2 Morava K-theories and their Hopf algebroids

Let $\kappa $ be a perfect field equipped with a choice of a height n formal group $\mathbf {G}_{0}$ . Associated to these data is the Lubin–Tate ring $E_{0}$ classifying deformations [Reference Lubin and TateLT66]. This is a complete local $W(\kappa )$ -algebra, such that there exists a noncanonical choice of regular generators inducing an isomorphism of rings $E_{0} \simeq W(\kappa )[\![ u_{1}, \ldots , u_{n-1}]\!]$ . We write $\mathfrak {m} = (p, u_{1} \ldots , u_{n-1})$ for the maximal ideal. The ring $E_{0}$ equipped with the universal deformation is Landweber exact and so can be lifted to a $2$ -periodic ring spectrum E with $E_{*} \simeq E_{0}[u^{\pm 1}]$ for some unit $u \in E_{2}$ , and the property that the formal group is the universal deformation of $\mathbf {G}_{0}$ . It is well-known that E admits a unique $\mathbf {E}_{\infty }$ -ring structure, functorial in the choice of $\kappa $ and the formal group (see [Reference Goerss and HopkinsGH05a]). More recently, it was proven by Lurie that E arises as a solution to a moduli problem involving formal groups over $\mathbf {E}_{\infty }$ -rings, see [Reference LurieLur18].

The canonical nature of the Lubin–Tate spectrum makes it a good starting point for the study of chromatic phenomena. Since $E_{0}$ is local, one would like to construct a spectrum which plays the role of the residue field of E. Following Hopkins and Lurie [Reference Hopkins and LurieHL17], we make the following definition.

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

1. (1) there are uncountably many Morava K-theories which are not equivalent as E-algebras,

2. (2) none of which is preferred and

3. (3) none of which can be promoted to an $\mathbf {E}_{2}$ -E-algebra

(see [Reference Hopkins and LurieHL17] for more information).

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

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

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

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

Notation 2.5. If $M, N$ are E-modules, we write

for their homology relative to E and

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

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

$$\begin{align*}K \otimes_{E} K \rightarrow K \otimes_{E} K \otimes_{E} K \end{align*}$$

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

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

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

where on the right-hand side, we have maps of E-modules. One can show that the latter is always isomorphic to an exterior algebra over a vector space of the needed dimension (see [Reference Hopkins and LurieHL17, Proposition 6.5.1]).

Remark 2.8. As both sides are $K_{*}$ -vector spaces of dimension $2^{n}$ , the latter by Lemma 2.6, the Künneth spectral sequence

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

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

The second, ‘understandable’ part of $K_{*}K$ is the image of the canonical map from $E_{*}E$ , which can be identified with $K_{*}E$ . The analogue of this Hopf algebroid for the minimal Morava K-theory of Section 3 is denoted in Ravenel’s book by $\Sigma (n)$ (see [Reference RavenelRav03, §6.2]). Notice that since $E_{*}E$ is flat over $E_{*}$ , we have canonical isomorphisms

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

which in turn have the following consequence.

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

$$\begin{align*}E_{*}E \simeq E_{*} \otimes_{BP_{*}} BP_{*}BP \otimes_{BP_{*}} E_{*}, \end{align*}$$

and by again invoking flatness, we see that

$$\begin{align*}K_{*}E \simeq K_{*} \otimes_{BP_{*}} BP_{*}BP \otimes_{BP_{*}} E_{*}. \end{align*}$$

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

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

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

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

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

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

which is exactly the claim.

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

Proof. We will show something stronger, namely, that the $E \otimes E$ -modules $K \otimes E$ and $E \otimes K$ are equivalent. Since the latter has the needed property, as it admits a structure of a K-module from its right factor, this will be enough. At the level of homotopy groups, the induced maps $E_{*}E \rightarrow E_{*}K$ and $E_{*}E \rightarrow K_{*}E$ are both quotients by $\mathfrak {m} E_{*}E = E_{*}E \mathfrak {m}$ , where the equality is the statement that $\mathfrak {m}$ is an invariant ideal. Since $\mathfrak {m}$ is generated by a regular sequence, the needed statement will follow from the following result: Let  be a commutative algebra in spectra, and let $B \in \operatorname {\mathrm {Alg}}(\operatorname {\mathrm {Mod}}_{A})$ be an algebra whose unit map induces an isomorphism, $B_{*} \simeq A_{*}/I$ , where I is an ideal generated by a regular sequence $x_{1}, \ldots x_{n}$ . Then we have an equivalence of A-modules

$$\begin{align*}B \simeq \bigotimes_{A} \mathrm{cofib}(x_{i} \colon A \rightarrow A). \end{align*}$$

In particular, as an A-module B is determined up to equivalence by I.

To see this, note that since $1 \in B_{*}$ is $x_{i}$ -torsion, the unit factors through a map $\mathrm {cofib}(x_{i}) \rightarrow A$ . Then the composite

$$\begin{align*}\bigotimes_{A} \mathrm{cofib}(x_{i}) \rightarrow \bigotimes_{A} B \rightarrow B, \end{align*}$$

where the second map is the algebra multiplication, is an isomorphism on homotopy groups and thus an equivalence.

Corollary 2.12. Consider $K \otimes E$ as an E-module with the module structure inherited only from the right. Then, the Künneth spectral sequence

$$\begin{align*}\operatorname{\mathrm{Tor}}^{E_{*}}(K_{*}E, K_{*}) \implies \pi_{*}((K \otimes E) \otimes_{E} K) \simeq K_{*}K \end{align*}$$

in E-modules collapses.

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

As a consequence of Corollary 2.12, the Künneth spectral sequence

$$\begin{align*}K_{*}E \otimes \operatorname{\mathrm{Tor}}^{E_{*}}(K_{*}, K_{*}) \simeq \operatorname{\mathrm{Tor}}^{E_{*}}(K_{*}E, K_{*}) \implies K_{*}K \end{align*}$$

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

Proposition 2.14. Passing to homotopy groups in the diagram

induces an isomorphism

$$\begin{align*}K _{*}K \otimes_{K_{*}E} K_{*} \simeq K_{*}^{E}K. \end{align*}$$

We are now ready to assemble the above information. By standard arguments, the pair $(K_{*}, K_{*}K)$ acquires the structure of a Hopf algebroid. More precisely, the two maps $K \rightarrow K \otimes K$ induce left and right units, both of which are central as a consequence of Proposition 2.13, as they factor through $K_{*}E$ . These make $K_{*}K$ into a $K_{*}$ -bimodule, and similarly, we get a suitable comultiplication and antipode.

Note that it is common in algebraic topology literature to assume in the definition of a Hopf algebroid that multiplication is commutative. However, this is not strictly necessary; in our case, as the units are central, $K_{*}K$ is a Hopf algebroid in the more general sense of Maltsiniotis [Reference MaltsiniotisMal92]. The category of comodules can be defined in the usual way, and it will be monoidal with the tensor product lifting that of $K_{*}$ -modules. It need not, in general, be symmetric monoidal.

