2.1 Topological Hochschild homology

Suppose that S is an $\mathbb E_\infty$ -ring spectrum and that R is an $\mathbb E_1$ -S-algebra. Then, we define the relative topological Hochschild homology as

$$ \operatorname{THH}(R/S) := R\otimes_{R\otimes_S R^{op}}R. $$

More generally, if M is a R-bimodule, or equivalently a $(R\otimes_S R^{op})$ -module, then the relative THH with coefficients in M is defined as

(2.1) \begin{equation} \operatorname{THH}(R/S;M) := M\otimes_{R\otimes_SR^{op}}R. \end{equation}

In this paper, we shall only consider $\mathbb E_\infty$ -ring spectra. More precisely, we shall only consider $\operatorname{THH}(R/S;M)$ when R is an $\mathbb E_\infty$ -S-algebra and M is an $\mathbb E_\infty$ -R-algebra. In this case, $\operatorname{THH}(R/S;M)$ naturally has a structure of an $\mathbb E_\infty$ -R-algebra and we have an equivalence

$$ \operatorname{THH}(R/S;M) = \operatorname{THH}(R/S)\otimes_RM $$

of $\mathbb E_\infty$ -R-algebras.

Suppose that the following is a commutative diagram in the category of $\mathbb E_\infty$ -ring spectra.

Then, we have an equivalence

\begin{equation*} \operatorname{THH}(R_2\otimes_{R_1}R_3/S_2\otimes_{S_1}S_3) = \operatorname{THH}(R_2/S_2)\otimes_{\operatorname{THH}(R_1/S_1)}\operatorname{THH}(R_3/S_3). \end{equation*}

This equivalence will be used several times without reference.

2.2 Suspension elements

Suppose that S is a connective $\mathbb E_\infty$ -ring spectrum and that R is a connective $\mathbb E_\infty$ -S-algebra. Let $F=\operatorname{fib}(S\to R)$ . We shall also assume that $S_\ast\to R_\ast$ is surjective, so that $F_\ast$ is an ideal of $S_\ast$ .

Then, we define the suspension map $\sigma$ to be the composite

(2.2) \begin{equation} \Sigma F = 0\coprod_F 0 \to R \coprod_S R \to R\otimes_S R. \end{equation}

We shall also use $\sigma$ to denote the induced map on the homotopy groups $\pi_\ast(F)\to\pi_{\ast+1}(R\otimes_SR)$ .

The suspension elements will play an important role as primitive elements in Hopf algebras.

Proposition 2.2. Let R and S be as above. In addition, assume that the odd homotopy groups of S are zero and that the unit map $S_\ast\to R_\ast$ is a quotient by a regular sequence

$$ R_\ast = S_\ast / (x_1,x_2,\dots). $$

Then, there is a $R_\ast$ -Hopf algebra structure on $\operatorname{THH}_\ast(R/S)$ such that:

1. (a) it is free as a $R_\ast$ -module;

2. (b) the submodule of primitive elements is the free module generated by the elements $\sigma^2 x_i$ ; and

3. (c) for each k and a prime p, $(\sigma^2 x_k)^p$ is a $R_\ast$ -linear combination of the elements $\sigma^2 x_i$ modulo p.

Proof. Let $T=R\otimes_SR$ . The Hopf algebra structure on $\pi_\ast\operatorname{THH}(R/S)=\pi_\ast(R\otimes_TR)$ is given, as usual, by the map

$$ R\otimes_TR\xrightarrow{\operatorname{id}\otimes1\otimes\operatorname{id}} R\otimes_TR\otimes_TR = (R\otimes_TR)\otimes_R(R\otimes_TR),$$

assuming that $\pi_\ast\operatorname{THH}(R/S)$ is free over $R_\ast$ . It is a Hopf algebra rather than just a Hopf algebroid because the left and the right units $R\rightrightarrows R\otimes_TR$ are homotopic. This can be seen, for example, by observing that $\operatorname{THH}(R/S)$ is the colimit of R over the circle $S^1$ in the category of $\mathbb E_\infty$ -S-algebras and that $S^1$ is a connected space.

By [Reference AngeltveitAng08, Proposition 3.6], the homotopy groups of T form a graded exterior algebra

$$ T_\ast = \Lambda_{R_\ast}(\sigma x_1,\sigma x_2,\dots). $$

Then, the $E_2$ -page of the Künneth spectral sequence for $\operatorname{THH}_\ast(R/S)$ ,

$$ E_2 = \operatorname{Tor}^{T_\ast}(R_\ast,R_\ast)\Rightarrow \operatorname{THH}_\ast(R/S), $$

can be computed as a divided power algebra

$$ E_2=\Gamma_{R_\ast}[\sigma^2 x_1,\sigma^2 x_2,\dots]. $$

The spectral sequence degenerates at the $E_2$ -page since everything is concentrated in even degrees. From the $E_2$ -page, we can see that $\operatorname{THH}_\ast(R/S)$ is free as a $R_\ast$ -module.

Taking the Whitehead tower of a spectrum defines a lax symmetric monoidal functor

$$ \tau_{\geq\ast}:\operatorname{Sp}\to \operatorname{Fun}(\mathbb Z^{op},\operatorname{Sp}), $$

so that

$$ \tau_{\geq\ast}R \otimes_{\tau_{\geq\ast}T}\tau_{\geq\ast}R$$

is a filtered $\mathbb E_\infty$ -ring spectrum whose underlying object is $\operatorname{THH}(R/S)$ . Taking the associated graded group, which is a symmetric monoidal process, we obtain

$$ \pi_\ast R\otimes_{\pi_\ast T}\pi_\ast R, $$

whose homotopy groups are

$$ \operatorname{Tor}^{T_\ast} (R_\ast,R_\ast).$$

This is the filtration that constructs the Künneth spectral sequence. We then see that the Künneth filtration is compatible with the coalgebra structure since the comultiplication is induced by the map

$$ \tau_{\geq\ast}R\otimes_{\tau_{\geq\ast}T}\tau_{\geq\ast}R\xrightarrow{\operatorname{id}\otimes1\otimes\operatorname{id}}\tau_{\geq\ast}R\otimes_{\tau_{\geq\ast}T}\tau_{\geq\ast}R\otimes_{\tau_{\geq\ast}T}\tau_{\geq\ast}R$$

of filtered spectra.

It follows that the element $\sigma^2 x_k$ is primitive, since $\Delta (\sigma^2 x_k)$ must have Künneth filtration 1, and the only possibility is $\sigma^2 x_k\otimes1 + 1\otimes\sigma^2 x_k$ . Conversely, any primitive element in $\operatorname{THH}_\ast(R/S)$ must be detected by a primitive element in the associated graded group. But in the associated graded group, the primitives are linear combinations of the $\sigma^2 x_i$ , since it is a divided power coalgebra. This proves part (b).

Lastly, the (underived) quotient $(\pi_\ast(R\otimes_TR))/p$ is a $(R_\ast/p)$ -Hopf algebra and has a filtration such that the associated graded group is a divided power Hopf algebra on the $\sigma^2 x_i$ . Therefore, as in the previous paragraph, the primitive elements form a free $(R_\ast/p)$ -module generated by the $\sigma^2 x_i$ . Then, part (c) follows since $(\sigma^2 x_k)^p$ is primitive.

Example 2.3. In this paper, we consider $\operatorname{THH}(\mathrm{ku}/\mathrm{MU})$ and

$$ \mathrm{MU}\otimes_{\operatorname{THH}(\mathrm{MU})}\mathrm{MU}=\operatorname{THH}(\mathrm{MU}/\mathrm{MU}\otimes\mathrm{MU}). $$

In the notation of Definition 3.11, $\operatorname{THH}_\ast(\mathrm{MU}/\mathrm{MU}\otimes\mathrm{MU})$ is a $\mathrm{MU}_\ast$ -Hopf algebra with the primitive elements $\sigma^2 b_i$ for $i\geq 1$ , and these elements will play an important role. For $\operatorname{THH}_\ast(\mathrm{ku}/\mathrm{MU})$ , we will need only part (c) of the previous proposition and not its Hopf algebra structure.

