& \simeq \Z/p^n[\Omega^2 Y_n] \otimes_{\Z/p^n[\Omega S^3\pdb{3}]} \Z/p^n[\Omega S^{2p+1} \times \BSU(p-1)].
\end{align*}

There is still a $p$-local Cartesian square

\Omega S^3\pdb{3} \ar[d] \ar[r] & \Omega^2 Y_n \ar[d] \\
\Omega S^{2p+1} \times \BSU(p-1) \ar[r] & \Omega S^{2p+1} \times \B^2(p^{n-1} \Z) \times \BSU(p-1),

which implies the calculation of $\THH(\Z/p^n/X(p))$. The calculation of $\THH(\Z/p^n/J(p))$ is similar.

One could also deduce <ref> for $n\geq 2$ from <ref> for $\FF_p$, using descent and the fact that $\HH(\FF_p/\Z/p^n) = \FF_p[K(\Z/p^{n-1}, 2)]$. Indeed, the composite $S^1 \xar{p^{n-1}} S^1 \xar{1-p} \BGL_1(S)$ detects the class $(1-p)^{p^{n-1}} = 1-p^n u \in \Z_p^\times$ for some $p$-adic unit $u$. Therefore, its Thom spectrum is equivalent to $\Z/p^n$. In turn, <cit.> (or [13]) and the fiber sequence

$$S^1 \xar{p^{n-1}} S^1 \to B\Z/p^{n-1}$$

imply that $\FF_p$ is the Thom spectrum of a map $B\Z/p^{n-1} \to \BGL_1(\Z/p^n)$ which detects $1-p\in (\Z/p^n)^\times$ on the bottom cell of the source. Applying [12] implies the desired calculation of $\HH(\FF_p/\Z/p^n)$.

There is a higher chromatic analogue of <ref>. To explain this, recall from <cit.> that there is an $\E{2}$-algebra $S\ls{\hbar}$ over the sphere spectrum with $|\hbar|=-2$. It follows from <cit.> that $S\ls{\hbar}$ can be upgraded to an $\Efr{2}$-algebra. Tensoring with $X(p^n)$ therefore defines an $\Efr{2}$-ring $X(p^n)\ls{\hbar}$; in particular, one can define THH relative to $X(p^n)\ls{\hbar}$. The $\E{2}$-map $X(p^n) \to \BP{n-1} \to \BP{n-1}^{tS^1}$ factors through an $\E{2}$-map $X(p^n)\ls{\hbar} \to \BP{n-1}^{tS^1}$, where $\hbar$ is sent to a complex orientation of $\BP{n-1}$ (viewed as a class in $\pi_{-2} \BP{n-1}^{tS^1}$). The calculation of <ref> implies that

$$\THH(\BP{n-1}^{tS^1}/X(p^n)\ls{\hbar}) \simeq \BP{n-1}^{tS^1}[\Omega S^{2p^n+1} \times \B\Delta_n].$$

The spectrum $\BP{n-1}^{t\Z/m}$ is the quotient $\BP{n-1}^{tS^1}/\tfrac{[m](\hbar)}{\hbar}$, where $[m](\hbar)$ denotes the $m$-series of the formal group law over $\BP{n-1}_\ast$. This can be viewed as the Thom spectrum of a map $S^1 \to \BGL_1(\BP{n-1}^{tS^1})$ detecting $1 + \tfrac{[m](\hbar)}{\hbar} \in \pi_0(\BP{n-1}^{tS^1})^\times$. It follows that

\begin{equation}\label{tZm-thh}
\THH(\BP{n-1}^{t\Z/m}/X(p^n)\ls{\hbar}) \simeq \BP{n-1}^{t\Z/m}[\B S^1 \times \Omega S^{2p^n+1} \times \B\Delta_n].
\end{equation}

When $n=1$, there is an equivalence $\BP{0}^{t\Z/m} \simeq (\Z/m)^{tS^1}$, and <ref> can be viewed as the equivalence of <ref>, base-changed along $\Z/m \to (\Z/m)^{tS^1}$.

Since $\B^2(p^{n-1} \Z) \cong \CP^\infty$ (more canonically, it is the total space of the line bundle $\co(p^{n-1})$ over the standard $\CP^\infty$), <ref> implies that $\pi_\ast \THH(\Z/p^n/J(p)) \cong \Z/p^n[x]\pdb{u_n}$ with $|u_n|=|x|=2$. Were <ref> to hold, <ref> would imply that $\diffr_{\Z/p^n}$ is a (discrete) divided power algebra over $\Z/p^n$. In <cit.>, it is shown that if $\GG_a^\sharp$ denotes the PD-completion of $\GG_a$ at the origin, then $\spec(\Z/p^n)^{\slashed{D}} \otimes \FF_p \cong \GG_a^\sharp \otimes \FF_p$ in the notation of [19]. This implies that $\diffr_{\Z/p^n} \otimes_{\Z/p^n} \FF_p$ is isomorphic to the divided power algebra $\FF_p\pdb{t_n}$ for $|t_n| = 0$. However, as predicted by <ref>, there is in fact no need to reduce modulo $p$: <ref> below says that $\diffr_{\Z/p^n}$ is indeed isomorphic to the divided power algebra $\Z/p^n\pdb{t_n}$ for $|t_n| = 0$.

I am grateful to Bhargav Bhatt for the statement of the following lemma, which is analogous to the calculation that if $R$ is a commutative ring and $x\in R$ is a regular element, then there is a $p$-complete equivalence $\dR_{R/x/R} \simeq R\pdb{x}/x$ (see <cit.>). The argument for <ref> below is my interpretation of Bhatt's explanation. The topological discussion above can be regarded as an analogue of the calculation that $\HH(R/x/R) \simeq R[\CP^\infty]/x$. We will freely use notation from [18, 19] below.

Let $(A, I)$ be a transversal prism (i.e., $A/I$ is $p$-torsionfree). Let $x\in A$ be an element such that $x\pmod{I}$ is regular in $\ol{A} := A/I$, and such that $(x)\subseteq A$ is $\phi$-stable. Then $\WCart^\HT_{A/(I,x)/A}$ is $p$-completely isomorphic to $\GG_a^\sharp \times \spf(A/(I,x))$, so that $\prismht_{A/(I,x)/A} \cong A/(I,x)\pdb{t}$ with $|t| = 0$.

By <cit.>, the map $\WCart^\HT_{A/(I,x)/A} \to \spf(A/(I,x))$ is a split gerbe, banded by $T_{A/(I,x)/\ol{A}}\{1\}^\sharp$. In this case, since $x\pmod{I}$ is a regular element of $\ol{A}$, we see that $L_{A/(I,x)/\ol{A}} = (x)/(x^2)[1]$, so that $T_{A/(I,x)/\ol{A}} = \spf \Sym_{A/(I,x)}(L_{A/(I,x)/\ol{A}})^\wedge_p$ is isomorphic to $\Omega \GG_a$ over $A/(I, x)$. It follows that $\WCart^\HT_{A/(I,x)/A}$ is isomorphic to a trivial $\GG_a^\sharp$-torsor over $\spf(A/(I,x))$. Since $\prismht_{A/(I,x)/A}$ is the global sections of the structure sheaf of $\WCart^\HT_{A/(I,x)/A}$, the lemma follows.

In fact, the conjugate filtration $\F^\conj_i \prismht_{A/(I,x)/A}$ is isomorphic to the divided power filtration on $A/(I,x)\pdb{t}$ under <ref>.

Sticking with the assumptions of <ref>, let us mention without proof that <ref> is also a consequence of <cit.>, which states that $\prism_{A/(I,x)/A} \cong A\{\frac{x}{I}\}^\wedge_{(p,I)}$.
If $I = (d)$ is principal, the $p$-complete isomorphism

$$\beta: A/(I,x)\pdb{t}^\wedge_p \xar{\sim} \prismht_{A/(I,x)/A} \cong A\left\{\frac{x}{I}\right\}^\wedge_p/I$$

leads to an $I$-adic Bockstein spectral sequence

$$E_1^{\ast,\ast} = A/(I,x)\pdb{t}^\wedge_p[\ol{d}] \cong A\left\langle \frac{x}{d}\right\rangle^\wedge_p[\ol{d}]/d \Rightarrow A\left\{\frac{x}{d}\right\}^\wedge_{(p,d)},$$

where $\ol{d}$ represents $d$ on the $E_1$-page.

The map $\beta$ sends $\gamma_{p^n}(t)\mapsto \delta^n(\frac{x}{d})$ (up to $p$-adic units). This can be proved by showing that in the setting of <ref>, $\phi(\delta^n(\frac{x}{d})) \in (d) \subseteq A\{\frac{x}{d}\}$ if $n\geq 0$ (see <ref> below). The fact that

$$\phi\left(\delta^n\left(\frac{x}{d}\right)\right) = \delta^n\left(\frac{x}{d}\right)^p + p\delta^{n+1}\left(\frac{x}{d}\right)$$

then implies that $\delta^n(\frac{x}{d})^p \equiv -p\delta^{n+1}(\frac{x}{d})\pmod{d}$. Therefore, the elements $\delta^n(\frac{x}{d})$ can be used to define divided powers of $\frac{x}{d}\pmod{d}$. In particular, we obtain the desired map $\beta: A/(I,x)\pdb{t} \to A\{\frac{x}{d}\}^\wedge_{(p,d)}/d$, but further work is required to show that it is a $p$-complete isomorphism.

Fix notation as in <ref>. Then $\phi(\delta^n(\frac{x}{d})) \in (d) \subseteq A\{\frac{x}{d}\}$.

Let $t = \frac{x}{d}$. The desired claim can be proved by induction on $n$. For the base case, we need to show that $\phi(t)\in I$. By reduction to the universal case, we may assume that $(p,d)$ is regular in $A$. Then <cit.> implies that the sequence $(d, \phi(d))$ is regular in $A$. Since $(x)$ is $\phi$-stable, we see that $d$ divides $\phi(x)$; it then follows from the formula $\phi(d) \phi(t) = \phi(x)$ that $d$ divides $\phi(t)$, as desired.
For the inductive step, observe that

$$p\phi(\delta^{n+1}(t)) = p\delta(\phi(\delta^n(t))) = \phi^2(\delta^n(t)) - \phi(\delta^n(t))^p.$$

The inductive hypothesis says that $\phi(\delta^n(t)) \in (d)$ for every $k\geq 1$, so that $d$ divides $p\phi(\delta^{n+1}(t))$. Since $(p,d)$ is a regular sequence, this implies that $d$ divides $\phi(\delta^{n+1}(t))$, as desired.

This implies the following result, which is also proved in <cit.>.

There is an isomorphism $\spec(\Z/p^n)^{\slashed{D}} \cong \GG_a^\sharp \times \spec(\Z/p^n)$ of $\Z/p^n$-schemes. In particular, the scaling action of $\GG_m^\sharp$ on $\GG_a^\sharp$ over $\Z/p^n$ gives an isomorphism $\WCart^\HT_{\Z/p^n} \cong \GG_a^\sharp/\GG_m^\sharp$ of $\Z/p^n$-stacks.

Recall that $\prismht_{\Z/p^n/\Z_p\pw{\ptl}} = \diffr_{\Z/p^n}$. <ref> implies that $\prismht_{\Z/p^n/\Z_p\pw{\ptl}} \cong \Z/p^n\pdb{t}$ with $|t| = 0$; this gives the desired claim. (It is useful to view $\gamma_{p^m}(t)$ as a $p$-adic unit multiple of $\delta^m(\frac{p^n}{\ptl})$, as described in <ref>.)

Alternatively, consider the transversal prism $(A,I) = (\Z_p\pw{q-1}, [p]_q)$, and let $x = (q-1)^{n(p-1)}$. Note that $\phi(x) \in (x)$, so $(x)$ is $\phi$-stable. Then $A/I \cong \Z_p[\zeta_p]$, and $A/(I,x)$ is isomorphic to $\Z_p[\zeta_p]/(\zeta_p - 1)^{n(p-1)} \cong \Z/p^n[\zeta_p]$ since the $p$-adic valuation of $(\zeta_p - 1)^{n(p-1)}$ is $n$. It follows from <ref> that $\prismht_{\Z/p^n[\zeta_p]/\Z_p\pw{q-1}} \cong \Z/p^n[\zeta_p]\pdb{t'}$ with $|t'| = 0$. There is an action of $\Z_p^\times$ (and hence $\FF_p^\times \subseteq \Z_p^\times$) on $(A, I)$; taking $\FF_p^\times$-fixed points produces an isomorphism

$$\prismht_{\Z/p^n/\Z_p\pw{\ptl}} \cong (\prismht_{\Z/p^n[\zeta_p]/\Z_p\pw{q-1}})^{h\FF_p^\times} \cong \Z/p^n\pdb{t}$$

with $|t| = 0$, as desired. Note that as described in <ref>, the divided power $\gamma_{p^m}(t')$ can be viewed as a $p$-adic multiple of $\delta^m(\frac{(q-1)^{n(p-1)}}{[p]_q}) = \delta^m(\frac{(q-1)^{np-n+1}}{q^p-1})$.

An alternative (and more hands-on) proof of <ref> is given in <ref>; this alternative argument is also presented as <cit.>.

Let us describe the topological Sen operator on $\THH(\Z/p^n/X(p))$ for $n\geq 2$ (recall that $p>2$). This is equivalent to describing the Serre spectral sequence in $\Z/p^n$-homology for the fibration

$$S^{2p-1} \to \Omega^2 Y_n \to \Omega S^{2p+1} \times \B(p^{n-1} \Z).$$

Note that this fibration is an analogue of the fibration <ref>.

