2. Hochschild homology and motivic filtrations

We first introduce filtrations on $\operatorname{THH}$ , $\operatorname{TC}^{-}$ , $\operatorname{TP}$ and $\operatorname{TC}$ . The reader is encouraged to read [Reference Bökstedt, Hsiang and MadsenBHM93, Reference Bökstedt and MadsenBM94, Reference Bökstedt and MadsenBM95, Reference Hesselholt and NikolausHN20] for background on these invariants and [Reference Hahn, Raksit and WilsonHRW22] for a thorough account of the filtrations that we use in this paper.

Recall that there is a map of $\mathbb{E}_\infty$ -rings $c \colon \mathrm{ko} \to \mathrm{ku}$ called the complexification map, where $\mathrm{ko}_* = \mathbb{Z}[\eta, A, B]/(2\eta,\eta^3, \eta A, A^2-4B)$ , $\mathrm{ku}_* = \mathbb{Z}[u]$ , $A \mapsto 2 u^2$ and $B\mapsto u^4$ , with $|\eta| = 1$ , $|A| = 4$ , $|B| = 8$ and $|u| =2$ .

We hereafter fix $\mathrm{MU} \to \mathrm{ku}$ and the $x_i$ as in the above lemma. This provides $\mathbb{E}_\infty$ -ring maps $C^q(\mathrm{MU}/\mathbb{S}) \to C^q(\mathrm{ku}/\mathrm{ko})$ , compatibly for all $q \geqslant 0$ . We write

\begin{align*}\tau_{\geqslant \star} \colon \mathrm{Sp}_2\longrightarrow \mathrm{Sp}_2^{\operatorname{fil}} \end{align*}

for the monoidal Whitehead filtration [Reference LurieLur17, Proposition 1.4.3.6, Example 2.2.1.10]. Note that the circle group $\mathbb{T}$ acts freely on $S^1 \subset S^{3} \subset \cdots \subset S^{\infty} = E\mathbb{T}$ . We write $F(S^{2\ell+1}_+, \operatorname{THH})^{\mathbb{T}}$ for the $\ell$ th-order approximate fixed points functor, taking $A/R$ to $F(S^{2\ell+1}_+,\operatorname{THH}(A/R))^{\mathbb{T}} \simeq \lim_{\mathbb{C}\mathrm{P}^\ell} \operatorname{THH}(A/R)$ , as in [Reference Bruner and RognesBR05, § 2] and [Reference Angelini-Knoll, Ausoni, Culver, Höning and RognesAAC⁺25, § 7]. Here, $\mathbb{C}\mathrm{P}^\ell \subset \mathbb{C}\mathrm{P}^\infty = B\mathbb{T}$ . We also write $\operatorname{THH}^{hC_2}$ and $\operatorname{THH}^{tC_2}$ for the functors obtained by applying $C_2$ -homotopy fixed points and the $C_2$ -Tate construction to $\operatorname{THH}$ , respectively.

Definition 2.3. For $F \in \{\operatorname{THH}, \operatorname{TC}^{-}, \operatorname{TP}, F(S^{2\ell+1}_+,\operatorname{THH})^{\mathbb{T}},\operatorname{THH}^{hC_2},\operatorname{THH}^{tC_2}\}$ we define $\mathbb{E}_\infty$ algebras

\begin{align*}\operatorname{fil}_{\mathrm{mot}}^{\star} F(\mathrm{ko}) & :=\operatorname{Tot} (\tau_{\geqslant 2\star}F(C^{\bullet}(\mathrm{ku}/\mathrm{ko})/C^{\bullet}(\mathrm{MU}/\mathbb{S}))), \\\operatorname{gr}_{\mathrm{mot}}^* F(\mathrm{ko}) & :=\operatorname{gr}^* (\operatorname{fil}_{\mathrm{mot}}^{\star}F(\mathrm{ko}))\end{align*}

in $\mathrm{Sp}_2^{\operatorname{fil}}$ and $\mathrm{Sp}_2^{\operatorname{gr}}$ , respectively. Here, $\operatorname{Tot}$ denotes the totalization of a cosimplicial object. We refer to $\operatorname{gr}_{\mathrm{mot}}^w\operatorname{TP}(\mathrm{ko})$ as the weight w prismatic cohomology spectrum of ko and to $\pi_* \operatorname{gr}_{\mathrm{mot}}^w \operatorname{TP}(\mathrm{ko})$ as the weight w prismatic cohomology groups.

Remark 2.4. We will show in Proposition 2.15 that

\begin{align*} \operatorname{THH}(C^q(\mathrm{ku}/\mathrm{ko}) / C^q(\mathrm{MU}/\mathbb{S}))=C^{q}(\operatorname{THH}(\mathrm{ku}/\mathrm{MU})/\operatorname{THH}(\mathrm{ko})) \end{align*}

is even and 2-torsion free. It follows that $F(C^q(\mathrm{ku}/\mathrm{ko}) / C^q(\mathrm{MU}/\mathbb{S}))$ is even for each functor $F \in \{\operatorname{THH}, \operatorname{TC}^{-},\operatorname{TP}, F(S^{2\ell+1}_+,\operatorname{THH})^{\mathbb{T}},\operatorname{THH}^{hC_2},\operatorname{THH}^{tC_2}\}$ and $q\geqslant 0$ . Hence the double-speed Whitehead filtration in the definition of $\operatorname{fil}_{\mathrm{mot}}^{\star} F(\mathrm{ko})$ will be well behaved for each F in the above list.

We also fix terminology for gradings.

Definition 2.7. Given $M^* \in \mathrm{Sp}_2^{\operatorname{gr}}$ and $x \in \pi_n M^w$ , we say that x has stem n, weight w and motivic filtration $2w-n$ , and write $\|x\| = (n,2w-n)$ . In [Reference Hahn, Raksit and WilsonHRW22, Definition 1.4.2], stem and motivic filtration are called degree and Adams weight, respectively. We refer to the (Adams indexed, Bousfield–Kan) spectral sequence

\begin{align*}E_2^{n,2w-n} = \pi_n \operatorname{gr}_{\mathrm{mot}}^w F(\mathrm{ko}) \Longrightarrow \pi_n F(\mathrm{ko}) \end{align*}

for each $F \in \{\operatorname{THH}, \operatorname{TC}^{-}, \operatorname{TP},F(S^{2\ell+1}_+,\operatorname{THH})^{\mathbb{T}},\operatorname{THH}^{hC_2}, \operatorname{THH}^{tC_2}\}$ as the motivic spectral sequence. We will show in Lemma 2.19 that the motivic spectral sequence converges in each case.

We plot each motivic spectral sequence with stem on the horizontal axis and motivic filtration on the vertical axis. Given this convention, if $\|x\| = (n,2w-n)$ then $\|d_r(x)\| = (n-1, 2w-n+r)$ . Note that we write $E_r^{n,2w-n}$ where it is also standard to write $E_r^{2w-n,2w}$ in the literature. With these conventions, the motivic spectral sequence $E_r^{n,s}$ -term is concentrated in internal degrees $n+s = 2w$ and, consequently, $d_r = 0$ for all even integers $r\geqslant 2$ .

Our first aim is to show that the filtrations from Definition 2.3 agree with the motivic filtrations considered in [Reference Hahn, Raksit and WilsonHRW22, Definition 4.2.1].

Lemma 2.10. The complexification map $\mathrm{ko} \to \mathrm{ku}$ is evenly free. Moreover, for each even $\mathbb{E}_\infty$ ko-algebra C, there is an isomorphism of $C_*$ -algebras

\begin{align*}\pi_* (\mathrm{ku} \otimes_{\mathrm{ko}} C) \cong C_*[b_1]/(\hat b_1^2) , \end{align*}

where $\hat b_1^2 = b_1^2+c_2b_1 +c_4$ for some $c_2, c_4 \in C_*$ with $|c_2| = 2$ and $|c_4| = 4$ .

Proof. Recall the Wood (homotopy) cofiber sequence

(2.1) \begin{align} \Sigma \mathrm{ko} \overset{\eta}\longrightarrow \mathrm{ko} \overset{c}\longrightarrow \mathrm{ku} \overset{R}\longrightarrow \Sigma^2\mathrm{ko}\end{align}

of ko-modules, where we write R for the map satisfying $R \circ(u\cdot -) = \Sigma^2 r$ , with $r \colon \mathrm{ku} \to \mathrm{ko}$ the realification map. The resulting cofiber sequence

\begin{align*}\Sigma C\overset{\eta}\longrightarrow C \longrightarrow \mathrm{ku} \otimes_{\mathrm{ko}} C \overset{R}\longrightarrow \Sigma^2 C \end{align*}

induces a short exact sequence of $C_*$ -modules

(2.2) \begin{align} 0 \to C_* \longrightarrow \pi_* (\mathrm{ku} \otimes_{\mathrm{ko}} C) \overset{R}\longrightarrow \Sigma^2 C_* \to 0 ,\end{align}

since C is even, so that $\eta \colon \Sigma C_* \to C_*$ is zero. We conclude that $ \pi_* (\mathrm{ku} \otimes_{\mathrm{ko}} C)$ is concentrated in even degrees.

A choice of a class $b_1 \in \pi_2(\mathrm{ku} \otimes_{\mathrm{ko}} C)$ with $R(b_1)= \Sigma^2 1$ is equivalent to a choice of a splitting of (2.2), so that $\pi_* (\mathrm{ku} \otimes_{\mathrm{ko}} C) \cong C_* \{1, b_1 \}$ is free over $C_*$ and nonzero if $C \ne 0$ . Writing $\hat b_1^2 = b_1^2+ c_2 b_1 + c_4$ for the monic polynomial that vanishes in $\pi_4(\mathrm{ku}\otimes_{\mathrm{ko}} C)$ , there is an isomorphism

\begin{align*}\pi_* (\mathrm{ku} \otimes_{\mathrm{ko}} C)\cong C_*[b_1]/(\hat b_1^2) \end{align*}

of $C_*$ -algebras.

Proof. Let C be a nonzero even $\mathbb{E}_\infty$ A-algebra of bounded 2-power torsion. Associated to the double-speed Whitehead filtration $\tau_{\geqslant2\star} C$ we have an algebra spectral sequence $E_1(C) \Longrightarrow \pi_*(C)$ with $E_1^{*,w}(C) =\pi_{*-2w}(H \pi_{2w} C)$ . Similarly, the induced filtration $B \otimes_A\tau_{\geqslant2\star} C$ yields a spectral sequence $E_1(B \otimes_A C)\Longrightarrow \pi_*(B \otimes_A C)$ with $E_1^{*,w}(B \otimes_A C) =\pi_{*-2w}(B \otimes_A H\pi_{2w} C)$ , which we view as a module spectral sequence over the first one. Both spectral sequences are conditionally convergent, since A and B are assumed to be connective. A choice of $\pi_0 A$ -module basis for $\pi_*(B \otimes_A H\pi_0 A)$ , and the equivalence

\begin{align*}B \otimes_A H\pi_{2w} C \simeq (B \otimes_A H\pi_0 A) \otimes_{H\pi_0 A} H\pi_{2w} C ,\end{align*}

show that $E_1(B \otimes_A C) \cong \pi_*(B \otimes_A H\pi_0 A)\otimes_{\pi_0 A} E_1(C)$ is a nonzero free $E_1(C)$ -module on generators in (filtration 0 and) even stems. Hence both spectral sequences are concentrated in even stems, collapse at $E_1 = E_\infty$ , and are strongly convergent by [Reference BoardmanBoa99, Theorem 8.10]. It follows that the $E_1(C)$ -module generators lift to the abutment, giving a $\pi_*(C)$ -module basis for $\pi_*(B \otimes_A C)$ .

Proof. Since ko, ku and MU are connective and have finite type, it follows that $\operatorname{THH}(\mathrm{ko})$ and $\operatorname{THH}(\mathrm{ku}/\mathrm{MU})$ have finite type, and therefore have bounded 2-power torsion. Hence, by Lemma 2.12, it suffices to show that

\begin{align*}\pi_*(\operatorname{THH}(\mathrm{ku}/\mathrm{MU}) \otimes_{\operatorname{THH}(\mathrm{ko})} H\mathbb{Z}) \end{align*}

is a nonzero, free and even $\mathbb{Z}$ -module. The $\mathbb{E}_\infty$ -ring map $\operatorname{THH}(\mathrm{ko}) \to H\mathbb{Z}$ can be rewritten as $\operatorname{THH}(\mathrm{ko}/\mathbb{S}) \to\operatorname{THH}(H\mathbb{Z}/H\mathbb{Z})$ , which is induced by $\mathrm{ko} \to H\mathbb{Z}$ over $\mathbb{S} \to H\mathbb{Z}$ . Then

\begin{align*}\operatorname{THH}(\mathrm{ku}/\mathrm{MU}) \otimes_{\operatorname{THH}(\mathrm{ko})} H\mathbb{Z} &\cong \operatorname{THH}(\mathrm{ku}/\mathrm{MU}) \otimes_{\operatorname{THH}(\mathrm{ko})} \operatorname{THH}(H\mathbb{Z}/H\mathbb{Z}) \\ &\simeq \operatorname{THH}(\mathrm{ku} \otimes_{\mathrm{ko}} H\mathbb{Z} / \mathrm{MU} \otimes H\mathbb{Z}) ,\end{align*}

where the last equivalence holds because $\operatorname{THH}$ commutes with pushouts of $\mathbb{E}_\infty$ -rings.

Recall that $\pi_*(\mathrm{MU} \otimes H\mathbb{Z}) \cong H_*(\mathrm{MU}; \mathbb{Z}) = \mathbb{Z}[b_k \mid k\geqslant1]$ as $\mathbb{Z}$ -algebras, with $|b_k| = 2k$ . Since $\mathrm{MU} \to \mathrm{ku}$ is 4-connected by Lemma 2.2, and $\mathbb{S} \to \mathrm{ko}$ is 3-connected, it follows that $\mathrm{MU} \otimes H\mathbb{Z} \to \mathrm{ku} \otimes_{\mathrm{ko}} H\mathbb{Z}$ is 4-connected. Hence the generators $b_k$ may be chosen so that $\pi_*(\mathrm{MU} \otimes H\mathbb{Z}) \to \pi_*(\mathrm{ku} \otimes_{\mathrm{ko}} H\mathbb{Z}) = \mathbb{Z}[b_1]/(b_1^2)$ satisfies $b_1 \mapsto b_1$ and $b_k \mapsto 0$ for $k\geqslant2$ , where $b_1$ in the target is fixed as in Lemma 2.10.

To determine $\operatorname{THH}$ of $\mathrm{ku} \otimes_{\mathrm{ko}} H\mathbb{Z}$ relative to $\mathrm{MU} \otimes H\mathbb{Z}$ , we first study the Künneth (or $\operatorname{Tor}$ ) spectral sequence

\begin{align*}E^2 &= \operatorname{Tor}_*^{\mathbb{Z}[b_k \mid k\geqslant1]} (\mathbb{Z}[b_1]/(b_1^2), \mathbb{Z}[b_1]/(b_1^2)) \\ &\Longrightarrow \pi_*( (\mathrm{ku} \otimes_{\mathrm{ko}} H\mathbb{Z}) \otimes_{(\mathrm{MU} \otimes H\mathbb{Z})} (\mathrm{ku} \otimes_{\mathrm{ko}} H\mathbb{Z})) .\end{align*}

Since $b_1^2$ and $b_k$ for $k\geqslant 2$ act trivially on $\mathbb{Z}[b_1]/(b_1^2)$ , we compute that

\begin{align*}E^2 \cong \mathbb{Z}[b_1]/(b_1^2) \otimes \Lambda(\sigma b_1^2, \sigma b_k \mid k\geqslant 2) \end{align*}

with $\sigma b_1^2, \sigma b_k \in \operatorname{Tor}_1$ , where the triviality of the square $(\sigma b_1^2)^2$ follows from the fact that the product in $\operatorname{Tor}$ is given by the shuffle product. Since the algebra generators are all in filtration $\leqslant 1$ , the Künneth spectral sequence collapses at the $E^2$ -term. Note that this is a homological spectral sequence associated to an increasing filtration. Since the $E^{\infty}$ -term is a nonzero free $\mathbb{Z}$ -module and the abutment is also a $\mathbb{Z}$ -module, we conclude that the abutment is a free and nonzero $\mathbb{Z}$ -module.

