodules
[HS05a], that all comodules finitely generated and projective over $E_{*}$ are
pure in the sense of Definition 14.7. We did not check whether this is true,
as we do not need this fact in our arguments.

######  Lemma 14.10.

The category of $E_{*}E$-comodules is generated under colimits by pure
comodules.

###### Proof.

Since $E\simeq\varinjlim E_{\alpha}$ is a filtered colimit of finite spectra,
it follows formally that the category of $E_{*}E$-comodules is generated under
colimits by shifts of comodules of the form $E_{*}E_{\alpha}$ [Hov04a,
Proposition 1.4.4].

Furthermore, we claim that since the $E_{\alpha}$ only have even cells, the
resulting comodules $E_{*}E_{\alpha}$ are pure. Using a cell decomposition of
$E_{\alpha}$ and the associated long exact sequences in homology, which will
be short exact here, we deduce that each $E_{*}E_{\alpha}$ is a comodule which
can be obtained using iterated extensions of shifts of $E_{*}$ and so is pure.
∎

Recall that a $E_{*}E$-comodule is dualizable if and only if the underlying
$E_{*}$-module is finitely generated and projective.

######  Theorem 14.11.

If $N$ is a dualizable $E_{*}E$-comodule, then

  1. (1)

$\operatorname{Ext}^{0}(N)$ is a finitely generated $\mathbb{Z}_{(p)}$-module
and

  2. (2)

the map
$\alpha_{N}\colon\operatorname{Ext}^{0}(N)\rightarrow\mathrm{H}^{0}(N)$ is a
$p$-completion, that is, it induces an isomorphism
$\operatorname{Ext}^{0}(N)\otimes_{\mathbb{Z}}\mathbb{Z}_{p}\simeq\mathrm{H}^{0}(N)$.

###### Proof.

For $N=E_{*}$ or its shifts, this is Lemma 14.5.

We first claim that the result is true for comodules $C$ which are pure in the
sense of Definition 14.7. By induction on rank, we can assume that we have a
short exact sequence

$0\rightarrow E_{*}\rightarrow C\rightarrow C^{\prime}\rightarrow 0$

and that the result already holds for $C^{\prime}$. The long exact sequence of
$\operatorname{Ext}$-groups implies that $\operatorname{Ext}^{0}(C)$ is a
finitely generated module over $\mathbb{Z}_{(p)}$. Thus, we have a commutative
diagram

${0}$${{\mathbb{Z}_{p}\otimes\operatorname{Ext}^{0}(E_{*})}}$${{\mathbb{Z}_{p}\otimes\operatorname{Ext}^{0}(C)}}$${{\mathbb{Z}_{p}\otimes\operatorname{Ext}^{0}(C^{\prime})}}$${{\mathbb{Z}_{p}\otimes\operatorname{Ext}^{1}(E_{*})}}$${0}$${{\mathrm{H}^{0}(E_{*})}}$${{\mathrm{H}^{0}(C)}}$${{\mathrm{H}^{0}(C^{\prime})}}$${{\mathrm{H}^{1}(E_{*}),}}$

where the top row is again exact, since $\mathbb{Z}_{p}$ is flat. By induction
on rank, counting from the left, the first and third vertical maps are
isomorphisms. By Lemma 14.6, the fourth one is a monomorphism, as
$\mathbb{Z}_{p}\otimes\operatorname{Ext}^{1}(E_{*})\simeq\operatorname{Ext}^{1}(E_{*})$
because the latter group is $p$-torsion. We deduce that the second one is an
isomorphism using the five-lemma, which was our claim.

Now suppose that $N$ is an arbitrary dualizable comodule; by Lemma 14.10 we
can find a surjection $C\rightarrow N$ from a pure comodule. By taking duals,
we can assume that we instead have a monomorphism $N\hookrightarrow C$, where
$C$ is again pure by Remark 14.8. Since we already know
$\operatorname{Ext}^{0}(C)$ and $\mathrm{H}^{0}(C)$ are free finitely
generated over respectively $\mathbb{Z}_{(p)}$ and $\mathbb{Z}_{p}$, we deduce
that the same is true for their submodules $\operatorname{Ext}^{0}(N)$ and
$\mathrm{H}^{0}(N)$, proving $(1)$.

Let us denote the cokernel of $N\hookrightarrow C$ by $C^{\prime}$. We then
have another commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{p}\otimes\operatorname{Ext}^{0}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{p}\otimes\operatorname{Ext}^{0}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{p}\otimes\operatorname{Ext}^{0}(C^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(C^{\prime}).}$

By what we have proven above, the middle map is an isomorphism as well. Since
the right one is injective, we deduce from the four-lemma that the left one is
surjective, ending the argument. ∎

######  Remark 14.12.

One would like to prove that Theorem 14.11 holds for all finitely generated
comodules, rather than only the finite projective ones. It is possible this
more general result can be deduced formally from the projective case, but we
were not able to do so.