It will be simpler to analyze the Serre spectral sequence in $\Z_p$-homology, since all the differentials in the Serre spectral sequence in $\Z/p^n$-homology arise from the Serre spectral sequence in $\Z_p$-homology. The analysis is similar to <ref>; the Serre spectral sequence runs

\begin{equation}\label{serre-yn}
E^2_{\ast,\ast} = \Z_p\pdb{u_n} \otimes_{\Z_p} \Z_p[\theta, \epsilon]/\epsilon^2 \Rightarrow \pi_\ast \Z_p[\Omega^2 Y_n],
\end{equation}

where $\epsilon$ lives in degree $2p-1$ and $u_n$ lives in degree $2$. There are several ways to determine the differentials in this spectral sequence. Our approach will be to describe the pattern of differentials by first calculating $\pi_\ast \Z_p[\Omega^2 Y_n]$; in turn, we will do this by computing $\pi_\ast C^\ast(\Omega^2 Y_n; \Z_p)$. For this, we use the Serre spectral sequence for the fibration

$$\B\Z/p^{n-1} \to \Omega^2 Y_n \to \Omega S^3.$$

Since $\H^\ast(\B\Z/p^{n-1}; \Z) \cong \Z[c]/p^{n-1} c$ with $|c|=2$, the Serre spectral sequence collapses on the $E_2$-page, and we find that $\pi_\ast C^\ast(\Omega^2 Y_n; \Z_p) \cong \Z_p\pdb{x}[c]/(x-p^{n-1}c)$ with $|x|=2$. (If $n=1$, then $\Omega^2 Y_n \simeq \Omega S^3$, and the cohomology ring is $\Z_p\pdb{x}$.) For $n\geq 2$, this is isomorphic to $\Z_p\pdb{y}[c]/y$, where $y = x - p^{n-1}c$. Indeed, observe that if $n\geq 2$, then

$$\gamma_j(y) := \sum_{i=0}^j \frac{p^{i(n-1)}}{i!} c^i \gamma_{j-i}(x)$$

is a well-defined class in $\Z_p\pdb{x}[c]/(x-p^{n-1}c)$ since $p$ has divided powers in $\Z_p$, and that these classes form a basis for $\Z_p\pdb{x}[c]/(x-p^{n-1}c)$ as a $\Z_p[c]$-module.
Recall that in homological grading, there is an equivalence:

$$\Z_p\pdb{y}/y \simeq \Z_p \oplus \bigoplus_{n\geq 1} \Z_p/n\{\gamma_n(y)\} [-2n],$$

which implies that if $n\geq 2$, then

$$\H^i(\Omega^2 Y_n; \Z_p) \cong \begin{cases}
\Z_p \oplus \bigoplus_{j=1}^k \Z_p/j \{\gamma_j(y) c^{k-j}\} & i = 2k\geq 0 \text{ even}\\
0 & \text{else}.
\end{cases}$$

Using the universal coefficients theorem, we find that if $n\geq 2$, then

$$\pi_i \Z_p[\Omega^2 Y_n] \cong \begin{cases}
\Z_p & i\in 2\Z_{\geq 0}\\
\bigoplus_{j=1}^k \Z_p/j & i = 2k-1.
\end{cases}$$

The generator of $\pi_{2j} \Z_p[\Omega^2 Y_n]$ is the linear dual to $c^j\in \Z_p\pdb{y}[c]/y$, while the generator of $\pi_{2k-1} \Z_p[\Omega^2 Y_n]$ which is killed by $j$ is dual to $\gamma_j(y) c^{k-j}$.
Note that the homotopy groups $\pi_\ast \Z_p[\Omega^2 Y_n]$ are independent of $n$ if $n\geq 2$ (but the generators of these groups do depend on $n$).

Let us now return to the Serre spectral sequence <ref>. Comparison with the Serre spectral sequence for the fibration <ref> (i.e., with the topological Sen operator on $\THH(\Z_p/X(p))$; see <ref>) forces the differentials in <ref> to be given by (up to $p$-adic units):

$$d^{2p}(\gamma_{p^k}(u_n)) = p^{n-1} \epsilon \prod_{j=1}^{k-1} \gamma_{p^j}(u_n)^{p-1} = p^{n} \epsilon \partial_{u_n^p}(\gamma_{p^k}(u_n)), \ d^{2p}(\theta^j) = jp\theta^{j-1} \epsilon.$$

Reducing modulo $p^n$, we get the topological Sen operator on $\THH(\Z/p^n/X(p))$ for $n\geq 2$:

$$\Theta: \gamma_{p^k}(u_n) \mapsto p^{n-1} \prod_{j=1}^{k-1} \gamma_{p^j}(u_n)^{p-1}, \ \Theta: \theta^j\mapsto jp \theta^{j-1}.$$

Observe that this acts as “$p^n \partial_{u_n^p}$”.
Of course, one can similarly deduce the action of the topological Sen operator on $\THH(\Z/p^n/J(p))$.
This recovers the calculation

$$\pi_j \THH(\Z/p^n) = \begin{cases}
\bigoplus_{i=0}^j \Z/\gcd(j,p^n) & \text{even }j\geq 0, \\
\bigoplus_{i=0}^j \Z/\gcd(j,p^n) & \text{odd }j\geq 0, \\
0 & j<0.
\end{cases}$$

Another example of the topological Sen operator comes from studying complete DVRs, where the relationship between $\THH$ relative to $J(p)$ and the diffracted Hodge complex predicted by <ref> can be seen directly.

Let $R$ be a $p$-torsionfree complete DVR of mixed characteristic $(0,p>0)$ whose residue field $k$ is perfect. Then we have

$$\pi_\ast \THH(R/X(p)) \cong \HH_\ast(R/\Z_p)[\theta] \otimes_{\Z_p} \epsilon^{\Z_p}_\ast,$$

and the map $\Theta: \pi_\ast \THH(R/X(p)) \to \pi_{\ast-2p} \THH(R/X(p))$ sends $\theta^j \mapsto jp \theta^{j-1}$. To compute the action of the topological Sen operator on the remainder of $\THH(R/X(p))$, it will be simpler to assume that $T(1)$ is an $\E{2}$-ring and work instead with $\THH(R/T(1))$; this is merely cosmetic, and it is not difficult to modify the below argument to use $\THH(R/X(p))$ instead. Then, we have $\pi_\ast \THH(R/T(1)) \cong \HH_\ast(R/\Z_p)[\theta]$. We will compute $\THH(R)$ using the topological Sen operator on $\THH(R/T(1))$ and <ref>. Let $\pi\in R$ be a uniformizer, let $E(u)\in W(k)\pw{u}$ be its minimal polynomial, and let $E'(u)\in W(k)\pw{u}$ denote its derivative with respect to $u$.
Recall that $R = W(k)\pw{u}/E(u)$, that $W(k)$ is étale over $\Z_p$, $\pi_\ast \HH(W(k)\pw{u}/W(k)) \cong \Lambda_{W(k)\pw{u}}(du)$ with $|du|=1$, and $\pi_\ast \HH(R/W(k)\pw{u}) \cong R\pdb{\sigma_E}$, where $\sigma_E := \sigma^2(E(u))$. The transitivity sequence for the composite $W(k) \to W(k)\pw{u} \to R$ implies that $\HH(R/W(k)) \simeq \HH(R/\Z_p)$ is the fiber of a map $R\pdb{\sigma_E} \to \Sigma^2 R\pdb{\sigma_E}$ sending $\gamma_n(\sigma_E) \mapsto E'(\pi) \gamma_{n-1}(\sigma_E)$. In particular,

$$\pi_n \HH(R/\Z_p) \cong \begin{cases}
R & n=0, \\
R/E'(\pi) & n=2j+1, \ j\geq 0,\\
0 & \text{else}.
\end{cases}$$

Let us denote the generator of $\pi_{2j-1} \HH(R/\Z_p)$ by $z_j$, so that $\gamma_{j-1}(\sigma_E) \in \pi_{2j} \Sigma^2 R\pdb{\sigma_E}$ is sent to $z_j$ under the boundary map $\Sigma^2 R\pdb{\sigma_E} \to \Sigma \HH(R/\Z_p)$. We then have

\begin{equation}\label{cdvr-t1}
\pi_n \THH(R/T(1))
%\cong \bigoplus_{i\geq 0} \pi_{n-2pi} \HH(R/\Z_p) \cdot \theta^i
\cong \begin{cases}
R\cdot \theta^j & n=2pj, j \geq 0\\
\bigoplus_{0\leq i < j/p} R/E'(\pi) \cdot z_{j-pi} \theta^i & n = 2j-1, j\geq 0,\\
0 & \text{else}.
\end{cases}
\end{equation}

From this, we can describe the topological Sen operator on $\THH(R/T(1))$. For this, it will be useful to rephrase the above calculations somewhat, and use $J(p)$ instead of $T(1)$. It is easy to compute that $\pi_\ast \THH(R/J(p)) \cong \HH_\ast(R/\Z_p)[x]$, where $x$ is the class in degree $2$ from <ref>. In other words,

\pi_n \THH(R/J(p)) \cong \begin{cases}
R \cdot x^j & n = 2j \text{ for }j \geq 0, \\
\bigoplus_{0\leq i < j} R/E'(\pi) \cdot z_{j-i} x^i & n = 2j-1, j\geq 1.
\end{cases}

Since $\HH(R/\Z_p)$ is the fiber of a map $R\pdb{\sigma_E} \to \Sigma^2 R\pdb{\sigma_E}$, it follows that there is a cofiber sequence

\begin{equation}\label{jp-dvr}
\THH(R/J(p)) \to R\pdb{\sigma_E}[x] \xar{\nabla} \Sigma^2 R\pdb{\sigma_E}[x],
\end{equation}

where we have denoted the second map by $\nabla$. The map $\nabla$ is given on homotopy by a derivation, sending $\sigma_E \mapsto E'(\pi)$. Informally, $\THH(R/J(p))$ can be written as $R\pdb{\sigma_E}[x]^{\nabla=0}$.

The topological Sen operator $\Theta: \THH(R/J(p)) \to \Sigma^2 \THH(R/J(p))$ is described on homotopy by the operator on $R\pdb{\sigma_E}[x]$ sending $x\mapsto nx^{n-1}$. Note that this operator commutes with $\nabla$ (so that it does indeed define an operator on $\pi_\ast \THH(R/J(p))$).
Observe that since $\THH(R)$ is the fiber of $\Theta: \THH(R/J(p)) \to \Sigma^2 \THH(R/J(p))$, and $\THH(R/J(p))$ is the fiber of $\nabla: R\pdb{\sigma_E}[x] \to \Sigma^2 R\pdb{\sigma_E}[x]$, we can write $\THH(R)$ as the total fiber of the square

\begin{equation}\label{total-fiber}
\xymatrix{
R\pdb{\sigma_E}[x] \ar[r]^-{\nabla} \ar[d]^-\Theta & \Sigma^2 R\pdb{\sigma_E}[x] \ar[d]^-\Theta \\
\Sigma^2 R\pdb{\sigma_E}[x] \ar[r]^-{\nabla} & \Sigma^4 R\pdb{\sigma_E}[x],
\end{equation}

where the map denoted $\Theta$ sends $x^n\mapsto nx^{n-1}$. In turn, it follows that $\THH(R)$ is also the total fiber of the square

\begin{equation}\label{theta-nabla}
\xymatrix{
R\pdb{\sigma_E}[x] \ar[r]^-{\nabla} \ar[d]^-{\Theta + \nabla} & \Sigma^2 R\pdb{\sigma_E}[x] \ar[d]^-{\Theta + \nabla} \\
\Sigma^2 R\pdb{\sigma_E}[x] \ar[r]^-\nabla & \Sigma^4 R\pdb{\sigma_E}[x].
\end{equation}

The operator $\nabla + \Theta$ acts on $R\pdb{\sigma_E}[x]$ by

\begin{equation}\label{psi-deg2nm}
\nabla + \Theta: x^n \gamma_m(\sigma_E) \mapsto nx^{n-1} \gamma_m(\sigma_E) + E'(\pi) x^n \gamma_{m-1}(\sigma_E).
\end{equation}

Let us now invert $x$, and write $y = \sigma_E x^{-1}$ in $R\pdb{\sigma_E}[x^{\pm 1}]$. Then $y$ has divided powers and lives in degree $0$, and there is an isomorphism $R\pdb{\sigma_E}[x^{\pm 1}] \cong R\pdb{y}^\wedge_{(y)}[x^{\pm 1}]$ on homotopy. We can formally define $\Theta$ on $x^n$ for $n\leq 0$ by the same formula: $\Theta(x^n)=nx^{n-1}$. It follows from <ref> that $\Psi := x(\nabla + \Theta)$ sends

$$\Psi: \gamma_n(y) \mapsto -nx^{-n} \gamma_n(\sigma_E) + E'(\pi) x^{-n+1} \gamma_{n-1}(\sigma_E) = E'(\pi) \gamma_{n-1}(y) - n\gamma_n(y).$$

We claim that the action of $-t\partial_t$ on $R\pdb{(1-t)E'(\pi)}$ agrees the action of $\Psi$ on $R\pdb{y}^\wedge_{(y)}$, if we identify $y = (1-t) E'(\pi)$. Indeed:

\begin{align*}
\gamma_n(y) & = \gamma_n((1-t) E'(\pi)) = -E'(\pi)^n \frac{(1-t)^n}{n!} \\
& \xmapsto{-t\partial_t} E'(\pi)^n \frac{t(1-t)^{n-1}}{(n-1)!} = E'(\pi)^n \left(\frac{(1-t)^{n-1}}{(n-1)!} - n\frac{(1-t)^n}{n!}\right) \\
& = E'(\pi) \gamma_{n-1}((1-t) E'(\pi)) - n\gamma_n((1-t) E'(\pi))\\
& = E'(\pi) \gamma_{n-1}(y) - n\gamma_n(y).
\end{align*}

In particular, we can rewrite the square <ref> after inverting $x$ as

R\pdb{(1-t)E'(\pi)}[x^{\pm 1}] \ar[r] \ar[d]^-{t\partial_t} & R\pdb{(1-t)E'(\pi)}[x^{\pm 1}] \ar[d]^-{t\partial_t}\\
R\pdb{(1-t)E'(\pi)}[x^{\pm 1}] \ar[r] & R\pdb{(1-t)E'(\pi)}[x^{\pm 1}],

where the horizontal maps act by sending $\gamma_n((1-t)E'(\pi)) \mapsto E'(\pi) \gamma_{n-1}((1-t)E'(\pi))$. The fiber of either of the horizontal maps in the above square can be identified with $\THH(R/J(p))[x^{-1}]$.

Using the above description of $\Theta$, one can calculate (with some tedium) that

$$\pi_n \THH(R) = \begin{cases}
R & n=0,\\
R/jE'(\pi) & n = 2j-1 \geq 0,\\
0 & \text{else}.
\end{cases}$$

This is exactly the calculation of $\pi_\ast \THH(R)$ from <cit.> (reproved in <cit.>).

The above discussion can be compared to <cit.>, which says that if $X = \spec R$, then $\WCart^\HT_X \cong X \times BG$, where $G = \{(a,t)\in \GG_a^\sharp \rtimes \GG_m^\sharp | t-1 = E'(\pi) a\}$. The canonical map $\WCart^\HT_X \to X \times \WCart^\HT \cong (B\GG_m^\sharp)_X$ can be identified with the map induced on classifying stacks by the quotient map $G \to \GG_m^\sharp$ of group schemes over $X$.
Recall that the diffracted Hodge stack $X^{\slashed{D}}$ can be identified with $\WCart^\HT_X \times_{\WCart^\HT} \spec(\Z_p) \cong \WCart^\HT_X \times_{\WCart^\HT \times X} X$. In particular, $X^{\slashed{D}} \cong (\GG_m^\sharp)_X/G$, i.e., the classifying stack of the group scheme $\GG_a^\sharp[E'(\pi)] = \{a\in \GG_a^\sharp | E'(\pi) a = 0\}$. One can show from this description that the $2$-periodification of the cohomology of the diffracted Hodge complex $\diffr_R \cong \Gamma(B\GG_a^\sharp[E'(\pi)]; \co)$ can be identified with $\pi_\ast \THH(R/J(p))[x^{-1}]$ (which is additively the $2$-periodification of $\pi_\ast \HH(R/\Z_p)$), as predicted by <ref>.