We can rule out all potential hidden multiplicative extensions, because $\operatorname{Tor}_0$ splits off from the abutment and all the classes in Künneth filtration 1 are in odd total degree. Therefore, we have an algebra isomorphism

\begin{align*}D_* := \pi_*((\mathrm{ku} \otimes_{\mathrm{ko}} H\mathbb{Z}) \otimes_{(\mathrm{MU} \otimes H\mathbb{Z})} (\mathrm{ku} \otimes_{\mathrm{ko}} H\mathbb{Z})) \cong \mathbb{Z}[b_1]/( b_1^2) \otimes \Lambda(\sigma b_1^2, \sigma b_k \mid k\geqslant2) . \end{align*}

Second, we apply the Künneth spectral sequence

\begin{align*}E^2 = \operatorname{Tor}_*^{D_*} (\mathbb{Z}[b_1]/( b_1^2), \mathbb{Z}[b_1]/( b_1^2)) \Longrightarrow \pi_* \operatorname{THH}(\mathrm{ku} \otimes_{\mathrm{ko}} H\mathbb{Z}/ \mathrm{MU} \otimes H\mathbb{Z}) . \end{align*}

The $E^2$ -term of this spectral sequence, namely,

\begin{align*}E^2 = \mathbb{Z}[b_1]/( b_1^2) \otimes \Gamma(\sigma^2 b_1^2, \sigma^2 b_k \mid k\geqslant2) , \end{align*}

is concentrated in even degrees, so the spectral sequence collapses at the $E^2$ -term and the abutment is also nonzero, free and concentrated in even degrees.

Proof. By Proposition 2.14 and induction, it suffices to show that $\pi_* \operatorname{THH}(\mathrm{ku}/\mathrm{MU})$ is free as a $\mathrm{ku}_*$ -module and concentrated in even degrees. This follows from [Reference Hahn and WilsonHW22, Theorem E], but we recall the argument in our context. The Künneth spectral sequence

\begin{align*}E^2 &= \operatorname{Tor}_*^{\mathrm{MU}_*}(\mathrm{ku}_*, \mathrm{ku}_*) = \operatorname{Tor}_*^{\mathbb{Z}[x_i \mid i\geqslant1]}(\mathbb{Z}[u], \mathbb{Z}[u]) \\ &= \mathbb{Z}[u] \otimes \Lambda(\sigma x_i \mid i\geqslant2) \Longrightarrow \pi_*(\mathrm{ku} \otimes_{\mathrm{MU}} \mathrm{ku})\end{align*}

collapses at the $E^2$ -term. Again, we can rule out all hidden multiplicative extensions, because $\operatorname{Tor}_0$ splits off from the abutment and all classes in Künneth filtration 1 are in odd total degrees. Next, we apply the Künneth spectral sequence

\begin{align*}E^2 &= \operatorname{Tor}_*^{\pi_*(\mathrm{ku} \otimes_{\mathrm{MU}} \mathrm{ku})}(\mathrm{ku}_*, \mathrm{ku}_*) = \operatorname{Tor}_*^{\mathbb{Z}[u] \otimes \Lambda(\sigma x_i \mid i\geqslant2)}(\mathbb{Z}[u], \mathbb{Z}[u]) \\ &= \mathbb{Z}[u] \otimes \Gamma(\sigma^2 x_i \mid i\geqslant2) \Longrightarrow \pi_* \operatorname{THH}(\mathrm{ku}/\mathrm{MU}) .\end{align*}

This $E^2$ -term is also concentrated in even degrees, and is free over $\mathbb{Z}[u] = \mathrm{ku}_*$ . Hence $E^2 = E^\infty$ , and the abutment is free over $\mathrm{ku}_*$ on even degree generators.

The following fact is stated without proof in [Reference Hahn, Raksit and WilsonHRW22, Example 1.3.3]. We follow this reference and write $L^{\mathrm{alg}}_{B/A}$ for the algebraic cotangent complex, with homological grading, of a homomorphism $A \to B$ of (ungraded) commutative rings.

Proof. Since $\mathrm{MU}_*$ is free over $\mathrm{BP}_*$ , we have a ring isomorphism $\mathrm{ko}_*\mathrm{BP} \otimes_{\mathrm{BP}_*} \mathrm{MU}_* \cong \mathrm{ko}_* \mathrm{MU}$ . Here, $H_*(\mathrm{ko}) = \mathcal{A}^\vee\mathbin{\square}_{\mathcal{A}(1)^\vee} \mathbb{F}_2$ , so $H_*(\mathrm{ko} \otimes \mathrm{BP}) \cong \mathcal{A}^\vee\mathbin{\square}_{\mathcal{A}(1)^\vee} H_*(\mathrm{BP})$ and the $E_2$ -term of the Adams spectral sequence

\begin{align*}{}^{\mathrm{Ad}} E_2 = \operatorname{Ext}_{\mathcal{A}^\vee}(\mathbb{F}_2, H_*(\mathrm{ko} \otimes \mathrm{BP})) \Longrightarrow \pi_*(\mathrm{ko} \otimes \mathrm{BP}) \end{align*}

can be rewritten as $\operatorname{Ext}_{\mathcal{A}(1)^\vee}(\mathbb{F}_2, H_*(\mathrm{BP}))$ . Since $H_*(\mathrm{BP}) = \mathbb{F}_2[\xi_k^2 \mid k\geqslant1]$ is concentrated in even degrees, its $\mathcal{A}(1)^\vee$ -coaction factors as

\begin{align*}H_*(\mathrm{BP}) &\overset{\nu}\longrightarrow \Lambda(\xi_1^2) \otimes H_*(\mathrm{BP}) \subset \mathcal{A}(1)^\vee \otimes H_*(\mathrm{BP}), \\\xi_k^2 &\longmapsto \begin{cases} 1 \otimes \xi_1^2 + \xi_1^2 \otimes 1 & \text{for $k=1$,} \\ 1 \otimes \xi_k^2 & \text{for $k\geqslant2$.} \end{cases}\end{align*}

To calculate the Adams $E_2$ -term, we use the Cartan–Eilenberg spectral sequence

\begin{align*}{}^{\mathrm{CE}} E_2 = \operatorname{Ext}_{\Lambda(\xi_1^2)}(\mathbb{F}_2, \operatorname{Ext}_{\Lambda(\xi_1, \xi_2)}(\mathbb{F}_2, H_*(\mathrm{BP}))) \Longrightarrow \operatorname{Ext}_{\mathcal{A}(1)^\vee}(\mathbb{F}_2, H_*(\mathrm{BP})) \end{align*}

(cf. [Reference Cartan and EilenbergCE56, Theorem XVI.6.1]) for the Hopf algebra extension $\Lambda(\xi_1^2) \to \mathcal{A}(1)^\vee\to \Lambda(\xi_1, \xi_2)$ . Its $E_2$ -term

\begin{align*}{}^{\mathrm{CE}} E_2 \cong \operatorname{Ext}_{\Lambda(\xi_1^2)}(\mathbb{F}_2, \mathbb{F}_2[v_0, v_1] \otimes H_*(\mathrm{BP})) = \mathbb{F}_2[v_0, v_1] \otimes \mathbb{F}_2[\xi_1^4, \xi_k^2 \mid k\geqslant2] \end{align*}

is concentrated in filtration degree 0, so ${}^{\mathrm{CE}} E_2 = {}^{\mathrm{CE}}E_\infty$ . Hence the Adams $E_2$ -term is concentrated in even stems, so we also have ${}^{\mathrm{Ad}} E_2 = {}^{\mathrm{Ad}} E_\infty$ . Thus $\pi_*(\mathrm{ko} \otimes \mathrm{BP})\cong \mathbb{Z}[u, t_1^2, \bar t_k \mid k\geqslant2]$ , with 2 detected by $v_0$ , u detected by $v_1$ , $t_1^2$ detected by $\xi_1^4$ , and $\bar t_k$ detected by $\xi_k^2$ . Base change along $\mathrm{BP} \to \mathrm{MU}$ replaces the degree $2(2^k-1)$ generators $\bar t_k$ for $k \geqslant 2$ with degree 2k polynomial generators $\bar b_k$ for $k \geqslant 2$ , while $t_1^2$ is replaced by $b_1^2$ .

Since $\mathrm{MU}_*$ and $\mathrm{ko}_*\mathrm{MU}$ are polynomial rings, the algebraic cotangent complexes $L^{\mathrm{alg}}_{\mathrm{MU}_*/\mathbb{Z}}$ and $L^{\mathrm{alg}}_{\mathrm{ko}_* \mathrm{MU}/\mathbb{Z}}$ are free modules over $\mathrm{MU}_*$ and $\mathrm{ko}_* \mathrm{MU}$ , respectively. By the transitivity cofiber sequence

\begin{align*}\mathrm{ko}_* \mathrm{MU} \otimes_{\mathrm{MU}_*} L^{\mathrm{alg}}_{\mathrm{MU}_*/\mathbb{Z}} \longrightarrow L^{\mathrm{alg}}_{\mathrm{ko}_* \mathrm{MU}/\mathbb{Z}} \longrightarrow L^{\mathrm{alg}}_{\mathrm{ko}_* \mathrm{MU}/\mathrm{MU}_*} \end{align*}

associated to the ring homomorphisms $\mathbb{Z} \to \mathrm{MU}_* \to \mathrm{ko}_* \mathrm{MU}$ (cf. [Reference QuillenQui70, 5.1]), it follows that $L^{\mathrm{alg}}_{\mathrm{ko}_* \mathrm{MU}/\mathrm{MU}_*}$ has 2-complete $\operatorname{Tor}$ -amplitude concentrated in homological degrees [0,1]. By definition [Reference Hahn, Raksit and WilsonHRW22, Definitions 4.1.1(2) and 4.1.6], this means that $\mathrm{MU} \to \mathrm{ko} \otimes \mathrm{MU}$ is 2-quasi-lci.

Proposition 2.17. The filtrations from Definition 2.3 for $F \in \{\operatorname{THH}, \operatorname{TC}^{-},\operatorname{TP}\}$ agree with the motivic filtrations considered in [Reference Hahn, Raksit and WilsonHRW22, Definition 4.2.1]. Moreover, there exist maps

\begin{align*}\mathrm{can} , \varphi \colon \operatorname{fil}_{\mathrm{mot}}^{\star} \operatorname{TC}^{-}(\mathrm{ko}) \longrightarrow \operatorname{fil}_{\mathrm{mot}}^{\star} \operatorname{TP}(\mathrm{ko}) \end{align*}

of $\mathbb{E}_\infty$ algebras in $\mathrm{Sp}_2^{\operatorname{fil}}$ , which converge to the canonical map and Frobenius map from [Reference Nikolaus and ScholzeNS18].

The map of connective $\mathbb{E}_\infty$ -rings $\mathbb{S} \to \mathrm{ko}$ is chromatically 2-quasi-lci in the sense of [Reference Hahn, Raksit and WilsonHRW22, Definition 4.1.11], because by Proposition 2.16 we know that both MU and $\mathrm{ko} \otimes \mathrm{MU}$ are even, and the map $\mathrm{MU} \to \mathrm{ko} \otimes \mathrm{MU}$ is 2-quasi-lci. Hence the even filtrations also define the motivic filtrations, as per [Reference Hahn, Raksit and WilsonHRW22, Definition 4.2.1], and this confirms the first statement.

The second statement then follows directly from [Reference Hahn, Raksit and WilsonHRW22, Theorem 1.3.6].

Definition 2.18. Motivated by Proposition 2.17, we define $\mathbb{E}_\infty$ algebras

\begin{align*}\operatorname{fil}_{\mathrm{mot}}^{\star} \operatorname{TC}(\mathrm{ko}) &:= \operatorname{eq} (\mathrm{can} , \varphi \colon \operatorname{fil}_{\mathrm{mot}}^{\star} \operatorname{TC}^{-}(\mathrm{ko}) \longrightarrow \operatorname{fil}_{\mathrm{mot}}^{\star} \operatorname{TP}(\mathrm{ko})), \\\operatorname{gr}_{\mathrm{mot}}^* \operatorname{TC}(\mathrm{ko}) &:= \operatorname{gr}^* (\operatorname{fil}_{\mathrm{mot}}^{\star} \operatorname{TC}(\mathrm{ko}))\end{align*}

in $\mathrm{Sp}_2^{\operatorname{fil}}$ and $\mathrm{Sp}_2^{\operatorname{gr}}$ , respectively. By taking the (homotopy) equalizer of can and $\varphi$ , we retain the multiplicative structure, but additively $\operatorname{fil}_{\mathrm{mot}}^{\star} \operatorname{TC}(\mathrm{ko})$ is also the (homotopy) fiber of $\mathrm{can} - \varphi$ . In light of [Reference Hahn, Raksit and WilsonHRW22, § 5] and [Reference Bhatt, Morrow and ScholzeBMS19], we refer to $\operatorname{gr}_{\mathrm{mot}}^w \operatorname{TC}(\mathrm{ko})$ and $\pi_*\operatorname{gr}_{\mathrm{mot}}^w \operatorname{TC}(\mathrm{ko})$ as the weight w syntomic cohomology spectrum and syntomic cohomology groups of ko, refer to the spectral sequence

\begin{align*}E_2^{n,2w-n} = \pi_n\operatorname{gr}_{\mathrm{mot}}^w \operatorname{TC}(\mathrm{ko}) \Longrightarrow \pi_n\operatorname{TC}(\mathrm{ko}) \end{align*}

as the motivic spectral sequence, and follow the same grading conventions as in Definition 2.7. This spectral sequence converges, as we now review.

Lemma 2.19. For $F \in \{\operatorname{THH}, \operatorname{TC}^{-}, \operatorname{TP}, \operatorname{TC}, F(S^{2\ell+1}_+,\operatorname{THH})^{\mathbb{T}}, \operatorname{THH}^{hC_2},\operatorname{THH}^{tC_2} \}$ , the filtration $\operatorname{fil}_{\mathrm{mot}}^{s} F(\mathrm{ko})$ is (derived) complete and exhaustive. In other words,

\begin{align*} \lim \operatorname{fil}_{\mathrm{mot}}^{\star}F(\mathrm{ko})=0,\quad \operatorname{colim} \operatorname{fil}_{\mathrm{mot}}^{\star} F(\mathrm{ko})=F(\mathrm{ko}) . \end{align*}

Proof. Since limits commute with limits, $\lim \operatorname{fil}_{\mathrm{mot}}^{\star} F(\mathrm{ko}) =0$ for each

\begin{align*} F \in \{\operatorname{THH}, \operatorname{TC}^{-}, \operatorname{TP}, \operatorname{TC}, F(S^{2\ell+1}_+,\operatorname{THH})^{\mathbb{T}},\operatorname{THH}^{hC_2},\operatorname{THH}^{tC_2}\}. \end{align*}

It follows from [Reference Hahn, Raksit and WilsonHRW22, Theorems 1.3.5 and 1.3.6, Corollary 2.2.17] and Propositions 2.14 and 2.17 that $\operatorname{colim} \operatorname{fil}_{\mathrm{mot}}^{\star} F(\mathrm{ko}) =F(\mathrm{ko})$ for $F \in \{\operatorname{THH}, \operatorname{TC}^{-}, \operatorname{TP},\operatorname{TC}\}$ . It therefore suffices to show that $\operatorname{colim} \operatorname{fil}_{\mathrm{mot}}^{\star} F(\mathrm{ko}) =F(\mathrm{ko})$ for each $F\in \{F(S^{2\ell+1}_+,\operatorname{THH})^{\mathbb{T}},\operatorname{THH}^{hC_2},\operatorname{THH}^{tC_2}\}$ . Here, we adapt the proof of [Reference Hahn, Raksit and WilsonHRW22, Proposition 2.3.5]. Let

\begin{align*}Y^{\bullet}:=C^{\bullet}(\operatorname{THH}(\mathrm{ku}/\mathrm{MU})/\operatorname{THH}(\mathrm{ko})) \end{align*}

and note that the canonical map

\begin{align*}\mathop{\hbox{colim}}\limits_w \operatorname{fil}_{\mathrm{mot}}^{w}G(\operatorname{THH}(\mathrm{ko})) &= \mathop{\hbox{colim}}\limits_w \operatorname{Tot} \tau_{\geqslant 2w} G(Y^{\bullet}) \overset{\simeq}{\longrightarrow } \operatorname{Tot} G(Y^{\bullet})\end{align*}

is an equivalence for each $G\in \{F(S^{2\ell+1}_+,-)^{\mathbb{T}}, (-)^{hC_2},(-)^{tC_2}\}$ since $\operatorname{Tot}$ preserves coconnectivity. Note also that the canonical maps

\begin{align*} F(S^{2\ell+1}_+,\operatorname{THH}(\mathrm{ko}))^{\mathbb{T}} \overset{\simeq}{\longrightarrow} \operatorname{Tot} (F(S^{2\ell+1}_+,Y^{\bullet})^{\mathbb{T}}) \end{align*}

and $ \operatorname{THH}(\mathrm{ko})^{hC_2} \overset{\simeq}{\longrightarrow} \operatorname{Tot} ((Y^{\bullet})^{hC_2})$ are equivalences, since limits commute with limits. This handles the cases $F=F(S^{2\ell+1}_+,\operatorname{THH})^{\mathbb{T}}$ and $F= \operatorname{THH}^{hC_2}$ .