The key ingredient in the above proof is the calculation of
$\mathrm{H}_{\operatorname{cts}}^{0}(\mathbb{G}_{n},E_{*})$; it seems likely
that the version for arbitrary comodules would follow from a more general
calculation of
$\mathrm{H}_{\operatorname{cts}}^{0}(\mathbb{G}_{n},E_{*}/I_{k})$ by Landweber
filtration arguments.

####  14.3. The comodule completion of Morava $E$-theory

We now come to the main theorem of this section. One may view the computation
of this inverse limit at all heights and primes as uniform evidence for the
validity for the algebraic chromatic splitting conjecture and hence of the
chromatic splitting conjecture itself.

######  Theorem 14.13.

We have

$\textstyle\varprojlim_{E_{*}E}E_{*}/\mathfrak{m}^{k}\simeq\mathrm{H}_{\operatorname{cts}}^{0}(\mathbb{G}_{n},E_{*}E)\simeq
E_{*}\otimes_{\mathbb{Z}}\mathbb{Z}_{p},$

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

###### Proof.

The first isomorphism is an instance of Proposition 12.11. By Corollary 12.8,
if $M_{\alpha}$ is a filtered diagram of dualizable $E_{*}E$-comodules such
that $\varinjlim M_{\alpha}\simeq E_{*}E$, then

$\mathrm{H}_{\operatorname{cts}}^{0}(\mathbb{G}_{n},E_{*}E)\simeq\varinjlim\mathrm{H}_{\operatorname{cts}}^{0}(\mathbb{G}_{n},M_{\alpha})\simeq\operatorname{Ext}^{0}_{E_{*}E}(E_{*},M_{\alpha})\otimes_{\mathbb{Z}}\mathbb{Z}_{p}\simeq
E_{*}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$

where the second isomorphism is Theorem 14.11. This ends the argument. ∎

######  Remark 14.14.

Since $E_{*}S^{0}_{p}\simeq E_{*}\otimes_{MU_{*}}MU_{*}S^{0}_{p}\simeq
E_{*}\otimes\mathbb{Z}_{p}$, one can rephrase Theorem 14.13 as saying that
$\varprojlim^{0}E_{*}/\mathfrak{m}^{k}\simeq E_{*}S^{0}_{p}$, which relates
this result to the topological chromatic splitting conjecture.

######  Remark 14.15.

There is a short exact sequence of $E_{*}E$-modules

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma^{|v_{n-1}|}E_{*}/I_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{n-1}}$$\textstyle{E_{*}/I_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{*}/I_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Applying the functor
$\varprojlim_{\mathcal{D}(E_{*}E)}E_{*}/I_{n}^{k}\otimes-$ and considering the
induced long exact sequence in cohomology, we deduce that $v_{n-1}$ acts on
$\varprojlim_{\mathcal{D}(E_{*}E)}^{s}E_{*}/(p,\ldots,v_{n-2},v_{n-1}^{i})$
injectively for $s=0$ and bijectively for $s>0$. This provides further
evidence for 14.2.

###  15\. Cooperations at height one

In the previous section, we discussed the importance of the derived functors
of the limit in the category of comodules and its relation to the chromatic
splitting conjecture. In Proposition 12.11, we gave an explicit formula for
these in the form of an isomorphism

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

where on the left hand side we have the $s$-th derived functor of the limit in
$E_{*}E$-comodules and the right hand side we have continuous cohomology of
the Morava stabilizer group. To make use of this, we would like to understand
$E_{*}E$ as a $\mathbb{G}_{n}$-representation; in this section, we will do so
for $n=1$.

######  Warning 15.1.

Let us warn the reader again that we are interested in the uncompleted
cooperations; that is, $E_{*}E=\pi_{*}(E\otimes E)$. The description of the
completed cooperations

$E^{\vee}_{*}E:=\pi_{*}(L_{K}(E\otimes
E))\simeq(E_{*}E)^{\vee}_{\mathfrak{m}}$

is well-known, as we have
$E^{\vee}_{*}E\simeq\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{n},E_{*})$
at any height by a result of Strickland [Str00] or [Hov04b].