3.1 Construction of the descent spectral sequence

Definition 3.1. Let $B\to C$ be a map of $\mathbb E_\infty$ -ring spectra. Then, we can form the augmented cosimplicial diagram

$$B \to C \rightrightarrows C \otimes_B C \mathrel{\raise6pt\hbox{$\rightarrow$}\mkern-18mu\raise-0pt\hbox{$\rightarrow$}\mkern-18mu\raise-6pt\hbox{$\rightarrow$}} \cdots$$

of $\mathbb E_\infty$ -ring spectra, called the descent diagram. If the above is a limit diagram, then we obtain the Bousfield–Kan spectral sequence

$$ E_1^{s,t} = \pi_t(C^{\otimes_B(s+1)})\Rightarrow \pi_{t-s}(B). $$

More generally, if M is a B-module, we have a descent diagram

$$ M\to M\otimes_BC\rightrightarrows M\otimes_BC\otimes_BC\mathrel{\raise6pt\hbox{$\rightarrow$}\mkern-18mu\raise-0pt\hbox{$\rightarrow$}\mkern-18mu\raise-6pt\hbox{$\rightarrow$}}\cdots $$

and, if it is a limit diagram, a spectral sequence

$$ E_1^{s,t} = \pi_t(M\otimes_B C^{\otimes_B(s+1)})\Rightarrow \pi_{t-s}(M). $$

This will be called the descent spectral sequence for $M\to M\otimes_BC$ (along $B\to C$ ). We will discuss a sufficient condition for the convergence in Lemma 3.5.

Definition 3.2. Suppose that $(k,\Gamma)$ is a graded Hopf algebroid such that $\Gamma$ is flat as a k-module and A is a right $\Gamma$ -comodule. By the cobar complex $\operatorname{CB}_{\Gamma}(A)$ , we mean the chain complex

$$ A\to A\otimes_k\Gamma\to A\otimes_k\Gamma\otimes_k\Gamma\to\cdots , $$

where the differentials are the alternating sum of comultiplication maps and the coaction maps. This is bigraded so that $\operatorname{CB}^{s,t}_{\Gamma}(A)$ is the degree-t part of $A\otimes_k \Gamma^{\otimes_ks}$ . The cohomology of $\operatorname{CB}_{\Gamma}(A)$ is denoted by $\operatorname{Ext}_{\Gamma}(A)$ , which is again bigraded. This agrees with the usual definition of $\operatorname{Ext}$ (see, for example, [Reference RavenelRav86, A.1.2.12]).

Proposition 3.3. In Definition 3.1, if $\pi_\ast(C\otimes_BC)$ is flat over $\pi_\ast C$ , then $\pi_\ast (C\otimes_BC)$ is a Hopf algebroid over $\pi_\ast C$ , $\pi_\ast(M\otimes_BC)$ is a comodule, and we have

\begin{align*} E_1 &= \operatorname{CB}_{\pi_\ast(C\otimes_BC)}(\pi_\ast(M\otimes_BC)),\\ E_2 &= \operatorname{Ext}_{\pi_\ast(C\otimes_BC)} (\pi_\ast(M\otimes_BC)). \end{align*}

Proof. Since the descent spectral sequence is constructed from the coskeletal filtration, the lemma follows from the fact that the fiber of the map

$$ M\to \operatorname{cosk}^s (M\otimes_B C^{\otimes_B(\bullet+1)}) $$

is $M\otimes_B I^{\otimes_B(s+1)}$ , where I is the fiber of $B\to C$ (see, for example, [Reference Mathew, Naumann and NoelMNN17, Proposition 2.14]).

Proof. It is enough to show that x is detected by $\sigma x$ in the equalizer

$$ \operatorname{Eq}(C\rightrightarrows C\otimes_BC). $$

This follows from chasing the diagram

from $x\in F_\ast$ .

3.2 Complex K-theory spectrum

Until the end of §4, we shall assume that p is a fixed odd prime number and that every spectrum is p-complete. For example, we shall write $\operatorname{THH}(\mathrm{ku})$ instead of $\operatorname{THH}(\mathrm{ku}_p^\wedge)_p^\wedge$ .

We write ku for the connective cover of the complex K-theory spectrum KU equipped with the standard complex orientation $\mathrm{MU}\to\mathrm{ku}$ , which can be lifted to be an $\mathbb E_\infty$ orientation according to [Reference JoachimJoa04].

We shall compute the descent spectral sequence for

$$ \operatorname{THH}(\mathrm{ku})\to\operatorname{THH}(\mathrm{ku}/\mathrm{MU})=\operatorname{THH}(\mathrm{ku})\otimes_{\operatorname{THH}(\mathrm{MU})}\mathrm{MU} $$

along $\operatorname{THH}(\mathrm{MU})\to \mathrm{MU}$ . More generally, for $M=\mathbb F_p$ or $\mathbb Z_p$ with a canonical ku-algebra structure, we shall compute the descent spectral sequence for

$$ \operatorname{THH}(\mathrm{ku};M)\to\operatorname{THH}(\mathrm{ku}/\mathrm{MU};M). $$

Let $E_r^{\ast,\ast}(\operatorname{THH}(\mathrm{ku};M))$ denote the $E_r$ -page of this descent spectral sequence. Until the end of §4, let us write

\begin{align*} \Gamma &:= \mathrm{MU}\otimes_{\operatorname{THH}(\mathrm{MU})}\mathrm{MU}=\operatorname{THH}(\mathrm{MU}/\mathrm{MU}\otimes\mathrm{MU}),\\ A&:= \operatorname{THH}(\mathrm{ku}/\mathrm{MU}). \end{align*}

Using Propositions 2.2 and 3.3, we note a few things about the homotopy groups of these ring spectra.

• – Both $A_\ast$ and $\Gamma_\ast$ are commutative even-graded rings.

• – We find that $\Gamma_\ast$ is a $\mathrm{MU}_\ast$ -Hopf algebra and $A_\ast$ is a right $\Gamma_\ast$ -comodule algebra. We shall write $\eta_R:A_\ast\to A_\ast\otimes\Gamma_\ast$ for the coaction map, for reasons to be explained in Proposition 3.10.

• – We can identify $E_1^{\ast,\ast}(\operatorname{THH}(\mathrm{ku}))$ with the cobar complex

  $$ \operatorname{CB}_{\Gamma_\ast}(A_\ast)= (A_\ast\xrightarrow{D^0} A_\ast\otimes_{\mathrm{MU}_\ast}\Gamma_\ast\xrightarrow{D^1} A_\ast\otimes_{\mathrm{MU}_\ast}\Gamma_\ast\otimes_{\mathrm{MU}_\ast}\Gamma_\ast\xrightarrow{D^2}\cdots),$$
  where we write $D^0, D^1, \dots$ for the differentials in $\operatorname{CB}_{\Gamma_\ast}(A_\ast)$ . These are the $d_1$ differentials in the descent spectral sequence.

Remark 3.7. Suppose that $B\to C$ is a map of $\mathbb E_\infty$ -ring spectra. Instead of the coskeletal filtration of the descent diagram for $B\to C$ , there is an alternative filtration using Whitehead covers. The kth filtration of B is given by

$$ \operatorname{Tot}(\tau_{\geq 2k} C^{\otimes_B (\bullet+1)}).$$

If we further assume that $C^{\otimes_B(s+1)}$ has no odd homotopy groups for all $s\geq0$ , then this filtration gives us a shearing of the descent spectral sequence in the sense that we have

$$ ^\backprime E_r^{k,3k-s} = E_{2r+1}^{s,2k}, $$

where the left-hand side is the spectral sequence associated with the new filtration and the right-hand side is the descent spectral sequence. More precisely, they are related by the décalage [Reference AntieauAnt24] and by doubling the speed of the filtration.