Theorem 2.16. The natural maps

(2.17) $$ \begin{align} (K_{*}, K_{*}E) \rightarrow (K_{*}, K_{*}K) \rightarrow (K_{*}, K_{*}^{E}K) \end{align} $$

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

Proof. We first recall the relevant notation and definitions from [Reference RavenelRav03, Section A.1]. If $(A, \Gamma )$ is a Hopf algebroid, M is a right $\Gamma $ -comodule and N is a left $\Gamma $ -comodule, then the cotensor product is defined as the equalizer

between the two possible comultiplications. Two particularly important cases are

$$\begin{align*}N \square_{\Gamma} A \simeq \operatorname{\mathrm{Ext}}^{0}_{\operatorname{\mathrm{Comod}}_{\Gamma}^{R}}(A, N), \end{align*}$$

the coinvariants in the category of right comodules and

$$\begin{align*}N \square_{\Gamma} A \simeq \operatorname{\mathrm{Ext}}^{0}_{\operatorname{\mathrm{Comod}}_{\Gamma}^{L}}(A, N), \end{align*}$$

the coinvariants in the category of left comodules. Since $K_{*}K \rightarrow K_{*}^{E}K$ is surjective, consulting Ravenel’s definitions of a normal map of Hopf algebroids and of an extension [Reference RavenelRav03, A.1.1.10, A.1.1.15], we see that in our case, we have to verify that

1. (1) $K_{*} \square _{K_{*}^{E}K} K_{*}K = K_{*}K \square _{K_{*}^{E}K} K_{*}$ ,

2. (2) $K_{*} \square _{K_{*}^{E}K} K_{*} = K_{*}$ ,

3. (3) $K_{*} \square _{K_{*}^{E}K} K_{*}K \square _{K_{*}^{E}K} K_{*} = K_{*}E$ .

We begin with the first part. If we write

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

then the left $K_{*}^{E}K$ -comodule structure on $K_{*}K$ given by the quotient map $K_{*}K \rightarrow K_{*}^{E}K$ corresponds to the standard comodule structure on the relative homology $K_{*}^{E}(E \otimes K)$ relative to E. This comodule structure map is determined by the structure of $E \otimes K$ as a left E-module with module structure coming only from the left factor. By Lemma 2.11, $E \otimes K$ is equivalent as an E-module to a direct sum of K, and so admits a compatible structure of a K-module. Choosing such a K-module structure determines an isomorphism

$$\begin{align*}K_{*}^{E}(E \otimes K) \simeq K_{*}^{E}K \otimes_{K_{*}} \pi_{*}(E \otimes K) \simeq K_{*}^{E}K \otimes_{K_{*}} E_{*}K \end{align*}$$

of left comodules, so that

$$\begin{align*}K_{*} \square_{K_{*}^{E}K} K_{*}K \simeq K_{*} \square_{K_{*}^{E}K} K_{*}^{E}(E \otimes K) \simeq K_{*} \square_{K_{*}^{E}K} (K_{*}^{E}K \otimes E_{*}K) \simeq E_{*}K \end{align*}$$

as needed.

Using symmetry, Lemma 2.11 similarly shows that

$$\begin{align*}K_{*}K \simeq K_{*}E \otimes_{K_{*}} K_{*}^{E}K \end{align*}$$

as right $K_{*}^{E}K$ -comodules, so that

$$\begin{align*}K_{*}K \square_{K_{*}^{E}K} K_{*} \simeq K_{*}E. \end{align*}$$

As $K_{*}E$ and $E_{*}K$ coincide as subobjects of $K_{*}K$ , this shows part $(1)$ and part $(3)$ . Part $(2)$ is an observation that $K_{*}^{E}K$ is a Hopf algebra rather than a Hopf algebroid.

3 Digression: Minimal Morava K-theory

Large parts of the literature on Morava K-theories are written in terms of the minimal Morava K-theory spectrum $K(n)$ , the unique up to equivalence $BP$ -module with $K(n)_{*} \simeq \mathbb {F}_{p}[v_{n}^{\pm 1}]$ . Note that it is not a Morava K-theory in the sense of Definition 2.1 as it is not $2$ -periodic, but as a consequence of nilpotence theorem, any Morava K-theory is equivalent as a spectrum to a direct sum of $K(n)$ [Reference Hopkins and SmithHS98, Lemma 1.8].

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

It is a result of Robinson that $K(n)$ can be made into an $\mathbf {E}_{1}$ -ring spectrum, but there is no canonical way to do so [Reference RobinsonRob89]. A surprising result of Angeltveit shows that all choices of $\mathbf {E}_{1}$ -multiplications yield equivalent $\mathbf {E}_{1}$ -algebras in spectra [Reference AngeltveitAng11].

One can also ask about less structured multiplication. By a theorem of Würgler [Reference WürglerWür91], in the homotopy category of spectra, the spectrum $K(n)$ admits,

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

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

which make it into an associative $BP$ -algebra in a way compatible with its $BP$ -module structure [Reference WürglerWür91, Theorem 1.5].

The ring of cooperations of $K(n)$ has been computed completely [Reference WürglerWür91, Theorem 2.4]. The map $K(n)_{*}BP \rightarrow K(n)_{*}K(n)$ is injective, and we have an isomorphism

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

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

$$\begin{align*}K(n)_{*}K(n) \simeq K(n)_{*}BP \otimes _{K(n)_{*}} \Lambda_{K(n)_{*}}(\tau_{0}, \ldots, \tau_{n-1}), \end{align*}$$

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

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

This is analogous to the variation in the dual Steenrod algebra depending on whether p is even or odd. Note that the latter computation implies that $K(n)_{*}K(n)$ is commutative also at $p=2$ , despite $K(n)$ not being homotopy commutative. The corresponding question in the $2$ -periodic case is open (see Remark 2.15).

4 Milnor modules

As we have seen in the previous section, there are many different Morava K-theory spectra, which makes any discussion of this subject dependent on these choices. However, as in the case of Proposition 2.9, one can show that many cohomological invariants are canonical and do not depend on any choices. In this section, we make this canonical nature transparent by introducing an absolute variant of Hopkins and Lurie’s categories of Milnor modules (see [Reference Hopkins and LurieHL17]).

Before we work in the absolute case, let us first describe the case relative to a fixed Lubin–Tate spectrum E. Note that by the uniqueness part of the Goerss–Hopkins–Miller theorem, see [Reference Goerss and HopkinsGH, Corollary 7.6] or [Reference Pstragowski and VanKoughnettPV22, Theorem 7.7], this is the same as choosing a perfect field equipped with a finite height formal group.

Definition 4.3. The category of E-based Milnor modules is the category

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

By construction, the category of Milnor modules is a compactly generated Grothendieck abelian category. Since molecular E-modules are stable under the E-tensor product, it acquires a nonunital symmetric monoidal structure via left Kan extension. One can show this extends uniquely to a unital symmetric monoidal structure [Reference Hopkins and LurieHL17, Propositions 4.4.1 and 4.4.10].