######  Notation 15.2.

Instead of the Honda formal group law, it will be convenient to work with the
multiplicative formal group law over $\mathbb{F}_{p}$, which we will denote by
$\Gamma$. This is of height one, so that the Lubin–Tate ring is isomorphic to
the $p$-adics, with the universal deformation the multiplicative formal group
law over $\mathbb{Z}_{p}$.

By uniqueness part of the Goerss–Hopkins–Miller theorem, the Lubin–Tate
spectrum associated to this universal deformation is given by

$E:=KU^{\vee}_{p},$

the $p$-complete complex $K$-theory. The action of $\mathbb{G}_{1}$ is given
by Adams operations.

######  Lemma 15.3.

There is an isomorphism

$E_{*}E\simeq E_{*}KU\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$

of $\mathbb{G}_{1}$-representations, where on the right hand side the action
is trivial on the $\mathbb{Z}_{p}$-factor.

###### Proof.

We have $KU_{*}\simeq\mathbb{Z}[u^{\pm 1}]$ and $E_{*}\simeq
KU_{*}\otimes\mathbb{Z}_{p}$, which is flat over $KU_{*}$. It then follows
from flatness that we have an isomorphism

$E_{*}E\simeq E_{*}KU\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$

induced by the multiplication map

$(E\otimes KU)\otimes_{KU}E\rightarrow E\otimes E$

Since this multiplication map is $\mathbb{G}_{1}$-equivariant with respect to
the action on the left $E$ factor, this is in fact an isomorphism of
representations. ∎

######  Corollary 15.4.

There is an isomorphism

$\mathrm{H}^{s}_{\operatorname{cts}}(\mathbb{G}_{1},E_{*}E)\simeq\mathrm{H}^{s}_{\operatorname{cts}}(\mathbb{G}_{1},E_{*}KU)\otimes_{\mathbb{Z}}\mathbb{Z}_{p}.$

###### Proof.

Let us write
$\mathrm{H}^{*}(M):=\mathrm{H}^{*}_{\operatorname{cts}}(\mathbb{G}_{1},M)$ for
brevity. As $\mathbb{Z}_{p}$ is flat, it can be written as a filtered colimit
$\mathbb{Z}_{p}\simeq F_{\alpha}$ of finitely generated, free abelian groups
by Lazard’s theorem. We then have

$\mathrm{H}^{s}(E_{*}E)\simeq\mathrm{H}^{s}(E_{*}KU\otimes_{\mathbb{Z}}\mathbb{Z}_{p})\simeq\mathrm{H}^{s}(\varinjlim
E_{*}KU\otimes_{\mathbb{Z}}F_{\alpha})\simeq\varinjlim\mathrm{H}^{s}(E_{*}KU\otimes_{\mathbb{Z}}F_{\alpha}),$

where the last equivalence used Corollary 12.8, and further

$\varinjlim\mathrm{H}^{s}(E_{*}KU\otimes_{\mathbb{Z}}F_{\alpha})\simeq\varinjlim\mathrm{H}^{s}(E_{*}KU)\otimes_{\mathbb{Z}}F_{\alpha}\simeq\mathrm{H}^{s}(E_{*}KU)\otimes_{\mathbb{Z}}\mathbb{Z}_{p},$

where the first equivalence used additivity of cohomology. ∎

It follows from Corollary 15.4 that to understand the cohomology with
coefficients in $E_{*}E$, it is enough to understand $E_{*}KU$.

As we are working with coefficients in the homology of a spectrum rather than
an arbitrary $\mathbb{G}_{1}$-module, we have a canonical family of
invariants. To be more precise, we have a Hurewicz homomorphism

$KU_{*}\rightarrow E_{*}KU$

whose image necessarily consists of $\mathbb{G}_{1}$-invariants. It follows
that the cohomology groups

$\mathrm{H}^{*}_{\operatorname{cts}}(\mathbb{G}_{1},E_{*}KU)$

are $2$-periodic. As they’re also even, as $E_{*}KU$ is, it is enough to
determine the cohomology with coefficients in $E_{0}KU$.

######  Lemma 15.5.

The composite

$E_{0}KU\rightarrow E_{0}^{\vee}KU\simeq
E_{0}^{\vee}E\simeq\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{Z}_{p})$

is injective.