In our case of $B=\operatorname{THH}(\mathrm{ku})$ and $C=\operatorname{THH}(\mathrm{ku}/\mathrm{MU})$ , the new filtration using Whitehead covers is an example of the even/motivic filtration of Hahn, Raksit, and Wilson [Reference Hahn, Raksit and WilsonHRW25, Definition 4.1.2], since C is evenly free over B in their sense.

Lemma 3.8. $\operatorname{CB}_{\Gamma_\ast}(A_\ast)$ is a chain complex of free $\mathrm{ku}_\ast$ -modules. Therefore, we have identifications

\begin{align*} E_1(\operatorname{THH}(\mathrm{ku};\mathbb Z_p)) &= \operatorname{CB}_{\Gamma_\ast}(A_\ast/\beta),\\ E_2(\operatorname{THH}(\mathrm{ku};\mathbb Z_p)) &= \operatorname{Ext}_{\Gamma_\ast}(A_\ast/\beta),\\ E_1(\operatorname{THH}(\mathrm{ku};\mathbb F_p)) &= \operatorname{CB}_{\Gamma_\ast}(A_\ast/(p,\beta)),\\ E_2(\operatorname{THH}(\mathrm{ku};\mathbb F_p)) &= \operatorname{Ext}_{\Gamma_\ast}(A_\ast/(p,\beta)). \end{align*}

There is an alternative description of the descent spectral sequence. The augmented cosimplicial diagram

$$ S^0 \to \mathrm{MU}\rightrightarrows \mathrm{MU}^{\otimes2} \mathrel{\raise6pt\hbox{$\rightarrow$}\mkern-18mu\raise-0pt\hbox{$\rightarrow$}\mkern-18mu\raise-6pt\hbox{$\rightarrow$}} \cdots$$

induces an augmented cosimplicial diagram

(3.1) \begin{equation} \operatorname{THH}(\mathrm{ku})\to\operatorname{THH}(\mathrm{ku}/\mathrm{MU})\rightrightarrows \operatorname{THH}(\mathrm{ku}/\mathrm{MU}^{\otimes2}) \mathrel{\raise6pt\hbox{$\rightarrow$}\mkern-18mu\raise-0pt\hbox{$\rightarrow$}\mkern-18mu\raise-6pt\hbox{$\rightarrow$}} \cdots \end{equation}

of $\mathbb E_\infty$ -ring spectra. The following proposition is immediate from the definitions.

Proposition 3.10. The augmented cosimplicial diagram (3.1) is equivalent to the descent diagram for $\operatorname{THH}(\mathrm{ku})\to\operatorname{THH}(\mathrm{ku}/\mathrm{MU})$ . Furthermore, under the identification $\operatorname{THH}_\ast(\mathrm{ku}/\mathrm{MU}^{\otimes2})=A_\ast\otimes\Gamma_\ast$ , the two maps

$$ \eta_L,\eta_R:\operatorname{THH}_\ast(\mathrm{ku}/\mathrm{MU})\to\operatorname{THH}_\ast(\mathrm{ku}/\mathrm{MU}^{\otimes2})$$

induced by the left and right units $\eta_L,\eta_R:\mathrm{MU}\to \mathrm{MU}^{\otimes2}$ can be identified with $\operatorname{id}\otimes1$ and the coaction map $A_\ast\to A_\ast\otimes\Gamma_\ast$ , respectively.

Definition 3.11. Following the classical notation, we write $x_1,x_2,\dots$ with $|x_i|=2i$ for the polynomial generators of the Lazard ring $\mathrm{MU}_\ast$ , and we write $b_1,b_2,\dots$ for the generators of $\mathrm{MU}_\ast\mathrm{MU} = \mathrm{MU}_\ast[b_1,b_2,\dots]$ as a $\mathrm{MU}_\ast$ -algebra, where $\mathrm{MU}_\ast\mathrm{MU}$ is given the algebra structure by the left unit $\mathrm{MU}_\ast\to\mathrm{MU}_\ast\mathrm{MU}$ and the $b_i$ vanish under the multiplication map $\mathrm{MU}_\ast\mathrm{MU}\to\mathrm{MU}_\ast$ . There are many choices for the generators, and for now, we only require that $x_1$ maps to $\beta$ under

$$ \mathrm{MU}_\ast\to \mathrm{ku}_\ast=\mathbb Z_p[\beta] $$

and that $x_i$ maps to zero for $i\geq 2$ . We shall give more specific choices of generators in Lemma 3.16.

For the lightness of notation, we shall often write $v_k$ instead of $x_{p^k-1}$ when $k\geq 2$ and $t_k$ instead of $b_{p^k-1}$ when $k\geq 1$ . Note that we do not write $v_1$ for $x_{p-1}$ , because $x_{p-1}$ maps to 0 in $\mathrm{ku}_\ast$ , while it is more natural that a class named $v_1$ maps to a nonzero class in $\mathrm{ku}_\ast$ .

The following two remarks hold for any choice of generators.

Remark 3.13. Consider the descent spectral sequence for $\operatorname{THH}(\mathrm{MU})\to\operatorname{THH}(\mathrm{MU}/\mathrm{MU})$ , whose $E_1$ -page is $\operatorname{CB}_{\Gamma_\ast}(\mathrm{MU}_\ast)$ . Since

$$ \operatorname{THH}_\ast(\mathrm{MU}) = \Lambda_{\mathrm{MU}_\ast}(\sigma b_1,\sigma b_2,\dots),$$

we can see, by Lemma 3.6, that the element $\sigma^2 b_i\in \operatorname{CB}_{\Gamma_\ast}^1(\mathrm{MU}_\ast)$ is a permanent cycle in the descent spectral sequence for any i and that it detects $\sigma b_i\in\operatorname{THH}_\ast(\mathrm{MU})$ .

Mapping to the descent spectral sequence for $\operatorname{THH}(\mathrm{ku})\to\operatorname{THH}(\mathrm{ku}/\mathrm{MU})$ , we can see that $\sigma^2 b_i\in \operatorname{CB}^1_{\Gamma_\ast}(A_\ast)$ is a permanent cycle in this descent spectral sequence and detects $\sigma b_i\in\operatorname{THH}_\ast(\mathrm{ku})$ . Here, $\sigma b_i$ is the suspension of the class $b_i\in\mathrm{ku}_\ast\mathrm{ku}$ , which is defined to be the image of $b_i\in\mathrm{MU}_\ast\mathrm{MU}$ .

Lemma 3.15. For any choice of generators in Definition 3.11, we have

$$ (\sigma^2 v_k)^p \equiv \sigma^2 v_{k+1} \pmod{p,\beta} $$

for $k\geq 2$ and

$$ (\sigma^2 x_{p-1})^p \equiv\sigma^2 v_2 \pmod{p,\beta}$$

in $A_\ast$ up to a p-adic unit. Similarly, we have

$$ (\sigma^2 b_i)^p \equiv \sigma^2 b_{pi+p-1} \pmod{p,x_1,x_2,\ldots} $$

for $i\geq 1$ in $\Gamma_\ast$ up to a p-adic unit.

Proof. For $A_\ast$ , the proof is the same as the proof of [Reference Hahn and WilsonHW22, Proposition 2.5.3]. For $\Gamma_\ast$ , it is similar and we shall sketch the proof. We wish to show that $(\sigma^2 v_k)^p=\sigma^2 v_{k+1}$ in

$$ \Gamma_\ast \otimes_{\mathrm{MU}_\ast} \mathbb F_p = \pi_\ast (\Gamma\otimes_\mathrm{MU}\mathbb F_p). $$

Since base changing along $\mathrm{MU}\to\mathbb F_p$ is a symmetric monoidal functor, we have

$$ \Gamma\otimes_{\mathrm{MU}}\mathbb F_p = \mathbb F_p\otimes_{\operatorname{THH}(\mathrm{MU};\mathbb F_p)}\mathbb F_p , $$

where $\operatorname{THH}(\mathrm{MU};\mathbb F_p) = \mathbb F_p\otimes_{\mathbb F_p\otimes\mathrm{MU}}\mathbb F_p$ , again by base change. By the stability of Dyer–Lashof operations, we have

$$ (\sigma^2 b_i)^p = Q_0(\sigma^2 b_i) = \sigma^2 (Q_2 b_i).$$

Then, the statement follows from the computation of the operation $Q_2$ in $(\mathbb F_p)_{\ast}\mathrm{MU}=H_\ast(BU;\mathbb F_p)$ , done in [Reference KochmanKoc71, Theorem 6].