Note that the extensions in the following long exact sequence in homotopy for <ref> are always nontrivial:

\begin{align}
\cdots \to \pi_{2j} \THH(R/T(1)) \to \pi_{2(j-p)} \THH(R/T(1)) \to \pi_{2j-1} \THH(R) \to \label{lexseq} \\
\pi_{2j-1} \THH(R/T(1)) \to \pi_{2(j-p)-1} \THH(R/T(1)) \to \pi_{2j-2} \THH(R) \to \cdots \nonumber
\end{align}

For example, when $j=p$, there is a long exact sequence

$$\pi_{2p} \THH(R/T(1)) \cong R \cdot \theta \to \pi_{0} \THH(R/T(1)) \cong R \to \pi_{2p-1} \THH(R) \to \pi_{2p-1} \THH(R/T(1)) \to 0,$$

which in particular gives a short exact sequence

$$0 \to R/p \to \pi_{2p-1} \THH(R) \to R/E'(\pi) \cong \pi_{2p-1} \THH(R/T(1)) \to 0.$$

Since $\pi_{2p-1} \THH(R) \cong R/pE'(\pi)$, this extension must be nontrivial.

§.§ Relation to the $\ptl$-de Rham complex

We now describe some additional calculations which give further evidence for <ref>.

Assume that $R$ is the $p$-completion of $\Z_p[t]$. Forthcoming work of Arpon Raksit ([95]) shows that (a completion of) $q\Omega_R$ arises as the associated graded of a motivic filtration on $\HP(\ku[t]/\ku)$. In fact, Raksit studies $\HP(A[t]/A)$ for a general $\Eoo$-ring $A$ with even homotopy groups.

Using <ref>, one can show that (a completion of) $\ptl\Omega_R$ arises as the associated graded of a motivic filtration on $\HP(\BP{1}[t]/\BP{1})$. Moreover, the class $\ptl$ is identified as the image of $v_1 \hbar^{p-1}$ in the associated graded.

For the sake of completeness, let us explicitly compute $\pi_\ast \TP(\Z_p[t]/X(p))$. As in <ref>, it will be convenient to assume that $T(1)$ is an $\Efr{2}$-ring and work instead with $\THH(R/T(1))$; again, this is merely cosmetic. We first need the following result, which is a special case of <cit.> and $S^1$-equivariant Poincaré duality for $S^1/\mu_n$.

Let $X$ be a bounded-below spectrum equipped with an action of $S^1$. Then there is an equivalence

$$\left(\bigoplus_{n \geq 1} X \otimes (S^1/\mu_n)_+\right)^{tS^1} \simeq \lim_{k\to \infty} \bigoplus_{n\geq 1} \left(\Sigma \tau_{\leq k} X\right)^{t\Cn}.$$

Let $S$ be the sphere spectrum. Recall that $\Z_p[t] \simeq \Z_p \otimes S[t]$, so that $\THH(\Z_p[t]/T(1)) \simeq \THH(\Z_p/T(1)) \otimes \THH(S[t])$. Let $\THH(S[t],(t))$ denote the fiber of the map $\THH(S[t]) \to \THH(S) \simeq S$ induced by the augmentation $S[t] \to S$ sending $t\mapsto 0$; note that the map $\THH(S[t]) \to S$ admits an $S^1$-equivariant splitting. Similarly, we write $\THH(\Z_p[t]/T(1), (t))$ to denote the fiber of the map $\THH(\Z_p[t]/T(1)) \to \THH(\Z_p/T(1))$ induced by the augmentation $\Z_p[t] \to \Z_p$. Then $\THH(S[t],(t)) \simeq \bigoplus_{n\geq 1} (S^1/\mu_n)_+$, so that

\begin{equation}\label{formula-poly-t1}
\THH(\Z_p[t]/T(1), (t)) \simeq \THH(\Z_p/T(1)) \otimes \THH(S[t],(t)) \simeq \bigoplus_{n\geq 1} (S^1/\mu_n)_+ \otimes \THH(\Z_p/T(1)).
\end{equation}

It follows from <ref> that there is an equivalence

$$\THH(\Z_p[t]/T(1), (t))^{tS^1} \simeq \lim_{k\to \infty} \bigoplus_{n\geq 1} \Sigma (\tau_{\leq k} \THH(\Z_p/T(1)))^{t\Cn}.$$

Using <ref>(a), we have $\tau_{\leq 2kp} \THH(\Z_p/T(1)) \simeq \Z_p[J_k(S^{2p})]$. A simple calculation using <ref>(a) shows that there is an isomorphism

$$\pi_\ast (\tau_{\leq 2kp} \THH(\Z_p/T(1)))^{t\Cn} \cong \pi_\ast (\BP{1}/v_1^{k+1})^{t\Cn} \cong \pi_\ast(\tau_{\leq 2k(p-1)} \BP{1})^{t\Cn}.$$

Let $\pdb{n}(\hbar) := \tfrac{[n](\hbar)}{\hbar}$, so that $\pi_\ast \BP{1}^{t\Cn} \cong \Z_p[v_1]\ls{\hbar}/\pdb{n}(\hbar)$. In analogy to $q = \beta\hbar + 1$, if we define $\ptl = v_1 \hbar^{p-1}$, then $\pdb{n}(\hbar)$ defines an element of $\Z_p\pw{\ptl}$ which we will denote $\pdb{n}_\ptl$.
We conclude that

$$\pi_\ast (\tau_{\leq 2kp} \THH(\Z_p/T(1)))^{t\Cn} \cong \Z_p\pw{\ptl}\ls{\hbar}/(\ptl^{k+1}, \pdb{n}(\hbar)).$$

It follows that

$$\pi_\ast \THH(\Z_p[t]/T(1), (t))^{tS^1} \cong \lim_{k\to \infty} \bigoplus_{n\geq 1} \Sigma \Z_p\pw{\ptl}\ls{\hbar}/(\ptl^{k+1}, \pdb{n}_\ptl),$$

i.e., that

$$\pi_\ast \TP(\Z_p[t]/T(1)) \cong \Z_p\pw{\ptl}\ls{\hbar} \times \lim_{k\to \infty} \bigoplus_{n\geq 1} \Sigma \Z_p\pw{\ptl}\ls{\hbar}/\pdb{n}_\ptl.$$

In a manner similar to <ref>, one calculates that if we write $\Z_p[\beta]\ls{\hbar} = \Z_p\pw{q-1}\ls{\hbar}$ by setting $q = 1+\beta \hbar$, and $\pdb{n}_{\GG_m}(\hbar) = \tfrac{[n]_{\GG_m}(\hbar)}{\hbar}$ is the divided $n$-series of the rescaled multiplicative formal group law $x + y + (q-1) xy$, then

\begin{equation}\label{HP-ku}
\pi_\ast \HP({\ku^\wedge_p}[t]/{\ku^\wedge_p}) \cong \Z_p\pw{q-1}\ls{\hbar} \times \lim_{k\to \infty} \bigoplus_{n\geq 1} \Sigma \Z_p\pw{q-1}\ls{\hbar}/((q-1)^{k+1},\pdb{n}_{\GG_m}(\hbar)).
\end{equation}

Moreover, $\pi_\ast \HP(\BP{1}[t]/\BP{1}) \cong \pi_\ast \HP({\ku^\wedge_p}[t]/{\ku^\wedge_p})^{\FF_p^\times}$, where $\FF_p^\times$ acts on $\HP({\ku^\wedge_p}[t]/{\ku^\wedge_p})$ via its action by Adams operations on ${\ku^\wedge_p}$. Note that since $\FF_p^\times$ has order coprime to $p$, taking $\FF_p^\times$-invariants preserves small limits and colimits after $p$-localization. In particular, $\pi_\ast \HP(\BP{1}[t]/\BP{1})$ is isomorphic to $\pi_\ast \TP(\Z_p[t]/T(1))$.

The following is also a consequence of the forthcoming work of Arpon Raksit ([95]) mentioned above.

There is an $\Z_p^\times$-equivariant isomorphism

$$\H^\ast(q\Omega_{\Z_p[t]})\ls{\hbar} \cong \Z_p\pw{q-1}\ls{\hbar} \times \bigoplus_{n\geq 1} \Sigma \Z_p\pw{q-1}\ls{\hbar}/\pdb{n}_{\GG_m}(\hbar).$$

For the formal group law over ${\ku^\wedge_p}$, we have

$$\pdb{n}_{\GG_m}(\hbar) = \sum_{i=1}^{n} \binom{n}{i} \hbar^{i-1} \beta^{i-1} = [n]_q \in \Z_p[\beta]\ls{\hbar},$$

where $q := 1+\beta\hbar$. The claim now follows from the fact that the differential $\nabla_q$ in $q\Omega_{\Z_p[t]}$ sends $t^n\mapsto [n]_q t^{n-1} dt$.

In particular, $\pi_\ast \HP(\BP{1}[t]/\BP{1}) \cong \pi_\ast \TP(\Z_p[t]/T(1))$ is a $2$-periodification of a completion of $\H^\ast(\ptl\Omega_{\Z_p[t]})$. This calculation leads to the following expectation related to <ref>:

Let $R$ be an animated $\Z_p$-algebra. Then $\TP(R/X(p))$ admits a motivic filtration $\F^\star_\mot \TP(R/X(p))$ such that $\gr^i_\mot \TP(R/X(p)) \simeq \hat{\prism}_{R/\Z_p\pw{\ptl}}[2i] \otimes_R \epsilon^R$, where $\hat{\prism}_{R/\Z_p\pw{\ptl}}$ is the Nygaard completion of $\ptl \Omega_R$.

We now turn to a higher chromatic analogue of (part of) this picture.

Let $R$ be an $\E{2}$-ring, and equip $\HH(R[t]/R) := \THH(S[t]) \otimes R$ with the $S^1$-action inherited from $\THH(S[t])$ and the trivial action on $R$.

If $R$ is only an $\E{2}$-ring, one cannot define Hochschild homology relative to $R$; in particular, the notation $\HH(R[t]/R)$ is rather abusive. As explained in <cit.>, if $R'$ is an $\E{1}$-$R$-algebra, then $\HH(R'/R)$ only exists (and has a natural $S^1$-action) when $R$ is a framed $\E{2}$-ring[Suppose that $R$ is an $\E{3}$-algebra, and $\cC$ is an $R$-linear $\infty$-category. The choice of a framed knot in $\RR^3$ also defines an $\E{n-3}$-map $\int_{S^1} R \to R$, and hence allows one to define relative Hochschild homology $\HH(\cC/R)$. However, this does not define an $S^1$-action on $\HH(\cC/R)$! Thanks to Robert Burklund for this point.]. In other words, if $R$ is merely an $\E{2}$-ring, it would not be clear how to define $\HH(R[t]/R)$, had we not known that $R[t]$ admits a lift to the sphere spectrum. This leads to the following unfortunate warning: if $R$ is an $\Efr{2}$-ring with a nontrivial $S^1$-action, then the (more natural) circle action on $\HH(R[t]/R)$ arising via the $S^1$-action on $R$ cannot be necessarily identified with the circle action from <ref>. However, for this article, we will only use the circle action from <ref>.

View $\BP{n-1}[t_1, \cdots, t_j]$ as a $\Z_{\geq 0}^j$-graded ring, where $t_i$ has weight $(0,\cdots,1,\cdots,0)$. Then, define $\HP^\gr(\BP{n}[t_1, \cdots, t_j]/\BP{n})$ to be the $S^1$-Tate construction of $\HH(\BP{n}[t_1, \cdots, t_j]/\BP{n})$ taken internally to $\Z_{\geq 0}^j$-graded $\BP{n}$-modules. Similarly, define $\TP^\gr(\BP{n-1}[t_1, \cdots, t_j]/X(p^n))$ to be the $S^1$-Tate construction of $\THH(\BP{n-1}[t_1, \cdots, t_j]/X(p^n))$ taken internally to $\Z_{\geq 0}^j$-graded $\BP{n-1}$-modules. Then, related to <ref>, we have the following result (which, when $n=0$, is a very special case of the main result of [93]):

There is a $p$-complete isomorphism of $\Z_{\geq 0}^j$-graded modules equipped with a map from $\pi_\ast \BP{n}^{tS^1}[\B\Delta_n] \cong \pi_\ast \TP(\BP{n-1}/X(p^n))$:

$$\pi_\ast \HP^\gr(\BP{n}[t_1, \cdots, t_j]/\BP{n})[\B\Delta_n] \cong \pi_\ast \TP^\gr(\BP{n-1}[t_1, \cdots, t_j]/X(p^n)).$$

The map $\TP^\gr(\BP{n-1}[t]/X(p^n)) \to \TP(\BP{n-1}/X(p^n))$ is an equivalence after $K(n)$-localization.

For simplicity, we assume that $j=1$ and write $t$ instead of $t_1$. In the graded setting, we may commute the $S^1$-Tate construction with the infinite direct sum (i.e., <ref> is not necessary). It follows that there are graded equivalences

\begin{align*}
\TP^\gr(\BP{n-1}[t]/X(p^n), (t)) & \simeq \bigoplus_{m\geq 1} \Sigma \THH(\BP{n-1}/X(p^n))^{t\Z/m} (m), \\
\HP^\gr(\BP{n}[t]/\BP{n}, (t)) & \simeq \bigoplus_{m\geq 1} \Sigma \BP{n}^{t\Z/m} (m).
\end{align*}

The desired result now follows from <ref>(a). The second statement follows from the above equivalences and the fact that $\Lk (\BP{n}^{t\Z/p^m}) = 0$ by <ref>.

The proof above used the following (well-known) fact.

Let $\BP{n}$ denote any form of the truncated Brown-Peterson spectrum. Then we have $\Lk (\BP{n}^{t\Z/p^m}) = 0$.

We first observe that $\BP{n}^{t\Z/p^m}$ depends only on the $p$-completion of $\BP{n}$; indeed, the obvious variant of <cit.> shows that if $X$ is a bounded-below spectrum with $\Z/p^m$-action, then $X^{t\Z/p^m}$ is $p$-complete, and the map $X^{t\Z/p^m} \to (X^\wedge_p)^{t\Z/p^m}$ is an equivalence. Since all forms of $\BP{n}$ are equivalent after $p$-completion by [3], we may therefore reduce to proving the claim for a single form of $\BP{n}$.