###### Proof.

Note that by Landweber exactness we have

$E_{0}KU\simeq\mathbb{Z}_{p}\otimes_{\mathbb{Z}}KU_{0}KU.$

This is flat over $\mathbb{Z}_{p}$ and we deduce from Proposition 8.6 that the
above composite can be identified with a $p$-completion map. By a result of
Adams, Harris, and Switzer, $KU_{0}KU$ is free as an abelian group [AHS71]. It
follows that the $p$-completion map

$\mathbb{Z}_{p}\otimes_{\mathbb{Z}}KU_{0}KU\rightarrow(\mathbb{Z}_{p}\otimes_{\mathbb{Z}}KU_{0}KU)^{\vee}_{p}\simeq(KU_{0}KU)^{\vee}_{p}\simeq
E_{0}^{\vee}E$

is injective as needed. ∎

The completion map of Lemma 15.5 is $\mathbb{G}_{1}$-equivariant, and we
deduce that $E_{0}KU$ is naturally a subrepresentation of
$\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{Z}_{p})$. On
the latter, $\mathbb{G}_{1}$ acts by right multiplication on the source. Our
goal is to describe the given subrepresentation.

By unwrapping the definitions, the map of Lemma 15.5 can be described
explicitly. By standard results about weakly periodic Landweber exact homology
theories, $KU_{0}KU$ is the algebra classifying automorphisms of the
multiplicative formal group law. If $g\in\mathbb{G}_{1}$ is an automorphisms
over the $p$-adics, then we can write

$g(x)=\sum_{i}b_{i}(g)x^{i},$

where the $b_{i}$ depend continuously on $g$. The above expression defines an
automorphism of the multiplicative formal group law over
$\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{Z}_{p})$; this
is the automorphism classified by the injective map

$KU_{0}KU\rightarrow\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{Z}_{p}).$

Combined with Lemma 15.5, this proves the following.

######  Proposition 15.6.

The ring $E_{0}KU$ is the $\mathbb{Z}_{p}$-subalgebra of
$\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{Z}_{p})$
generated by the functions
$b_{i}\colon\mathbb{G}_{1}\rightarrow\mathbb{Z}_{p}$ for $i\geq 1$ as well as
$b_{1}^{-1}$.

As we work with the multiplicative formal group law, these functions $b_{i}$
can be made explicit. Assume first that $a\in\mathbb{Z}$, so that the
corresponding multiple of the identity of the multiplicative formal group law
is given by

$[a](x)=(1+x)^{a}-1=\sum^{\infty}_{k>0}{a\choose k}x^{k}.$

The last expression makes sense even if $a$ is a $p$-adic integer, where we
take

${a\choose k}=\frac{a(a-1)\ldots(a-k+1)}{k!},$

and the sum is now possibly infinite. One can check that these are
automorphisms of the multiplicative formal group law even if $a$ is $p$-adic,
so that the identification

$\mathbb{Z}_{p}^{\times}\simeq\mathbb{G}_{1}$

is given by $a\mapsto[a](x)=\sum^{\infty}_{k>0}{a\choose k}x^{k}$. Thus,
Proposition 15.6 can be stated in the following more explicit form.

######  Proposition 15.7.

The algebra $E_{0}KU$ is the $\mathbb{Z}_{p}$-subalgebra of
$\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{Z}_{p})$
generated by the functions $b_{i}(a)={a\choose i}$ for $i\geq 1$ as well as
$b_{1}^{-1}(a)=\frac{1}{a}$.

######  Remark 15.8.

The functions $b_{i}$ appearing in Proposition 15.7 are well-known to $p$-adic
analysts. By a fundamental theorem of Mahler, any continuous $p$-adic function
on $\mathbb{Z}_{p}$ can be written as an infinite convergent sum of the
$b_{i}$-s in a unique way [Mah58].

This immediately implies the following calculation.

######  Proposition 15.9.

We have
$\mathrm{H}^{0}_{\operatorname{cts}}(\mathbb{G}_{1},E_{0}KU)\simeq\mathbb{Z}_{p}$.

###### Proof.

This is immediate from Proposition 15.7, as the only functions
$\mathbb{G}_{1}\rightarrow\mathbb{Z}_{p}$ invariant under translation are the
constants. ∎

To compute higher cohomology groups, it is convenient to observe that they are
necessarily rational.

######  Lemma 15.10.

The groups $\mathrm{H}^{i}_{\operatorname{cts}}(\mathbb{G}_{1},E_{0}KU)$ are
rational vector spaces for $i>0$.