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

Proposition 4.4. Let K be a Morava K-theory of E. Then, the left Kan extension of the functor

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

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

$$\begin{align*}\operatorname{\mathrm{Mil}}_{E} \simeq \operatorname{\mathrm{Comod}}_{K_{*}^{E}K} \end{align*}$$

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

Remark 4.7. The category $\operatorname {\mathrm {Mil}}_{E}$ is not only canonical but, in a certain, sense gives the right answer even if $K_{*}^{E}K$ -comodules do not. As an example, the restricted Yoneda embedding

$$\begin{align*}y\colon \operatorname{\mathrm{Mod}}_{E} \rightarrow \operatorname{\mathrm{Mil}}_{E}, \end{align*}$$

which can be thought of as a way of taking relative K-homology without choosing K, is always a symmetric monoidal functor [Reference Hopkins and LurieHL17, Variant 4.4.11].

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

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

need not be symmetric monoidal.

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

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

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

As a consequence of the nilpotence theorem, all Morava K-theories at a fixed prime and height are, as spectra, direct sums of the minimal Morava K-theory spectrum of Section 3. It follows that they all determine the same class of epimorphisms, and hence isomorphic Adams spectral sequences. In fact, the class of epimorphisms is enough to describe canonically the abelian category whose $\operatorname {\mathrm {Ext}}$ -groups form the $E_{2}$ -page. To do so, we consider sheaves with respect to a certain topology, as in the construction of synthetic spectra [Reference PstragowskiPst22].

Definition 4.10. The category of absolute Milnor modules is the category

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

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

The following is the absolute analogue of Proposition 4.4.

Proposition 4.12. If K is a Morava K-theory, then the left Kan extension of the functor

$$\begin{align*}K_{*}: \operatorname{\mathrm{Sp}}^{fin} \rightarrow \operatorname{\mathrm{Comod}}_{K_{*}K} \end{align*}$$

induces a monoidal equivalence

$$\begin{align*}\operatorname{\mathrm{Mil}}_{abs} \simeq \operatorname{\mathrm{Comod}}_{K_{*}K} \end{align*}$$

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

As a consequence, we deduce the following result proven earlier by Hovey and Strickland using slightly different methods [Reference Hovey and StricklandHS05a, Section 8].

Remark 4.16 (Milnor modules as sheaves).

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

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

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

$$\begin{align*}\operatorname{\mathrm{Mil}}_{abs} \rightarrow \operatorname{\mathrm{Mil}}_{E}. \end{align*}$$

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

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

$$\begin{align*}\operatorname{\mathrm{Comod}}_{K_{*}K} \rightarrow \operatorname{\mathrm{Comod}}_{K_{*}^{E}K}. \end{align*}$$

The image of the monoidal unit under the corresponding right adjoint can be identified with the Hopf algebroid $K_{*}E$ studied previously. As the Adams spectral sequence based on $K_{*}E$ in the appropriate derived $\infty $ -categories can be identified with the Cartan–Eilenberg spectral sequence associated to this extension of Hopf algebroids [Reference BelmontBel20, Theorem 1.1], we deduce that the Cartan–Eilenberg spectral sequence itself can be constructed using only the categories of Milnor modules, so it only depends on E.

5 Cohomology

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

We have seen in Theorem 2.16 that we have an extension of Hopf algebroids

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

which induces a Cartan–Eilenberg spectral sequence of signature

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

(see [Reference RavenelRav03, A.1.3.14]). This can be thought of as approximating the cohomology of $K_{*}K$ using simpler pieces, namely, the cohomology of $K_{*}E$ and $K_{*}^{E}K$ . To make the best use of this, we need to begin by understanding the latter two. We have seen in Section 2 that the Hopf algebroid $(K_{*}, K_{*}E)$ is an even graded presentation of the moduli of formal groups of height exactly n (see Proposition 2.10). To identify its cohomology, it is convenient to choose a particularly small presentation of the latter moduli stack, which is what we will do.

Notation 5.1. We follow the conventions of [Reference Goerss, Henn, Mahowald and RezkGHMR05, Section 2]. Let $\mathbf {G}_{0}$ be the Honda formal group law over $\mathbb {F}_{q}$ , where $q = p^{n}$ , and let $E_{n}$ be the associated Lubin–Tate spectrum and $K_{n}$ a Morava K-theory associated to $E_{n}$ . Note that we have a noncanonical isomorphism

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

and a canonical isomorphism

$$\begin{align*}\pi_{*} E_{n} \simeq (\pi_{0} E_{n})[u^{\pm 1}] \end{align*}$$

with $|u| = -1$ . We write $\mathbb {G}_{n} \simeq \mathbb {S}_{n} \rtimes \text {Gal}(\mathbb {F}_{q} / \mathbb {F}_{p})$ for the extended Morava stabiliser group, the automorphism group of the pair $(\operatorname {\mathrm {Spec}}(\mathbb {F}_{q}), \mathbf {G}_{0})$ ; see [Reference HennHen17, Section 4.1] for an explicit description of this group. By the functoriality of the Goerss–Hopkins–Miller theorem [Reference Goerss and HopkinsGH05b], the canonical action of $\mathbb {G}_{n}$ on $(\operatorname {\mathrm {Spec}}(\mathbb {F}_{q}), \mathbf {G}_{0})$ induces a left action on the Lubin–Tate spectrum  $E_{n}$ .

Lemma 5.2. The association $\mathbb {G}_{n} \times (K_{n})_{*}E_{n} \rightarrow (K_{n})_{*}$ sending an element $g \in \mathbb {G}_{n}$ and a homotopy class $S^{k} \rightarrow K_{n} \otimes E_{n}$ to the homotopy class of the composite

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

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

Corollary 5.5. For any Morava K-theory, there is a canonical isomorphism

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

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

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

The action of the Morava stabiliser group on $\mathbb {F}_{q}[u^{\pm 1}]$ is easy to describe. The Galois group acts by its standard action on $\mathbb {F}_{q}$ . The action of $\mathbb {S}_{n}$ factors through the leading coefficient homomorphism $\rho \colon \mathbb {S}_{n} \rightarrow \mathbb {F}_{q}^{\times }$ and is determined by the formula $s \cdot u = p(s)u$ for $s \in \mathbb {S}_{n}$ . The kernel of $\rho $ is the strict Morava stabiliser group $S_{n}$ ; it is a pro-p-group acting trivially. Morava [Reference MoravaMor85] essentially proved:

Theorem 5.6 (Morava).

We have isomorphisms of continuous cohomology groups

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

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

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

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

$$\begin{align*}E \rightarrow E/x \rightarrow \Sigma E \end{align*}$$

of E-modules which induces a short exact sequence

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

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

Proposition 5.8. The association $x \mapsto \psi (x)$ induces a map

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

which extends to an isomorphism of bigraded rings

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

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

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

the cohomology in Milnor modules, where $y(E)$ is the Milnor module associated to E. For the right-hand side, the needed isomorphism is proven in [Reference Hopkins and LurieHL17, Proposition 7.2.7].