To see that the canonical map $\operatorname{THH}(\mathrm{ko})^{tC_2} \overset{\simeq}{\longrightarrow} \operatorname{Tot} ((Y^{\bullet})^{tC_2})$ is an equivalence, it now suffices to show that the canonical map

\begin{align*} \operatorname{THH}(\mathrm{ko})_{hC_2} \overset{\simeq}{\longrightarrow} \operatorname{Tot} ((Y^{\bullet})_{hC_2}) \end{align*}

is an equivalence. This will follow from the fact that the fiber I of the map

\begin{align*} A:=\operatorname{THH}(\mathrm{ko})\to \operatorname{THH}(\mathrm{ku}/\mathrm{MU}) \end{align*}

is 1-connective and that $(-)_{hC_2}$ preserves connectivity. In particular, by [Reference Mathew, Naumann and NoelMNN17, Proposition 2.14], the fiber of the map

\begin{align*} \operatorname{THH}(\mathrm{ko})_{hC_2} \longrightarrow \operatorname{Tot}_n ((Y^{\bullet})_{hC_2}) \end{align*}

can be identified with $(I^{\otimes_{A} (n+1)})_{hC_2}$ , which is $(n+1)$ -connective, implying the claim after passing to the limit as n grows. Here, we use the fact that $\operatorname{Tot}_n$ is a finite limit and therefore commutes with $(-)_{hC_2}$ .

We now introduce the relevant type 2 finite coefficient spectra. Let $V(0)= C2$ denote the cofiber of the map $2\colon \mathbb{S}\to \mathbb{S}$ and let $C\eta$ denote the cofiber of the Hopf map $\eta \colon \Sigma \mathbb{S}\to \mathbb{S}$ . By [Reference Davis and MahowaldDM81, Proposition 2.1], there are precisely four equivalence classes of finite spectra $X = (V(0) \otimes C\eta)/v_1$ with the property that $H^*(X) \cong \mathcal{A}(1)$ as an $\mathcal{A}(1)$ -module, each characterized by the $Sq^4$ -action in its mod 2 cohomology.

In view of Proposition 2.15 and the fact that $A(1) \otimes \mathrm{ku}$ is even, we know that $A(1) \otimes C^{\bullet}(\operatorname{THH}(\mathrm{ku}/\mathrm{MU}) / \operatorname{THH}(\mathrm{ko}))$ is even. This motivates the following definition.

Definition 2.21. Let

\begin{align*}\operatorname{fil}_{\mathrm{mot}}^{\star} (A(1) \otimes \operatorname{THH}(\mathrm{ko})) := \operatorname{Tot} ( \tau_{\geqslant 2\star} ( A(1) \otimes C^{\bullet}(\operatorname{THH}(\mathrm{ku}/\mathrm{MU}) / \operatorname{THH}(\mathrm{ko})) ) ) , \end{align*}

and write $\operatorname{gr}_{\mathrm{mot}}^* (A(1) \otimes \operatorname{THH}(\mathrm{ko})) := \operatorname{gr}^* (\operatorname{fil}_{\mathrm{mot}}^{\star} (A(1) \otimes \operatorname{THH}(\mathrm{ko})) )$ for the associated graded spectrum.

Remark 2.22. The spectrum $A(1) \otimes \mathrm{ko} \simeq H\mathbb{F}_2$ admits a unique $\mathbb{E}_\infty$ ko-algebra structure. Hence, for each $\mathbb{E}_\infty$ ko-algebra R, the identification

\begin{align*}A(1) \otimes R = A(1) \otimes \mathrm{ko} \otimes_{\mathrm{ko}} R \simeq H\mathbb{F}_2 \otimes_{\mathrm{ko}} R \end{align*}

lets us regard $A(1) \otimes R$ as an $\mathbb{E}_\infty$ $H\mathbb{F}_2$ -algebra. For example, we will identify $\operatorname{THH}(\mathrm{ko}, H\mathbb{F}_2) := H\mathbb{F}_2 \otimes_{\mathrm{ko}}\operatorname{THH}(\mathrm{ko})$ with $A(1) \otimes \operatorname{THH}(\mathrm{ko})$ , and we may therefore consider the latter as an $\mathbb{E}_\infty$ $H\mathbb{F}_2$ -algebra.

Lemma 2.23. Let $A := A(1) \otimes \operatorname{THH}(\mathrm{ko})$ and let $B := A(1)\otimes \operatorname{THH}(\mathrm{ku}/\mathrm{MU})$ . Then there is an isomorphism

\begin{align*} A_* \cong \Lambda(\lambda'_1, \lambda_2) \otimes \mathbb{F}_2[\mu] \end{align*}

of graded $\mathbb{F}_2$ -algebras, where $|\lambda'_1| = 5$ , $|\lambda_2|= 7$ and $|\mu| = 8$ . There is also an isomorphism

\begin{align*}B_* \cong \Lambda(\hat\xi_1^2) \otimes \mathbb{F}_2[\mu] \otimes P \end{align*}

of $A_*$ -algebras, where $|\hat\xi_1^2| = 2$ , $|\mu| = 8$ and P is a polynomial algebra with generators in even degrees. The $A_*$ -algebra structure is determined by the map $A_* \to B_*$ sending $\lambda'_1$ and $\lambda_2$ to zero and mapping $\mu$ to $\mu$ .

Proof. The complexification map $c \colon \mathrm{ko} \to \mathrm{ku}$ and the unique $\mathbb{E}_\infty$ -ring map $\mathrm{ku} \to H\mathbb{F}_2$ induce monomorphisms

\begin{align*}H_*(\mathrm{ko}) = \mathbb{F}_2[\xi_1^4, \bar\xi_2^2, \bar\xi_k \mid k\geqslant 3]\longrightarrow H_*(\mathrm{ku}) = \mathbb{F}_2[\xi_1^2, \bar\xi_2^2, \bar\xi_k \mid k\geqslant 3] \longrightarrow \mathcal{A}^\vee \end{align*}

of $\mathcal{A}^\vee$ -comodule algebras. By Milnor’s construction of the $\xi_i$ , the composite map $\mathrm{MU} \to \mathrm{ku}\to H\mathbb{F}_2$ induces the homomorphism $H_*(\mathrm{MU}) = \mathbb{F}_2[b_i \mid i\geqslant1]\to H_*(H\mathbb{F}_2) = \mathcal{A}^\vee$ given by $b_{2^i-1} \mapsto \xi_i^2$ and $b_j\mapsto 0$ for $j \ne 2^i-1$ . Hence these formulas also hold in $H_*(\mathrm{ku})$ . Let $\bar b_i = \chi b_i$ denote the conjugate classes in $H_*(\mathrm{MU})$ , so that $\bar b_{2^i-1}\mapsto \bar\xi_i^2$ and $\bar b_j \mapsto 0$ for $j \ne 2^i-1$ . Standard Hochschild homology computations (cf. [Reference McClure and StaffeldtMS93, Proposition 2]) yield

\begin{align*}\operatorname{HH}_* (H_* (\mathrm{ko})) &= H_* (\mathrm{ko}) \otimes \Lambda(\sigma\xi_1^4, \sigma\bar\xi_2^2, \sigma\bar\xi_k \mid k\geqslant 3 ), \\\operatorname{HH}_* (H_* (\mathrm{ku})) &= H_* (\mathrm{ku}) \otimes \Lambda(\sigma\xi_1^2, \sigma\bar\xi_2^2, \sigma\bar\xi_k \mid k\geqslant 3 ) .\end{align*}

The usual argument for hidden extensions in the Bökstedt spectral sequence (see, e.g., [Reference Angeltveit and RognesAR05, Theorem 6.2]) implies that

\begin{align*}(\sigma\bar\xi_3)^{2^{k-3}} = \sigma\bar\xi_k , \end{align*}

and produces identifications

\begin{align*}H_*(\operatorname{THH}(\mathrm{ko})) & \cong H_* (\mathrm{ko}) \otimes \Lambda(\sigma\xi_1^4, \sigma\bar\xi_2^2) \otimes \mathbb{F}_2[\sigma\bar\xi_3], \\H_*(\operatorname{THH}(\mathrm{ku})) & \cong H_* (\mathrm{ku}) \otimes \Lambda(\sigma\xi_1^2, \sigma\bar\xi_2^2) \otimes \mathbb{F}_2[\sigma\bar\xi_3] .\end{align*}

As noted above, $H_*(A(1) \otimes \mathrm{ko}) \cong \mathcal{A}^\vee$ . By Remark 2.22 and the evident collapsing Künneth spectral sequence, we have

\begin{align*}H_* (A(1) \otimes \mathrm{ku}) \cong H_* (H\mathbb{F}_2 \otimes_{\mathrm{ko}} \mathrm{ku}) \cong \mathcal{A}^\vee \otimes_{H_*(\mathrm{ko})} H_*(\mathrm{ku}) \cong \mathcal{A}^\vee \otimes \Lambda(\hat\xi_1^2) , \end{align*}

where $\hat\xi_1^2 := 1 \otimes \xi_1^2 + \xi_1^2 \otimes 1$ denotes the $\mathcal{A}^\vee$ -comodule primitive class, so that $A(1)_*(\mathrm{ku}) = \pi_*(A(1) \otimes \mathrm{ku}) = \Lambda(\hat\xi_1^2)$ . We conclude that

\begin{align*}A_* = A(1)_* \operatorname{THH}(\mathrm{ko}) &= \Lambda(\lambda'_1, \lambda_2) \otimes \mathbb{F}_2[\mu], \\A(1)_* \operatorname{THH}(\mathrm{ku}) &= \Lambda(\hat\xi_1^2, \lambda_1, \lambda_2) \otimes \mathbb{F}_2[\mu] ,\end{align*}

with $\lambda'_1$ , $\lambda_2$ and $\mu$ in $A(1)_* \operatorname{THH}(\mathrm{ko})$ having Hurewicz images $\sigma\xi_1^4$ , $\sigma\bar\xi_2^2 + \xi_1^2 \cdot\sigma\xi_1^4$ and $\sigma\bar\xi_3 + \xi_1 \cdot \sigma\bar\xi_2^2 +\xi_2 \cdot \sigma\xi_1^4$ , respectively, while $\lambda_1$ , $\lambda_2$ and $\mu$ in $A(1)_* \operatorname{THH}(\mathrm{ku})$ have Hurewicz images $\sigma\xi_1^2$ , $\sigma\bar\xi_2^2$ and $\sigma\bar\xi_3 + \xi_1 \cdot \sigma\bar\xi_2^2$ (cf. [Reference Angeltveit and RognesAR05, Proposition 8.7]). Note that $\sigma\xi_1^4 =\sigma \xi_1^2 \cdot \xi_1^2 + \xi_1^2 \cdot \sigma\xi_1^2 = 0$ in the ku-case. Hence the induced homomorphism $A(1)_* \operatorname{THH}(\mathrm{ko}) \to A(1)_*\operatorname{THH}(\mathrm{ku})$ is given by $\lambda'_1 \mapsto 0$ , $\lambda_2 \mapsto\lambda_2$ and $\mu \mapsto \mu$ .

Next, we compute $A(1)_* \operatorname{THH}(\mathrm{ku}/\mathrm{MU})$ , using the fact that

(2.3) \begin{align} A(1) \otimes \operatorname{THH}(\mathrm{ku}/\mathrm{MU}) \simeq (A(1) \otimes \operatorname{THH}(\mathrm{ku})) \otimes_{\operatorname{THH}(\mathrm{MU})} \mathrm{MU} .\end{align}

We know that

\begin{align*}\pi_* \operatorname{THH}(\mathrm{MU}) \cong \mathrm{MU}_* \otimes \Lambda(\sigma\bar b_i \mid i\geqslant 1) \end{align*}

by [Reference McClure and StaffeldtMS93, Remark 4.3] (cf. [Reference RognesRog20, Proposition 4.5]). We expand

\begin{align*}\operatorname{THH}(\mathrm{MU}) \longrightarrow \operatorname{THH}(\mathrm{ku})\longrightarrow A(1) \otimes \operatorname{THH}(\mathrm{ku}) \end{align*}

as the composite

\begin{align*}\mathrm{MU} \otimes_{\mathrm{MU} \otimes \mathrm{MU}} \mathrm{MU} & \longrightarrow \mathrm{ku} \otimes_{\mathrm{ku} \otimes \mathrm{ku}}\mathrm{ku} \\ & \longrightarrow A(1) \otimes \mathrm{ku} \otimes_{A(1) \otimes \mathrm{ku} \otimes \mathrm{ku}} A(1) \otimes \mathrm{ku} ,\end{align*}

where

\begin{align*}\mathrm{MU}_*[\bar b_i \mid i\geqslant1] =\pi_*(\mathrm{MU} \otimes \mathrm{MU}) & \longrightarrow \pi_*(\mathrm{ku} \otimes \mathrm{ku}) \\ & \longrightarrow \pi_*(A(1) \otimes \mathrm{ku} \otimes \mathrm{ku}) = \Lambda(\hat\xi_1^2) \otimes H_*(\mathrm{ku})\end{align*}

takes $\bar b_{2^i-1}$ to $\bar\xi_i^2$ for $i\geqslant1$ and $\bar b_j$ to 0 for $j \ne 2^i-1$ . Hence $\pi_* \operatorname{THH}(\mathrm{MU}) \to A(1)_* \operatorname{THH}(\mathrm{ku})$ is given by $\sigma\bar b_1 \mapsto \sigma\xi_1^2 = \lambda_1$ , $\sigma\bar b_3 \mapsto \sigma\bar\xi_2^2 = \lambda_2$ and $\sigma\bar {b}_i \mapsto0$ for $i \notin \{1,3\}$ . (This uses that $\sigma\bar b_{2^i-1}\mapsto \sigma\bar\xi_i^2 = \sigma\bar\xi_i \cdot \bar\xi_i + \bar\xi_i\cdot \sigma\bar\xi_i = 0$ for $i\geqslant3$ , while $\xi_1$ and $\bar\xi_2$ do not exist in $H_*(\mathrm{ku})$ .) The $\pi_* \operatorname{THH}(\mathrm{MU})$ -algebra structure on $\mathrm{MU}_*$ is given by mapping $x_i$ to $x_i$ and mapping $\sigma\bar b_i$ trivially for all $i\geqslant 1$ . The Künneth spectral sequence associated to (2.3) therefore has $E^2$ -term

\begin{align*} \Lambda(\hat\xi_1^2) \otimes \mathbb{F}_2[\mu] \otimes \Gamma(\sigma^2 \bar b_{2^i-1} \mid i\geqslant3) \otimes \Gamma(\sigma^2 \bar b_j \mid j \ne 2^i-1) . \end{align*}

This spectral sequence is concentrated in even total degrees and therefore collapses at the $E^2$ -term. We resolve the hidden multiplicative extensions using Steinberger’s computation [Reference Bruner, May, McClure and SteinbergerBMM⁺86, III.2] of the Dyer–Lashof operations on $H_*(H\mathbb{F}_2) = \mathcal{A}^\vee$ and Kochman’s computation [Reference KochmanKoc73, Theorem 6] of the Dyer–Lashof operations on $H_*(\mathrm{BU})\cong H_*(\mathrm{MU})$ , as in the proof of [Reference Hahn and WilsonHW22, Lemma 2.4.1]. (Note that Steinberger’s result is used for the $\sigma^2 \bar b_{2^i-1}$ , while Kochman’s theorem is used for the remaining $\sigma^2\bar b_j$ .) This produces the identification

\begin{align*}B_* = A(1)_* \operatorname{THH}(\mathrm{ku}/\mathrm{MU}) \cong \Lambda(\hat\xi_1^2) \otimes \mathbb{F}_2[\mu] \otimes P , \end{align*}

where

\begin{align*}P := \mathbb{F}_2[w_i \mid i\geqslant0] \otimes \mathbb{F}_2[y_{j,i} \mid \text{$j\geqslant2$ even, $i\geqslant0$}] \end{align*}

is a polynomial algebra with algebra generators in even degrees. Here, $w_i$ is any choice of lift of $\gamma_{2^i}(\sigma^2 \bar b_7)$ and $y_{j,i}$ is any choice of lift of $\gamma_{2^i}(\sigma^2 \bar b_j)$ . We conclude that $B_*$ is an even $\mathbb{E}_\infty$ -ring, and the $A_*$ -algebra structure is determined by $\lambda'_1$ and $\lambda_2$ mapping trivially for degree reasons and $\mu$ mapping to $\mu$ by the first half of this proof.