To show that $\Lk (\BP{n}^{t\Z/p^m}) = 0$, it suffices to show (since $\BP{n}^{t\Z/p^m}$ is an $\MU$-module) that $(\pi_\ast \BP{n}^{t\Z/p^m}[\frac{1}{v_n}])/(p,\cdots,v_{n-1}) \cong \pi_\ast k(n)^{t\Z/p^m}[\frac{1}{v_n}] = 0$. Recall that $\pi_\ast \BP{n}^{t\Z/p^m} \cong \BP{n}_\ast\ls{\hbar}/[p^m](\hbar)$. We will work with the form of $\BP{n}$ such that the associated formal group law over $\pi_\ast \BP{n}$ induces the Honda formal group law over $\pi_\ast k(n)$. Then, the $p^m$-series of the formal group law over $\pi_\ast k(n)$ satisfies $[p^m](\hbar) = v_n^{\frac{p^{nm}-1}{p^n-1}} \hbar^{p^{mn}}$; so $\pi_\ast k(n)^{t\Z/p^m} \cong \FF_p[v_n]\ls{\hbar}/v_n^{\frac{p^{nm}-1}{p^n-1}}$. In particular, $v_n$ is nilpotent in $k(n)^{t\Z/p^m}$, so that $\pi_\ast k(n)^{t\Z/p^m}[\frac{1}{v_n}] = 0$.

In general, $\pi_\ast \HP(\BP{n}[t]/\BP{n})$ looks like a completion of the $2$-periodification of the cohomology of the following two-term complex:

\begin{equation}\label{two-term-bpn}
\BP{n}_\ast\pw{\hbar}[t] \xar{\nabla} \BP{n}_\ast\pw{\hbar}[t] dt, \ \nabla: t^m \mapsto \tfrac{[m]_{\BP{n}}(\hbar)}{\hbar} t^{m-1} dt.
\end{equation}

This is a variant of the $q$-de Rham complex, and was first considered by Arpon Raksit (in forthcoming work). Note that an analogue of <ref> can be defined for a formal group law $F(x,y)$ over any commutative ring $A$:

\begin{equation}\label{any-ring-two-term}
F\Omega_{A[t]/A} := \left(A\pw{\hbar}[t] \xar{\nabla} A\pw{\hbar}[t] dt\right), \ \nabla: t^m\mapsto \tfrac{[m](\hbar)}{\hbar} t^{m-1} dt;
\end{equation}

we will study basic combinatorial properties of such complexes in [43].
After base-changing to $\QQ$, the operator $\nabla$ can be characterized by the formula $\hbar t\nabla = \exp_F(t\partial_t \log_F(\hbar))$.

We also have:

If $\cC$ is a left $\BP{n-1}$-linear $\infty$-category, and $\cC[t]$ denotes $\cC \otimes_{\BP{n-1}} \BP{n-1}[t]$, then <ref> implies that the map $\Lk \TP^\gr(\cC[t]/X(p^n)) \to \Lk \TP(\cC/X(p^n))$ is an equivalence.

Observe that

$$\THH(\cC[t]/X(p^n)) \simeq \THH(\cC/X(p^n)) (0) \oplus \bigoplus_{m\geq 1} (S^1/\mu_m)_+ \otimes \THH(\cC/X(p^n)) (m),$$

so that

$$\TP^\gr(\cC[t]/X(p^n)) \simeq \TP(\cC/X(p^n)) (0) \oplus \bigoplus_{m\geq 1} \Sigma \THH(\cC/X(p^n))^{t\Z/p^m} (m).$$

Now, <ref> implies that $\THH(\cC/X(p^n))^{t\Z/p^m}$ is a $\BP{n}^{t\Z/p^m}$-module. But $\Lk (\BP{n}^{t\Z/p^m}) = 0$ by <ref>, so that $\Lk \TP^\gr(\cC[t]/X(p^n)) \simeq \Lk \TP(\cC/X(p^n))$, as desired.

Let $n=0$, and suppose $\cC$ is the $\infty$-category of quasicoherent sheaves on an $\FF_p$-scheme $X$. Then <ref> says that the map $\TP^\gr(\AA^1\times X) \to \TP(X)$ is a rational equivalence. This is generally not true in the non-graded setting.

Note that the functor $L_{K(0)} \TP$ is not nil-invariant; the same is true of the functor $\Lk \TP(-/T(n))$ on $\BP{n-1}$-algebras. Indeed, <cit.> says that the map $L_{K(0)} \TP(\FF_p[t]/t^k) \to L_{K(0)} \TP(\FF_p) \simeq \QQ_p^{tS^1}$ is an isomorphism if and only if $k$ is a power of $p$. We can also see this at the level of algebra by calculating the crystalline cohomology of $\FF_p[t]/t^k$. If $R$ denotes the $p$-completion of the PD-envelope of the quotient map $\Z_p[t] \to \FF_p[t]/t^k$ (so that $R = \Z_p\left[t, \frac{t^{kj}}{j!} | j\geq 1\right]^\wedge_p$), then <cit.> implies that $\Gamma_\crys((\FF_p[t]/t^k) / \Z_p)$ is quasi-isomorphic to the de Rham complex $\Omega^\bull_{R/\Z_p}$. Note that $R$ is additively isomorphic to the $p$-completion of $\bigoplus_{0\leq i \leq k-1} \bigoplus_{j\geq 0} \Z_p \{\frac{t^{kj+i}}{j!}\}$.

Since the derivative of $\frac{t^{kj+i}}{j!}$ is $(kj+i) \frac{t^{kj+i-1}}{j!}$, which simplifies to $k \frac{t^{k(j-1)+k-1}}{(j-1)!}$ when $i=0$, we find that $\pi_0 \Gamma_\crys((\FF_p[t]/t^k) / \Z_p) \cong \Z_p$, and

\begin{align*}
\pi_{-1} \Gamma_\crys((\FF_p[t]/t^k) / \Z_p) & \cong \left(\bigoplus_{j\geq 0} \Z_p/k \cdot \left\{ \frac{t^{k(j+1)-1}}{j!}\right\} \right)^\wedge_p\\
& \ \ \ \oplus \left(\bigoplus_{0\leq i < k-1} \bigoplus_{j\geq 0} \Z_p/(kj+i+1) \cdot \left\{\frac{t^{kj+i}}{j!}\right\}\right)^\wedge_p.
\end{align*}

For instance, suppose $k=p$. Then $p j+i+1 \equiv i+1\pmod{p}$, which is never zero since $0\leq i< p-1$. Therefore, the second summand is zero since $pj + i + 1$ is a $p$-adic unit, and we find that $\pi_{-1} \Gamma_\crys((\FF_p[t]/t^p) / \Z_p) \cong \left(\bigoplus_{j\geq 0} \Z/p \cdot \left\{ \frac{t^{p(j+1)-1}}{j!}\right\} \right)^\wedge_p$. However, if $k$ is not a power of $p$, the second summand contains a non-torsion piece; for example, if $k=2$ and $p$ is odd, the second summand contains the $p$-completion of $\bigoplus_{m\geq 0} \Z/p^m$, which is non-torsion.

Let $n=1$; then, <ref> says that <ref> implies that up to a Nygaard-type completion, $\Lone \TP^\gr(R[t]/X(p)) \xar{\sim} \Lone \TP(R/X(p))$ for $R$ being an $\E{1}$-$\Z_p$-algebra. In the non-graded setting, this is generally not true; this is in contrast to <cit.> (for instance), which says that $K(1)$-local algebraic K-theory is $\AA^1$-invariant on connective $K(1)$-acyclic ring spectra (in particular, on connective $\E{1}$-$\Z_p$-algebras).

Let us now pivot somewhat to a slightly different topic, working at the famed prime $p=2$. Then $\ptl\Omega_R = q\Omega_R$, and there is an interesting action of $\Z/2\subseteq \Z_2^\times$ on $q\Omega_R$ sending $q\mapsto q^{-1}$. If we view $q$ as the Chern class (in K-theory) of the tautological line bundle on $\CP^\infty$, this corresponds to the action of $\Z/2$ on $\CP^\infty$ given by complex conjugation. This motivates the following discussion:

We expect that most of the results and conjectures in this article continue to hold with $\Z/2$-equivariance, where “real” topological Hochschild homology $\THH_\RR$ is interpreted to mean the construction described in [44, 49]. Recall that $\Z/2$ acts on $\SU(n)$ by complex conjugation; we will denote this $\Z/2$-space by $\SU(n)_\RR$. Let $\sigma$ (resp. $\rho = \sigma+1$) denote the sign representation (resp. regular representation), and if $X$ is a $\Z/2$-space, let $\Omega^\sigma X$ denote the space of maps $\Map(S^\sigma, X)$. There is a $\Z/2$-equivariant $\E{\rho}$-map

$$\Omega^\sigma \SU(n)_\RR \simeq \Omega^\rho \BSU(n)_\RR \to \Omega^\rho \BSU_\RR \simeq \BU_\RR,$$

which equips its Thom spectrum $X(n)_\RR$ with the structure of an $\E{\rho}$-ring. One can show that the equivariant Quillen idempotent on $\MU_\RR$ restricts to an idempotent on $X(n)_\RR$, and we will write $T(n)_\RR$ to denote the resulting summand of $X(2^n)_\RR$. Moreover, $\Phi^{\Z/2} T(n)_\RR \simeq y(n)$ as $2$-local $\E{1}$-algebras.

We then expect:

The following are true:

* $T(n)_\RR$ admits the structure of an $\E{\rho} \rtimes \U(1)_\RR$-algebra, and $X(2^n)_\RR$ splits as a direct sum of shifts of $T(n)_\RR$ such that the inclusion $T(n)_\RR \to X(2^n)_\RR$ of the unit summand is a map of $\E{\rho}$-algebras. In particular, $\THH_\RR(\BP{n-1}_\RR/T(n)_\RR)$ exists and admits an $\U(1)_\RR$-action[Note that $\U(1)_\RR = S^\sigma$.].

* Let $\BP{n}_\RR$ denote the Real truncated Brown-Peterson spectrum. Then there are equivalences

\begin{align}
\THH_\RR(\BP{n-1}_\RR/T(n)_\RR) & \simeq \BP{n-1}_\RR[\Omega S^{2^n\rho+1}], \label{equiv-thh-tn} \\
\THH_\RR(\BP{n-1}_\RR/T(n-1)_\RR) & \simeq \BP{n-1}_\RR \oplus \bigoplus_{j\geq 1} \Sigma^{2^{n-1}j\rho-1} \BP{n}_\RR/j \label{equiv-thh-tn1}
\end{align}

of $\BP{n-1}_\RR$-modules. The second equivalence requires $n\geq 1$.
Furthermore, the class in $\pi_{2^n\rho} \THH_\RR(\BP{n-1}_\RR/T(n)_\RR)$ induced by the map $E: S^{2^n\rho} \to \Omega S^{2^n\rho+1}$ detects $\sigma^\rho(\ul{v}_n)$.

* There is a $\Z/2$-equivariant space $\tilde{K}_n$ and an equivariant fibration

$$S^{2^n\rho-1}\to \tilde{K}_n \to \Omega S^{2^n\rho+1}$$

such that $\tilde{K}_0 = \Omega S^\rho$ and $\tilde{K}_1 = \Omega S^{\rho+1}\pdb{\rho+1}$. For $n=0$, this is simply the EHP sequence for $S^\sigma$.
The boundary map $\Omega^2 S^{2^{n+1}+1} \to S^{2^{n+1}-1}$ of the underlying fibration is degree $2$ on the bottom cell of the source, and $(\tilde{K}_n)^{\Z/2} = K_{n-1}$ as $(S^{2^n\rho-1})^{\Z/2} = S^{2n-1}$-fibrations over $(\Omega S^{2^n\rho+1})^{\Z/2} = \Omega S^{2^n + 1}$.

* For any $\Z/2$-equivariant $\E{\sigma}$-$T(n)_\RR$-algebra $R$, there is an equivariant cofiber sequence

$$\THH_\RR(R/T(n-1)_\RR) \to \THH_\RR(R/T(n)_\RR) \to \Sigma^{2^n\rho} \THH_\RR(R/T(n)_\RR),$$

where the second map is a $\Z/2$-equivariant analogue of the topological Sen operator.

* Let

$$\TP_\RR(\BP{n-1}_\RR/T(n)_\RR) := \THH_\RR(\BP{n-1}_\RR/T(n)_\RR)^{t_{C_2} \U(1)_\RR},$$

where the notation “$t_{C_2} \U(1)_\RR$” means the parametrized Tate construction from <cit.>.
Then there is a $\Z/2$-equivariant equivalence

$$\TP_\RR(\BP{n-1}_\RR/T(n)_\RR) \simeq \BP{n}_\RR^{t_{C_2} \U(1)_\RR},$$

where $\U(1)_\RR$ acts trivially on $\BP{n}_\RR$.

* Let $R$ be a $2$-complete animated commutative ring, equipped with the trivial $\Z/2$-action. Then there is a $\Z/2$-equivariant filtration on $\TP_\RR(R/T(1)_\RR)$ such that

$$\gr^i_\mot \TP_\RR(R/T(1)_\RR) \simeq (\hat{\prism}_{R/\Z\pw{q-1}})^\wedge_2[2i] = (\hat{q\Omega}_R)^\wedge_2[2i],$$

where the $\Z/2$-action on the right-hand side is obtained by viewing $\Z/2\subseteq \Z_2^\times \cong \Z/2 \times (1+4\Z_2)$ and using the $\Z_2^\times$-action on the $2$-completed $q$-de Rham complex.

Note that <cit.> and <cit.> prove <ref> for $n=0$ and <ref> for $n=1$, respectively.

We expect that $\BP{n}_\RR^{\Z/2} \otimes T(n)$ is concentrated in even degrees.

The equivalence <ref> is true for $n=1$.

This can be proved analogously to <ref>(a) for $n=1$ using the equivariant Toda fiber sequence

$$S^{\rho+\sigma}\to \Omega S^{\rho+1}\pdb{\rho+1} \to \Omega S^{2\rho+1}$$

of <cit.>.[See also <cit.>, which says that $\ul{\Z}$ is the Thom spectrum of a map $\Omega^\rho S^{2\rho+1} \to \BGL_1(X(2)_\RR)$ whose bottom cell detects $\ul{v}_1\in \pi_\rho X(2)_\RR$.] Indeed, recall from <cit.> that $X(2)_\RR$ is the Thom spectrum of the map $\Omega^\sigma S^{\rho+\sigma} \to \BGL_1(S)$ detecting $\tilde{\eta}\in \pi_\sigma S$. By an argument similar to [12], this implies that $\THH_\RR(X(2)_\RR) \simeq X(2)_\RR[S^{\rho+\sigma}]$. Therefore:

\begin{align*}
\THH_\RR(\ul{\Z}/X(2)_\RR) & \simeq \ul{\Z}[\Omega S^{\rho+1}\pdb{\rho+1}] \otimes_{X(2)_\RR[S^{\rho+\sigma}]} X(2)_\RR \\
& \simeq \ul{\Z}[\Omega S^{\rho+1}\pdb{\rho+1}] \otimes_{\ul{\Z}[S^{\rho+\sigma}]} \ul{\Z} \simeq \Z[\Omega S^{2\rho+1}],
\end{align*}

where the last equivalence uses the equivariant Toda fiber sequence.