By the naturality of $\sigma$ , we have $\eta_R(\sigma^2 \alpha) = \sigma^2 \eta_R(\alpha)$ for any $\alpha\in \pi_\ast\operatorname{fib}(\mathrm{MU}\to\mathrm{ku})$ . Therefore, the lemma will be proved using properties of the right unit $\mathrm{MU}_\ast\to\mathrm{MU}_\ast\mathrm{MU}$ .

Setup. Let us outline our strategy. Starting from some set of polynomial generators for $\mathrm{MU}_\ast$ and $\mathrm{MU}_\ast\mathrm{MU}$ as in Definition 3.11, we shall try to modify the generators so that they satisfy the lemma.

Let us write $I_0=(x_2,x_3,\dots)\subseteq \mathrm{MU}_\ast$ and $I_1=(x_2,x_3,\dots,b_1,b_2,\dots)\subseteq \mathrm{MU}_\ast\mathrm{MU}$ for the kernel ideals of maps to $\mathrm{ku}_\ast$ . Being kernels of maps to $\mathrm{ku}_\ast$ implies that $\eta_L$ and $\eta_R$ map $I_0$ to $I_1$ . In addition, by Lemma 2.1, $\sigma^2$ annihilates $I_0^2$ and $I_1^2$ . In other words, there is the following commutative diagram for $?=L,R$ , where $\overline{\eta_?}$ denotes the top arrow.

The quotients $I_0/I_0^2$ and $I_1/I_1^2$ are naturally modules over $\mathrm{MU}_\ast/I_0=\mathrm{MU}_\ast\mathrm{MU}/I_1=\mathrm{ku}_\ast$ . As a $\mathrm{ku}_\ast$ -module, $I_0/I_0^2$ is a free graded module generated by $\{\overline{x_2},\overline{x_3},\dots\}$ , where $\overline{x_i}$ is the image of $x_i\in I_0$ . Similarly, $I_1/I_1^2$ is free and there is a natural splitting

$$ I_1/I_1^2=\overline{\eta_L}(I_0/I_0^2) \oplus \mathrm{ku}_\ast\{\overline{b_1},\overline{b_2},\dots\} $$

induced by the multiplication map $\mathrm{MU}_\ast\mathrm{MU}\to\mathrm{MU}$ .

We also observe that $\overline{\eta_L}$ and $\overline{\eta_R}$ are $\mathrm{ku}_\ast$ -linear. For $\overline{\eta_L}$ , this is clear since $\eta_L(x_1)=x_1$ . For $\overline{\eta_R}$ , this can be seen by either observing that the vertical arrows in the above square are injective or by a direct computation:

$$ \eta_R(x_1\alpha)-x_1\eta_R(\alpha)=b_1\eta_R(\alpha)\in I_1^2\qquad\forall\alpha\in I_0.$$

For the purpose of proving this lemma, we only need to choose a set of generators of $I_0/I_0^2$ and $I_1/I_1^2$ as free $\mathrm{ku}_\ast$ -modules. Any set of generators lift to a set of polynomial generators for $\mathrm{MU}_\ast$ and $\mathrm{MU}_\ast\mathrm{MU}$ , and the choice of a lift is irrelevant, since $\sigma^2$ annihilates any difference.

The reduction argument. For any $k\geq 2$ , we have

(3.2) \begin{equation} \overline{\eta_R}(\overline{x_k}) = \overline{x_k} +\sum_{i=1}^k c_{k,i} \beta^{k-i}\overline{b_i},\end{equation}

for some $c_{k,0},\dots,c_{k,k}\in\mathbb Z_p$ . It is a standard fact [Reference RavenelRav86, Theorem 3.1.5] that the top coefficient $c_{k,k}$ is a p-adic unit unless $k+1$ is a power of p and that we have $c_{k,k}\in p\mathbb Z_p^\times$ if $k+1$ is a power of p. Note that by counitality of the Hopf algebroid $\mathrm{MU}_\ast\mathrm{MU}$ , there are no terms of the form $\beta^{k-i}\overline{x_i}$ on the right-hand side.

Let us consider $(c_{k,i})$ as a matrix, indexed by $\{2,3,\dots\}\times\{1,2,\dots\}$ , i.e. it is an infinite square matrix with the first row missing. Then, it is ‘lower-triangular’ in the sense that $c_{k,i}=0$ for $i \gt k$ , and the ‘diagonal’ entries $c_{k,k}$ are units except for those such that $k+1$ is a power of p.

Since $\overline{\eta_R}$ is $\mathrm{ku}_\ast$ -linear, modifying a generator of $I_0/I_0^2$ , by adding multiples of $\beta^{k-j}x_j$ to $x_k$ for example, has an effect of adding the same multiple of the jth row to the kth row. As the diagonal entries are mostly units, we can perform the row operations to make the matrix diagonal except on the columns with a non-unit diagonal entry and the first column. More precisely, suppose that we have

$$ \overline{\eta_R}(\overline{x_k})=\overline{x_k}+\sum_{i=1}^k c_{k,i}\beta^{k-i}\overline{b_i}.$$

Then, we take the greatest index $i_0\in \{2,\dots,k-1\}\backslash\{p^j-1:j\geq1\}$ such that $c_{k,i_0}\neq 0$ , if one exists. Then, if we replace $\overline{x_k}$ by $\overline{x_k}-c_{i_0,i_0}^{-1}c_{k,i_0}\beta^{k-i_0}\overline{x_{i_0}}$ , we effectively make $c_{k,i_0}=0$ . Repeating this, we arrive at a generator $\overline{x_k}$ such that

$$ \overline{\eta_R}(\overline{x_k}) = \overline{x_k} +c_{k,k}\overline{b_k} +\bigg(\sum_{p^j-1<k}c_{k,p^j-1}\beta^{k-p^j+1}\overline{t_j}\bigg) + c_{k,1}\beta^{k-1}\overline{b_1}. $$

This reduction argument will be used multiple times in the following proof.

Step 1: choosing $x_{p-1},t_1,$ and $b_1$ to satisfy property (d). Let $i_Q:(\mathrm{BP}_\ast,\mathrm{BP}_\ast\mathrm{BP})\to(\mathrm{MU}_\ast,\mathrm{MU}_\ast\mathrm{MU})$ be the map of Hopf algebroids induced by the Quillen idempotent and let $v_1',v_2',\dots,t_1',t_2',\ldots$ be the Hazewinkel generators [Reference RavenelRav86, A2.2] for $\mathrm{BP}_\ast=\mathbb Z_p[v_1',\dots]$ and $\mathrm{BP}_\ast\mathrm{BP}=\mathrm{BP}_\ast[t_1',\dots]$ . Note that $i_Q(v_1')\in\mathrm{MU}_\ast$ is the coefficient of $X^p$ in the p-series of the universal formal group law mod p. This means that the image of $i_Q(v_1')$ in $\mathrm{ku}_\ast$ is $\varepsilon\beta^{p-1}$ for some p-adic unit $\varepsilon$ . We define $x_{p-1}\in\mathrm{MU}_\ast$ to be

$$ x_{p-1}:=i_Q(v_1') - \varepsilon x_1^{p-1}\in I_0 $$

and define $t_1\in\mathrm{MU}_\ast\mathrm{MU}$ to be $i_Q(t_1')$ . If we choose $b_1$ so that $\eta_R(x_1)=x_1+b_1$ at least for now, then we have

\begin{align*} \eta_R(x_{p-1}) &= i_Q(\eta_R v_1') - \varepsilon \eta_R(x_1^{p-1}) \\ &= i_Q(v_1'+pt_1') - \varepsilon(x_1+b_1)^{p-1} \\ &= x_{p-1}+pt_1 - \varepsilon x_1^{p-2}b_1 \pmod{I_1^2}.\end{align*}

Therefore, after scaling $b_1$ by a p-adic unit, we have

$$ \overline{\eta_R}(\overline{x_{p-1}})=\overline{x_{p-1}}+p\overline{t_1}+\beta^{p-2}\overline{b_1} $$

inducing the second equation of property (d) after taking $\sigma^2$ . After this scaling, we also have $\eta_R(x_1)=x_1-\varepsilon^{-1}b_1$ .