###### Proof.

Since the inclusion
$E_{0}KU\hookrightarrow\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{Z}_{p})$
is an isomorphism after $p$-completion, as we observed in the proof of Lemma
15.5, we deduce that it is an isomorphism modulo $p$. It follows that

$\mathrm{H}^{i}_{\operatorname{cts}}(\mathbb{G}_{1},E_{0}KU\otimes_{\mathbb{Z}_{p}}\mathbb{F}_{p})\simeq\mathrm{H}^{i}_{\operatorname{cts}}(\mathbb{G}_{1},\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{F}_{p}))$

and the right hand side vanishes for $i>0$. The long exact sequence of
cohomology associated to

$0\rightarrow E_{0}KU\rightarrow E_{0}KU\rightarrow
E_{0}KU\otimes_{\mathbb{Z}_{p}}\mathbb{F}_{p}\rightarrow 0,$

where the first map is multiplication by $p$, gives the required result. ∎

######  Proposition 15.11.

We have

$\mathrm{H}^{1}_{\operatorname{cts}}(\mathbb{G}_{1},E_{0}KU)\simeq\mathbb{Q}_{p}$

and the higher cohomology groups vanish.

###### Proof.

By Lemma 15.10, all cohomology groups of positive degree are rational, so that
we can replace $E_{0}KU$ by $\mathbb{Q}_{p}\otimes_{\mathbb{Z}_{p}}E_{0}KU$
without changing these. The cohomology with the coefficients in the latter can
be computed directly.

As each binomial coefficient $b_{i}(a)={a\choose i}$ is a polynomial in $a$
with rational coefficients, we deduce from Proposition 15.7 that
$\mathbb{Q}_{p}\otimes_{\mathbb{Z}_{p}}E_{0}KU$ can be identified with the
$\mathbb{Q}_{p}$-subalgebra of
$\operatorname{map}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{Q}_{p})$
generated by $b_{1}(a)=a$ and its inverse.

It follows that there is an isomorphism of representations

$\mathbb{Q}_{p}\otimes_{\mathbb{Z}_{p}}E_{0}KU\simeq\mathbb{Q}_{p}[b^{\pm
1}],$

where $b=b_{1}$ and the action is determined by $g\cdot b=gb$ for
$g\in\mathbb{G}_{1}$. This decomposes as a direct sum of character
representations

$\mathbb{Q}_{p}[b^{\pm 1}]\simeq\bigoplus_{k\in\mathbb{Z}}\mathbb{Q}_{p}(k),$

where we identify $\mathbb{Q}_{p}(k)$ with the subrepresentation generated by
$b^{k}$. It follows that

$\mathrm{H}^{i}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{Q}_{p}[b^{\pm
1}])\simeq\bigoplus_{k\in\mathbb{Z}}\mathrm{H}^{i}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{Q}_{p}(k)).$

Let $U\subseteq\mathbb{G}_{1}\simeq\mathbb{Z}_{p}^{\times}$ be a normal
subgroup of finite index such that $U\simeq\mathbb{Z}_{p}$. As finite groups
have no rational cohomology, Lyndon–Hochschild–Serre spectral sequence
collapses and gives an isomorphism

$\mathrm{H}^{i}(\mathbb{G}_{1},\mathbb{Q}_{p}(k))\simeq\mathrm{H}^{i}(U,\mathbb{Q}_{p}(k))^{\mathbb{G}_{1}/U}.$

It follows that the cohomology groups vanish above degree $1$, as
$U\simeq\mathbb{Z}_{p}$ is of cohomological dimension one.

As $U\simeq\mathbb{Z}_{p}$ is cyclic as a profinite group, its cohomology can
be computed using a two-step cochain complex, as we did in Section 10. It
follows that the only one-dimensional representation of $U$ over
$\mathbb{Q}_{p}$ with non-trivial cohomology is the trivial one. As the
characters $\mathbb{Q}_{p}(k)$ restrict to non-trivial representations for
$k\neq 0$, we deduce that

$\mathrm{H}^{1}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{Q}_{p}[b^{\pm
1}])\simeq\mathrm{H}^{1}_{\operatorname{cts}}(\mathbb{G}_{1},\mathbb{Q}_{p})\simeq\mathbb{Q}_{p}$

as claimed. ∎

######  Theorem 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$;\\\
0&\mbox{otherwise}.\end{cases}$

###### Proof.

As we observed previously that the cohomology with coefficients in $E_{*}E$ is
concentrated in even internal degrees and $2$-periodic, it is enough to
establish this in internal degree zero, where this is a combination of
Corollary 15.4, Proposition 15.9 and Proposition 15.11. ∎

######  Remark 15.13.

As we observed at the beginning, the cohomology groups of Theorem 15.12
coincide with derived functors of completion in comodules by Proposition
12.11. Using the latter perspective, a sketch of calculation of the
rationalization of the above groups at $p>2$ appears in a talk by Hopkins,
recorded in [Pet, Talk 14]. We were independently informed by Hopkins that he
first made this calculation in the 1990s and that it was part of the
motivation towards the algebraic chromatic splitting conjecture.

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