Let us note some additional evidence for <ref>(a): if $X$ is a $\Z/2$-space, then the cofiber sequence $(\Z/2)_+ \to S^0 \to S^\sigma$ of spaces implies that $(\Omega^\sigma X)^{\Z/2}$ is equivalent to the fiber of the canonical map $X^{\Z/2} \to X$. In particular, since $(\SU(n)_\RR)^{\Z/2} = \SO(n)$, we see that $(\Omega^\sigma \SU(n)_\RR)^{\Z/2} \simeq \Omega(\SU(n)/\SO(n))$. Since geometric fixed points preserves colimits, this implies that $\Phi^{\Z/2} X(n)_\RR$ is the Thom spectrum of the map

$$(\Omega^\sigma \SU(n)_\RR)^{\Z/2} \simeq \Omega(\SU(n)/\SO(n)) \to \Omega(\SU/\SO) \simeq \BO \simeq \BU_\RR^{\Z/2}.$$

Since $\Phi^{\Z/2} T(n)_\RR \simeq y(n)$ as $2$-local $\E{1}$-algebras, <ref>(a) would imply that $\Phi^{\Z/2} X(2^n)_\RR$ (i.e., the Thom spectrum of the map $\Omega(\SU(n)/\SO(n)) \to \BO$) is a direct sum of shifts of $y(n)$ such that the inclusion $y(n) \to \Phi^{\Z/2} X(2^n)_\RR$ of the unit summand is an $\E{1}$-map. This is indeed true, and was proved in [104].

The strongest evidence for <ref>(d) is the following. It follows from <cit.> that there is a map $\Omega S^{n\rho-1} \to \BGL_1(X(n-1)_\RR)$ whose Thom spectrum is $X(n)_\RR$. The same construction used to prove <ref> then shows that for any $\Z/2$-equivariant $\E{\sigma}$-$X(n)_\RR$-algebra $R$, there is an equivariant cofiber sequence

\begin{equation}\label{equiv-sen}
\THH_\RR(R/X(n-1)_\RR) \to \THH_\RR(R/X(n)_\RR) \to \Sigma^{n\rho} \THH_\RR(R/X(n)_\RR),
\end{equation}

where the second map is a $\Z/2$-equivariant analogue of the topological Sen operator. It is not difficult to see that given the first half of <ref>(a), <ref>(d) can be easily proved using the construction of <ref>.

For example, we have $X(2)_\RR = T(1)_\RR$, and the cofiber sequence of <ref> is precisely <ref>(d). For $R = \ul{\Z}$, <ref> becomes a cofiber sequence

$$\ul{\Z}[\Omega S^{\rho+1}\pdb{\rho+1}] \simeq \THH_\RR(\ul{\Z}) \to \THH_\RR(\ul{\Z}/X(2)_\RR) \simeq \ul{\Z}[\Omega S^{2\rho+1}] \simeq \bigoplus_{n\geq 0} \Sigma^{2n\rho} \ul{\Z} \to \bigoplus_{m\geq 1} \Sigma^{2m\rho} \ul{\Z}.$$

A version of this fiber sequence was in fact already studied in <cit.>.

<ref>(a) and <ref>(d) together imply that

$$\pi_\ast \Phi^{\Z/2} \THH_\RR(\BP{n-1}_\RR) \simeq \FF_2[t, \sigma^2(v_{n-1}), \sigma(v_{j-1}) | 1\leq j \leq n]/(\sigma(v_j)^2),$$

where $|t| = 2^n = |\sigma^2(v_{n-1})|$ and $|\sigma(v_{j-1})| = 2^j-1$.
This can also be proved unconditionally using methods similar to that of <cit.>, by writing

$$\Phi^{\Z/2} \THH_\RR(\BP{n-1}_\RR) \simeq \Phi^{\Z/2} \BP{n-1}_\RR \otimes_{\BP{n-1}_\RR} \Phi^{\Z/2} \BP{n-1}_\RR,$$

and using that $\pi_\ast \Phi^{\Z/2} \BP{n-1}_\RR \cong \FF_2[t]$.

Note that if we assume <ref>(c), then $\THH_\RR(\BP{n-1}_\RR/T(n-1)_\RR) \simeq \BP{n-1}_\RR[\tilde{K}_n]$; the conjectural equivalence $\tilde{K}_n^{\Z/2} = K_{n-1}$ then gives an equivalence

\begin{equation}\label{real-thh-fixed}
\Phi^{\Z/2} \THH_\RR(\BP{n-1}_\RR/T(n-1)_\RR) \simeq \Phi^{\Z/2} \BP{n-1}_\RR[K_{n-1}].
\end{equation}

Observe that

$$\pi_\ast \Phi^{\Z/2} \BP{n-1}_\RR[K_{n-1}] \cong \FF_2[t, x, e]/e^2,$$

where $|x| = 2^n$ and $|y| = 2^n - 1$. For instance, when $n=1$, there is an equivalence $(\Omega S^{\rho+1}\pdb{\rho+1})^{\Z/2} = \Omega S^2$, and <ref> reduces to the equivalence

$$\Phi^{\Z/2} \THH_\RR(\ul{\Z}) \simeq \Phi^{\Z/2} \ul{\Z}[\Omega S^2] \simeq (\tau_{\geq 0} \FF_2^{tS^1})[\Omega S^2].$$

§.§ Aside: the Segal conjecture

In this section, we make some brief remarks regarding the Segal conjecture; the reader is referred to <cit.> and <cit.> for a discussion of its algebraic interpretation and a review of the literature on this topic.

An $\Eoo$-ring $R$ is said to satisfy the Segal conjecture if the cyclotomic Frobenius $\THH(R) \to \THH(R)^{t\Cp}$ is an equivalence in large degrees.

Let $R$ be a commutative $\FF_p$-algebra. If $R$ is Cartier smooth in the sense of <cit.> and $\Omega^n_{R/\FF_p} = 0$ for $n\gg 0$, then $R$ satisfies the Segal conjecture in the sense of <ref> (see <cit.>).

For instance, suppose $R = k$ is a field of characteristic $p>0$. Then $\THH(k) \simeq \HH(k/\FF_p)[\sigma]$ as a module over $\THH(\FF_p) \simeq \FF_p[\sigma]$, and $\pi_i \HH(k/\FF_p) = 0$ for $i>\log_p [k:k^p] = \dim_k \Omega^1_{k/\FF_p}$. This implies that the localization map $\THH(k) \to \THH(k)[\frac{1}{\sigma}] \simeq_\varphi \THH(k)^{t\Cp}$ is an equivalence in degrees $>\log_p [k:k^p] - 2$.

The proof of <cit.> can be used to show that the map $\THH(\BP{n-1}) \otimes_{\BP{n-1}} \FF_p \to \THH(\BP{n-1})^{t\Cp} \otimes_{\BP{n-1}} \FF_p$ is an equivalence in degrees $>n+\sum_{i=0}^{n-1} |v_i| = \sum_{i=0}^{n-1} (2p^i-1) = 2\frac{p^n-1}{p-1} - n$. Note that $2\frac{p^n-1}{p-1} - n$ is also precisely the shift appearing in Mahowald-Rezk duality for $\BP{n}$ (see <cit.>).

Assume <ref>, so that we can define the $\THH$ of a left $T(n)$-linear $\infty$-category relative to $T(n)$.
Since we do not know if $\THH$ relative to $T(n)$ admits the structure of a cyclotomic spectrum (presumably it does not), it does not seem possible to state a direct analogue of <ref> in this context. However, recall that if $k$ is a perfect field of characteristic $p>0$ and $R$ is an animated $k$-algebra, the cyclotomic Frobenius $\varphi: \THH(R) \to \THH(R)^{t\Cp}$ is the Frobenius-linear map given by inverting $\sigma\in \pi_2 \THH(k)$: this is a consequence of the observation that the map $\varphi: \THH(k) \to \THH(k)^{t\Cp}$ is given by composing the localization $\THH(k) \to \THH(k)[\sigma^{-1}]$ with a Frobenius-linear equivalence $\THH(k)[\sigma^{-1}] \simeq_\Fr \THH(k)^{t\Cp}$.

This observation motivates the following terminology: we say that an $\E{1}$-$\BP{n-1}$-algebra $R$ satisfies the “$T(n)$-Segal conjecture” if the base-change of the localization map $\THH(R/T(n)) \to \THH(R/T(n))[\theta_n^{-1}]$ along $\BP{n-1} \to \FF_p = \BP{n-1}/(p,\cdots,v_{n-1})$ is an equivalence in large degrees. Note that if $n=1$, this is equivalent to saying that the $p$-completion of the map $\THH(R/T(1)) \to \THH(R/T(1))[\theta^{-1}]$ is an equivalence in large degrees. One can similarly say that an $\E{1}$-$\Z_p$-algebra $R$ satisfies the “$J(p)$-Segal conjecture” if the map $\THH(R/J(p)) \to \THH(R/J(p))[x^{-1}]$ is an equivalence in large degrees.

If we assume <ref>, the localization map

$$\gamma: \THH(\BP{n-1}[x_1,\cdots,x_d]/T(n)) \to \THH(\BP{n-1}[x_1,\cdots,x_d]/T(n))[\theta_n^{-1}]$$

is an equivalence in degrees $>d-2p^n$ after base-changing along $\BP{n-1} \to \FF_p$. In particular, the flat polynomial algebra $\BP{n-1}[x_1,\cdots,x_d]$ satisfies the $T(n)$-Segal conjecture.

Write $T := \THH(\BP{n-1}/T(n))$ for notational simplicity.
Using <ref>, we have

$$\THH(\BP{n-1}[t]/T(n))[\theta^{-1}] \simeq T[\theta^{-1}] \oplus \bigoplus_{n\geq 1} T[\theta^{-1}] \otimes (S^1/\mu_n)_+.$$

Since the map $T \to T[\theta^{-1}]$ is an equivalence in degrees $>-2p^n$ after base-changing along $\BP{n-1} \to \FF_p$, the map $T \otimes (S^1/\mu_n)_+ \to T[\theta^{-1}] \otimes (S^1/\mu_n)_+$ is an equivalence in degrees $>-2p^n+1$ after base-changing along $\BP{n-1} \to \FF_p$. Because the map $\gamma: \THH(\BP{n-1}[t]/T(n)) \to \THH(\BP{n-1}[t]/T(n))[\theta^{-1}]$ preserves the summands, we see that $\gamma$ is an equivalence in degrees $>-2p^n+1$ after base-changing along $\BP{n-1} \to \FF_p$. Inducting on the number of variables, we find that the map $\gamma$ is an equivalence in degrees $>d-2p^n$ after base-changing along $\BP{n-1} \to \FF_p$, as desired.

When $d=0$, <ref> should be compared to <cit.>. In fact, we expect it is possible to recover their result using <ref>. We also note the following variant. Let $R := \BP{n-1}[t_1,\cdots,t_d]$ denote the flat polynomial $\E{2}$-$\BP{n-1}$-algebra on classes $t_i$ in even degrees (i.e., the base-change of the $\Eoo$-$\MU$-algebra $\MU[t_1,\cdots,t_d]$ along the $\E{3}$-map $\MU \to \BP{n-1}$). The argument of <ref> then shows that after base-change along the composite $R \to \BP{n-1} \to \FF_p$, the localization map $\gamma: \THH(R/T(n)) \to \THH(R/T(n))[\theta_n^{-1}]$ is an equivalence in degrees $>-2p^n + \sum_{j=1}^d (|t_j|+1)$. When $n=0$, this is <cit.>.

Let $R$ be a $p$-torsionfree discrete commutative ring such that $R/p$ is regular Noetherian. Suppose $L\Omega^n_R = 0$ for $n>d$. Then <ref> implies that $R$ satisfies the $J(p)$-Segal conjecture: in fact, the map $\THH(R/J(p)) \to \THH(R/J(p))[x^{-1}]$ is an equivalence in degrees $>d-2$.

Recall that <ref> asserts that $\THH(R/J(p))$ has a filtration such that $\gr^i_\mot \THH(R/J(p)) \simeq (\F^\conj_i \diffr_R)[2i]$, and such that the map $\gamma: \THH(R/J(p)) \to \THH(R/J(p))[x^{-1}]$ induces the map $\F^\conj_i \diffr_R \to \diffr_R$ on $\gr^i_\mot[-2i]$. By <cit.>, $\diffr_R/\F^\conj_i \diffr_R$ is concentrated in cohomological degrees $\geq i+1$, so that the cofiber of $\gr^i(\gamma)$ is concentrated in degrees $\leq 2i-(i+1) = i-1$. Moreover, the hypothesis that $L\Omega^n_R = 0$ for $n>d$ implies that $\gamma$ induces an equivalence on $\gr^i_\mot$ for $i\geq d$. Combining these observations gives the desired statement (see also the proof of <cit.>).

§.§ Aside: Cartier isomorphism

In this section, we study a topological analogue of the Cartier isomorphism for the two-term complexes from <ref>; we will study basic algebraic properties of these complexes in future work. To avoid dealing with completion issues, we use the following (see <ref> for a remark about the notation $\HH(R[t]/R)$):

Let $R$ be an $\E{2}$-ring. The polynomial $\E{1}$-$R$-algebra $R[t] = R[\NN]$ acquires a natural $\Z$-grading, and we will write $\HH(R[t]/R)_{\leq m}$ to denote the graded left $R$-module given by truncating $\HH(R[t]/R) := R \otimes \THH(S[t])$ in weights $\geq m+1$. Explicitly, $\HH(R[t]/R)_{\leq m}$ is equivalent to $R \oplus \left(\bigoplus_{1\leq n\leq m} R \otimes (S^1/\mu_n)_+\right)$.

If $X\in \Sp^{\B S^1}$, the following composite is an equivalence:

$$X^{t\Cp} \otimes (S^1/\mu_n)_+ \xar{\sim} (X \otimes (S^1/\mu_n)_+)^{t\Cp} \xar{\psi} (X \otimes (S^1/\mu_{np})_+)^{t\Cp}.$$

Moreover, if $p\nmid m$, then $(X \otimes (S^1/\mu_m)_+)^{t\Cp} = 0$.