Step 2: choosing the $b_i$ for $i\neq1,p-1$ to satisfy properties (b) and (c). Make arbitrary choices for all of the $\overline{b_i}$ that have not already been defined. We shall redefine the $\overline{t_k}$ inductively in $k(\geq 2)$ so that property (b) holds. Suppose that we have chosen $\overline{t_{k-1}}$ . By Proposition 2.2, we have

$$ (\sigma^2 t_{k-1})^p \equiv \alpha_0 \sigma^2 t_k + \sum_{i=1}^{p^k-2} \alpha_i\beta^i \sigma^2 b_{p^k-1-i}\pmod p, $$

for some constants $\alpha_i$ . By Lemma 3.15, $\alpha_0$ is a p-adic unit. Therefore, we may redefine $\overline{t_k}$ to be

$$ \alpha_0\overline{t_k}+\sum_{i=1}^{p^k-2}\alpha_i\beta^i \overline{b_{p^k-1-i}}, $$

and then $(\sigma^2 t_{k-1})^p$ would be a unit multiple of $\sigma^2 t_k$ modulo p.

By the same argument, we can redefine $\overline{b_{2p-1}}$ to satisfy property (c).

Step 3: choosing $v_2$ to satisfy property (d). We shall now choose $\overline{v_2}$ . Recall that $v_2'\in\mathrm{BP}_\ast\subseteq\mathrm{MU}_\ast$ is the Hazewinkel generator. Suppose that the image of $v_2'$ in $\mathrm{ku}_\ast$ is $c\beta^{p^2-1}$ for some $c\in\mathbb Z_p$ . For now, let us define

$$ v_2= i_Q(v_2')-cx_1^{p^2-1}.$$

Using the definition of Hazewinkel generators and formulas in [Reference RavenelRav86, A.2.1.27], it can be computed that

$$ \eta_R(v_2')=v_2'+pt_2'-(p+1)(v_1')^{p+1}t_1'\pmod{(t_1')^2}$$

in $\mathrm{BP}_\ast\mathrm{BP}$ . Therefore,

\begin{align*} &\eta_R(v_2) \\ &\quad = i_Q(v_2')+pi_Q(t_2')-(p+1)i_Q(v_1')^{p+1}i_Q(t_1') - c(x_1-\varepsilon^{-1} b_1)^{p^2-1} \pmod{I_1^2}\\ &\quad = v_2 + pi_Q(t_2')-(p+1)\varepsilon^{p+1}x_1^{p^2-p}i_Q(t_1') + (p^2-1)c\varepsilon^{-1} x_1^{p^2-2}b_1 \pmod{I_1^2} , \end{align*}

where we have used that $i_Q(v_1')=\varepsilon x_1^{p-1}\pmod{I_1}$ from step 1. After writing $i_Q(t_1')$ and $i_Q(t_2')$ in terms of our generators $b_i$ from steps 1 and 2, we conclude that in Equation (3.2),

$$ \overline{\eta_R}(\overline{v_2}) = \overline{v_2} + \sum_{i=1}^{p^2-1} c_{p^2-1,i}\beta^{p^2-1-i}\overline{b_i}, $$

we have that the coefficients $c_{p^2-1,i}$ are multiples of p for $p-1<i\leq p^2-1$ and is a p-adic unit for $i=p-1$ .

Then, using the reduction argument, we can modify $\overline{v_2}$ so that its $\overline{\eta_R}$ is of the form

$$ \overline{\eta_R}(\overline{v_2})=\overline{v_2}+c_{p^2-1,p^2-1}\overline{t_2}+c_{p^2-1,p-1}\beta^{p^2-p}\overline{t_1}+\beta^{p^2-2}c_{p^2-1,1}\overline{b_1} $$

for some $c_{p^2-1,p^2-1}\in p\mathbb Z_p^\times$ . Since the previous coefficients $c_{p^2-1,i}$ were multiples of p for $p-1<i<p^2-1$ , this implies that the coefficient $c_{p^2-1,p-1}$ is still a p-adic unit after the reduction argument. Then, by redefining $\overline{v_2}$ to be $\overline{v_2}-c_{p^2-1,1}\beta^{p^2-p}\overline{x_{p-1}}$ , we can also make $c_{p^2-1,1}=0$ while retaining that $c_{p^2-1,p-1}\in\mathbb Z_p^\times$ . Finally, by scaling $\overline{v_2}$ and $\overline{t_2}$ if necessary, we obtain the desired formula

$$ \overline{\eta_R}(\overline{v_2})=\overline{v_2}+p\overline{t_2} + \beta^{p^2-p}\overline{t_1}. $$

Step 4: choosing $x_{2p-1}$ to satisfy property (e). The argument is similar to the one in step 3. Starting from an arbitrary choice of $\overline{x_{2p-1}}$ , we can modify it with the reduction argument to obtain

$$ \eta_R(\overline{x_{2p-1}}) = \overline{x_{2p-1}}+c_{2p-1,2p-1}\overline{b_{2p-1}} + c_{2p-1,p-1}\beta^p\overline{t_1}+c_{2p-1,1}\beta^{2p-2}\overline{b_1}.$$

By adding a multiple of $\beta^p\overline{x_{p-1}}$ to $\overline{x_{2p-1}}$ , we may also assume that $c_{2p-1,1}=0$ . Finally, by scaling $x_{2p-1}$ , we may assume that $c_{2p-1,2p-1}=1$ and set $\delta':=c_{2p-1,p-1}$ .

The difference from the previous step is that we cannot scale $\delta'$ to a fixed number because $b_{2p-1}$ cannot be scaled considering the consistency with property (c).

Step 5: choosing the $v_k$ for $k\geq3$ . By the same argument as in step 2, we can choose $v_3,v_4,\dots$ satisfying property (a).

Then, we shall modify the $v_k$ inductively for $k\geq 3$ so that property (d) is true. Suppose that we have chosen $v_{k-1}$ so that property (d) is true. Consider the following two equations:

\begin{align*} \eta_R \sigma^2 v_{k-1} &= \sigma^2 v_{k-1} + p\sigma^2 t_{k-1} + \beta^{p^{k-1}-p^{k-2}}\sigma^2 t_{k-2} , \\ \eta_R\sigma^2 v_k &=\sigma^2v_k + \sum_{i=1}^{p^k-1}c_{p^k-1,i}\beta^{p^k-1-i}\sigma^2 b_i. \end{align*}

Taking the pth power mod p of the first and comparing with the second using properties (a) and (b), we see that the $c_{p^k-1,i}$ are multiples of p for $i\neq p^{k-1}-1$ and that $c_{p^k-1,p^{k-1}-1}$ is a unit.