Remark 2.26. By [Reference Hahn, Raksit and WilsonHRW22, Corollary 2.2.17], Remark 2.22 and Corollary 2.24, we can identify

\begin{align*}\operatorname{fil}_{\mathrm{mot}}^{\star} (A(1) \otimes \operatorname{THH}(\mathrm{ko})) \simeq \operatorname{fil}_{\mathrm{ev}}^{\star} (A(1) \otimes \operatorname{THH}(\mathrm{ko})) , \end{align*}

in the sense of Definition 2.21 and [Reference Hahn, Raksit and WilsonHRW22, Construction 2.1.11].

Theorem 2.27. There is an isomorphism

\begin{align*}\pi_* \operatorname{gr}_{\mathrm{mot}}^* (A(1) \otimes \operatorname{THH}(\mathrm{ko})) \cong \Lambda(\lambda'_1, \lambda_2) \otimes \mathbb{F}_2[\mu] \end{align*}

of bigraded $\mathbb{F}_2$ -algebras, with $\|\lambda'_1\| = (5,1)$ , $\|\lambda_2\| = (7,1)$ and $\|\mu\| = (8,0)$ .

Proof. We closely follow [Reference Hahn and WilsonHW22] and [Reference Hahn, Raksit and WilsonHRW22]. Starting with the proof of [Reference Hahn and WilsonHW22, Proposition 6.1.6], let $A := A(1) \otimes\operatorname{THH}(\mathrm{ko})$ and $B := A(1) \otimes \operatorname{THH}(\mathrm{ku}/\mathrm{MU})$ , so that $A_* \cong\Lambda(\lambda'_1, \lambda_2) \otimes \mathbb{F}_2[\mu]$ and $B_* \cong\Lambda(\hat\xi_1^2) \otimes \mathbb{F}_2[\mu] \otimes P$ by Lemma 2.23. The descent spectral sequence associated to the cosimplicial Amitsur resolution $C^{\bullet}(B/A) = B^{\otimes_A\bullet+1}$ for $A \to B$ has $E_1$ -term

\begin{align*}E^q_1(B/A) = \pi_*(B^{\otimes_A q+1}) \end{align*}

for $q\geqslant0$ , and converges to $A_*$ . Since $B_*$ is concentrated in even stems, Corollary 2.24 implies that $\Sigma :=\pi_*(B \otimes_A B)$ is even and free over $B_*$ , so that $(B_*, \Sigma)$ is a flat Hopf algebroid. Let $C^*_\Sigma(B_*, B_*)$ denote the associated cobar complex. It follows by induction on q that the natural homomorphism

\begin{align*}C^q_{\Sigma}(B_*, B_*) &= \Sigma \otimes_{B_*} \cdots \otimes_{B_*} \Sigma \overset{\cong}\longrightarrow\pi_*((B \otimes_A B) \otimes_B \cdots \otimes_B (B \otimes_A B)) \\ &\cong \pi_*(B \otimes_A \cdots \otimes_A B) = E^q_1(B/A)\end{align*}

is an isomorphism for each $q\geqslant0$ , since the relevant Künneth spectral sequences collapse. Passing to cohomology, we obtain an isomorphism

\begin{align*}\operatorname{Ext}_{\Sigma}^*(B_*, B_*) \cong E_2^*(B/A) , \end{align*}

identifying the descent spectral sequence $E_2$ -term with the Hopf algebroid cohomology of $(B_*, \Sigma)$ . We claim that in each stem this $E_2$ -term has the same finite order as $A_*$ , so that the descent spectral sequence for $A \to B$ must collapse at $E_2 = E_\infty$ .

By convergence, the descent $E_2$ -term is an upper bound for $A_*$ . To show that the bound is exact, we consider the multiplicative Whitehead filtrations $\tau_{\geqslant\star} A$ and $\tau_{\geqslant\star} B$ of A and B, respectively. For each $q\geqslant0$ , we equip $B^{\otimes_A q+1}$ with the relative convolution filtration

\begin{align*}\operatorname{fil}^{\star} B^{\otimes_A q+1} = (\tau_{\geqslant\star} B)^{\otimes_{(\tau_{\geqslant\star} A)} q+1} , \end{align*}

having associated graded $\mathbb{E}_\infty$ -ring

\begin{align*}\operatorname{gr}^* B^{\otimes_A q+1} = (H\pi_* B)^{\otimes_{(H\pi_* A)} q+1} . \end{align*}

Here, $H\pi_* A$ and $H\pi_* B$ are to be interpreted as the graded $\mathbb{E}_\infty$ -rings $\operatorname{gr}^*(\tau_{\geqslant\star} A)$ and $\operatorname{gr}^*(\tau_{\geqslant\star}B)$ , respectively. We have proved in Lemma 2.23 that $A_* \to B_*$ is given by $\lambda'_1 \mapsto 0$ , $\lambda_2 \mapsto 0$ and $\mu \mapsto \mu$ , so that

\begin{align*}\overline{\Sigma} := \pi_* (H\pi_* B \otimes_{H\pi_* A} H\pi_* B) \cong \Gamma(\sigma\lambda'_1, \sigma\lambda_2) \otimes \Lambda(\hat\xi_1^2) \otimes P \otimes \Lambda(\hat\xi_1^2) \otimes P \otimes \mathbb{F}_2[\mu] \end{align*}

is even and free over $B_*$ . Hence $(B_*, \overline{\Sigma})$ is a flat Hopf algebroid, and as above we have compatible isomorphisms

\begin{align*}C^q_{\overline{\Sigma}}(B_*, B_*) = \overline{\Sigma} \otimes_{B_*} \cdots \otimes_{B_*} \overline{\Sigma}\overset{\cong}\longrightarrow (H\pi_* B)^{\otimes_{(H\pi_* A)} q+1} \end{align*}

for all $q\geqslant0$ . Since these bigraded groups are concentrated in even stems, and differentials reduce the stem by one, the convolution filtration spectral sequence

\begin{align*}\pi_* ((H\pi_* B)^{\otimes_{(H\pi_* A)} q+1}) \Longrightarrow \pi_*(B^{\otimes_A q+1}) \end{align*}

collapses at this term. This proves that $\pi_*(B^{\otimes_A q+1}) =E^q_1(B/A)$ has a descending filtration with associated graded given by $C^q_{\overline{\Sigma}}(B_*, B_*)$ . These filtrations are compatible for varying $q\geqslant0$ , so the descent $E_1$ -term is a filtered differential graded algebra with associated graded $E_1 = C^*_{\overline{\Sigma}}(B_*, B_*)$ . Passing to cohomology, we obtain the May–Ravenel spectral sequence

\begin{align*}E_2 = \operatorname{Ext}_{\overline{\Sigma}}(B_*, B_*) \Longrightarrow \operatorname{Ext}_\Sigma(B_*, B_*) \end{align*}

converging to the descent $E_2$ -term (cf. [Reference RavenelRav86, Theorem A1.3.9]).

We now view the Hopf algebroid $(B_*, \overline{\Sigma})$ as the tensor product of the three Hopf algebroids

\begin{align*}(\mathbb{F}_2, \Gamma(\sigma\lambda'_1, \sigma\lambda_2)),\quad(\Lambda(\hat\xi_1^2) \otimes P, \Lambda(\hat\xi_1^2) \otimes P \otimes \Lambda(\hat\xi_1^2) \otimes P), \quad(\mathbb{F}_2[\mu], \mathbb{F}_2[\mu]) . \end{align*}

These have cohomology algebras $\Lambda(\lambda'_1, \lambda_2)$ , $\mathbb{F}_2$ and $\mathbb{F}_2[\mu]$ , respectively, with $\lambda'_1, \lambda_2 \in \operatorname{Ext}^1$ and $\mu \in \operatorname{Ext}^0$ . This confirms that the May–Ravenel $E_2$ -term

\begin{align*}\operatorname{Ext}_{\overline{\Sigma}}(B_*, B_*) \cong \Lambda(\lambda'_1, \lambda_2) \otimes \mathbb{F}_2[\mu] \end{align*}

has the same finite order in each stem as $A_*$ , which implies that the May–Ravenel spectral sequence and the descent spectral sequence both collapse at their $E_2$ -terms. Moreover, there is no room for hidden multiplicative extensions, since $\lambda'_1$ and $\lambda_2$ both square to zero in $A_*$ .

We have now established that the descent spectral sequence

\begin{align*}E_1^q(B/A) = \pi_*(B^{\otimes_A q+1}) \Longrightarrow A_* \end{align*}

is concentrated in even internal degrees $n+q = 2w$ , having $E_2$ -term

\begin{align*}E_2(B/A) \cong \operatorname{Ext}_{\Sigma}(B_*, B_*) \cong\Lambda(\lambda'_1, \lambda_2) \otimes \mathbb{F}_2[\mu] \end{align*}

with (n,q)-bidegrees $\|\lambda'_1\| = (5,1)$ , $\|\lambda_2\| = (7,1)$ and $\|\mu\| = (8,0)$ . Following [Reference Hahn, Raksit and WilsonHRW22, Example 4.2.3], we apply [Reference Hahn, Raksit and WilsonHRW22, Corollary 2.2.17(1)] to the evenly free map $A \to B$ to see that

\begin{align*}\operatorname{fil}_{\mathrm{ev}}^{\star} A \overset{\simeq}\longrightarrow \operatorname{Tot} ( \operatorname{fil}_{\mathrm{ev}}^{\star} B^{\otimes_A \bullet+1} ) = \operatorname{Tot} ( \tau_{\geqslant 2\star}(B^{\otimes_A \bullet+1}) ) \end{align*}

is an equivalence. For each integer weight w, there is a spectral sequence converging to

\begin{align*} \pi_* \operatorname{Tot} ( \tau_{\geqslant 2w}(B^{\otimes_A\bullet+1}) ) , \end{align*}

with the $E_1$ -term given by the part of the descent spectral sequence $E_1(B/A)$ that is located in internal degrees $n+q\geqslant 2w$ . The $d_1$ -differential preserves this part, so the $E_2$ -term for weight w is given by the part of $E_2(B/A)$ in the same range of internal degrees. By naturality, this spectral sequence must collapse at the $E_2$ -term, since the full descent spectral sequence does so. It follows that

\begin{align*}\pi_* \operatorname{fil}_{\mathrm{ev}}^w A \longrightarrow A_* \end{align*}

maps the source isomorphically to the subgroup of classes in internal degree $\geqslant 2w$ , and $\pi_*\operatorname{gr}_{\mathrm{ev}}^w A$ is isomorphic to the summand in $A_*$ consisting of classes in internal degree $= 2w$ . Hence

\begin{align*}\pi_*\operatorname{gr}_{\mathrm{ev}}^* A \cong \Lambda(\lambda'_1, \lambda_2) \otimes \mathbb{F}_2[\mu] \end{align*}

as bigraded algebras, with $(n, 2w-n)$ -bidegrees $\|\lambda'_1\| =(5,1)$ , $\|\lambda_2\| = (7,1)$ and $\|\mu\| = (8,0)$ .

By [Reference Hahn, Raksit and WilsonHRW22, Corollary 2.2.21(1)], there are equivalences

\begin{align*} \operatorname{fil}_{\mathrm{ev}}^{\star}\mathbb{S} \simeq \operatorname{Tot} ( \tau_{\geqslant 2\star}C^\bullet(\mathrm{MU}/\mathbb{S}) ),\quad\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S} \simeq \operatorname{Tot} \Sigma^{2*} H\pi_{2*} C^{\bullet}(\mathrm{MU}/\mathbb{S}) . \end{align*}

The even filtration is extended to module spectra in [Reference Hahn, Raksit and WilsonHRW22, Appendix A]. As such the even filtration is lax symmetric monoidal [Reference Hahn, Raksit and WilsonHRW22, Remark A.1.6] and the associated graded functor is symmetric monoidal, so $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ is an $\mathbb{E}_\infty$ algebra in graded 2-complete spectra, and $\operatorname{gr}_{\mathrm{ev}/\mathbb{S}}^*$ is a lax symmetric monoidal functor from 2-complete spectra to $\operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ -modules. Note that when R is an $\mathbb{E}_{\infty}$ -ring and $\mathrm{MU}_*R$ is concentrated in even degrees, then $\operatorname{fil}_{\mathrm{ev}}^{\star}R$ agrees with $\operatorname{fil}_{\mathrm{ev}/\mathbb{S}}^{\star}R$ by [Reference Hahn, Raksit and WilsonHRW22, Corollary 1.1.6 and Remark 1.1.7].

By [Reference Gheorghe, Isaksen, Krause and RickaGIK⁺22], we can identify $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ with $C\tau$ in the $\mathbb{C}$ -motivic homotopy category $\operatorname{SH}(\mathbb{C})$ (cf. [Reference Hahn, Raksit and WilsonHRW22, Recollection 6.0.1]). Consequently, the $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -module $\overline{V}$ corresponds to the even $\mathrm{MU}_*\mathrm{MU}$ -comodule $\mathrm{MU}_*(V)$ under the equivalence of [Reference Gheorghe, Wang and XuGWX21, Theorem 1.13(2)]. Moreover, by [Reference Gheorghe, Wang and XuGWX21, Remark 4.15], this equivalence is symmetric monoidal.