If $\varphi: (S^1/\mu_n)_+ \to ((S^1/\mu_{np})_+)^{h\Cp}$ denotes the unstable Frobenius (sending $x\mapsto x^{1/p}$), the cofiber of the composite

$$\psi: (S^1/\mu_n)_+ \to ((S^1/\mu_{np})_+)^{h\Cp} \to (S^1/\mu_{np})_+$$

has induced $\Cp$-action, where $(S^1/\mu_n)_+$ and $((S^1/\mu_{np})_+)^{h\Cp}$ are equipped with the trivial $\Cp$-action.
Therefore, the canonical map $X^{t\Cp} \otimes (S^1/\mu_n)_+ \to (X \otimes (S^1/\mu_n)_+)^{t\Cp}$ is an equivalence (since $(S^1/\mu_n)_+$ is a finite spectrum with trivial $\Cp$-action). This gives the first claim.
Finally, if $p\nmid m$, then the $\Cp$-action on $S^1/\mu_m$ is free, so that $(X \otimes (S^1/\mu_m)_+)^{t\Cp} = 0$, as desired.

Let $R$ be an $\E{2}$-ring. Then:

* There is an $S^1$-equivariant map $\fr{C}: \HH(R^{t\Cp}[t]/R^{t\Cp}) \to \HH(R[t]/R)^{t\Cp}$, where $\HH(R[t]/R)^{t\Cp}$ is endowed with the residual $S^1/\mu_p$-action and $\HH(R^{t\Cp}[t]/R^{t\Cp})$ is endowed with the diagonal $S^1$-action arising from the $S^1$-action on $\HH$ and the residual $S^1/\mu_p$-action on $R^{t\Cp}$. Moreover, the map $\fr{C}$ sends $t\mapsto t^p$.

* For each $m\geq 1$, the map $\fr{C}$ induces an equivalence $\fr{C}_{\leq m}: \HH(R^{t\Cp}[t]/R^{t\Cp})_{\leq m} \xar{\sim} \left(\HH(R[t]/R)_{\leq mp}\right)^{t\Cp}$.

Recall that there is an equivalence $\HH(R[t]/R) \simeq R \otimes \THH(S[t])$. Since the $\Cp$-Tate construction is lax symmetric monoidal, we obtain the map $\fr{C}$ via the composite

\begin{align*}
\HH(R^{t\Cp}[t]/R^{t\Cp}) & \simeq R^{t\Cp} \otimes \THH(S[t]) \\
& \xar{\id\otimes \varphi} R^{t\Cp} \otimes \THH(S[t])^{t\Cp} \\
& \to (R \otimes \THH(S[t]))^{t\Cp} \simeq \HH(R[t]/R)^{t\Cp}.
\end{align*}

For each $m\geq 1$, there is an equivalence

\begin{align*}
\left(\HH(R[t]/R)_{\leq m}\right)^{t\Cp} & \simeq R^{t\Cp} \oplus \left(\bigoplus_{1\leq n\leq m} R \otimes (S^1/\mu_n)_+\right)^{t\Cp}.
\end{align*}

Since the maps $\varphi: (S^1/\mu_n)_+ \to ((S^1/\mu_{np})_+)^{h\Cp}$ define the Frobenius on $\THH(S[t]) \simeq S \oplus \bigoplus_{n\geq 1} (S^1/\mu_n)_+$, we see from <ref> that for each $m\geq 1$, the map $\fr{C}_{\leq m}$ defines an equivalence

$$\bigoplus_{1\leq j\leq m} R^{t\Cp} \otimes (S^1/\mu_j)_+ \xar{\sim} \left(\bigoplus_{1\leq n\leq mp} R \otimes (S^1/\mu_n)_+\right)^{t\Cp}.$$

The left-hand side is $\HH(R^{t\Cp}[t]/R^{t\Cp})_{\leq m}$, and the right-hand side is $\left(\HH(R[t]/R)_{\leq mp}\right)^{t\Cp}$.

When $R$ is an $\Eoo$-ring, the map $\fr{C}: \HH(R^{t\Cp}[t]/R^{t\Cp}) \to \HH(R[t]/R)^{t\Cp}$ of <ref> can also be constructed using (a simple case of) <cit.>. The cited result says the following. Suppose $k$ is an $\Eoo$-ring, so that the Tate-valued Frobenius $k \to k^{t\Cp}$ admits an extension $\THH(k) \to k^{t\Cp}$ to an $S^1$-equivariant map of $\Eoo$-rings.
If $A$ is an $\E{1}$-$k$-algebra, and $M$ is an $A$-bimodule in $\Mod_k$, then there is a relative Tate diagonal

$$k^{t\Cp} \otimes_{\THH(k)} \THH(A, M) \to \THH^k(A, M^{\otimes_A p})^{t\Cp},$$

where $\THH^k$ denotes $\THH$ relative to $k$. To construct $\fr{C}$, take $k = R$ and $A = M = k[t]$. Then

$$k^{t\Cp} \otimes_{\THH(k)} \THH(A, M) \simeq k^{t\Cp} \otimes \THH(S^0[t]) \simeq \HH(k^{t\Cp}[t]/k^{t\Cp}),$$

since $\THH(A,M) \simeq \THH(A) \simeq \THH(S^0[t]) \otimes \THH(k)$. Similarly, $\THH^k(A, M^{\otimes_A p}) \simeq \HH(A/k)$, and it is straightforward to check that Lawson's relative Tate diagonal agrees with the map $\fr{C}$.

One advantage of the construction of $\fr{C}$ in <ref> is that it is manifestly $S^1$-equivariant, and does not rely on $R$ being an $\Eoo$-ring. More generally, one finds that if $\cC$ is a stable $\infty$-category and $R$ is any $\E{2}$-ring, the cyclotomic Frobenius on $\THH(\cC)$ defines an $S^1$-equivariant map $\fr{C}: \HH(\cC \otimes R^{t\Cp}/R^{t\Cp}) \to \HH(\cC \otimes R/R)^{t\Cp}$ which generalizes the map of <ref>. This map is furthermore an equivalence if $\cC$ is smooth and proper.

In <ref>, the map $\fr{C}: \HH(R^{t\Cp}[t]/R^{t\Cp}) \to \HH(R[t]/R)^{t\Cp}$ is itself almost an equivalence: the main issue is that the canonical map

$$\colim_m \left(\HH(R[t]/R)_{\leq mp}\right)^{t\Cp} \to \left(\colim_m \HH(R[t]/R)_{\leq mp}\right)^{t\Cp} \simeq \HH(R[t]/R)^{t\Cp}$$

may not be an equivalence. However, <ref> implies that the graded map $\fr{C}^\gr: \HH(R[t]/R)^{t\Cp} \to \HH(R[t]/R)^{t\Cp}$ is itself an equivalence.

If $R$ is a complex-oriented $\E{2}$-ring, let $[p](\hbar)\in \pi_{-2} R^{hS^1}$ denote the $p$-series of the formal group law over $R$. If $M\in \LMod_R^{\B S^1}$, then it is not difficult to show that there is an equivalence $M^{tS^1}/\frac{[p](\hbar)}{\hbar} \xar{\sim} M^{t\Cp}$. (Although certainly well-known, the only reference in the literature for a statement in this generality seems to be <cit.>.) In particular, $\HH(R[t]/R)^{t\Cp} \simeq \HP(R[t]/R)/\frac{[p](\hbar)}{\hbar}$, so that <ref> and <ref> imply that there is an $S^1$-equivariant graded equivalence

$$\fr{C}: \HH(R^{t\Cp}[t]/R^{t\Cp}) \to \HP(R[t]/R)/\tfrac{[p](\hbar)}{\hbar} \simeq \HH(R[t]/R)^{t\Cp}.$$

In future work, we will show that if $R$ is further assumed to be an $\Eoo$-ring and $\cC$ is a $R$-linear $\infty$-category, then the $(R^{t\Cp})^{hS^1} \simeq (R^{tS^1})^\wedge_p$-module $(\THH(\cC) \otimes_{\THH(R)} R^{t\Cp})^{hS^1}$ behaves as a noncommutative analogue of $L\eta_{[p](\hbar)/\hbar}$ applied to $\HP(\cC/R)$. Here, $\hbar \in \pi_{-2} R^{hS^1}$ is the complex-orientation of $R$, and $[p](\hbar)/\hbar\in \pi_0 R^{tS^1}$ is the quotient of the $p$-series of the associated formal group law.

There is no reason to restrict to polynomial rings in a single variable in the equivalence of <ref>(b); we leave the details of the resulting statement to the reader.

Let $R = \Z$. Then $\Z^{t\Cp}$ is an $\Eoo$-$\Z$-algebra, and has homotopy groups given by $\FF_p\ls{\hbar}$ with $|\hbar|=2$. Therefore, $\Z^{t\Cp} \simeq \FF_p^{tS^1}$ as $\E{2}$-rings[In fact, they are equivalent as $\Eoo$-rings. Although seemingly innocuous, even the weaker claim that $\Z^{t\Cp}$ admits the structure of an $\Eoo$-$\FF_p$-algebra is surprisingly difficult to prove from first principles (see <cit.>). One might try to argue as follows: since $p=0\in \pi_0 \Z^{t\Cp}$, there is a map from the free $\Eoo$-algebra with $p=0$ to $\Z^{t\Cp}$. However, oddly enough, the free $\Eoo$-algebra with $p=0$ is not an $\Eoo$-$\FF_p$-algebra; this dashes any hopes of proving that $\Z^{t\Cp}$ is an $\Eoo$-$\FF_p$-algebra through this argument.

More generally, the free $\E{n}$-algebra $R$ with $p=0$ is not an $\E{n}$-$\FF_p$-algebra unless $n=2$: indeed, applying the $\E{n}$-cotangent complex to the composite $\FF_p \to R \to \FF_p$ of $\E{n}$-algebra maps shows that $L^{\E{n}}_{\FF_p}$ is a retract of $\FF_p \otimes_R L^{\E{n}}_R \simeq \FF_p$. This forces $L^{\E{n}}_{\FF_p} = \FF_p$, i.e., $\FF_p$ would be built from the sphere by attaching a single $\E{n}$-cell in degree $1$ – but this is impossible, since $\FF_p \otimes \FF_p \not \simeq \FF_p[\Omega^n S^{n+1}]$ unless $n=2$.], and <ref> (combined with <ref>) specializes to the statement that there is a Frobenius-linear equivalence

$$\fr{C}: \HH(\FF_p[t]/\FF_p)\ls{\hbar}_{\leq m} \simeq \HH(\FF_p^{tS^1}[t]/\FF_p^{tS^1})_{\leq m} \xar{\sim} (\HH(\Z[t]/\Z)_{\leq mp})^{t\Cp}.$$

Note that $\HH(\Z[t]/\Z)^{t\Cp} \simeq \HP(\Z[t]/\Z)/p \simeq \TP(\FF_p[t])/p$.
Since the HKR theorem implies that $\HH(\FF_p[t]/\FF_p)^{tS^1}$ is a $2$-periodification of the Hodge cohomology of $\AA^1_{\FF_p}$, and $\HP(\Z[t]/\Z)/p$ is a $2$-periodification of the de Rham cohomology of $\AA^1_{\Z_p}$ modulo $p$ (which is the de Rham cohomology of $\AA^1_{\FF_p}$), one can view $\fr{C}$ as a topological analogue of the Cartier isomorphism for the affine line. It reduces to the usual Cartier isomorphism on graded pieces. In this case, the statement of <ref> should also be compared to [56, 57, 78]. Taking homotopy fixed points for the $S^1$-equivariance of $\fr{C}$ from <ref>(a), we obtain a Frobenius-linear equivalence

\begin{equation}\label{ogus}
\fr{C}^{hS^1}: (\HH(\Z^{t\Cp}[t]/\Z^{t\Cp})^{hS^1})_{\leq m} \xar{\sim} ((\HH(\Z[t]/\Z)_{\leq mp})^{tS^1})^\wedge_p.
\end{equation}

More succinctly, there is a graded equivalence

$$(\fr{C}^\gr)^{hS^1}: \HH(\Z^{t\Cp}[t]/\Z^{t\Cp})^{hS^1} \xar{\sim} \HP^\gr(\Z[t]/\Z)^\wedge_p.$$

Using the HKR filtration on $\HH(\Z[t]/\Z)$, one can prove that $\HH(\Z^{t\Cp}[t]/\Z^{t\Cp})^{hS^1}$ admits a filtration whose graded pieces are given by even shifts of $L\eta_p \Gamma_\dR(\Z_p[t]/\Z_p) \simeq L\eta_p \Gamma_\crys(\FF_p[t]/\Z_p)$. We will explain this in greater detail in a future article.
Since $\HP(\Z_p[t]/\Z_p) \simeq \TP(\FF_p)$, <ref> can be regarded as a $2$-periodification of the “Cartier isomorphism” $L\eta_p \Gamma_\crys(\FF_p[t]/\Z_p) \simeq \Gamma_\crys(\FF_p[t]/\Z_p)$ for the crystalline cohomology of $\FF_p[t]$ (see <cit.> for the general case).

Let $R = \ku$. Then $\pi_\ast \ku^{t\Cp} \cong \Z[\zeta_p]\ls{\hbar}$, and it is expected that this lifts to an equivalence $\ku^{t\Cp} \simeq \Z[\zeta_p]^{tS^1}$ of $\Eoo$-rings (see also <ref> below).
Nevertheless, there is an equivalence $\ku^{t\Cp} \simeq \Z[\zeta_p]^{tS^1}$ of $\E{2}$-rings (one can show this by using <cit.>; thanks to Arpon Raksit for pointing this out).
Therefore, <ref> and <ref> give an equivalence

$$\fr{C}: \HH(\Z[\zeta_p][t]/\Z[\zeta_p])\ls{\hbar}_{\leq m} \simeq \HH(\Z[\zeta_p]^{tS^1}[t]/\Z[\zeta_p]^{tS^1})_{\leq m} \xar{\sim} (\HH(\ku[t]/\ku)_{\leq mp})^{t\Cp}.$$

Note that $\HH(\ku[t]/\ku)^{t\Cp} \simeq \HP(\ku[t]/\ku)/[p]_q$.
Here, we identify $\frac{[p]_{\GG_m}(\hbar)}{\hbar}\in \pi_0 \ku^{tS^1} \cong \Z\pw{q-1}$ with the $q$-integer $[p]_q$.

By the HKR theorem, one can view $\HH(\Z[\zeta_p][t]/\Z[\zeta_p])\ls{\hbar}$ as a $2$-periodification of the Hodge cohomology of $\AA^1_{\Z}$ base-changed along the map $\Z \to \Z[\zeta_p]$. Similarly, the aforementioned work of Raksit (see [95] and <ref>, as well as <ref>) implies that $\HP(\ku[t]/\ku)$ can be viewed as a $2$-periodification of the $q$-de Rham complex of $\Z[t]$. Since killing $[p]_q\in \Z\pw{q-1}$ amounts to specializing $q$ to a primitive $p$th root of unity, one can view <ref> as a topological analogue of the Cartier isomorphism for the $q$-de Rham complex of the affine line (see, e.g., <cit.>).