Then, as in step 3, we can perform the reduction argument to obtain

$$ \overline{\eta_R}(\overline{v_k})=\overline{v_k} + \sum_{j=1}^k c_{p^k-1,p^j-1} \beta^{p^k-p^j} \overline{t_j} + c_{p^k-1,1}\beta^{p^k-2}\overline{b_1} $$

for some $c_{p^k-1,p^k-1}\in p\mathbb Z_p^\times$ . Since the $c_{p^k-1,i}$ were previously multiples of p for $i\neq p^{k-1}-1$ , the reduction argument does not change $\overline{v_k}$ mod p, so that property (a) still holds. In addition, it implies that the reduction argument does not change the coefficients $c_{p^k-1,i}$ mod p, so that $c_{p^k-1,p^{j}-1}$ is a unit for $j=k-1$ and is a multiple of p if $j<k-1$ , and also that $c_{p^k-1,1}$ is a multiple of p.

By adding a multiple of $\beta^{p^k-p}\overline{x_{p-1}}$ to $\overline{v_k}$ , we can make $c_{p^k-1,1}=0$ . This does not change anything mod p, since $c_{p^k-1,1}$ was previously a multiple of p. Then, inductively in $j\in\{2,\dots, k-1\}$ , we can add a multiple of $\beta^{p^k-p^j}\overline{v_j}$ to make $c_{p^k-1,p^{j-1}-1}=0$ . In the end, we are left with

$$ \overline{\eta_R}(\overline{v_k}) = \overline{v_k} + c_{p^k-1,p^k-1}\overline{t_k}+c_{p^k-1,p^{k-1}-1}\beta^{p^k-p^{k-1}}\overline{t_{k-1}}, $$

where $c_{p^k-1,p^k-1}\in p\mathbb Z_p^\times$ and $c_{p^k-1,p^{k-1}-1}\in\mathbb Z_p^\times$ . After scaling $\overline{v_k}$ and $\overline{t_k}$ , we obtain

$$ \overline{\eta_R}(\overline{v_k})=\overline{v_k}+p\overline{t_k}+\beta^{p^k-p^{k-1}}\overline{t_{k-1}}. $$

Step 6: proof of property (f). By Proposition 2.2 and Lemma 3.15, we have

$$ (\sigma^2 x_{p-1})^p \equiv \alpha_0 \sigma^2 v_2 + \sum_{i=1}^{p^2-2} \alpha_i \beta^i \sigma^2 x_{p^2-1-i} \pmod p $$

for some constants $\alpha_0,\dots,\alpha_{p^2-2}$ with $\alpha_0$ a unit. We shall compare both sides after applying $\eta_R$ . Since

$$ \eta_R((\sigma^2 x_{p-1})^p) \equiv (\sigma^2 x_{p-1})^p + \beta^{p(p-2)}\sigma^2 b_{2p-1}\pmod p ,$$

we have $\alpha_i\equiv0\pmod p$ for $i\neq 0,p^2-2p,p^2-p$ , since it would otherwise introduce a nonzero term $\beta^i \sigma^2 b_{p^2-1-i}$ when $\eta_R$ is applied. If $\delta'\equiv0\pmod p$ , then on the right-hand side, we would not be able to cancel out $\beta^{p^2-p}\sigma^2 t_1$ appearing in $\eta_R(\alpha_0 \sigma^2 v_2)$ .

4.1 $v_0$ -Bockstein

Proposition 4.1. We have an isomorphisms of $\mathbb F_p$ -algebras

$$ \operatorname{Ext}_{\Gamma_\ast}(A_\ast/(p,\beta)) = \mathbb F_p[\mu]\otimes_{\mathbb F_p} \Lambda_{\mathbb F_p}(\lambda_1,\lambda_2), $$

where $\mu,\lambda_1,\lambda_2$ are represented by $\sigma^2 x_{p-1}, \sigma^2 t_1,\sigma^2 b_1$ , respectively, in $\operatorname{CB}_{\Gamma_\ast}(A_\ast/(p,\beta))$ .

Proof. We can mimic the proof of [Reference Hahn and WilsonHW22, Proposition 6.1.6]. The necessary ingredient is [Reference AusoniAus05, Theorem 6.8], which states that

$$ \operatorname{THH}_\ast(\mathrm{ku};\mathbb F_p) \simeq \mathbb F_p[\sigma^2 x_{p-1}] \otimes \Lambda_{\mathbb F_p}(\sigma t_1,\sigma b_1).$$

Corollary 4.2. The descent spectral sequence for

$$ \operatorname{THH}(\mathrm{ku})\to\operatorname{THH}(\mathrm{ku}/\mathrm{MU}) $$

degenerates at the $E_2$ -page.

Proposition 4.5. We have

$$ \operatorname{Ext}_{\Gamma_\ast}(A_\ast/\beta) \simeq \bigg[\mathbb Z_p\{1\}\oplus\bigoplus_{k=1}^\infty \mathbb Z/p^{\operatorname{val}_p(k)+1}\{a_k\}\bigg]\otimes_{\mathbb Z_p}\Lambda_{\mathbb Z_p}(\lambda_2) $$

as $\mathbb Z_p$ -modules. The generators are represented in the $\operatorname{CB}_{\Gamma_\ast}(A_\ast/\beta)$ as the following:

• – $\mathbb Z_p$ is generated by 1;

• – the generator $a_k$ of $\mathbb Z/p^{\operatorname{val}_p(k)+1}$ is represented by the bidegree-(1,2pk) cycle (also denoted by $a_k$ by abuse of notation)

  $$ a_k:=\frac{D^0((\sigma^2 x_{p-1})^k)}{p^{\operatorname{val}_p(k)+1}} = \frac{(\sigma^2 x_{p-1}+p\sigma^2 b_{p-1})^k-(\sigma^2 x_{p-1})^k}{p^{\operatorname{val}_p(k)+1}}; $$

• – $\lambda_2$ is represented by $\sigma^2 b_1$ .

Multiplication by $\lambda_2$ should be interpreted as in Remark 3.14.

Proof. We work in $\operatorname{CB}_{\Gamma_\ast}(A_\ast/\beta)$ . Then, by Lemma 4.3(b), we have

\begin{align*} D^0((\sigma^2 x_{p-1})^k) &= (\sigma^2 x_{p-1}+p\sigma^2 t_1)^k - (\sigma^2 x_{p-1})^k \\ &\equiv kp(\sigma^2 x_{p-1})^{k-1}\sigma^2 t_1 \pmod{p^{\operatorname{val}_p(k)+2}}, \end{align*}

which gives us the differentials

$$ d_{\operatorname{val}_p(k)+1}(\mu^k) = v_0^{\operatorname{val}_p(k)+1}\mu^{k-1}\lambda_1$$

up to a p-adic unit in the $v_0$ -Bockstein spectral sequence (4.2). Since $\lambda_2$ is a permanent cycle represented by $\sigma^2 b_1$ , we have the differentials

$$ d_{\operatorname{val}_p(k)+1}(\mu^k \lambda_2) = v_0^{\operatorname{val}_p(k)+1}\mu^{k-1}\lambda_1\lambda_2. $$

The generators in the statement can be derived from this computation.

Corollary 4.6. We have

$$ \operatorname{THH}_\ast(\mathrm{ku};\mathbb Z_p) = \bigg[\mathbb Z_p\oplus\bigoplus_{k=1}^\infty \Sigma^{2pk-1}\mathbb Z/p^{\operatorname{val}_p(k)+1}\bigg]\otimes_{\mathbb Z_p}\Lambda_{\mathbb Z_p}(\lambda_2) $$

as graded $\mathbb Z_p$ -modules with $|\lambda_2|=3$ .

Proof. Since

$$ E_2^{s,t}(\operatorname{THH}(\mathrm{ku};\mathbb Z_p)) = \operatorname{Ext}_{\Gamma_\ast}^{s,t}(A_\ast/\beta)$$

is concentrated in $0\leq s\leq 2$ and $t\in 2\mathbb Z$ , there is no room for any further differential in the descent spectral sequence computing $\operatorname{THH}(\mathrm{ku};\mathbb Z_p)$ . There is no extension problem since $E_2^{0,\ast}$ is free as a $\mathbb Z_p$ -module.

4.2 $\beta$ -Bockstein

The differentials of the $\beta$ -Bockstein spectral sequence are given as the following.