Remark 5.9. If we again let $K_{n}$ be any Morava K-theory associated to the Honda formal group over $\mathbb {F}_{q}$ , it follows that the Cartan–Eilenberg spectral sequence for cohomology of the unit has signature

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

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

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

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

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

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

6 Milnor filtration

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

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

A nonnegatively filtred spectrum is a functor , where we consider the natural numbers as a partially ordered set. If X is a spectrum, then a filtration on X is a filtred spectrum $P_{\bullet }$ together with an equivalence $\varinjlim P_{\bullet } \simeq X$ .

To any filtration $P_{\bullet }$ , we can attach an associated nonnegatively graded spectrum

$$\begin{align*}\mathrm{gr}(P_{\bullet}) = \bigoplus_s\mathrm{gr}_{s}(P_{\bullet}) \end{align*}$$

with the degree s part defined by

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

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

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

where the first differential is induced by the chain complex structure on the associated graded [Reference LurieLur16, Section 1.2.2]. The increasing filtration on $\pi _{*}X$ associated to this spectral sequence is given by the images of the maps $\pi _{*} P_{s} \rightarrow \pi _{*}X$ .

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

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

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

In fact, the latter equivalence can be upgraded to an isomorphism in the category of chain complexes in the homotopy category of spectra, where we consider the latter as symmetric monoidal with the differential on the tensor product given by the usual Leibniz rule [Reference RaksitRak20, Remark 3.2.15].

Example 6.3. We will need certain special filtrations of E-modules, the main reference here being Lurie [Reference LurieLur16, Section 7.2.1], who works with simplicial objects. By the stable Dold–Kan correspondence of [Reference LurieLur16, Section 1.2.4], this is equivalent to working with filtred objects as we do.

If $P_{\bullet }$ is an E-module filtration of some M, then we say $P_{\bullet }$ is E-free if the associated simplicial object is S-free with respect to the set $S = \{ \ \Sigma ^{n} E \ | \ n \in \mathbb {Z} \ \}$ . We say it is an E-resolution if it is E-free and an S-hypercover. For precise definitions of these terms, see [Reference LurieLur16, Definition 7.2.1.2.]. The two main facts we need are that:

1. (1) if $P_{\bullet }$ is E-free, then the associated graded E-module is levelwise free,

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

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

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

This is the Künneth spectral sequence which we previously mentioned in Corollary 2.12. Note that in the particular case of $X = S^{0}$ , this spectral sequence is concentrated on the $s = 0$ line, as $E_{*}$ is flat over itself, so that the filtration on the homotopy groups $M_{*}$ induced by $P_{\bullet }$ is trivial.

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

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

Lemma 6.5. The Milnor filtration coincides with the filtration induced by the Künneth spectral sequence of signature

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

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

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

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

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

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

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

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

Remark 6.7. By symmetry, Lemma 6.5 implies that the Milnor filtration also coincides with the filtration induced by the other Künneth spectral sequence, namely, that of signature

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

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

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

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

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

$$\begin{align*}F_{0}^{Mil}(K_{*}K) = E_{*}K, \end{align*}$$

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

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

Observe that since K is an E-algebra, the multiplication factors canonically through a map $K \otimes _{E} K \rightarrow K$ . Observe that $P_{\bullet } \otimes _{E} P_{\bullet }$ is an E-free filtration of $K \otimes _{E} K$ (although not a resolution, but this is not needed). Then, since $P_{\bullet }$ is an E-module resolution of K, it follows from [Reference LurieLur16, Proposition 7.2.1.5] (where $C = K$ ) that the map $K \otimes _{E} K \rightarrow K$ of E-modules lifts to a map $P_{\bullet } \otimes _{E} P_{\bullet } \rightarrow P_{\bullet }$ of filtrations. The needed lift of $K \otimes K \rightarrow K$ is then given as composition

$$\begin{align*}P_{\bullet} \otimes P_{\bullet} \rightarrow P_{\bullet} \otimes_{E} P_{\bullet} \rightarrow P_{\bullet}, \end{align*}$$

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

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

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

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

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

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

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

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

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

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

Consider the composite

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

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

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

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

Observe that since $F_{0}^{\operatorname {\mathrm {Mil}}}(K_{*}K) \simeq E_{*}K$ is concentrated in even internal degrees, the canonical projection

$$\begin{align*}F_{1}^{\operatorname{\mathrm{Mil}}}(K_{*}K) \rightarrow \mathrm{gr}_{1}^{\operatorname{\mathrm{Mil}}}(K_{*}K) \simeq \operatorname{\mathrm{Tor}}_{1}^{E_{*}}(K_{*}, K_{*}) \otimes E_{*}K \end{align*}$$

is an isomorphism in odd internal degrees. It follows that the inclusion

$$\begin{align*}i \colon \operatorname{\mathrm{Tor}}_{1}^{E_{*}}(K_{*}, K_{*}) \hookrightarrow \operatorname{\mathrm{Tor}}_{1}^{E_{*}}(K_{*}, K_{*}) \otimes E_{*}K \end{align*}$$

can be uniquely lifted to a map

$$\begin{align*}j \colon \operatorname{\mathrm{Tor}}_{1}^{E_{*}}(K_{*}, K_{*}) \rightarrow F_{1}^{\operatorname{\mathrm{Mil}}} K_{*}K. \end{align*}$$

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

$$\begin{align*}\operatorname{\mathrm{Tor}}_{*, *}^{E_{*}}(K_{*}, K_{*}) \rightarrow K_{*}K \end{align*}$$

with the following property:

Proposition 6.10. Let $p> 2$ , and assume that $K_{*}K$ is commutative. Then, the induced map of $E_{*}K$ -algebras

(6.11) $$ \begin{align} \operatorname{\mathrm{Tor}}^{E_{*}}_{*, *}(K_{*}, K_{*}) \otimes_{K_{*}} E_{*}K \rightarrow K_{*}K \end{align} $$

is an isomorphism.

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

Theorem 6.17. Suppose that $p> 2$ and that $K_{*}K$ is commutative. Then, the Cartan–Eilenberg spectral sequence associated to the extension

$$\begin{align*}K_{*}E \rightarrow K_{*}K \rightarrow K_{*}^{E}K \end{align*}$$

collapses on the second page, inducing a canonical isomorphism

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

Proof. Under the given assumptions, $K_{*}K$ is bigraded using its internal grading and the Milnor grading of Definition 6.12. This grading is compatible with the above extension, with all of $K_{*}E$ of Milnor degree zero.

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

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

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

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

Part 2. The K -based Adams spectral sequence

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

Remark 6.18 (Adams spectral sequence based on Milnor modules).

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

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

$$\begin{align*}y\colon \operatorname{\mathrm{Sp}} \rightarrow \operatorname{\mathrm{Mil}}_{abs} \end{align*}$$

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

By construction, any injective object in Milnor modules can be lifted to a representing object in spectra, and this allows one to construct a ‘y-based’ Adams spectral sequence based on injectives, where the $E_{2}$ -terms are canonically given by $\operatorname {\mathrm {Ext}}$ -groups in absolute Milnor modules [Reference Patchkoria and PstragowskiPP21]. Through the equivalence of Proposition 4.12, this spectral sequence is canonically isomorphic to the K-based Adams spectral sequence for any choice of Morava K-theory.