Taking homotopy fixed points for the $S^1$-equivariance of $\fr{C}$ from <ref>(a), we obtain an equivalence

\begin{equation}\label{q-ogus}
\fr{C}^{hS^1}: (\HH(\ku^{t\Cp}[t]/\ku^{t\Cp})^{hS^1})_{\leq m} \xar{\sim} ((\HH(\ku[t]/\ku)_{\leq mp})^{tS^1})^\wedge_p.
\end{equation}

More succinctly, there is a graded equivalence

$$(\fr{C}^\gr)^{hS^1}: \HH(\ku^{t\Cp}[t]/\ku^{t\Cp})^{hS^1} \xar{\sim} \HP^\gr(\ku[t]/\ku)^\wedge_p.$$

In future work, we show that Raksit's filtration on $\HP(\ku[t]/\ku)$ can be refined to construct a filtration on $\HH(\ku^{t\Cp}[t]/\ku^{t\Cp})^{hS^1}$ whose graded pieces are given by even shifts of $L\eta_{[p]_q} q\Omega_{\Z_p[t]}$.
Then, <ref> can be regarded as a $2$-periodification of the “Cartier isomorphism” $\phi_{\Z_p\pw{q-1}}^\ast q\Omega_{\Z_p[t]} \simeq L\eta_{[p]_q} q\Omega_{\Z_p[t]}$ for the $q$-de Rham cohomology of $\Z_p[t]$. (See <cit.> applied to the $q$-crystalline prism $(\Z_p\pw{q-1}, [p]_q)$.)

<ref> admits a mild generalization. Namely, if $\ku^{\Z/p^{n-1}}$ denotes the strict fixed points (so $\ku^{\Z/p^{n-1}} = \tau_{\geq 0}(\ku^{h\Z/p^{n-1}})$), then one can calculate $\pi_\ast (\ku^{\Z/p^{n-1}})^{t\Cp} \cong \pi_\ast \Z_p[\zeta_{p^n}]^{tS^1}$.

One can show that this can be extended to an equivalence $(\ku^{\Z/p^{n-1}})^{t\Cp} \simeq \Z_p[\zeta_{p^n}]^{tS^1}$ of $\E{2}$-rings.
<ref> and <ref> give a graded equivalence

$$\fr{C}: \HH(\Z[\zeta_{p^n}][t]/\Z[\zeta_{p^n}])\ls{\hbar} \xar{\sim} \HH(\ku^{\Z/p^{n-1}}[t]/\ku^{\Z/p^{n-1}})^{t\Cp}.$$

Here, the action of $\Cp$ on $\HH(\ku^{\Z/p^{n-1}}[t]/\ku^{\Z/p^{n-1}}) = \THH(S[t]) \otimes \ku^{\Z/p^{n-1}}$ is via the diagonal action on $\THH$ and $\ku^{\Z/p^{n-1}}$. In this case, one can therefore view <ref> as a topological analogue of the Cartier isomorphism for Hodge-Tate cohomology relative to the prism $(\Z_p\pw{q^{1/p^{n-1}}-1}, [p]_q)$ of the affine line.

More generally, let $R = \BP{n}$. As recalled in <ref>, <cit.> proved that there is an isomorphism $\pi_\ast \BP{n}^{t\Cp} \cong \pi_\ast \BP{n-1}^{tS^1}$, and this was conjectured to lift to an equivalence of spectra in <cit.>. If we assume that there is in fact an equivalence $\BP{n}^{t\Cp} \simeq \BP{n-1}^{tS^1}$ of $\E{2}$-rings, <ref> and <ref> give an equivalence

$$\fr{C}: \HH(\BP{n-1}[t]/\BP{n-1})\ls{\hbar}_{\leq m} \xar{\sim} (\HH(\BP{n}[t]/\BP{n})_{\leq mp})^{t\Cp}.$$

Therefore, <ref> in this case can be viewed as an analogue of the Cartier isomorphism for the affine line in the setting of “$v_n$-adic Hodge theory”. Taking homotopy fixed points for the $S^1$-equivariance of $\fr{C}$ from <ref>(a), we obtain an equivalence

\begin{equation}\label{c-ts1}
\fr{C}^{hS^1}: (\HH(\BP{n}^{t\Cp}[t]/\BP{n}^{t\Cp})^{hS^1})_{\leq m} \xar{\sim}
((\HH(\BP{n}[t]/\BP{n})_{\leq mp})^{tS^1})^\wedge_p.
\end{equation}

More succinctly, there is a graded equivalence

$$(\fr{C}^\gr)^{hS^1}: \HH(\BP{n}^{t\Cp}[t]/\BP{n}^{t\Cp})^{hS^1} \xar{\sim} \HP^\gr(\BP{n}[t]/\BP{n})^\wedge_p.$$

Note that <ref> in particular implies that if $T(n)$ admits the structure of an $\Efr{2}$-ring, then $\HP(\BP{n}[t]/\BP{n})$ is closely related to $\TP(\BP{n-1}[t]/T(n))$ by <ref>. In this form, <ref> holds when $\BP{n}$ is replaced by any complex-oriented $\E{2}$-ring $R$. As in the preceding examples, we believe that when $R$ is connective, this can be regarded as a $2$-periodification of a “Cartier isomorphism” for the two-term complex <ref>. See [43] for further discussion.

§ RELATIONSHIP TO THE MODULI STACK OF FORMAL GROUPS

§.§ Incarnation of the topological Sen operator over $\Mfg$

In <ref>, we showed that the descent spectral sequence of <ref> admits a generalization given by the topological Sen operator (<ref>). This has an incarnation over $\Mfg$, as we now explain. The analogues of <ref>, <ref>, etc., that we discuss in this section are useful for making topological predictions since the calculations involved are easier.
[Even filtration]
Let $\F^\star_\ev: \CAlg \to \CAlg(\Sp^\fil)$ be the even filtration of [53]: if $\CAlg^\ev$ denotes the full subcategory of $\CAlg$ spanned by the $\Eoo$-rings with even homotopy, then $\F^\star_\ev$ is the right Kan extension of the functor $\tau_{\geq 2\star}: \CAlg^\ev \to \CAlg(\Sp^\fil)$ along the inclusion $\CAlg^\ev \hookrightarrow \CAlg$. Note that since $\tau_{\geq 2\star}$ is lax symmetric monoidal and $\F^\star_\ev$ is defined by a right Kan extension, it is also a lax symmetric monoidal functor. We will need the following result from [53]: if $R$ is an $\Eoo$-ring such that $\MU \otimes R \in \CAlg^\ev$, then $\F^\star_\ev R$ is $p$-completely equivalent to the underlying filtered $\Eoo$-ring of its Adams-Novikov tower $\nu(R)\in \Syn_\MU^\ev(\Sp) = \Mod_{\Tot(\tau_{\geq 2\star} \MU^{\otimes \bull+1})}(\Sp^\fil)$. (Also see [92, 48].) In this case, the associated graded Hopf algebroid $(\MU_\ast(R), \MU_\ast(\MU\otimes R))$ defines a stack over $B\GG_m$. If $R$ is complex-oriented, then this stack is isomorphic to $\spec(\pi_\ast R)/\GG_m$, where the $\GG_m$-action encodes the grading on $\pi_\ast R$.

In order to define the stack $\tilde{\cM}_R$ associated to the graded Hopf algebroid $(\MU_\ast(R), \MU_\ast(\MU\otimes R))$, one does not need $R$ to be an $\Eoo$-ring: it only needs to admit the structure of a homotopy commutative ring such that $\MU_\ast(R)$ is concentrated in even degrees. This perspective is explained in Hopkins' lecture in <cit.>. In particular, one can define the stack associated to $X(n)$: this is the moduli stack of graded formal groups equipped with a coordinate of order $\leq n$, and strict isomorphisms between them. (See, e.g., <cit.>.)

We will find it convenient to work with the $p$-typical variant of the graded Hopf algebroid $(\MU_\ast(R), \MU_\ast(\MU\otimes R))$. Namely, if $R$ is a $p$-local homotopy commutative ring such that $\BPP_\ast(R)$ is concentrated in even degrees, then we will write $\cM_R$ to denote the graded stack associated to the graded Hopf algebroid $(\BPP_\ast(R), \BPP_\ast(\BPP\otimes R))$. For example, $\cM_{T(n)}$ is the moduli stack of $p$-typical graded formal groups equipped with a coordinate up to order $\leq p^{n+1}-1$; by $p$-typicality, this is further isomorphic to the moduli stack of $p$-typical graded formal groups equipped with a coordinate up to order $\leq p^n$. In particular, $\cM_{S^0}$ is isomorphic to the moduli stack $\Mfg$ of $p$-typical graded formal groups. Similarly, if $R$ is a $p$-local complex-oriented homotopy commutative ring, then $\cM_R$ is isomorphic to $\spec(\pi_\ast R)/\GG_m$.

The unit map $S^0 \to \MU$ induces the map $\tilde{\cM}_\MU \cong \spec(\MU_\ast)/\GG_m \to \tilde{\cM}_{S^0}$ which describes the flat cover of the moduli stack of graded formal groups given by the graded Lazard ring. This map exhibits $\tilde{\cM}_{S^0}$ as the quotient of $\spec(\MU_\ast)/\GG_m$ by the group scheme $\spec(\pi_\ast(\MU \otimes \MU))/\GG_m$. Note that $\MU \otimes \MU \simeq \MU[\BU]$; since $\pi_\ast \Z[\BU]$ is the coordinate ring of the big Witt ring scheme, we see that $\spec(\pi_\ast(\MU \otimes \MU))/\GG_m$ is a lift of the big Witt ring scheme to $\spec(\MU_\ast)/\GG_m$. Similarly, $\cM_{S^0} = \Mfg$ is the quotient of $\cM_{\BPP} = \spec(\BPP_\ast)/\GG_m$ by a lift of the $p$-typical Witt ring scheme $\W$ to $\spec(\BPP_\ast)/\GG_m$.

If $A \to B$ is a map of $p$-local $\Eoo$-rings such that $\MU_\ast(A)$ and $\MU_\ast(B)$ are even, then there is an induced map $\cM_B \to \cM_A$ of graded stacks. Recall that $\THH(B/A)$ is the geometric realization of the simplicial $A$-algebra $B^{\otimes_A \bull+1}$. Applying $\F^\star_\ev$ levelwise to $B^{\otimes_A \bull+1}\in \Fun(\Deltab^\op, \CAlg_A)$ produces an Adams-Novikov analogue of the Bökstedt spectral sequence:

$$\pi_\ast \HH(\cM_B/\cM_A) \Rightarrow \pi_\ast \gr^\bull_\ev \THH(B/A).$$

In particular, note that $\HH(\cM_B/\Mfg)$ is an approximation to $\gr^\bull_\ev \THH(B)$. For this spectral sequence to exist, it is not necessary that $A$ and $B$ be $\Eoo$-rings: for example, it suffices that $A \to B$ be a map of $p$-local $\E{2}$-rings such that $\MU_\ast(A)$ and $\MU_\ast(B)$ are even, and such that $\THH(B/A)$ is bounded below and has even $\MU$-homology. Then, $\gr^\bull_\ev \THH(B/A)$ must be interpreted as the associated graded of the Adams-Novikov filtration on $\THH(B/A)$; see <cit.>.

Let $\Mfg^s$ denote the total space of the canonical line bundle over $\Mfg$ (determined by the map $\Mfg \to B\GG_m$).
If $R$ is a $p$-quasisyntomic ring, then <cit.> and <ref> give a spectral sequence

$$\pi_\ast \HH(\spec(R)/\Mfg^s) \Rightarrow \pi_\ast \cN^\ast \hat{\prism}_R[2\ast].$$

Indeed, $\cM_R = \spec(R)/\GG_m \cong \spec(R) \times B\GG_m$, so that the underlying $R$-algebra of $\HH(\cM_R/\Mfg)$ is $\HH(\spec(R)/\Mfg^s)$.

The complex orientation $\BPP \to \BP{n}$ induces a map $\cM_{\BP{n}} \to \cM_{\BPP}$ which factors the structure map $\cM_{\BP{n}} \to \Mfg$. Explicitly, we have the following composite map of stacks over $B\GG_m$:

$$\spec(\BP{n}_\ast)/\GG_m \to \spec(\BPP_\ast)/\GG_m \to \Mfg.$$

Taking cotangent complexes gives the following transitivity cofiber sequence in $\Mod_{\BP{n}_\ast}^\gr$:

$$\BP{n}_\ast \otimes_{\BPP_\ast} L_{\spec(\BPP_\ast)/\Mfg} \to L_{\BP{n}_\ast/\Mfg} \to L_{\BP{n}_\ast/\BPP_\ast}.$$

Since $\BPP_\ast/(v_{n+1}, v_{n+2}, \cdots) \cong \BP{n}_\ast$, observe that $L_{\BP{n}_\ast/\BPP_\ast}$ is a free $\BP{n}_\ast$-module generated by classes $\sigma(v_{n+1}), \sigma(v_{n+2}), \cdots$.

Similarly, the discussion in <ref> implies that $L_{\spec(\BPP_\ast)/\Mfg}$ is a free $\BPP_\ast$-module generated by classes $d(t_i)$.

From this, one can deduce that $L_{\spec(\BP{n}_\ast)/\GG_m/\Mfg}$ is a free $\BP{n}_\ast$-module generated by classes $\sigma(v_j)$ with $j\geq n+1$ and $d(t_i)$ with $i\geq 1$. By the HKR theorem, $\pi_\ast \HH(\spec(\BP{n}_\ast)/\GG_m/\Mfg)$ is isomorphic to $\Sym_{\BP{n}_\ast}(L_{\BP{n}_\ast/\Mfg}[1])$,

which can be identified as

$$\pi_\ast \HH(\spec(\BP{n}_\ast)/\GG_m/\Mfg) \cong \BP{n}_\ast\pdb{\sigma^2 v_j | j\geq n+1} \otimes_{\BP{n}_\ast} \Lambda_{\BP{n}_\ast}(dt_i | i\geq 1).$$

Since $v_j$ lives in degree $2p^j-2$ and weight $p^j-1$, the class $\sigma^2 v_j$ lives in degree $2p^j = |v_j| + 2$ and weight $p^j$; similarly, since $t_i$ lives in degree $2p^i-2$ and weight $p^j-1$, the class $dt_i$ lives in degree $2p^i-1$ and weight $p^j$.