Hence, for each $k\geqslant0$ , there is an essentially unique $\mathbb{E}_\infty$ $\operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ -algebra $\overline{V}(k)$ that corresponds to the commutative $\mathrm{MU}_*\mathrm{MU}$ -comodule algebra $\mathrm{MU}_*/(2, v_1, \dots, v_k)$ . For $k=0$ this gives the previously defined $\operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ -module $\overline{V}(0)$ an $\mathbb{E}_\infty$ algebra structure, even though $V(0) = C2$ is not a ring spectrum, and for $k\geqslant1$ it defines the $\mathbb{E}_\infty$ algebras $\overline{V}(k)$ , in spite of V(k) not existing as a spectrum. Furthermore, the isomorphism

(2.4) \begin{align} \mathrm{MU}_*(A(1)) \cong \mathrm{MU}_*/(2,v_1) \otimes_{\mathrm{MU}_*} \mathrm{MU}_*(C\eta)\end{align}

of $\mathrm{MU}_*\mathrm{MU}$ -comodules exhibits $\mathrm{MU}_*(A(1))$ as an $\mathrm{MU}_*/(2,v_1)$ -module in that category. It follows that we have an equivalence $\overline{A}(1) \simeq \overline{V}(1) \otimes \overline{C}\eta$ , exhibiting $\overline{A}(1)$ as a $\overline{V}(1)$ -module in the category of $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -modules. Here, we use the fact that $\eta \in \pi_1 \mathbb{S}$ is detected in Novikov filtration 1, so the cofiber of $\eta \colon \Sigma^{1,1} \operatorname{gr}_{\mathrm{ev}}^* \mathbb{S} \to \operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ is equivalent to $\overline{C}\eta$ as defined above. Hence we have a cofiber sequence

(2.5) \begin{align} \Sigma^{1,1} \overline{V}(1) \overset{\eta}\longrightarrow \overline{V}(1) \overset{i}\longrightarrow \overline{A}(1) \overset{j}\longrightarrow \Sigma^{2,0} \overline{V}(1)\end{align}

of $\overline{V}(1)$ -modules, mapping to the cofiber sequence

(2.6) \begin{align} \Sigma^{1,1} \overline{V}(2) \overset{\eta}\longrightarrow \overline{V}(2) \overset{i}\longrightarrow \overline{V}(2) \otimes \overline{C}\eta \overset{j}\longrightarrow \Sigma^{2,0} \overline{V}(2)\end{align}

of $\overline{V}(2)$ -modules.

When $\overline{V}$ and $M^*$ are $\operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ -modules, we shall write

\begin{align*}\overline{V}_* M^* := \pi_* (\overline{V} \otimes M^*) \end{align*}

for the homotopy groups of the graded spectrum $\overline{V} \otimes M^*$ , keeping in mind that this is a bigraded abelian group. Note that we are applying Convention 2.28 throughout this construction.

Proof. In view of (2.4), it suffices to prove this for $\mathrm{MU}_*(A(1))$ . Writing $\mathrm{MU}_*(C\eta) = \mathrm{MU}_*\{1, b_1\}$ , so that $\mathrm{MU}_*(A(1)) \cong\mathrm{MU}_*/(2,v_1) \{1, b_1\}$ , the coaction

\begin{align*}\nu \colon \mathrm{MU}_*(A(1)) \longrightarrow \mathrm{MU}_*\mathrm{MU} \otimes_{\mathrm{MU}_*} \mathrm{MU}_*(A(1)) \end{align*}

satisfies $\nu(b_1) = b_1 \otimes 1 + 1 \otimes b_1$ and $\nu(x_2) = b_2 \otimes1 + 1 \otimes x_2$ (see, e.g., [Reference RognesRog20, (2.1)]). If $\mathrm{MU}_*(A(1))$ were an $\mathrm{MU}_*\mathrm{MU}$ -comodule algebra, we would have $\nu(b_1^2) = (b_1\otimes 1 + 1 \otimes b_1)^2 = b_1^2 \otimes 1 +1 \otimes b_1^2$ . This contradicts each of the possibilities $b_1^2 = 0$ or $b_1^2 = x_2$ , and there are no other alternatives.

Lemma 2.31. There is an equivalence

\begin{align*}\overline{A}(1) \otimes \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \simeq \operatorname{gr}_{\mathrm{mot}}^* (A(1) \otimes \operatorname{THH}(\mathrm{ko})) \end{align*}

of $\operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ -modules.

Since $v_1$ acts injectively on $\mathrm{MU}_*(V(0) \otimes C\eta)$ and on $\mathrm{MU}_*(V(0) \otimes C\eta \otimes \mathrm{ku})$ it follows, by [Reference Gheorghe, Isaksen, Krause and RickaGIK⁺22, Proposition 3.18] again, that the columns in the following diagram are cofiber sequences.

Hence the lower horizontal arrow is also an equivalence.

Using Proposition 2.15, it follows that there are equivalences

\begin{align*}\overline{A}(1) \otimes \operatorname{gr}_{\mathrm{ev}}^* C^q(\operatorname{THH}(\mathrm{ku}/\mathrm{MU})/\operatorname{THH}(\mathrm{ko})) \overset{\simeq}\longrightarrow\operatorname{gr}_{\mathrm{ev}}^* ( A(1) \otimes C^q(\operatorname{THH}(\mathrm{ku}/\mathrm{MU})/\operatorname{THH}(\mathrm{ko})) ) \end{align*}

for all $q\geqslant0$ , compatible with the cosimplicial structure maps. Passing to totalizations, and using that $\overline{A}(1)$ is a finite $\operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ -module, we obtain

\begin{align*}&\overline{A}(1) \otimes \operatorname{Tot} ( \operatorname{gr}_{\mathrm{ev}}^* C^\bullet(\operatorname{THH}(\mathrm{ku}/\mathrm{MU})/\operatorname{THH}(\mathrm{ko})) ) \\&\quad\simeq \operatorname{Tot} ( \overline{A}(1) \otimes \operatorname{gr}_{\mathrm{ev}}^* C^\bullet(\operatorname{THH}(\mathrm{ku}/\mathrm{MU})/\operatorname{THH}(\mathrm{ko})) ) \\&\quad\simeq \operatorname{Tot} \operatorname{gr}_{\mathrm{ev}}^* ( A(1) \otimes C^\bullet(\operatorname{THH}(\mathrm{ku}/\mathrm{MU})/\operatorname{THH}(\mathrm{ko})) ) .\end{align*}

In view of Definitions 2.3 and 2.21, this establishes the asserted equivalence.

Remark 2.32. A consequence of Lemma 2.30 is that $\overline{C}\eta$ and $\overline{A}(1)$ are not $\mathbb{E}_1$ algebras in $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -modules. However, by Lemma 2.31, there is an identification of $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -modules

\begin{align*}\overline{A}(1) \otimes \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \simeq \operatorname{gr}_{\mathrm{mot}}^* (A(1) \otimes \operatorname{THH}(\mathrm{ko})) , \end{align*}

where the right-hand side is an $\mathbb{E}_\infty$ $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -algebra. We therefore use this to equip the left-hand side with an $\mathbb{E}_\infty$ $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -algebra structure. Note that the left-hand side also has a canonical action of the circle $\mathbb{T}$ , but this $\mathbb{T}$ -action is not an action through $\mathbb{E}_\infty$ -ring maps, because the right-hand side is not equipped with a compatible $\mathbb{T}$ -action. See Remark 5.7 for an algebraic incarnation of this.

Lemma 2.33. Let $M^*$ be a $\operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}$ -module. Then $\overline{C}\eta \otimes M^*\simeq \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ku} \otimes_{\operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}} M^*$ and there is a natural, trigraded, $\eta$ -Bockstein spectral sequence

\begin{align*}E_1 = (\overline{C}\eta_* M^*) \, [\eta] \Longrightarrow \pi_*(M^*) . \end{align*}

If $M^*$ is uniformly bounded below, then this spectral sequence is conditionally convergent. If $M^*$ is a $\operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}$ -algebra, then this is an algebra spectral sequence.

Proof. The Wood cofiber sequence (2.1) induces a cofiber sequence

\begin{align*}\Sigma^{1,1} \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} \overset{\eta}\longrightarrow \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} \overset{c}\longrightarrow \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ku} \overset{R}\longrightarrow \Sigma^{2,0} \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} \end{align*}

in the category of $\operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}$ -modules, where c is a map of $\mathbb{E}_\infty$ algebras. This follows as in [Reference Gheorghe, Isaksen, Krause and RickaGIK⁺22, Proposition 3.18], since $\mathrm{MU}_*(\mathrm{ko})$ is concentrated in even degrees by Proposition 2.16. Hence $\overline{C}\eta\otimes M^*$ and $\operatorname{gr}_{\mathrm{ev}}^* \mathrm{ku} \otimes_{\operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}} M^*$ are both the cofiber of $\eta \colon \Sigma^{1,1} M^* \to M^*$ .

The Bousfield–Kan homotopy (= descent) spectral sequence for the cosimplicial $\mathbb{E}_\infty$ $\operatorname{gr}_{\mathrm{ev}}^*\mathrm{ko}$ -algebra

\begin{align*}C^\bullet( \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ku} / \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} ) \end{align*}

is well known to be multiplicative, and converges conditionally (and strongly) to $\pi_* \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}$ . The normalized $\operatorname{Tot}$ -filtration is the same as the $\eta$ -adic tower

\begin{align*}\cdots \overset{\eta}\longrightarrow \Sigma^{2,2} \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} \overset{\eta}\longrightarrow \Sigma^{1,1} \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} \overset{\eta}\longrightarrow \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} , \end{align*}

so we can equally well call this the $\eta$ -Bockstein spectral sequence. In particular, its $E_1$ -term is

\begin{align*}E_1^q = \Sigma^{q,q} \pi_* \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ku} \cong (\pi_* \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ku}) \{\eta^q\} \end{align*}

for each $q\geqslant0$ , and it is concentrated in even internal degrees (= integer weights).

Tensoring over $\operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}$ with $M^*$ , the Bousfield–Kan spectral sequence for the cosimplicial $\operatorname{gr}_{\mathrm{ev}}^*\mathrm{ko}$ -module

\begin{align*}C^\bullet( \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ku} / \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} ) \otimes_{\operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}} M^* \end{align*}

has abutment $\pi_* M^*$ , and is multiplicative if $M^*$ is a $\operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}$ -algebra. The normalized $\operatorname{Tot}$ -filtration is the same as the $\eta$ -adic tower

\begin{align*}\cdots \overset{\eta}\longrightarrow \Sigma^{2,2} M^* \overset{\eta}\longrightarrow \Sigma^{1,1} M^* \overset{\eta}\longrightarrow M^* , \end{align*}

and the $E_1$ -term is

\begin{align*}E_1^* \cong \pi_* (\overline{C}\eta \otimes M^*) \, [\eta] . \end{align*}

If $M^*$ is uniformly bounded below, then its $\eta$ -adic tower has trivial (homotopy) limit, which ensures conditional convergence.

Example 2.34. The $\eta$ -Bockstein spectral sequence

\begin{align*}E_1 = (\pi_* \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ku}) \, [\eta] \Longrightarrow \pi_* \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} \end{align*}

has $E_1 = \mathbb{Z}[\eta, u]$ , $d_1(u) = 2 \eta$ and $E_2 = E_\infty =\mathbb{Z}[\eta, u^2]/(2\eta) \cong \pi_* \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}$ . The motivic (= Novikov) spectral sequence

\begin{align*}E_2 = \pi_* \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} \Longrightarrow \pi_* \mathrm{ko} \end{align*}

has $E_2 = \mathbb{Z}[\eta, u^2]/(2\eta)$ with $\|\eta\| = (1,1)$ , $\|u^2\|= (4,0)$ , $d_3(u^2) = \eta^3$ and $E_4 = E_\infty = (\mathbb{Z}\{1, 2u^2\}\oplus \mathbb{Z}/2\{\eta, \eta^2\}) \otimes \mathbb{Z}[u^4]$ . Here, $A \in \pi_4(\mathrm{ko})$ and $B \in \pi_8(\mathrm{ko})$ are detected by $2 u^2$ and $u^4$ , respectively. The bigraded homotopy rings of the $\mathbb{E}_\infty$ $\operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}$ -algebras

are thus as follows.

Definition 2.35. Let $v_2 \colon \Sigma^{6,0} \overline{V}(1) \to \overline{V}(1)$ be the $\operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ -module map corresponding to the $\mathrm{MU}_*\mathrm{MU}$ -comodule homomorphism $v_2 \colon \Sigma^6 \mathrm{MU}_*/(2,v_1) \to \mathrm{MU}_*/(2,v_1)$ , so that there is a cofiber sequence

\begin{align*}\Sigma^{6,0} \overline{V}(1) \overset{v_2}\longrightarrow \overline{V}(1) \overset{i_2}\longrightarrow \overline{V}(2) \overset{j_2}\longrightarrow \Sigma^{7,-1} \overline{V}(1) \end{align*}

of $\operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ -modules. The induced map

\begin{align*}v_2 \colon \Sigma^{6,0} \overline{V}(1) \otimes \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} \longrightarrow \overline{V}(1) \otimes \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} \end{align*}

of $\operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}$ -modules is null-homotopic, and there is a unique class

\begin{align*}\varepsilon_2 \in \overline{V}(2)_* \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} \end{align*}

in bidegree $\|\varepsilon_2\| = (7,-1)$ with $j_2(\varepsilon_2) =\Sigma^{7,-1} 1$ . We have $\varepsilon_2^2 = 0$ , since the group in bidegree $(14,-2)$ is trivial. Then $\overline{V}(2)_* \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko} \cong\Lambda(\varepsilon_2) \otimes \overline{V}(1)_* \operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}$ , and in general we have a natural algebra isomorphism

\begin{align*}\Lambda(\varepsilon_2) \otimes \overline{V}(1)_* M^* \cong \overline{V}(2)_* M^* \end{align*}

for any $\operatorname{gr}_{\mathrm{ev}}^* \mathrm{ko}$ -algebra $M^*$ .

Corollary 2.36. There are preferred isomorphisms of bigraded $\mathbb{F}_2$ -algebras

\begin{align*}\overline{A}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \cong \Lambda(\lambda'_1, \lambda_2) \otimes \mathbb{F}_2[\mu] \end{align*}

and

\begin{align*}(\overline{V}(2) \otimes \overline{C}\eta )_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \cong \Lambda(\varepsilon_2, \lambda'_1, \lambda_2) \otimes \mathbb{F}_2[\mu] . \end{align*}

In the following proposition, we evaluate the spectral sequence from Lemma 2.33 in the case when $M^* = \overline{V}(1) \otimes\operatorname{gr}_{\mathrm{mot}}^*\operatorname{THH}(\mathrm{ko})$ and $\overline{C}\eta \otimes M^* \cong \overline{A}(1) \otimes\operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})$ : for an illustration, see Figure 2. The authors thank an anonymous referee for pointing out how to remove an indeterminacy in the following statement, which we had previously only resolved later in the paper, in a more indirect manner.

Figure 2.

The $\eta$ -Bockstein $E_{1}\Longrightarrow \overline{V}(1)_{*}\operatorname{gr}_{\mathrm{mot}}^{*}\operatorname{THH}(\mathrm{ko})$ .

Proposition 2.37. The $\eta$ -Bockstein spectral sequence

\begin{align*}E_1 = \overline{A}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \, [\eta] \Longrightarrow \overline{V}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \end{align*}

has differentials

\begin{align*} d_1 (\lambda_2) &= \eta \lambda'_1, \\ d_3 (\lambda'_1\lambda_2) &= \eta^3 \mu\end{align*}

and no further differentials besides those generated by the Leibniz rule. Moreover, there is no room for $\eta$ -extensions. Consequently, we can identify

\begin{align*}\overline{V}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \cong \frac{\mathbb{F}_2[\eta, \lambda'_1, \mu]}{(\eta\lambda'_1,(\lambda'_1)^2 = \eta^2 \mu)} \end{align*}

as a bigraded $\mathbb{F}_2$ -algebra, where $\|\eta\| = (1,1)$ , $\|\mu\| =(8,0)$ and $\|\lambda'_1\| = (5,1)$ .