7 Convergence of the K-based Adams spectral sequence

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

By standard arguments, the K-based Adams spectral sequence is isomorphic to the totalisation spectral sequence obtained by mapping into the canonical K-Adams resolution

$$\begin{align*}X \rightarrow K \otimes X \rightrightarrows K \otimes K \otimes X \Rrightarrow \ldots .\end{align*}$$

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

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

Proposition 7.2. For any spectrum X, the K-nilpotent completion of X is equivalent to the K-localisation of X; that is, the canonical map

(7.3) $$ \begin{align} L_{K}X \rightarrow \operatorname{\mathrm{Tot}}(K^{\otimes \bullet+1} \otimes X) \end{align} $$

is an equivalence.

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

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

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

as both sides of (7.3) define exact functors in X and so commute with tensoring with a finite spectrum. Thus, it is enough to check the result holds for $X \otimes F(n)$ .

The proof of the smash product theorem in [Reference RavenelRav92, Chapter 8] shows that $S^0$ is E-prenilpotent; that is, $L_ES^0 \in \operatorname {\mathrm {Thick}}^{\otimes }(E)$ , where $\operatorname {\mathrm {Thick}}^{\otimes }(Y)$ denotes the smallest full subcategory of which contains Y, is thick and is closed under smashing with arbitrary objects in . It follows that

$$\begin{align*}L_{K}F(n) \simeq L_E F(n) \in \operatorname{\mathrm{Thick}}^{\otimes}(E\otimes F(n)) \subseteq \operatorname{\mathrm{Thick}}^{\otimes}(K). \end{align*}$$

In other words, $F(n)$ is K-prenilpotent. By definition, $X \otimes F(n)$ is thus also K-prenilpotent for all . Therefore, Bousfield’s theorem [Reference BousfieldBou79, Theorem 6.10] applies to provide an equivalence $L_{K}(X \otimes F(n)) \simeq \operatorname {\mathrm {Tot}} (K^{\otimes \bullet +1} \otimes X \otimes F(n))$ , which is what we wanted.

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

$$\begin{align*}E_2^{s,t}(Y, X) \simeq \operatorname{\mathrm{Ext}}_{K_*K}^{s,t}(K_*Y,K_*X) \implies [Y, L_{K}X]^{t-s}. \end{align*}$$

Indeed, if we use the canonical Adams resolution, then the first page of the Adams spectral sequence is given by the cobar complex, which in the projective case computes the $\operatorname {\mathrm {Ext}}$ -groups in comodules [Reference RavenelRav03, Corollary A.1.2.12].

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

$$\begin{align*}d_r\colon E_r^{s,t}(Y, X) \to E_r^{s+r,t+r-1}(Y, X). \end{align*}$$

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

$$\begin{align*}E_{\infty}^{s,t}(Y, X) = \operatorname{\mathrm{lim}}_rE_{r}^{s,t}(Y, X). \end{align*}$$

Let $F^*[Y,L_{K}X]_*$ denote the filtration on $[Y,L_{K}X]_*$ induced by the spectral sequence. Following Bousfield [Reference BousfieldBou79, Section 6], we say that the K-based Adams spectral sequence for X converges completely if the canonical maps

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

and

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

are isomorphisms. By the work of Bousfield, see [Reference BousfieldBou79, Proposition 6.3], this is equivalent to the condition that $\operatorname {\mathrm {lim}}_{r>s}^1E_{r}^{s,t}(Y, X)$ vanishes.

Proposition 7.5. The K-based Adams spectral sequence of signature

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

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

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

$$\begin{align*}(E_r^{s,t}(Y, X))_{r>s} \end{align*}$$

is Mittag-Leffler, so that Bousfield’s convergence criterion [Reference BousfieldBou79, Proposition 6.3] applies.

8 The finite height Miller square

In his seminal paper [Reference MillerMil81] on relations between Adams spectral sequences, Miller uses the relationship between the Adams–Novikov and classical Adams spectral sequences to prove the telescope conjecture at height 1 and odd primes. In this section, we apply his axiomatic framework to the case of Lubin–Tate and Morava K-theory spectra.

In Miller’s framework, one starts with a map $A \rightarrow B$ of homotopy ring spectra; that is, an arrow in the category $\operatorname {\mathrm {Alg}}(\operatorname {\mathrm {ho}}(\operatorname {\mathrm {Sp}}))$ . In this context, it is not hard to show that a map $X \rightarrow Y$ of spectra which is A-monic, in the sense that $A \otimes X \rightarrow A \otimes Y$ is a split inclusion, is also B-monic. Thus, any B-injective spectrum is also A-injective (see [Reference MillerMil81, Lemma 2.1]).

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

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

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

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

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

Our first goal is to show that the K-based Adams spectral sequence has a particularly simple form for E-modules with flat homotopy groups. To do so, we will reduce to the profree case using the work of Hovey [Reference HoveyHov04c].

Before we proceed, let us recall that the completion functor is neither left nor right exact, but it has left derived functors $L_{s}\colon \operatorname {\mathrm {Mod}}_{E_{*}} \rightarrow \operatorname {\mathrm {Mod}}_{E_{*}}$ . For more properties of these functors, see [Reference Hovey and StricklandHS99, Appendix B] or [Reference Barthel and FranklandBF15, Appendix A].

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

$$\begin{align*}L_{s} M_{t} \rightarrow \pi_{t+s} L_{K} M. \end{align*}$$

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

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

We have a base-change isomorphism

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

and, by Proposition 5.8, a further isomorphism

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

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

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

We are left with showing that the K-based Adams filtration on $E_{*}$ coincides with the $\mathfrak {m}$ -adic one. Observe that since $E \rightarrow K$ is a $K_{*}$ -monomorphism by Proposition 2.13, the elements of positive Adams filtration are precisely given by $\mathfrak {m} E_{*} = \mathrm {ker}(E_{*} \rightarrow K_{*})$ .

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

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

Since $E_{*}E$ is flat over $E_{*}$ , this, in particular, implies that $E_{*}^{\vee } E$ is profree, which can be also proven directly using evenness (see [Reference Hovey and StricklandHS99, Theorem 8.6]). It follows (as in [Reference Barthel and HeardBH16, Corollary 1.24], for example) that

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

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

satisfies the primarity condition.

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

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

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

Corollary 8.8. If $E_{*}^{\vee }X$ is profree, then Miller’s May spectral sequence associated to $(E, K)$ coincides with the spectral sequence of signature

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

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

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

(8.10)

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

Proposition 8.11. If $E_{*}^{\vee }X$ is profree, then Miller’s Mahowald spectral sequence based on $(E, K)$ can be identified with the Cartan–Eilenberg spectral sequence associated to the extension of Hopf algebroids

$$\begin{align*}K_{*}E \rightarrow K_{*}K \rightarrow K_{*}^{E}K. \end{align*}$$

Proof. Taking the standard K-local E-Adams resolution

$$\begin{align*}X \rightarrow L_{K}(E \otimes X) \rightarrow L_{K}(E \otimes \overline{E} \otimes X) \rightarrow \ldots \end{align*}$$

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

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

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

Note that from the description of the $E_{1}$ -term, we see that this is just the $K_{*}E$ -based Adams spectral sequence in the derived category of $K_{*}K$ -comodules, sometimes called the Margolis–Palmieri spectral sequence. This is known to be isomorphic to the Cartan–Eilenberg spectral sequence associated to the extension (see [Reference BelmontBel20, Theorem 4.2]).