To prove the theorem, let us first construct some elements in the cobar complex. Define

$$ y := (\delta')^{-1}(-(\sigma^2 x_{p-1})^p+\beta^{p(p-2)}\sigma^2 x_{2p-1})\in A_\ast. $$

Next, fix a pair of positive integers (m,e) such that $1\leq e\leq \operatorname{val}_p(m)$ . Then, we define constants $\varepsilon_1,\dots,\varepsilon_e$ inductively as

(4.3) \begin{align} \varepsilon_1 = -\frac{\delta_{e-1}}{p^{\operatorname{val}_p(m)-e}}\binom{m/p}{p^{e-1}},\quad \varepsilon_{k+1} = -\varepsilon_k \bigg(1+\binom{(m-p^e)/p}{p^{e-k-1}}\bigg). \end{align}

Then, define

\begin{align*} f_{m,e} := &\frac1{p^{\operatorname{val}_p(m)-e+1}}D^0(y^{m/p}) \\[5pt] &+ \sum_{k=1}^{e} \frac{\varepsilon_k}{p^{k+1}}\beta^{p^{e+1}-p^{e-k+1}} D^0(y^{(m-p^e)/p}\sigma^2 x_{p^{e-k+1}-1}) , \end{align*}

which is an element in $(A_\ast\otimes\Gamma_\ast)[p^{-1}]$ . We also define

$$ f_{m,0} := \frac1{p^{\operatorname{val}_p(m)+1}} D^0((\sigma^2 x_{p-1})^m). $$

Lemma 4.11. The image of $f_{m,e}$ in $(A_\ast\otimes\Gamma_\ast)/(p^\infty, \beta^{p^{e+1}-1})$ is

$$ \frac1{p^{e+1}}\beta^{p^{e+1}-2}(\sigma^2 x_{p-1})^{m-p^e}\sigma^2 b_1 $$

up to a p-adic unit. Here, we write $M/p^\infty:=M[p^{-1}]/M$ for an abelian group M.

Proof. For $e=0$ , we have

\begin{align*} &D^0((\sigma^2 x_{p-1})^m) \\ &\quad \equiv (\sigma^2 x_{p-1} +\beta^{p-2}\sigma^2b_1)^m - (\sigma^2 x_{p-1})^m \pmod{p^{\operatorname{val}_p(m)+1}}\\[5pt] &\quad \equiv m\beta^{p-2}(\sigma^2x_{p-1})^{m-1}\sigma^2 b_1\quad\quad\quad\quad\quad \pmod{\beta^{p-1},p^{\operatorname{val}_p(m)+1}}, \end{align*}

by Lemmas 4.3 and 3.16, and the statement follows.

For $e \gt 0$ , we similarly have

\begin{align*} D^0(y^{m/p}) &\equiv (y + \beta^{p(p-1)}\sigma^2 t_1)^{m/p} - y^{m/p} \quad \,\,\,\,\pmod{p^{\operatorname{val}_p(m)+1}}\\[5pt] &\equiv \binom{m/p}{p^{e-1}} \beta^{p^e(p-1)}y^{(m-p^e)/p}\sigma^2 t_1^{p^{e-1}} \quad \,\,\,(\!\bmod{\beta^{p^{e+1}-1},p^{\operatorname{val}_p(m)-e+1}})\\[5pt] &\equiv \delta_{e-1}\binom{m/p}{p^{e-1}} \beta^{p^e(p-1)}y^{(m-p^e)/p}\sigma^2 t_e \,\,\,\, (\!\bmod{\beta^{p^{e+1}-1},p^{\operatorname{val}_p(m)-e+1}}). \end{align*}

In addition, using Lemma 4.3 we can check that

\begin{align*} &D^0(y^{(m-p^e)/p}\sigma^2 v_{e-k+1}) \\ &\quad\equiv py^{(m-p^e)/p}\sigma^2 t_{e-k+1} \\ &\qquad+ \bigg(1+\binom{(m-p^e)/p}{p^{e-k-1}}\bigg)\beta^{p^{e-k}(p-1)} y^{(m-p^e)/p}\sigma^2 t_{e-k}(\!\bmod{p^{k+1},\beta^{p^{e-k+1}-1}}) \end{align*}

and similarly,

\begin{align*} &D^0(y^{(m-p^e)/p}\sigma^2 x_{p-1})\\ &\quad =py^{(m-p^e)/p}\sigma^2 t_1+\beta^{p-2}y^{(m-p^e)/p}\sigma^2 b_1 \pmod{p^{e+1},\beta^{p-1}}.\end{align*}

Combining these calculations, we obtain the result.

In addition, we have

$$ f^+_{m,e}\equiv f_{m,e}\equiv a_m \pmod\beta,$$

and since $D^1(f_{m,e})=0$ by construction, we have

$$ D^1(f^+_{m,e}) = \beta^{p^{e+1}-2}D^1(g_{m,e}), $$

which implies the Bockstein differentials.

Using Theorem 4.8, we can compute $\operatorname{gr}_\beta\operatorname{Ext}_{\Gamma_\ast}(A_\ast)$ , the associated graded group with respect to the $\beta$ -adic filtration. We shall discuss these groups and extension problems in §4.3, but we have a rough description as follows:

• – $\operatorname{gr}_\beta\operatorname{Ext}^0$ is $\mathrm{ku}_\ast$ and is generated by an element represented by the cycle 1;

• – $\operatorname{gr}_\beta\operatorname{Ext}^1$ is generated by the elements detected by $p^ea_{p^e}$ for $e\geq 0$ and $\lambda_2$ , where these generators are p-torsions and support infinite $\beta$ -towers;

• – $\operatorname{gr}_\beta\operatorname{Ext}^2$ is generated by elements detected by $a_m\lambda_2$ in $\operatorname{Ext}^1_{\Gamma_\ast}(A_\ast)$ , which is $p^j\beta^{p^{\operatorname{val}_p(m)-j+1}-2}$ -torsion for each $0\leq j\leq \operatorname{val}_p(m)$ and $p^{\operatorname{val}_p(m)+1}$ -torsion.

4.3 Extension problems

We shall resolve the extension problems with respect to the $\beta$ -adic filtration, thereby proving Theorem 1.4.

Proof of Theorem 1.5. By Theorem 4.8, $\operatorname{gr}_\beta\operatorname{Ext}^1_{\Gamma_\ast}(A_\ast)$ is a free $\mathbb Z/p[\beta]$ -module generated by classes represented by the cycles $\sigma^2b_1,\sigma^2t_1,\sigma^2t_2,\dots$ From the following differentials in $\operatorname{CB}_{\Gamma_\ast}(A_\ast)$ ,

\begin{align*} D^0(\sigma^2 v_{e+1}) &= p\sigma^2 t_{e+1}+\beta^{p^{e+1}-p^e}\sigma^2 t_e,\\ D^0(\sigma^2 x_{p-1}) &= p\sigma^2 t_1 + \delta'\beta^{p-2}\sigma^2 b_1,\end{align*}

we can see that $\operatorname{Ext}^1_{\Gamma_\ast}(A_\ast)$ is isomorphic to $F_p^\wedge$ , defined in §1.2, by identifying the class in $\operatorname{Ext}^1$ represented by $\sigma^2 t_e$ with $(\beta^{p^e-2}/f(p^e-2))\in F_p^\wedge$ up to a p-adic unit.