Proof. We deduce these differentials using a small part of the known (implicitly 2-complete) computation of $\pi_*\operatorname{THH}(\mathrm{ko})$ from [Reference Angeltveit, Hill and LawsonAHL10, § 7]. The unit $\mathrm{ko} \to\operatorname{THH}(\mathrm{ko})$ and augmentation $\epsilon \colon\operatorname{THH}(\mathrm{ko})\to \mathrm{ko}$ exhibit ko as a retract of $\operatorname{THH}(\mathrm{ko})$ in the category of $\mathbb{E}_\infty$ -rings. We write $\operatorname{THH}(\mathrm{ko})/\mathrm{ko}$ for the complementary summand in ko-modules. In degrees $* \lt 12$ we have

\begin{align*} H_*(\mathrm{ko})\{\sigma\bar\xi_1^4, \sigma\bar\xi_2^2, \sigma\bar\xi_3\} \cong H_*(\operatorname{THH}(\mathrm{ko})/\mathrm{ko}), \end{align*}

so there is an 11-connected map $\Sigma^5 \mathrm{ksp}\simeq \mathrm{ko} \otimes (S^5 \cup_\eta e^7 \cup_2 e^8) \to\operatorname{THH}(\mathrm{ko})/\mathrm{ko}$ . By [Reference Angeltveit, Hill and LawsonAHL10, Corollary 7.3, Figure 5], the $\eta^2$ -multiple in $\pi_6\mathrm{ksp} \cong \pi_{11} \Sigma^5 \mathrm{ksp}$ maps to zero in $\operatorname{THH}$ , so

\begin{align*} \pi_*(\operatorname{THH}(\mathrm{ko})/\mathrm{ko}) \cong (\mathbb{Z}, 0, 0, 0, \mathbb{Z}, \mathbb{Z}/2, 0) \end{align*}

for $5 \leqslant * \leqslant 11$ .

We consider the $\eta$ -Bockstein spectral sequence

\begin{align*}E_1 = \Lambda(\lambda'_1, \lambda_2) \otimes \mathbb{F}_2[\eta, \mu] \Longrightarrow \overline{V}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \end{align*}

with $\|\lambda'_1\| = (5,1)$ , $\|\lambda_2\| = (7,1)$ , $\|\mu\| = (8,0)$ and $\|\eta\| = (1,1)$ , the $v_1$ -Bockstein spectral sequence

\begin{align*}E_1 = \overline{V}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \, [v_1] \to \overline{V}(0)_*\operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \end{align*}

with $\|v_1\| = (2,0)$ , the $v_0$ -Bockstein spectral sequence

\begin{align*}E_1 = \overline{V}(0)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \, [v_0] \Longrightarrow \pi_*\operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \end{align*}

with $\|v_0\| = (0,0)$ , and the motivic spectral sequence

\begin{align*}E_2 = \pi_*\operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \Longrightarrow \pi_*\operatorname{THH}(\mathrm{ko}) . \end{align*}

In each case the spectral sequence for ko splits off as a direct summand. Taking this into account, there is no possible source or target for a differential affecting $\lambda'_1$ in any of these spectral sequences. Hence $\lambda'_1$ survives in bidegree (5,1) to detect the generator of $\pi_5(\operatorname{THH}(\mathrm{ko})/\mathrm{ko}) \cong \mathbb{Z}$ . Since $\pi_6(\operatorname{THH}(\mathrm{ko})/\mathrm{ko}) = 0$ , it follows that $\eta \lambda'_1$ in bidegree (6,2) is an infinite cycle that detects zero, i.e., a boundary in one of these spectral sequences. Since $\eta \lambda'_1$ is not a $v_1$ - or $v_0$ -multiple, it cannot be a $v_1$ -Bockstein or $v_0$ -Bockstein boundary. Since the motivic $E_2$ -term is readily seen to be zero in bidegree (7,0), it can also not be a motivic boundary. Hence $d_1(\lambda_2) = \eta \lambda'_1$ in the $\eta$ -Bockstein spectral sequence is the only remaining possibility.

By construction, this $d_1$ -differential can be rewritten as $i(j(\lambda_2)) = \Sigma^{2,0} \lambda'_1$ , with i and j induced by the maps in (2.5). Here,

\begin{align*}i \colon \overline{V}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \longrightarrow \overline{A}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \end{align*}

must be injective (in fact, bijective) in bidegree (5,1), since there are no nonzero $\eta$ -multiples there, so $\Sigma^{-2,0} j(\lambda_2)$ in its source is the unique lift over i of $\lambda'_1$ , which we also denote by $\lambda'_1$ . We shall return to this insight at the end of the proof.

There is no room for other $\eta$ -Bockstein $d_1$ -differentials, so the next differential to be determined is $d_3(\lambda'_1 \lambda_2)\in \mathbb{F}_2\{\eta^3 \mu\}$ . On one hand, if $d_3(\lambda'_1 \lambda_2)= \eta^3 \mu$ , then the $\eta$ -Bockstein $E_\infty$ -term (modulo the summand for ko) will be

\begin{align*}\mathbb{F}_2\{\lambda'_1, \mu, \eta \mu, \eta^2 \mu\} \end{align*}

in stems $\leqslant 12$ . On the other hand, if $d_3(\lambda'_1 \lambda_2) =0$ then it will be

\begin{align*}\mathbb{F}_2\{\lambda'_1, \mu, \eta \mu, \eta^2 \mu, \eta^3 \mu\} \end{align*}

in stems $\leqslant 11$ , with the 12-stem concentrated in motivic filtrations $\geqslant 2$ . In either case this determines $\overline{V}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})$ in these stems.

The first nonzero $v_1$ -Bockstein differential is $d_1(\mu) =v_1 \lambda'_1$ . If it were not there, then $v_1 \lambda'_1$ would survive to $\overline{V}(0)_*\operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})$ and $\pi_*\operatorname{gr}_{\mathrm{mot}}^*\operatorname{THH}(\mathrm{ko})$ to detect a nonzero class in the trivial group $\pi_7(\operatorname{THH}(\mathrm{ko})/\mathrm{ko})$ , which is impossible. There is no room for other $v_1$ -Bockstein differentials affecting stems $\leqslant 11$ , so if $d_3(\lambda'_1 \lambda_2) = \eta^3 \mu$ , then the $v_1$ -Bockstein $E_\infty$ -term (modulo the summand for ko) will be

\begin{align*}\mathbb{F}_2\{\lambda'_1, \eta \mu, \eta^2 \mu, v_1 \eta \mu\} \end{align*}

in stems $\leqslant 11$ , while if $d_3(\lambda'_1 \lambda_2) = 0$ , then it will be

\begin{align*}\mathbb{F}_2\{\lambda'_1, \eta \mu, \eta^2 \mu, \eta^3 \mu, v_1 \eta \mu\} \end{align*}

in these stems. In either case, the 12-stem is concentrated in motivic filtrations $\geqslant 2$ , and these expressions determine $\overline{V}(0)_*\operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})$ in this range of stems.

In the $v_0$ -Bockstein spectral sequence, there is no room for differentials on ( $\lambda'_1$ and) $\eta \mu$ . Multiplying by $\eta^2$ , it follows that $\eta^3 \mu$ is an infinite cycle (but possibly zero). Since it is not a $v_0$ -multiple, it cannot be a $v_0$ -Bockstein boundary, and since it is in motivic filtration 3, and the motivic $E_2$ -term is now known to be zero in bidegrees (12,0) and (12,1), it cannot be a motivic $d_r$ -boundary for $r\geqslant2$ . Hence if $d_3(\lambda'_1 \lambda_2)$ were zero, then $\eta^3 \mu$ would survive to $\overline{V}(0)_*\operatorname{gr}_{\mathrm{mot}}^*\operatorname{THH}(\mathrm{ko})$ and $\pi_*\operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})$ to detect a nonzero class in $\pi_{11}(\operatorname{THH}(\mathrm{ko})/\mathrm{ko}) = 0$ , which is impossible.

This contradiction shows that $d_3(\lambda'_1 \lambda_2) = \eta^3\mu$ , as claimed. This leaves the $\eta$ -Bockstein $E_4$ -term

\begin{align*}\frac{ \Lambda(\lambda'_1) \otimes \mathbb{F}_2[\eta, \mu] } { (\eta \lambda'_1, \eta^3 \mu) } . \end{align*}

There is no room for further differentials, so this is also the $E_\infty$ -term. The only multiplicative ambiguity in the abutment $\overline{V}(1)_* \operatorname{gr}_{\mathrm{mot}}^*\operatorname{THH}(\mathrm{ko})$ is whether $(\lambda'_1)^2$ is 0 or $\eta^2 \mu$ . To resolve this indeterminacy, observe that

\begin{align*}j \colon \overline{A}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \longrightarrow \Sigma^{2,0} \overline{V}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \end{align*}

is a map of $\overline{V}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})$ -modules, so $j(\lambda_2) = \Sigma^{2,0} \lambda'_1$ implies that

\begin{align*}j(\lambda'_1 \lambda_2) = \Sigma^{2,0} (\lambda'_1)^2 . \end{align*}

If $(\lambda'_1)^2$ were 0, then $\lambda'_1 \lambda_2$ would lift over i to a class in $\overline{V}(1)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})$ , but this cannot occur because it would contradict the nonzero differential $d_3(\lambda'_1\lambda_2) = \eta^3 \mu$ . Hence $(\lambda'_1)^2 = \eta^2 \mu$ .

Corollary 2.38. We can identify

\begin{align*}\overline{V}(2)_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \cong \Lambda(\varepsilon_2) \otimes \frac{\mathbb{F}_2[\eta, \lambda'_1, \mu]}{ (\eta\lambda'_1, (\lambda'_1)^2 = \eta^2 \mu ) } \end{align*}

as a bigraded $\mathbb{F}_2$ -algebra.

3. Filtering the motivic associated graded

In subsequent sections, we will use various filtrations on the associated graded of the motivic filtration on topological negative cyclic homology, topological periodic cyclic homology, and the $C_2$ -homotopy fixed points and $C_2$ -Tate construction on topological Hochschild homology. Here, we discuss two filtrations that one might consider and show in what sense they agree.

We first discuss a collection of increasing filtrations $\operatorname{fil}_{\star}^{\mathrm{GM}}(-)$ . Here, we use (subscripted) homological indexing. Recall that the circle group $\mathbb{T}$ acts freely on $S^1 \subset S^3 \subset \cdots \subset S^\infty = E\mathbb{T}$ and $C_2$ acts freely on $S^0 \subset S^1 \subset \cdots \subset S^{\infty}=EC_2$ .

Definition 3.1. Define a filtered $\mathbb{T}$ -spectrum $\widetilde{E\mathbb{T}}_{\star}$ by $\widetilde{E\mathbb{T}}_{2k}=\widetilde{E\mathbb{T}}_{2k+1}= S^{k\mathbb{C}}$ , and a filtered $C_2$ -spectrum $(\widetilde{EC_2})_{\star}$ by $(\widetilde{EC_2})_k=S^{k\mathbb{R}}$ , for all integers k. Following Greenlees and May [Reference Greenlees and MayGM95], for each spectrum X with $\mathbb{T}$ -action, we define a filtered spectrum

\begin{align*} \mathrm{GM}_{\star}^{\mathbb{T}}(X):= [ \widetilde{E\mathbb{T}}_{\star}\otimes F(E\mathbb{T}_+,X) ]^{\mathbb{T}} \end{align*}

and for each spectrum X with $C_2$ -action, we define a filtered spectrum

\begin{align*} \mathrm{GM}_{\star}^{C_2}(X):= [(\widetilde{EC_2})_{\star}\otimes F(EC_{2 +},X)]^{C_2} . \end{align*}

In filtration $\star = 0$ , these specialize to the homotopy fixed point spectra $ \mathrm{GM}_{0}^{\mathbb{T}}(X)= X^{h\mathbb{T}}$ and $\mathrm{GM}_{0}^{C_2}(X)= X^{hC_2}$ . More generally, we have

\begin{align*}\mathrm{GM}_{-2k}^{\mathbb{T}}(X)=\mathrm{GM}_{-2k+1}^{\mathbb{T}}(X)\simeq F(E\mathbb{T}/S^{2k-1},X)^{\mathbb{T}} \end{align*}

and $\mathrm{GM}_{-k}^{C_2}(X)\simeq F(EC_2/S^{k-1},X)^{C_2}$ for all $k\geqslant 0$ (cf. [Reference GreenleesGre87, p. 437] or [Reference Hedenlund and RognesHR24, Lemma 6.32]).

For brevity, let

\begin{align*} Y^{\bullet}=C^{\bullet}(\operatorname{THH}(\mathrm{ku}/\mathrm{MU})/\operatorname{THH}(\mathrm{ko})) . \end{align*}

This is naturally an $\mathbb{E}_\infty$ algebra in $C^\bullet(\mathrm{MU}/\mathbb{S})$ -modules with $\mathbb{T}$ -action. Define a filtered graded spectrum

\begin{align*}\operatorname{fil}_{\star}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^{\ast}\operatorname{TP}(\mathrm{ko}) := \operatorname{Tot} \Sigma^{2\ast}H\pi_{2\ast } \mathrm{GM}_{\star}^{\mathbb{T}}(Y^{\bullet}) , \end{align*}

and define $\operatorname{fil}_{\star}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^{\ast}\operatorname{TC}^{-}(\mathrm{ko})$ and $\operatorname{fil}_{\star}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^{\ast}F(S^{2\ell+1}_+,\operatorname{THH}(\mathrm{ko}))^{\mathbb{T}}$ for $\ell\geqslant 0$ by truncating this filtration to live in the ranges $\star \leqslant 0$ and $-2\ell \leqslant \star \leqslant 0$ , respectively. Similarly, define a filtered graded spectrum

\begin{align*}\operatorname{fil}_{\star}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^{\ast}\operatorname{THH}(\mathrm{ko})^{tC_2} := \operatorname{Tot} \Sigma^{2\ast}H\pi_{2\ast} \mathrm{GM}_{\star}^{C_2}(Y^{\bullet}) , \end{align*}

and define $\operatorname{fil}_{\star}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^{\ast}\operatorname{THH}(\mathrm{ko})^{hC_2}$ by truncating this filtration to the range $\star \leqslant 0$ .

We know from [Reference Bökstedt, Bruner, Lunøe-Nielsen and RognesBBL⁺14, Proposition 3.8] (cf. [Reference Nikolaus and ScholzeNS18, Lemma II.4.2]) that there is a natural equivalence $G \colon (Y^q)^{t\mathbb{T}} \simeq((Y^q)^{tC_2})^{h\mathbb{T}}$ for each $q\geqslant0$ . Since each $Y^q$ is even and 2-torsion free by Proposition 2.15, we know that each $(Y^q)^{tC_2}$ is even, and we obtain a second filtration of $\operatorname{gr}_{\mathrm{mot}}^*\operatorname{TP}(\mathrm{ko}) \simeq \operatorname{gr}_{\mathrm{mot}}^*(\operatorname{THH}(\mathrm{ko})^{tC_2})^{h\mathbb{T}}$ by truncating the filtration $\operatorname{Tot} \Sigma^{2\ast} H\pi_{2\ast} \mathrm{GM}_{\star}^{\mathbb{T}}((Y^{\bullet})^{tC_2})$ to the range $\star\leqslant 0$ . We write $\operatorname{fil}_{\star}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^{\ast}(\operatorname{THH}(\mathrm{ko})^{tC_2})^{h\mathbb{T}}$ for this filtration. Letting the weight w vary, these are modules over $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S} \simeq \operatorname{Tot} \Sigma^{2*} H\pi_{2*} C^\bullet(\mathrm{MU}/\mathbb{S})$ , regarded as a filtered graded spectrum with $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ in filtrations $\star \geqslant 0$ and 0 in filtrations $\star \lt0$ , where the structure maps are identities for $\star \geqslant 0$ and the zero map for $\star \lt0$ .