Remark 8.13. Note that applying Proposition 8.11 to the starting terms of the respective spectral sequences implies, in particular, that

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

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

Remark 8.14. If we again pick $E = E_{n}$ , the Lubin–Tate spectrum associated to the Honda formal group law of height n over $\mathbb {F}_{p^{n}}$ , then the relevant $\operatorname {\mathrm {Ext}}$ -groups appearing in the Miller square can be described in terms of the cohomology of the Morava stabiliser group (see [Reference Barthel and HeardBH16, Theorem 4.3]). Keeping this choice of E in mind, we can rewrite the Miller square as

9 The K-based Adams spectral sequence at large primes

It is well-known that if p is large compared to the height, then the E-based Adams–Novikov spectral sequence for the K-local sphere collapses. As the same is true for the Mahowald spectral sequence, as we have shown in Remark 8.12, we deduce that under these assumptions, two of the sides of the Miller square of (8.10) represent collapsing spectral sequences.

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

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

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

To prove our comparison result, we use an algebraicity result for homotopy categories of E-local spectra at large primes due to the second author, strengthened in recent work with Patchkoria [Reference Patchkoria and PstragowskiPP21], which compares it to the following algebraic construction.

Notation 9.3. Let us write

for the derived $\infty $ -category of the abelian category of comodules. This is a symmetric monoidal stable $\infty $ -category, with monoidal unit given by $E_{*}$ itself, considered as a chain complex concentrated in degree zero. The unit defines for any

its homotopy groups

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

1. (1) the homological one given by the suspension and corresponding above to s and

2. (2) the internal one induced from the grading shift functor on $\operatorname {\mathrm {Comod}}_{E_*E}$ and corresponding above to t.

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

Definition 9.4. The periodicity algebra $P(E_{*})$ is the $\mathbf {E}_{\infty }$ -algebra in determined by the commutative algebra

$$\begin{align*}E_{*}[\tau^{\pm 1}], \text{where } | \tau | = (1, -1) \end{align*}$$

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

Notation 9.5. For brevity, let us write

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

Theorem 9.6. If $2p-2> n^{2}+n$ , there exists an equivalence

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

Remark 9.8. For any $M, N \in \operatorname {\mathrm {Mod}}_{P(E_{*})}$ , we have an Adams spectral sequence of signature

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

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

This observation, due to Franke [Reference FrankeFra], is the starting point for the comparison of $\operatorname {\mathrm {Mod}}_{P(E_{*})}$ to the E-local category. The general form of Franke’s conjecture implies that all stable $\infty $ -categories admitting a suitably convergent Adams-type spectral sequence of this signature must have equivalent homotopy categories [Reference Patchkoria and PstragowskiPP21].

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

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

Remark 9.9. Since the periodicity algebra is a sum of shifts of a unit as an object of the derived $\infty $ -category, the homotopy groups of the free module are simple to understand. More precisely, for any

, we have

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

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

Lemma 9.10. Let X be either the E-local sphere or a Morava K-theory; that is, $X = S^{0}_{E}$ or $X = K$ . Then, there exists a canonical isomorphism

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

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

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

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

Both this ring and $E(n)_{*}E(n)$ are concentrated in degrees divisible by $2p-2$ , which is a key ingredient in the construction of the equivalence $\phi $ [Reference PstragowskiPst18, Section 2].

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

$$\begin{align*}E_{*}X \simeq E_{*} \otimes_{E(n)_{*}} E(n)_{*}X. \end{align*}$$

The induced functor $\operatorname {\mathrm {Comod}}_{E(n)_{*}E(n)} \rightarrow \operatorname {\mathrm {Comod}}_{E_{*}E}$ is an equivalence of categories by a result of Hovey and Strickland [Reference Hovey and StricklandHS05a].

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

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

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

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

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

$$\begin{align*}\operatorname{\mathrm{Ext}}^{s, t}(E(n)_{*}X, E(n)_{*}X) \simeq \operatorname{\mathrm{Ext}}^{s, t}(E_{*}X, E_{*}X). \end{align*}$$

This vanishes above $n^{2}+n$ by [Reference PstragowskiPst18, Remark 2.5] and is concentrated in degrees t divisible by $2p-2$ . Since $2p-2> n^{2}+n$ by assumption, we deduce that the spectral sequence collapses on the second page, so that the identity of $E_{*}X$ descends to an equivalence $\phi (X) \simeq P(E_{*}) \otimes _{E_{*}} E_{*}X$ .

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

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

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

Let us write

for the algebraic model for the Morava K-theory spectrum. Note that its homotopy groups satisfy

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

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

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

This forms a class of monomorphisms and so determines an Adams spectral sequence [Reference Patchkoria and PstragowskiPP21, Section 3.1].

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

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

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

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

which is an image of the $E_{*}K$ -Adams resolution of $E_{*}$ in under the periodicisation functor. The statement follows from the isomorphism of homotopy groups of Remark 9.9 applied to the exact couple giving rise to the spectral sequence.

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

We will need to recall some results on décalage, following Deligne and Levine [Reference LevineLev15].

Construction 9.15. If $X^{\bullet }$ is a cosimplicial spectrum, we have an associated conditionally convergent spectral sequence

The décalage of this cosimplicial object is the tower

obtained by totalising the Postnikov truncations of X. Since limits commute with limits, we have

$$\begin{align*}\varprojlim \operatorname{\mathrm{Dec}}_{n} \simeq \varprojlim \operatorname{\mathrm{Tot}}(\tau_{\leq n} X^{\bullet}) \simeq \operatorname{\mathrm{Tot}}(\varprojlim \tau_{\leq n} X^{\bullet}) \simeq \operatorname{\mathrm{Tot}}(X^{\bullet}). \end{align*}$$

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

Levine proves that the evident isomorphisms between the $E^{2}$ -term of the $\operatorname {\mathrm {Tot}}$ -spectral sequence and the $E_{1}$ -term of the spectral sequence associated to the décalage extends to an isomorphism of spectral sequences [Reference LevineLev15].

In our case, we will be working with cosimplicial objects in , rather than in spectra. Thus, the role of the Postnikov towers will be played by the truncations in the standard t-structure on the derived category.

Definition 9.16. Let

be a cosimplicial object in the derived $\infty $ -category of comodules. The homological décalage of $X^{\bullet }$ is the tower

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

Warning 9.17. Note that if $X^{\bullet }$ is a cosimplicial object in , then the homological décalage tower induces a spectral sequence obtained by applying $\operatorname {\mathrm {Ext}}(E_{*}, -)$ . Since Postnikov towers converge in , this is conditionally convergent to $\operatorname {\mathrm {Ext}}(E_{*}, \operatorname {\mathrm {Tot}}(X^{\bullet }))$ .