Next, let us describe the extension problems in $\operatorname{Ext}^2_{\Gamma_\ast}(A_\ast)$ . From Theorem 4.8, we see that the class $p^ja_m\lambda_2\in\operatorname{Ext}^1_{\Gamma_\ast}(A_\ast/\beta)$ is hit by the Bockstein differential

$$ d_{p^{\operatorname{val}_p(m)-j+1}-2}(p^{\operatorname{val}_p(m)-j}a_{m+p^{\operatorname{val}_p(m)-j}}). $$

From the proof of that theorem, we see that the class $p^ja_m\lambda_2$ can be lifted to a $\beta^{p^{\operatorname{val}_p(m)-j+1}-2}$ -torsion class $h_{m,j}\in\operatorname{Ext}^1_{\Gamma_\ast}(A_\ast)$ represented by the cycle

$$ D^1(g_{m+p^{\operatorname{val}_p(m)-j},\operatorname{val}_p(m)-j}).$$

Let $d=\operatorname{val}_p(m)$ and let j be an integer such that $0\leq j<d$ . Then, let $\varepsilon_1,\dots,\varepsilon_{d-j}$ be the sequence defined in Equation (4.3) for the pair $(m+p^{d-j},d-j)$ . If, in addition $j<d-1$ , then let $\varepsilon_1',\dots,\varepsilon_{d-j-1}'$ be the sequence for the pair $(m+p^{d-j-1},d-j-1)$ .

Case 1. If $j<d-1$ , we have

\begin{align*} &pf_{m+p^{d-j},d-j} - \frac{\varepsilon_{d-j}}{\varepsilon_{d-j-1}'} \beta^{p^{d-j}(p-1)}f_{m+p^{d-j-1},d-j-1}\\ &\quad=\frac{1}{p^{\operatorname{val}_p(m+p^{d-j})-d+j}}D^0(y^{(m+p^{d-j})/p}) \\ &\qquad+ \frac1p\beta^{p^{d-j}(p-1)}\bigg(\varepsilon_1D^0(y^{m/p}\sigma^2v_{d-j})-\frac{\varepsilon_{d-j}}{\varepsilon_{d-j-1}'}D^0(y^{(m+p^{d-j-1})/p})\bigg)\\ &\qquad+\sum_{k=1}^{d-j-1}\frac{1}{p^{k+1}}\beta^{p^{d-j+1}-p^{d-j-k}}\bigg(\varepsilon_{k+1} - \frac{\varepsilon_{d-j}}{\varepsilon_{d-j-1}'}\varepsilon_k'\bigg)D^0(y^{m/p}\sigma^2 x_{p^{d-j-k}-1}). \end{align*}

We shall show that this is integral, i.e. an element of $A_\ast\otimes\Gamma_\ast$ , possibly except for the first term. For the second term, we have

$$ D(y^{(m+p^{d-j-1})/p}) \equiv \delta_{d-j-2} D(y^{m/p}\sigma^2 v_{d-j})\pmod p, $$

so that we need to show

$$ \varepsilon_1 - \frac{\varepsilon_{d-j}}{\varepsilon_{d-j-1}'}\delta_{d-j-2} \equiv 0\pmod p. $$

This is true since we have

$$ \varepsilon_1 - \frac{\varepsilon_{d-j}}{\varepsilon_{d-j-1}'}\delta_{d-j-2} \equiv\varepsilon_1 + \frac{\varepsilon_1}{\varepsilon_{1}'}\delta_{d-j-2}\pmod p $$

by Lemma 4.10 and

$$ \varepsilon_1' + \delta_{d-j-2} = \delta_{d-j-2}\bigg(-\frac{m+p^{d-j-1}}{p^{d-j-1}}+1\bigg) \equiv 0\pmod p $$

by Lemma 4.3. Next, for the summation part, we wish to show that

$$ \varepsilon_{k+1} \equiv \frac{\varepsilon_{d-j}}{\varepsilon_{d-j-1}'}\varepsilon_k'\pmod{p^{k+1}}, $$

and this follows from Lemma 4.10. Therefore, we have proved that

(4.4) \begin{align} &pf_{m+p^{d-j},d-j} - \frac{\varepsilon_{d-j}}{\varepsilon_{d-j-1}'} \beta^{p^{d-j}(p-1)}f_{m+p^{d-j-1},d-j-1} \end{align}

(4.5) \begin{align} &\quad = \frac{1}{p^{\operatorname{val}_p(m+p^{d-j})-d+j}}D^0(y^{(m+p^{d-j})/p}) + (\text{integral element}). \end{align}

We divide into three cases.

Case 1-1. If $\operatorname{val}_p(m+p^{d-j})=d-j$ , then the whole right-hand side of Equation (4.4) is integral. In this case, we see that

\begin{align*} pg_{m+p^{d-j},d-j} - \frac{\varepsilon_{d-j}}{\varepsilon_{d-j-1}'} g_{m+p^{d-j-1},d-j-1} \end{align*}

is integral, so that by taking $D^1$ , we have

$$ ph_{m,j} = h_{m,j+1} $$

in $\operatorname{Ext}^2_{\Gamma_\ast}(A_\ast)$ up to a p-adic unit.

Case 1-2. If $m=p^d(p-1)$ and $j=0$ , then from the proof of Theorem 4.8(a)

$$ \frac{D^0(y^{p^d})}{p} = \delta_d\beta^{p^{d+2}-p^{d+1}}\frac{\sigma^2 t_{e+1}}p + \text{(integral element)}. $$

Therefore,

$$ pg_{p^{d+1},d} - \frac{\varepsilon_{d}}{\varepsilon_{d-1}'} g_{m+p^{d-1},d-1} - \delta_d\beta^{p^{d+2}-2p^{d+1}+2}\frac{\sigma^2 t_{e+1}}{p}$$

is integral, and by taking $D^1$ , we obtain

$$ ph_{p^d(p-1),0} = h_{p^d(p-1),1}. $$

Case 1-3. The remaining case is when $\operatorname{val}_p(m+p^{d-j})\gt d-j$ but not $(m,j)\neq(p^d(p-1),0)$ . This can happen only if $j=0$ . In this case, we have

$$ f_{m+p^d,d+1} = \frac{1}{p^{\operatorname{val}_p(m+p^d)-d}}D^0(y^{(m+p^d)/p}) + \beta^{p^{d+2}-p^{d+1}}\times\text{(cycle)}, $$

where (cycle) means some cycle in $A_\ast\otimes\Gamma_\ast[p^{-1}]$ . Comparing with Equation (4.4), we see that

$$ pg_{m+p^d,d} - \frac{\varepsilon_{d}}{\varepsilon_{d-1}'}g_{m+p^{d-1},d-1} - \beta^{p^{d+2}-p^{d+1}} g_{m+p^d,d+1} + \beta^{p^{d+2}-2p^{d+1}+2} \times\text{(cycle)} $$

is integral. Taking $D^1$ , we have

$$ ph_{m,0} = h_{m,1} + \beta^{p^{d+2}-p^{d+1}}h_{m-p^d(p-1),d'-d-1}, $$

where $d'=\operatorname{val}_p(m-p^d(p-1))$ . This completes the proof when $j<d-1$ .

Case 2 If $j=d-1$ , then we have

\begin{align*} &pf_{m+p,1} - c\beta^{p(p-1)}f_{m+1,0} \\ &\quad = \frac{1}{p^{\operatorname{val}_p(m+p)-1}} D^0(y^{(m+p)/p}) \\ &\qquad + \frac1p\beta^{p(p-1)}(\varepsilon_1D^0(y^{m/p}\sigma^2 x_{p-1})-cD^0((\sigma^2 x_{p-1})^{m+1})) \end{align*}

and we can choose a p-adic unit c so that

$$ \frac1p\beta^{p(p-1)}(\varepsilon_1D^0(y^{m/p}\sigma^2 x_{p-1})-cD^0((\sigma^2 x_{p-1})^{m+1})) $$

is $\beta^{2p^2-3p}$ times a cycle by the definition of y. The rest of the argument is similar to the previous cases.

Case 3. For $j=d$ , we have

$$ pf_{m+1,0} = \frac{1}{p^{\operatorname{val}_p(m+1)}}D^0((\sigma^2 x_{p-1})^{m+1})$$

and the rest of the argument is similar to the previous cases.

This completes the proof of the description of $\operatorname{Ext}_{\Gamma_\ast}(A_\ast)$ . The rest of the statement immediately follows.