We refer to these as Greenlees–May filtrations. Each one is complete and exhaustive, and hence induces a conditionally convergent spectral sequence. We write

\begin{align*} \operatorname{gr}_{s}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^wF(\mathrm{ko})= \operatorname{fil}_{s}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^wF(\mathrm{ko})/ \operatorname{fil}_{s-1}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^wF(\mathrm{ko}) \end{align*}

for each $F\in \{\operatorname{TC}^{-},\operatorname{TP}, F(S^{2\ell+1}_+,\operatorname{THH})^{\mathbb{T}},\operatorname{THH}^{hC_2},\operatorname{THH}^{tC_2},(\operatorname{THH}^{tC_2})^{h\mathbb{T}}\}$ . Letting s and w vary, these are modules over $\operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ , with action mapping $\operatorname{gr}_{s}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^wF(\mathrm{ko})$ to $\operatorname{gr}_{s}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^{w+*}F(\mathrm{ko})$ .

Alternatively, we can take the approach of [Reference Hahn, Raksit and WilsonHRW22, Construction 2.1.14] and define the following filtrations of the associated graded of the motivic filtration. Here, we return to our standard conventions and use (superscripted) cohomological indexing.

Definition 3.2. Let $Y^{\bullet}=C^{\bullet}(\operatorname{THH}(\mathrm{ku}/\mathrm{MU})/\operatorname{THH}(\mathrm{ko}))$ . We define filtered graded objects

\begin{align*}\operatorname{fil}_+^{\star}\operatorname{gr}_{\mathrm{mot}}^*\operatorname{TC}^{-}(\mathrm{ko}) & := \operatorname{Tot} \Sigma^{2*}H\pi_{2*} ( \tau_{\geqslant \star} Y^{\bullet})^{h\mathbb{T}} , \\\operatorname{fil}_+^{\star}\operatorname{gr}_{\mathrm{mot}}^*\operatorname{TP}(\mathrm{ko})& := \operatorname{Tot} \Sigma^{2*}H\pi_{2*} ( \tau_{\geqslant \star} Y^{\bullet})^{t\mathbb{T}} ,\\\operatorname{fil}_+^{\star}\operatorname{gr}_{\mathrm{mot}}^*F(S^{2\ell+1}_+,\operatorname{THH}(\mathrm{ko}))^{\mathbb{T}}& := \operatorname{Tot} \Sigma^{2*}H\pi_{2*}F(S^{2\ell+1}_+, \tau_{\geqslant \star} Y^{\bullet} )^{\mathbb{T}} ,\\\operatorname{fil}_+^{\star}\operatorname{gr}_{\mathrm{mot}}^*\operatorname{THH}(\mathrm{ko})^{hC_2} & := \operatorname{Tot} \Sigma^{2*}H\pi_{2*} ( \tau_{\geqslant \star} Y^{\bullet} )^{hC_2} ,\\\operatorname{fil}_+^{\star}\operatorname{gr}_{\mathrm{mot}}^*\operatorname{THH}(\mathrm{ko})^{tC_2}& := \operatorname{Tot} \Sigma^{2*}H\pi_{2*} ( \tau_{\geqslant\star} Y^{\bullet})^{tC_2},\\\operatorname{fil}_+^{\star}\operatorname{gr}_{\mathrm{mot}}^*(\operatorname{THH}(\mathrm{ko})^{tC_2})^{h\mathbb{T}} & := \operatorname{Tot} \Sigma^{2*}H\pi_{2*} ( \tau_{\geqslant \star} ((Y^{\bullet})^{tC_2}) )^{h\mathbb{T}} ,\end{align*}

where $\ell\geqslant 0$ . Each one is complete and exhaustive, by proofs similar to that of Lemma 2.19, and hence induces a conditionally convergent spectral sequence. We refer to these as Whitehead filtrations. They come equipped with an action of the $\mathbb{E}_\infty$ algebra in filtered graded spectra

\begin{align*} \operatorname{Tot} \Sigma^{2*}H\pi_{2*} ( \tau_{\geqslant \star} C^{\bullet}(\mathrm{MU}/\mathbb{S} ) ) \simeq\begin{cases}\operatorname{gr}_\mathrm{ev}^* \mathbb{S} & \text{for $\star \leqslant 2*$,} \\0 & \text{for $\star \gt 2*$.}\end{cases} \end{align*}

We write

\begin{align*} \operatorname{gr}_+^s\operatorname{gr}_{\mathrm{mot}}^wF(\mathrm{ko})= \operatorname{fil}_+^s\operatorname{gr}_{\mathrm{mot}}^wF(\mathrm{ko})/\operatorname{fil}_+^{s+1}\operatorname{gr}_{\mathrm{mot}}^wF(\mathrm{ko}) \end{align*}

for $F\in \{\operatorname{TC}^{-},\operatorname{TP}, F(S^{2\ell+1}_+,\operatorname{THH})^{\mathbb{T}}, \operatorname{THH}^{hC_2}, \operatorname{THH}^{tC_2},(\operatorname{THH}^{tC_2})^{h\mathbb{T}}\}$ . Letting s and w vary, these are modules over $\operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ , with action mapping $\operatorname{gr}_+^s \operatorname{gr}_{\mathrm{mot}}^w F(\mathrm{ko})$ to $\operatorname{gr}_+^{s+2*} \operatorname{gr}_{\mathrm{mot}}^{w+*} F(\mathrm{ko})$ .

The filtrations from Definition 3.2 are clearly multiplicative, whereas our identification of the spectral sequence starting pages is more clear for the filtrations of Definition 3.1. Luckily, we can prove that they agree, up to an indexing shift that depends upon the weight grading.

Proposition 3.3. There is an equivalence of filtered $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -modules

\begin{align*} \operatorname{fil}_{-2n}^{\mathrm{GM}}\operatorname{gr}_{\mathrm{mot}}^{\ast}F(\mathrm{ko}) \longrightarrow \operatorname{fil}_+^{2n+2\ast}\operatorname{gr}_{\mathrm{mot}}^{\ast}F(\mathrm{ko}) \end{align*}

for $F\in\{\operatorname{TC}^{-},\operatorname{TP},F(S^{2\ell+1}_+,\operatorname{THH})^{\mathbb{T}},(\operatorname{THH}^{tC_2})^{h\mathbb{T}}\}$ .

Proof. Again, let $Y^{\bullet}=C^{\bullet}(\operatorname{THH}(\mathrm{ku}/\mathrm{MU})/\operatorname{THH}(\mathrm{ko}))$ . We first discuss the case $F=\operatorname{TP}$ . We have

\begin{align*}\operatorname{gr}_{\mathrm{mot}}^w \operatorname{THH}(\mathrm{ko}) = \operatorname{Tot} \Sigma^{2w}H\pi_{2w}Y^\bullet\end{align*}

and

\begin{align*}\operatorname{gr}_{\mathrm{mot}}^w \operatorname{TP}(\mathrm{ko}) = \operatorname{Tot} \Sigma^{2w}H\pi_{2w} ((Y^\bullet)^{t\mathbb{T}}) .\end{align*}

The latter is filtered by

\begin{align*}\operatorname{fil}_{m}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^w \operatorname{TP}(\mathrm{ko}) = \operatorname{Tot} \Sigma^{2w}H\pi_{2w}\mathrm{GM}_{m}^{\mathbb{T}} (Y^\bullet)\end{align*}

for $m\in \mathbb{Z}$ , with associated graded

\begin{align*}\operatorname{gr}_{-2n}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^w \operatorname{TP}(\mathrm{ko}) &= \operatorname{Tot} \Sigma^{2w}H\pi_{2w} (\Omega^{2n} Y^\bullet) \\ &\cong (\operatorname{Tot} \Sigma^{2w+2n}H\pi_{2w+2n} Y^\bullet) \, \{t^n\}\end{align*}

in even gradings, where $t^n$ reduces stem by 2n. In odd gradings, $\operatorname{gr}_{-2n+1}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^w \operatorname{TP}(\mathrm{ko})= 0$ .

The Whitehead filtration

\begin{align*}\operatorname{fil}^{2w+2n}_+ \operatorname{gr}_{\mathrm{mot}}^w \operatorname{TP}(\mathrm{ko}) = \operatorname{Tot} \Sigma^{2w}H\pi_{2w} ((\tau_{\geqslant 2w+2n} Y^\bullet)^{t\mathbb{T}})\end{align*}

has associated graded

\begin{align*}\operatorname{gr}^{2w+2n}_+ \operatorname{gr}_{\mathrm{mot}}^w \operatorname{TP}(\mathrm{ko}) &= \operatorname{Tot} \Sigma^{2w}H\pi_{2w} ((\Sigma^{2w+2n}H\pi_{2w+2n} Y^\bullet)^{t\mathbb{T}}) \\ &\cong (\operatorname{Tot} \Sigma^{2w+2n}H\pi_{2w+2n} Y^\bullet) \, \{t^n\} .\end{align*}

Again, $\operatorname{gr}^{2w+2n-1}_+ \operatorname{gr}_{\mathrm{mot}}^w \operatorname{TP}(\mathrm{ko})= 0$ . These constructions are lax symmetric monoidal, by [Reference Nikolaus and ScholzeNS18, Corollary I.4.3].

The composite map

\begin{align*}\Sigma^{2w}H\pi_{2w}([\widetilde{E\mathbb{T}}_{-2n} \otimes F(E\mathbb{T}_+, Y^\bullet)]^{\mathbb{T}}) \longrightarrow \Sigma^{2w}H\pi_{2w} ((Y^\bullet)^{t\mathbb{T}}) \longrightarrow \Sigma^{2w}H\pi_{2w} ((\tau_{\lt 2w+2n} Y^\bullet)^{t\mathbb{T}}) \end{align*}

factors through

\begin{align*}\Sigma^{2w}H\pi_{2w}( [ \widetilde{E\mathbb{T}}_{-2n} \otimes F(E\mathbb{T}_+, \tau_{\lt 2w+2n} Y^\bullet)]^{\mathbb{T}}) = 0 ,\end{align*}

and hence the first map in the composite lifts to induce a map of filtrations

\begin{align*}\operatorname{fil}_{-2n}^{\mathrm{GM}} \operatorname{gr}_{\mathrm{mot}}^w \operatorname{TP}(\mathrm{ko}) \longrightarrow \operatorname{fil}^{2w+2n}_+ \operatorname{gr}_{\mathrm{mot}}^w \operatorname{TP}(\mathrm{ko}) ,\end{align*}

both with limit 0 and colimit $\operatorname{gr}_{\mathrm{mot}}^w \operatorname{TP}(\mathrm{ko})$ . From the definition of the filtered $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -module structure on either side, it is clear that this is a map of filtered $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -modules. The associated graded spectra are equivalent as $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -modules, and hence so are the filtrations, up to the reindexing from $\operatorname{fil}_{-2n}^{\mathrm{GM}}$ to $\operatorname{fil}^{2w+2n}_+$ . Hence the filtrations agree, up to reindexing. By restricting to filtrations $m\leqslant 0$ , or $-2\ell \leqslant m\leqslant 0$ , the same proof applies when replacing $\operatorname{TP}$ with $\operatorname{TC}^{-}$ or $F(S^{2\ell+1}_+,\operatorname{THH})^{\mathbb{T}}$ . Since $Y^q$ is even and 2-torsion free, we know $(Y^{q})^{tC_2}$ is even for each $q\geqslant 0$ , and again the same proof applies to deduce the result for $F=(\operatorname{THH}^{tC_2})^{h\mathbb{T}}$ .

Corollary 3.4. Let $\overline{V}$ be a $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -module. There are trigraded spectral sequences

\begin{align*}\overline{V}_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \, [t] & \Longrightarrow \overline{V}_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{TC}^{-}(\mathrm{ko}) , \\\overline{V}_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \, [t^{\pm 1}] & \Longrightarrow \overline{V}_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{TP}(\mathrm{ko}) , \\\overline{V}_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \, [t]/(t^{\ell+1}) & \Longrightarrow \overline{V}_* \operatorname{gr}_{\mathrm{mot}}^* F(S^{2\ell+1}_+,\operatorname{THH}(\mathrm{ko}))^{\mathbb{T}},\\\overline{V}_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})^{tC_2} \, [t] & \Longrightarrow \overline{V}_* \operatorname{gr}_{\mathrm{mot}}^* (\operatorname{THH}(\mathrm{ko})^{tC_2})^{h\mathbb{T}} .\end{align*}

The first is the (algebraic) homotopy fixed point spectral sequence, the second is the (algebraic) Tate spectral sequence, the third is the approximate fixed point spectral sequence, and the last is the $\mu$ -inverted homotopy fixed point spectral sequence.

When $\overline{V}$ is a $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -algebra, these are algebra spectral sequences. When $\overline{V}$ is a module over a given $\operatorname{gr}_\mathrm{ev}^* \mathbb{S}$ -algebra, they are module spectral sequences over the corresponding algebra spectral sequences. When $\overline{V}$ is a finite $\operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ -module, they are conditionally convergent.

Proof. We apply $\overline{V}\otimes(-)$ to the filtered $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -modules $\operatorname{fil}_{\star}^{\mathrm{GM}}\operatorname{gr}_{\mathrm{mot}}^{\ast}F(\mathrm{ko})$ for

\begin{align*} F\in \{\operatorname{TC}^{-}, \operatorname{TP}, F(S^{2\ell+1}_+,\operatorname{THH})^{\mathbb{T}}, (\operatorname{THH}^{tC_2})^{h\mathbb{T}}\}. \end{align*}

In the case of $F=\operatorname{TP}$ , we have observed in the proof of Proposition 3.3 that there is an equivalence $\operatorname{gr}^{\mathrm{GM}}_{-2n} \operatorname{gr}_{\mathrm{mot}}^* \operatorname{TP}(\mathrm{ko}) \simeq \operatorname{gr}_{\mathrm{mot}}^{*+n} \operatorname{THH}(\mathrm{ko})\{t^n\}$ , so that we have an equivalence

\begin{align*} \overline{V} \otimes \operatorname{gr}^{\mathrm{GM}}_{-2n} \operatorname{gr}_{\mathrm{mot}}^* \operatorname{TP}(\mathrm{ko}) \simeq\overline{V} \otimes \operatorname{gr}_{\mathrm{mot}}^{*+n} \operatorname{THH}(\mathrm{ko}) \{t^n\} \end{align*}

for each integer n. The cases $F = \operatorname{TC}^{-}$ and $F = F(S^{2\ell+1}_+, \operatorname{THH})^{\mathbb{T}}$ are obtained by restricting the filtrations. The case of $F = (\operatorname{THH}^{tC_2})^{h\mathbb{T}}$ is similar to that of $\operatorname{TC}^{-}$ , since each $(Y^q)^{tC_2}$ is even by Proposition 2.15. The fact that these are algebra spectral sequences when $\overline{V}$ is a $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -algebra follows by Proposition 3.3 and the fact that the filtrations in Definition 3.2 are multiplicative.

Proposition 3.5. Suppose that $\overline{V}$ is a $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -module such that 2 acts trivially on $\overline{V}$ . Then we can identify the $C_2$ -Tate spectral sequence associated to the filtered $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -module

\begin{align*} \overline{V}\otimes \operatorname{fil}_+^{\star}\operatorname{gr}_{\mathrm{mot}}^*\operatorname{THH}(\mathrm{ko})^{tC_2} \end{align*}

with the spectral sequence associated to the filtered $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -module

\begin{align*} \overline{V}\otimes \operatorname{fil}_{\star}^{\mathrm{GM}}\operatorname{gr}_{\mathrm{mot}}^*\operatorname{THH}(\mathrm{ko})^{tC_2} . \end{align*}

The spectral sequence is of the form

\begin{align*}\pi_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \, [t^{\pm 1}]\otimes \Lambda(u_1) & \Longrightarrow \pi_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})^{tC_2} ,\end{align*}

where t in Tate filtration $-2$ has bidegree $\|t\| = (-2, 0)$ and $u_1$ in Tate filtration $-1$ has bidegree $\|u_1\| = (-1, -1)$ .