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

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

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

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

Lemma 9.18. Let be a cosimplicial object, and assume that each $X^{m}$ is a direct sum of shifts of objects in the heart, each of which is a cofree $E_{*}E$ -comodule. Then

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

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

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

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

and

$$\begin{align*}K_{*}^{\otimes_{E_*} n} \simeq \operatorname{\mathrm{Tor}}^{E_{*}}(K_{*}, K_{*}) \otimes_{K_{*}} \ldots \otimes_{K_{*}} \operatorname{\mathrm{Tor}}^{E_{*}}(K_{*}, K_{*}), \end{align*}$$

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

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

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

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

The map is obtained by tensoring the map

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

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

Proposition 9.21. From the second page on, the Adams spectral sequence associated to $E_{*}K$ is isomorphic to the spectral sequence obtained from the tower

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

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

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

We’re interested in the cosimplicial object

$$\begin{align*}K_{*} \rightrightarrows K_{*} \otimes _{E_{*}} K_{*} \Rrightarrow \ldots, \end{align*}$$

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

$$\begin{align*}K_{0} \rightrightarrows K_{0} \otimes_{E_0} K_{0} \Rrightarrow \ldots, \end{align*}$$

working in the derived category of $E_{0}$ -modules. As an object of the latter, $K_{0} \otimes _{E_0} K_{0}$ is a direct sum of its homology groups, and we have

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

and more generally

(9.22) $$ \begin{align} H_{*}(K_{0}^{\otimes_{E_0} n}) \simeq \operatorname{\mathrm{Tor}}_{*}^{E_{0}}(K_{0}, K_{0})^{\otimes_{K_{0}} n-1}, \end{align} $$

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

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

with the tangent space extends to a grading-preserving isomorphism

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

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 (9.22) encodes the cobar complex computing

$$\begin{align*}\text{Cotor}_{\Lambda_{K_{0}}((\mathfrak{m} / \mathfrak{m}^{2})^{\vee})}(K_{0}, K_{0}) \simeq \operatorname{\mathrm{Sym}}_{K_{0}}(\mathfrak{m}/\mathfrak{m}^{2}) \simeq \bigoplus_{n \geq 0} \mathfrak{m}^{n}/\mathfrak{m}^{n+1}. \end{align*}$$

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

Using the isomorphism in Equation (9.22) and passing to cohomology, we deduce that

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

where $H^{s}$ is the cohomology of the given cosimplicial abelian group. Thus, the totalisation 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

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

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

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

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

We believe the following question related to the above result is interesting, but we do not pursue it in this paper.

Remark 9.25. 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

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

Thus, the spectral sequence of Theorem 9.23 is isomorphic to the conditionally convergent spectral sequence

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

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

10 Height 1 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

$$\begin{align*}2p-2 \geq 4> 3 = n^{2}+n+1. \end{align*}$$

Theorem 9.23 and Remark 9.25 imply that the relevant spectral sequence is isomorphic to the one computing the cohomology $\mathrm {H}_{\operatorname {\mathrm {cts}}}^{*}(\mathbb {G}_{n}, E_{*})$ 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 again follow the conventions of [Reference Goerss, Henn, Mahowald and RezkGHMR05, Section 2]. We have $E_{*} \simeq \mathbb {Z}_{p}[u^{\pm 1}]$ with $|u| = -2$ , $\mathfrak {m} = (p)$ and $\mathbb {G}_{1} \simeq \mathbb {Z}_{p}^{\times }$ with the action determined by $\lambda _{*}(u) = \lambda ^{-1} u$ for $\lambda \in \mathbb {Z}_{p}^{\times }$ . We denote the associated graded of $E_{*}$ by

$$\begin{align*}A_{*} \simeq \mathbb{F}_{p}[b][u^{\pm1}], \end{align*}$$

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

(10.2) $$ \begin{align} \partial\colon M \rightarrow M, \end{align} $$

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

The complex of (10.2) is a quasiisomorphic quotient of the standard group cohomology cochain complex

(10.3) $$ \begin{align} M \rightarrow \operatorname{\mathrm{Map}}_{\operatorname{\mathrm{cts}}}(\mathbb{Z}_{p}^{\times}, M) \rightarrow \operatorname{\mathrm{Map}}_{\operatorname{\mathrm{cts}}}(\mathbb{Z}^{\times}_{p} \times \mathbb{Z}_{p}^{\times}, M) \rightarrow \ldots, \end{align} $$

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 filtred complexes, where we consider both (10.2) and (10.3) with the filtration induced from that of M. In particular, both filtrations induced isomorphic spectral sequences.

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 $\psi _{*} u \equiv \sigma ^{-1} u \pmod p$ , we have $\psi _{*} u = [\sigma ]^{-1} u$ 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 10.5. The $E_{2}$ -term of the K-based Adams for $S^{0}$ at $n=1$ and $p> 2$ is given by

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

where $v_{1} = u^{1-p}$ is of degree $(2p-2, 0, 0)$ and $\zeta $ is the class of $1$ of degree $(0, 0, 1)$ , where the three degrees are, respectively, the internal degree, the degree of the associated graded and the cohomological degree.

By Remark 9.25, if the K-based Adams is isomorphic to the spectral sequence computing $\mathrm {H}_{\operatorname {\mathrm {cts}}}^{*}(\mathbb {Z}_{p}^{\times }, \mathbb {Z}_{p}[u^{\pm 1}])$ induced by the filtration of (10.2) 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 now analyse 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 $v_{1}^{k}$ . If we write $k = p^{n}m$ , where $p \nmid m$ , then one can calculate

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

from which we deduce that

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

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

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

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

with even longer differentials supported on $v_{1}^{p^{k}}$ for $k \geq 1$ . The above picture is made in the Adams grading $(t-s, s)$ , with t the internal degree and s the cohomological one, while the one inherited from passing to the associated graded not visible.

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

Example 10.7 (Noncompletely 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 construct a spectrum X for which $\operatorname {\mathrm {lim}}_r^1E_r^{0,0}(X)$ does not vanish at height $n =1$ and $p> 2$ . Set $\lambda _k = 2(p-1)p^k$ , and consider the spectrum

$$\begin{align*}X = L_{K}\bigoplus_{k\ge 0}S^{-\lambda_k}. \end{align*}$$

In bidegree $(s,t) = (0,0)$ , the $E_2$ -page is given by

$$\begin{align*}E_2^{0,0}(X) \simeq \bigoplus_{k\ge 0} \mathbb{F}_p\{v_1^{p^k}\}, \end{align*}$$

where the generators correspond to the maps $v_1^{p^k}\colon K_* \to K_*S^{\lambda _k}$ . The formulas for the differentials of Equation (10.6) show that (up to reindexing) the filtration on $E_2^{0,0}(X)$ is given by

$$\begin{align*}\ldots \subset \bigoplus_{k \ge 2}\mathbb{F}_p\{v_1^{p^k}\} \subset \bigoplus_{k \ge 1}\mathbb{F}_p\{v_1^{p^k}\} \subset \bigoplus_{k \ge 0}\mathbb{F}_p\{v_1^{p^k}\} = E_2^{0,0}(X). \end{align*}$$

Therefore, we obtain an exact sequence

$$\begin{align*}0 \to \operatorname{\mathrm{lim}}_rE_r^{s,t}(X) \to E_2^{0,0}(X) = \bigoplus_{k\ge 0} \mathbb{F}_p\{v_1^{p^k}\} \to \prod_{k \ge 0}\mathbb{F}_p\{v_1^{p^k}\} \to \operatorname{\mathrm{lim}}_r^1E_r^{s,t}(X) \to 0. \end{align*}$$

In particular, we see that $\operatorname {\mathrm {lim}}_rE_r^{s,t}(X) = 0$ , while $\operatorname {\mathrm {lim}}_r^1E_r^{s,t}(X) \neq 0$ .

Part 3. 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-localisation is a form of completion, and so we first focus on homology of inverse limits, constructing a spectral sequence computing these in the generality of an arbitrary adapted homology theory.

In the case of K-localisation, 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 stabiliser group, and use it to compute the zeroth one at all heights and primes. At height 1, 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 [Reference DevinatzDev97]. 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 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 is most close to [Reference Patchkoria and PstragowskiPP21, Section 2].

A locally graded $\infty $ -category is an $\infty $ -category equipped with a distinguished autoequivalence . A locally graded functor is functor equipped with a natural isomorphism .

Definition 11.1. Let be a stable $\infty $ -category, considered as a locally graded $\infty $ -category using the suspension functor, and an arbitrary locally graded abelian category. We say a locally graded functor is a homology theory if, for any cofibre sequence

$$\begin{align*}X \rightarrow Y \rightarrow Z \end{align*}$$

in , the induced diagram

$$\begin{align*}H(X) \rightarrow H(Y) \rightarrow H(Z) \end{align*}$$

in is exact in the middle.

Remark 11.2. The requirement that H is locally graded amounts to specifying an isomorphism

$$\begin{align*}H(\Sigma X) \simeq H(X)[1] \end{align*}$$

natural in .

In cases of interest to us, H is Grothendieck in the sense that

1. (1) is presentable,

2. (2) is Grothendieck abelian and

3. (3) H preserves arbitrary direct sums.

In particular, this means that

has enough injectives. Any injective object I of

gives rise to a cohomological functor

Since

is presentable and stable, Brown’s representability theorem holds [Reference Patchkoria and PstragowskiPP21, Proposition 2.15], so that there exists some

representing

. It follows that, for any object

, there is a natural equivalence

Note that the object $D(I)$ is H-local, in the sense that for any object

with $H(Y) = 0$ , we have $\operatorname {\mathrm {Hom}}(Y,D(I)) =0$ . 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 is adapted if the counit map $H(D(I)) \to I$ is an equivalence for any injective .

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 (generalised) 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 ${\mathcal {A}}_p$ which is larger than $\mathbb {F}_p$ .

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

$$\begin{align*}H_*(-;\mathbb{F}_p)\colon \operatorname{\mathrm{Sp}} \to \operatorname{\mathrm{Comod}}_{{\mathcal{A}}_p} \end{align*}$$

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 to encode all available homological information. This generalises considerably: one can show that any homology theory factors uniquely through an adapted one followed by an exact comonadic functor of abelian categories [Reference Patchkoria and PstragowskiPP21, Section 3.3]

Let R be a homotopy commutative ring spectrum which satisfies the Adams condition, that is, it can be written as a filtred 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 $R^{*}X_{\alpha } \rightarrow \operatorname {\mathrm {Hom}}_{R_{*}}(R_{*}X_{\alpha }, R_{*})$ 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 [Reference HoveyHov04a, Theorems 1.4.7–1.4.9], but $H\mathbb {Z}$ does not [Reference Goerss and HopkinsGH, Page 17]. This condition guarantees that the associated Hopf algebroid $(R_*,R_*R)$ is Adams and thus has a well-behaved category of comodules $\operatorname {\mathrm {Comod}}_{R_*R}$ (see [Reference HoveyHov04a, Section 1.4] for details).

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

Lemma 11.7 [Reference DevinatzDev97, 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 $D(I)$ corresponds (in the sense explained above) to an isomorphism $R_*D(I) \simeq I$ . In other words, the functor

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 . We collect its properties in the next result:

Lemma 11.8. Consider an adapted homology theory

. Let

, and suppose

is an injective resolution of $H(X)$ in

, where $C_i = \ker (\tau _i)$ for all i. There exists a tower of objects in

over X of the form

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

2. (2) $H(X_i) \simeq C_i[{-i}]$ ;

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 (noncanonical) map of towers $(X_i) \to (Y_i)$ .

Proof. This lemma is an axiomatisation of the results proven in [Reference DevinatzDev97, 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 .

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

Lemma 11.10. Let be a Grothendieck abelian category. An object 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 [Reference JannsenJan88, 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 $I=(I_{\alpha }) \in \operatorname {\mathrm {Inj}}({\mathcal {A}}^{\mathbb {N}})$ is an injective tower in

, then there is a preferred equivalence

of objects in

, well-defined up to homotopy.

Proof. For brevity, let us write

; note that this is an injective object of

. The canonical structure maps

give arrows

$$\begin{align*}D(I_{\infty}) \rightarrow D(I_{\alpha}), \end{align*}$$

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

in

. 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

, there are isomorphisms

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

(11.12)

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

are split epimorphisms, so the $\varprojlim ^1$ -term in (11.12) vanishes. Thus, the Yoneda lemma implies that $\phi $ is an equivalence, as claimed.

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

Notation 11.13. If

is a Grothendieck abelian category and

is a diagram, then we write

for the limit taken in

itself. Formation of limits defines a functor

and we denote its right derived functors by

Alternatively,

has an associated derived $\infty $ -category, and we write

for the limit of the composite of F with the inclusion with the heart. These two different notions of limits, one in

and the other in the derived $\infty $ -category, are related by the formula

where on the right-hand side, we have the homology of an object of the derived $\infty $ -category. Thus,

can be thought of as the total derived functor of the limit.

Theorem 11.14. Let be an adapted homology theory. If is a tower in , then there exists a natural spectral sequence in of the form

(11.15)

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

between the $\infty $ -categories of towers [Reference Patchkoria and PstragowskiPP21, 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

of the form

For brevity, let us write

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

of the form

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

and

. This gives the $E_2$ -term

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

.

Warning 11.16. Note that since the local grading

is an equivalence, it commutes with derived functors of the limit, and in the spectral sequence of Theorem 11.14, we have

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

consists of permanent cycles; that is, it is in the kernel of all the differentials in the spectral sequence of Theorem 11.14.

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

The $E_2$ -page of the top spectral sequence is concentrated in

so that the spectral sequence collapses. The comparison map is readily identified with the induced morphism

, so the claim follows.