Under the same hypotheses, we can similarly identify the $C_2$ -homotopy fixed point spectral sequence associated to the filtered $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -module

\begin{align*} \overline{V}\otimes \operatorname{fil}_+^{\star}\operatorname{gr}_{\mathrm{mot}}^*\operatorname{THH}(\mathrm{ko})^{hC_2} \end{align*}

with the spectral sequence associated to the filtered $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -module

\begin{align*} \overline{V}\otimes \operatorname{fil}_{\star}^{\mathrm{GM}}\operatorname{gr}_{\mathrm{mot}}^*\operatorname{THH}(\mathrm{ko})^{hC_2} . \end{align*}

In this case, the spectral sequence is of the form

\begin{align*}\pi_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko}) \, [t]\otimes \Lambda(u_1) & \Longrightarrow \pi_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})^{hC_2} ,\end{align*}

where t in homotopy fixed point filtration $-2$ has bidegree $\|t\| = (-2,0)$ and $u_1$ in homotopy fixed point filtration $-1$ has bidegree $\|u_1\| = (-1, -1)$ .

In each case, the spectral sequence is an algebra spectral sequence when $\overline{V}$ is a $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -algebra, and is conditionally convergent when $\overline{V}$ is a finite $\operatorname{gr}_{\mathrm{ev}}^* \mathbb{S}$ -module.

Proof. Let $Y^{\bullet}=C^{\bullet}(\operatorname{THH}(\mathrm{ku}/\mathrm{MU})/\operatorname{THH}(\mathrm{ko}))$ . We write $\tau_{[a,b]}=\operatorname{cof} (\tau_{\geqslant b+1} \to \tau_{\geqslant a})$ for integers $a\lt b$ and similarly we write $\mathrm{GM}^{C_2}_{[m,m+1]}(X)=\operatorname{cof}(\mathrm{GM}_{m-1}^{C_2}(X)\to \mathrm{GM}_{m+1}^{C_2}(X))$ . When m is even, there are equivalences

\begin{align*}\tau_{[2w,2w+1]}(\mathrm{GM}_{[m,m+1]}^{C_2} (Y^q)) \simeq \tau_{[2w,2w+1]} ((\Sigma^{2w-m}H\pi_{2w-m} Y^q)^{tC_2})\end{align*}

since $Y^q$ is even for each q. These are compatible with the cosimplicial structure maps, and totalize to an equivalence

\begin{align*}\operatorname{fil}^\mathrm{GM}_{[m,m+1]} \operatorname{gr}_{\mathrm{mot}}^w \operatorname{THH}(\mathrm{ko})^{tC_2} \simeq \operatorname{fil}_+^{2w-m} \operatorname{gr}_{\mathrm{mot}}^w \operatorname{THH}(\mathrm{ko})^{tC_2} .\end{align*}

Here, we write $\operatorname{fil}^\mathrm{GM}_{[m,m+1]} \operatorname{gr}_{\mathrm{mot}}^w \operatorname{THH}(\mathrm{ko})^{tC_2}$ for the cofiber of

\begin{align*}\operatorname{fil}^\mathrm{GM}_{m-1} \operatorname{gr}_{\mathrm{mot}}^w \operatorname{THH}(\mathrm{ko})^{tC_2} \longrightarrow \operatorname{fil}^\mathrm{GM}_{m+1} \operatorname{gr}_{\mathrm{mot}}^w \operatorname{THH}(\mathrm{ko})^{tC_2} .\end{align*}

The free $C_2$ -equivariant $(m+1)$ -cell in $\widetilde{EC}_2= S^\infty$ is attached to the free $C_2$ -equivariant m-cell by a pinch map $S^m \to S^m/S^{m-1} \cong C_{2+} \wedge S^m$ . Hence there are cofiber sequences

\begin{align*}\Sigma^{2w} H\pi_{2w}(\Sigma^m Y^q) \longrightarrow \tau_{[2w,2w+1]} \mathrm{GM}_{[m,m+1]}(Y^q) \longrightarrow \Sigma^{2w+1} H\pi_{2w+1}(\Sigma^{m+1} Y^q) \overset{N}\longrightarrow \Sigma(-) , \end{align*}

with $N = (+1)_*+(-1)_*$ the norm (or trace) map given by the sum of the actions by the elements in $C_2$ , having totalization a cofiber sequence

\begin{align*}\operatorname{gr}^\mathrm{GM}_m \operatorname{gr}_{\mathrm{mot}}^w \operatorname{THH}(\mathrm{ko})^{tC_2} \longrightarrow \operatorname{fil}^\mathrm{GM}_{[m,m+1]} \operatorname{gr}_{\mathrm{mot}}^w \operatorname{THH}(\mathrm{ko})^{tC_2} \longrightarrow \operatorname{gr}^\mathrm{GM}_{m+1} \operatorname{gr}_{\mathrm{mot}}^w \operatorname{THH}(\mathrm{ko})^{tC_2} \overset{N}\longrightarrow \Sigma(-) . \end{align*}

Since the $C_2$ -action on each $Y^q$ is the restriction of a $\mathbb{T}$ -action, the map N is homotopic to multiplication by 2.

Using the earlier equivalence, and letting w vary, we get a cofiber sequence

\begin{align*}\operatorname{gr}^\mathrm{GM}_m \operatorname{gr}_{\mathrm{mot}}^{\ast} \operatorname{THH}(\mathrm{ko})^{tC_2} \longrightarrow \operatorname{fil}_+^{2\ast-m} \operatorname{gr}_{\mathrm{mot}}^{\ast}\operatorname{THH}(\mathrm{ko})^{tC_2} \longrightarrow \operatorname{gr}^\mathrm{GM}_{m+1} \operatorname{gr}_{\mathrm{mot}}^{\ast}\operatorname{THH}(\mathrm{ko})^{tC_2} \overset{N}\longrightarrow \Sigma(-)\end{align*}

of $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -modules, where the $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -actions are as in Definitions 3.1 and 3.2. For each $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -module $\overline{V}$ , we get another cofiber sequence

\begin{align*}\overline{V} \otimes \operatorname{gr}^\mathrm{GM}_m \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})^{tC_2} &\longrightarrow \overline{V} \otimes \operatorname{fil}_+^{2\ast-m} \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})^{tC_2} \\ &\longrightarrow \overline{V} \otimes \operatorname{gr}^\mathrm{GM}_{m+1} \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})^{tC_2} \overset{\overline{V} \otimes N}\longrightarrow \cdots ,\end{align*}

where we again consider the $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -actions as in Definitions 3.1 and 3.2. If multiplication by 2 acts trivially on $\overline{V}$ , e.g., if this is a $\overline{V}(0)$ -module, then the connecting map $\overline{V} \otimes N$ is null-homotopic, and the cofiber sequence splits. Then the spectral sequence associated to the filtration $\overline{V} \otimes \operatorname{fil}_+^{\star} \operatorname{gr}_{\mathrm{mot}}^*\operatorname{THH}(\mathrm{ko})^{tC_2}$ of $\overline{V} \otimes \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})^{tC_2}$ has starting term the sum over $n \in \mathbb{Z}$ of

\begin{align*}&\overline{V}_* \operatorname{gr}_+^{2*+2n} \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})^{tC_2} \\&\quad\cong \overline{V}_* \operatorname{gr}^\mathrm{GM}_{-2n} \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})^{tC_2}\oplus \overline{V}_* \operatorname{gr}^\mathrm{GM}_{-2n+1} \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})^{tC_2} \\&\quad\cong\overline{V}_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})^{tC_2} \{t^n, u_1 t^{n-1}\} ,\end{align*}

with $t^n$ and $u_1 t^{n-1}$ in (stem, motivic filtration) bigrading equal to $(-2n, 0)$ and $(-2n+1, -1)$ , respectively. Truncating at $\star \leqslant 0$ produces the result for $(-)^{hC_2}$ in place of $(-)^{tC_2}$ .

Suppose that $\overline{V}$ is a $\operatorname{gr}_{\mathrm{ev}}^*\mathbb{S}$ -algebra. To confirm the formula

\begin{align*} \overline{V}_*\operatorname{gr}^{\mathrm{GM}}_*\operatorname{gr}_{\mathrm{mot}}^*\operatorname{THH}(\mathrm{ko})^{tC_2} \cong \overline{V}_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})\, [t^{\pm1}] \otimes \Lambda(u_1) \end{align*}

for the $E^2$ -page of the $C_2$ -Tate spectral sequence as a trigraded $\mathbb{F}_2$ -algebra, it suffices to check that multiplication by $u_1t^{-1}$ takes the copy of $\overline{V}_* \operatorname{gr}_{\mathrm{mot}}^* \operatorname{THH}(\mathrm{ko})$ in Tate filtration 0 to the copy in Tate filtration 1, since we know that t, in the case of the $\mathbb{T}$ -Tate spectral sequence, acts invertibly on this spectral sequence. However, this is already built into the notation since, in the $n=0$ case, the two module generators there are $1 = t^0$ and $u_1 t^{-1}$ . Note that $u_1^2$ lands in a bidegree that is trivial, so the relation $u_1^2=0$ follows. In the case of the $C_2$ -homotopy fixed point spectral sequence, the algebra structure on the $E^2$ -page can be determined from the fact that it is a subalgebra of the $E^2$ -page of the $C_2$ -Tate spectral sequence.

4. Detection

The classical mod 2 Adams spectral sequence

\begin{align*}{}^{\mathrm{Ad}} E_2(X) = \operatorname{Ext}_{\mathcal{A}^\vee}(\mathbb{F}_2, H_*(X)) \Longrightarrow \pi_*(X^\wedge_2) \end{align*}

is strongly convergent for bounded-below spectra X with $H_*(X)$ of finite type. Its $E_2$ -term can be calculated as the cohomology of the normalized cobar complex

\begin{align*}0 \longrightarrow H_*(X) \overset{d_1^0} \longrightarrow \bar{\mathcal{A}}^\vee \otimes H_*(X) \overset{d_1^1} \longrightarrow \bar{\mathcal{A}}^\vee \otimes \bar{\mathcal{A}}^\vee \otimes H_*(X) \longrightarrow \cdots . \end{align*}

Here, $\bar{\mathcal{A}}^\vee = \operatorname{cok}(\mathbb{F}_2 \to \mathcal{A}^\vee)$ , and we will use the notation $[a]m = a \otimes m \in \bar{\mathcal{A}}^\vee \otimes H_*(X)$ . Recall that $d_1^0$ is given by the normalized $\mathcal{A}^\vee$ -coaction on $H_*(X)$ , while $d_1^1$ also involves the coproduct $\psi \colon \mathcal{A}^\vee\to \mathcal{A}^\vee \otimes \mathcal{A}^\vee$ .

When $X = A(1)[ij]$ as in Notation 2.20, the Adams $E_2$ -terms

\begin{align*}{}^{\mathrm{Ad}} E_2(A(1)[ij]) = \operatorname{Ext}_{\mathcal{A}}(H^*(A(1)[ij]), \mathbb{F}_2) \Longrightarrow \pi_* A(1)[ij] \end{align*}

are readily calculated in a finite range using Bruner’s ext software [Reference BrunerBru93, Reference Bruner and RognesBR21]. The results in stems $* \leqslant 28$ are shown in Figure 3, with the usual (stem, Adams filtration) bigrading. Lines of bidegree (0,1), (1,1) and (3,1) (dashed) indicate multiplications by $h_0$ , $h_1$ and $h_2$ , respectively.

Figure 3.

Adams $E_2$ -terms for A(1)[00], A(1)[10], A(1)[01] and A(1)[11] (from top to bottom).

In each case, the three 1-cochains

(4.1) \begin{align} [\xi_1^4]1,\quad[\xi_2^2]1 + [\xi_1^4]\xi_1^2, \quad[\xi_3]1 + [\xi_2^2]\xi_1 + [\xi_1^4]\xi_2\end{align}

in $\bar{\mathcal{A}}^\vee \otimes \mathcal{A}(1)^\vee$ are cocycles, but not coboundaries; hence they represent nonzero classes in ${}^{\mathrm{Ad}} E_2(A(1)[ij])$ in bidegrees (3, 1), (5, 1) and (6, 1), respectively. For sparsity reasons, these classes all survive to ${}^{\mathrm{Ad}} E_\infty(A(1)[ij])$ , and detect nonzero homotopy classes in stems 3, 5 and 6, denoted

\begin{align*}\nu , \quad w , \quad v_2 \end{align*}

in $\pi_* A(1)[ij]$ , for each $i,j \in \{0,1\}$ . Observe that $v_2$ is only defined modulo $\nu^2$ , and $2 v_2 = 0$ if $i=0$ , while $2 v_2 =\nu^2$ if $i=1$ .

In each case, the map ${}^{\mathrm{Ad}} E_2(\mathbb{S}) \to {}^{\mathrm{Ad}} E_2(A(1))$ of Adams $E_2$ -terms takes the bidegree (23,5) class $h_2 g$ detecting $\nu\bar\kappa$ to zero, as can be checked with ext, and the target has no classes in stem 23 and Adams filtration $\geqslant 6$ . It follows that $\nu \bar\kappa \mapsto 0$ .

To calculate the Novikov $E_2$ -term

\begin{align*}{}^{\mathrm{Nov}} E_2(A(1)) = \operatorname{Ext}_{\mathrm{MU}_*\mathrm{MU}}(\mathrm{MU}_*, \mathrm{MU}_* A(1)) \cong \operatorname{Ext}_{\mathrm{BP}_*\mathrm{BP}}(\mathrm{BP}_*, \mathrm{BP}_* A(1)) \end{align*}

for these spectra, we can note that $\mathrm{BP}_* A(1) = \mathrm{BP}_*/I_2 \{1, t_1\}$ and use the long exact sequence obtained by applying $\operatorname{Ext}_{\mathrm{BP}_*\mathrm{BP}}(\mathrm{BP}_*,-)$ to the $\mathrm{BP}_*\mathrm{BP}$ -comodule extension

\begin{align*}0 \longrightarrow \mathrm{BP}_*/I_2 \longrightarrow \mathrm{BP}_* A(1) \longrightarrow \Sigma^2 \mathrm{BP}_*/I_2 \to 0 \end{align*}

classified by

\begin{align*}h_{10} = [t_1] \in \operatorname{Ext}_{\mathrm{BP}_*\mathrm{BP}}(\mathrm{BP}_*, \mathrm{BP}_*) \end{align*}

in (stem, Novikov filtration) bidegree (1,1). The groups

\begin{align*}\operatorname{Ext}_{\mathrm{BP}_*\mathrm{BP}}(\mathrm{BP}_*, \mathrm{BP}_*/I_2) \end{align*}

are calculated in a range as in [Reference RavenelRav86, § 4.4, p. 162], starting with the isomorphism

\begin{align*}\operatorname{Ext}_{\mathcal{A}}(\mathbb{F}_2, \mathbb{F}_2) \cong \operatorname{Ext}_{\mathrm{BP}_*\mathrm{BP}}(\mathrm{BP}_*, \mathrm{BP}_*/I_\infty) \end{align*}

that doubles internal degrees, followed by the $v_n$ -Bockstein spectral sequences

\begin{align*}E_1 = \operatorname{Ext}_{\mathrm{BP}_*\mathrm{BP}}(\mathrm{BP}_*, \mathrm{BP}_*/I_{n+1}) \, [v_n] \Longrightarrow \operatorname{Ext}_{\mathrm{BP}_*\mathrm{BP}}(\mathrm{BP}_*, \mathrm{BP}_*/I_n) \end{align*}

for descending $n\geqslant2$ . The $v_2$ -Bockstein spectral sequence $E_\infty$ -term for $\mathrm{BP}_*/I_2$ in stems $* \leqslant26$ is shown in Figure 4, corresponding to [Reference RavenelRav86, Figure 4.4.23(c)]. Lines of bidegree (1,1), (3,1) and (7,1) (dashed) indicate multiplications by $h_{10} = [t_1]$ , $h_{11} =[t_1^2]$ and $h_{12} = [t_1^4]$ , respectively. (Some) hidden extensions are shown as dotted lines. We emphasize the relation