The same discussion for the following composite map of stacks over $B\GG_m$

$$\spec(\BP{n-1}_\ast)/\GG_m \to \spec(\BPP_\ast)/\GG_m \to \cM_{T(n)}$$

shows that $L_{\spec(\BP{n-1}_\ast)/\GG_m/\cM_{T(n)}}$ is a free $\BP{n-1}_\ast$-module generated by classes $\sigma(v_j)$ with $j\geq n$ and $d(t_i)$ with $i\geq n+1$. Therefore, the HKR theorem implies that $\pi_\ast \HH(\spec(\BP{n-1}_\ast)/\GG_m/\M_{T(n)})$ is isomorphic to a symmetric algebra over $\BP{n-1}_\ast$ on classes $\sigma^2(v_i)$ for $i \geq n$,

and $d(t_i)$ for $i\geq n+1$.

$$\pi_\ast \HH(\spec(\BP{n-1}_\ast)/\GG_m/\M_{T(n)}) \cong \BP{n-1}_\ast\pdb{\sigma^2 v_j | j\geq n} \otimes_{\BP{n-1}_\ast} \Lambda_{\BP{n-1}_\ast}(dt_i | i\geq n+1).$$

The class $\sigma^2 v_j$ lives in degree $2p^j = |v_j| + 2$ and weight $p^j$, and the class $dt_i$ lives in degree $2p^i-1$ and weight $p^j$.
This mirrors the calculation of the $E^2$-term of the Bökstedt spectral sequence in <ref>.

In fact, one can recover <ref> in this way by running the Adams-Novikov-Bökstedt spectral sequence (<ref>) and using the $\E{2}$-Dyer-Lashof argument of <ref> to resolve the extension problems on the $E^\infty$-page. We use the term “recover” in a very weak sense here: the differentials in the Adams-Novikov-Bökstedt spectral sequence are forced by the differentials in the usual Bökstedt spectral sequence (<ref>). Explicitly, we have

$$d^{p-1}(\gamma_j(\sigma^2 v_m)) = \gamma_{j-p}(\sigma^2 v_m) dt_m$$

modulo decomposables, and the spectral sequence collapses on the $E_p$-page. There are topologically determined extensions $(\sigma^2 v_m)^p = \sigma^2 v_{m+1}$ modulo decomposables, which give an isomorphism (as implied by <ref>)

$$\pi_\ast \gr^\bull_\ev \THH(\BP{n-1}/T(n)) \cong \BP{n-1}_\ast[\sigma^2(v_n)].$$

Let $Y$ be a scheme, and let $q: X \to \AA^1\times Y$ be a morphism, so that $X$ is a scheme over $Y$ via the projection $\pr: \AA^1\times Y \to Y$. Then the transitivity cofiber sequence in $\QCoh(X)$ runs

$$q^\ast L_{\AA^1\times Y/Y} \to L_{X/Y} \to L_{X/\AA^1\times Y}.$$

Since $q^\ast L_{\AA^1\times Y/Y}$ is a free $\co_X$-module of rank $1$ generated by $dt$ (where $t$ is a coordinate on $\AA^1$), we obtain a cofiber sequence

$$\dR^\ast_{X/Y} \to \dR^\ast_{X/\AA^1\times Y} \xar{\nabla} \dR^\ast_{X/\AA^1\times Y} dt,$$

where $\dR^\ast_{X/Y} = \bigoplus_{i\geq 0} (\wedge^i L_{X/Y})[-i]$ denotes the underlying derived commutative algebra of the downwards-shearing of $\Sym_{\co_X}(L_{X/Y}[1](1))$. The map $\nabla$ is the Gauss-Manin connection for the morphism $q$. Note that $\nabla$ satisfies Griffiths transversality: it sends the $n$th piece of the Hodge filtration to the $(n-1)$st piece.

Observe that if $q$ is taken to be the morphism $Y \to \AA^1\times Y$ given by the inclusion of the origin into $\AA^1$, then $\dR_{Y/\AA^1\times Y}$ is $p$-completely isomorphic to the divided power algebra $\co_Y\pdb{t}$. Using the fact that $\dR_{Y/Y} \cong \co_Y$, it is not difficult to see that the Gauss-Manin connection $\nabla$ must send $\gamma_j(t) \mapsto \gamma_{j-1}(t) dt$. Here, we set $\gamma_{-1}(t) = 0$. In particular, $\nabla$ is a PD-derivation.

[The topological Sen operator and $\Mfg$]
The map $T(n-1) \to T(n)$ of homotopy commutative rings induces a map $\cM_{T(n)} \to \cM_{T(n-1)}$ of graded stacks, which sends a $p$-typical graded formal group equipped with a coordinate up to order $\leq p^n$ to the underlying $p$-typical graded formal group equipped with a coordinate up to order $\leq p^n-1$. The map $\cM_{T(n)} \to \cM_{T(n-1)}$ is an affine bundle: in other words, it exhibits $\cM_{T(n-1)}$ as the quotient of $\cM_{T(n)}$ by the group scheme $\GG_a^{(p^n-1)}/\GG_m$ over $B\GG_m$, where $\GG_a^{(p^n-1)}$ denotes the affine line with $\GG_m$-action of weight $p^n-1$. This follows, for instance, from <cit.>. If $f: X \to \cM_{T(n)}$ is a stack over $\cM_{T(n)}$, the transitivity cofiber sequence in $\QCoh(X)$ is given by

$$f^\ast L_{\cM_{T(n)}/\cM_{T(n-1)}} \to L_{X/\cM_{T(n-1)}} \to L_{X/\cM_{T(n)}}.$$

Since $\cM_{T(n)} \to \cM_{T(n-1)}$ is a $\GG_a$-bundle, we see that $L_{\cM_{T(n)}/\cM_{T(n-1)}}$ is a free $\co_{\cM_{T(n)}}$-module of rank $1$ generated by the class $dt_i$.

It follows that there is a cofiber sequence

\begin{equation}\label{hh-mtn}
\HH(X/\cM_{T(n-1)}) \to \HH(X/\cM_{T(n)}) \xar{\Theta_\mot} \Sigma^{2p^n,p^n} \HH(X/\cM_{T(n)})
\end{equation}

of quasicoherent sheaves on $X$, where $\Sigma^{n,w}$ denotes a shift by degree $n$ and weight $w$. As indicated by the notation, the map $\Theta_\mot$ behaves as an analogue on $\Mfg$ of the topological Sen operator of <ref>; more precisely, it is the effect of the topological Sen operator at the level of the $E^2$-page of the Adams-Novikov-Bökstedt spectral sequence of <ref>. Moreover, the discussion in <ref> says that $\Theta_\mot$ can be understood as an analogue of the Gauss-Manin connection.

The topological Sen operator on $\THH(\Z_p/J(p)) \cong \Z_p[x]$ sends $x^j\mapsto jx^{j-1}$, so that the action of the Sen operator is precisely the action of $\GG_a^\sharp$ on $\GG_a = \spec \Z_p[x]$ given by $\partial_x: \Z_p[x] \to \Z_p[x]$. Therefore, there is a $p$-complete graded isomorphism $\gr^\bull_\ev \THH(\Z_p) \cong \Gamma(\GG_a/\GG_a^\sharp; \co)$. In the same way, one can argue that there is a $p$-complete isomorphism $\gr^\bull_\ev \THH(\Z_p)^{t\Cp} \cong \Gamma(\GG_m/\GG_m^\sharp; \co)$.

This perspective is related to the stacky approach to Hodge-Tate cohomology à la [47, 18] in the following way. By <cit.>, there is an isomorphism $\GG_a/\GG_a^\sharp \cong \GG_a^\dR$; similarly, $\GG_m/\GG_m^\sharp \cong \GG_m^\dR$. Therefore:

\begin{align}
\gr^\bull_\ev \THH(\Z_p) & \cong \Gamma(\GG_a^\dR; \co), \label{grmot-thhz}\\
\gr^\bull_\ev \THH(\Z_p)^{t\Cp} & \cong \Gamma(\GG_m^\dR; \co). \label{grmot-thhz-tcp}
\end{align}

Since $\gr^\bull_\ev \THH(\Z_p)^{t\Cp}$ is supposed to arise as the cohomology of the total space $\Tot(\co_{\WCart^\HT}\{1\})$ of the Breuil-Kisin twisting line bundle $\co_{\WCart^\HT}\{1\}$ over $\WCart^\HT$, the isomorphism <ref> suggests that $\Tot(\co_{\WCart^\HT}\{1\}) \cong \GG_m^\dR$. In turn, this suggests that $\WCart^\HT$ should be $\GG_m^\dR/\GG_m = B\GG_m^\sharp$. This is indeed true: it is precisely <cit.>.

Similarly, $\gr^\bull_\ev \THH(\Z_p)$ is supposed to arise as the cohomology of the total space of the Breuil-Kisin twisting line bundle over the “extended Hodge-Tate locus” $\Delta_0'$ in Drinfeld's $\Sigma'$. (The stack $\Delta_0'$ is defined in <cit.>.) In [16], the stack $\Sigma'$ is denoted by $\spf(\Z_p)^\cN$, and one might therefore denote $\Delta_0'$ by $\spf(\Z_p)^{\cN, \HT}$. The isomorphism <ref> then suggests that the total space of the Breuil-Kisin line bundle over $\spf(\Z_p)^{\cN, \HT}$ is $\GG_a^\dR$, which in turn suggests that $\spf(\Z_p)^{\cN, \HT}$ should be $\GG_a^\dR/\GG_m \cong \GG_a/(\GG_a^\sharp \rtimes \GG_m)$. This is indeed true: it is precisely <cit.>.

Had we worked with the evenly faithfully flat cover $\gr^\bull_\ev \THH(\Z_p) \to \gr^\bull_\ev \THH(\Z_p/S[t])$ (where $t\mapsto p$) instead, the stack associated to the even filtration on $\THH(\Z_p)$ would in fact be presented by (and is therefore isomorphic to) $\GG_a^\dR/\GG_m$.

One can also study the stack $\cM_{J(p)}$ associated to the $\Efr{2}$-ring $J(p)$. It is not difficult to show that the morphism $\cM_{J(p)} \to \Mfg$ exhibits $\cM_{J(p)}$ as a $\GG_m$-bundle over $\Mfg$; for example, the fiber product $\spec(\MU_\ast)/\GG_m \times_{\Mfg} \cM_{J(p)}$ is isomorphic to $\spec(\pi_\ast(\MU \otimes J(p)))/\GG_m$, but there is an equivalence of $\E{2}$-$\MU$-algebras $\MU \otimes J(p) \simeq \MU[t^{\pm 1}]$ with $|t|=0$.

Since $\cM_{J(p)}$ is a $\GG_m$-bundle over $\Mfg$, descent in Hochschild homology is controlled by a Gauss-Manin connection. If $Y$ is a scheme and $q: X \to \GG_m\times Y$ is a morphism, then there is a cofiber sequence

$$\dR^\ast_{X/Y} \to \dR^\ast_{X/\GG_m\times Y} \xar{\nabla} \dR^\ast_{X/\GG_m\times Y} d\log(t).$$

If $X$ is a stack over $\cM_{J(p)}$, we then obtain a cofiber sequence

$$\HH(X/\Mfg) \to \HH(X/\cM_{J(p)}) \xar{\Theta_\mot} \Sigma^{2,1} \HH(X/\cM_{J(p)})$$

of quasicoherent sheaves on $X$. This is an analogue on $\Mfg$ of the topological Sen operator of <ref>.

Suppose that $T(1)$ admits the structure of an $\Efr{2}$-ring (this is true at $p=2$). The unit map on $T(1)$ defines a map $\TP(\Z_p) \to \TP(\Z_p/T(1))$. Since $\TP(\Z_p/T(1))$ is concentrated in even degrees by <ref>, one can define the motivic filtration on $\TP(\Z_p/T(1))$ using the double-speed Postnikov filtration. Under the isomorphism $\pi_\ast \TP(\Z_p/T(1)) \cong \pi_\ast \BP{1}^{tS^1} \cong \Z_p\pw{\ptl}^{tS^1}$, one can view $\gr^0$ of the motivic filtration $\TP(\Z_p/T(1))$ as $\Z_p\pw{\ptl}$. Recall that $\TP(\Z_p)$ is a homotopical analogue of the Cartier-Witt stack $\WCart_{\Z_p}$ from [19]. One can then view the map $\TP(\Z_p) \to \TP(\Z_p/T(1))$ as an analogue of the following map induced by the $q$-de Rham point:

$$\rho_{\ptl\dR}:\spf \Z_p\pw{\ptl} \cong (\spf \Z_p\pw{q-1})/\FF_p^\times \to (\spf \Z_p\pw{q-1})/\Z_p^\times \xar{\rho_{q\dR}} \WCart_{\Z_p}.$$

This map classifies the prism $(\Z_p\pw{\ptl}, (\ptl))$, and can reasonably be called the $\ptl$-de Rham point.

As explained in the end of the introduction to [53], one hopes that the unit map $S^0 \to \TP(\Z_p)$ induces the map $\WCart_{\Z_p} \to \Mfg$ classifying Drinfeld's formal group over $\WCart_{\Z_p} = \Sigma$ from [46] on the associated graded of the motivic filtration. If <ref> were true (i.e., there is an equivalence $\TP(\Z_p/T(1)) \simeq \BP{1}^{tS^1}$ of spectra), the resulting unit map $S^0 \to \TP(\Z_p/T(1)) \to \BP{1}^{tS^1}$ would just be the unit of the $\Eoo$-ring $\BP{1}^{tS^1}$. Since $\BP{1}$ is complex-oriented, the formal group over $\pi_0 \BP{1}^{tS^1} \cong \Z_p\pw{\ptl}$ must be isomorphic to the formal group of $\BP{1}$, i.e., the $p$-typification of the multiplicative formal group. In particular, the aforementioned expectation about the formal group over $\WCart_{\Z_p}$ and its relation to $\TP(\Z_p)$ would predict that the pullback of Drinfeld's formal group over $\WCart_{\Z_p}$ along the map $\rho_{\ptl\dR}$ is the $p$-typification of the multiplicative formal group over $\Z_p\pw{\ptl}$. This is indeed true, and was proved in <cit.>. This lends further evidence to the idea that the map $\TP(\Z_p) \to \TP(\Z_p/T(1))$ is a homotopical analogue of the $\ptl$-de Rham point of $\WCart_{\Z_p}$.

§.§ Comparing $\THH$ relative to $T(n)$ and $T(n+1)$

Recall from <ref> that $\pi_\ast(\THH(\BP{n}/T(n+1)) \otimes_{\BP{n}} \BP{n-1})$ is (additively) equivalent to the “subalgebra” $\BP{n-1}_\ast[\theta_{n-1}^p]$ of $\pi_\ast \THH(\BP{n-1}/T(n)) \cong \BP{n-1}_\ast[\theta_{n-1}]$. This picture has an analogue over $\Mfg$, as we now explain. We first need a simple calculation.

Let $R$ be a commutative ring, and let $x\in R$ be a regular element. Then there is a $p$-completed equivalence $\dR^\ast_{R[t]/x / R} \simeq R[t]\pdb{x'}/x \otimes_{R[t]/x} \Lambda_{R[t]/x}(dt)$ with $|x'|=0$. Indeed, this follows from combining the observation that $R[t]/x \cong R[t] \otimes_R R/x$ with the following $p$-completed equivalences: $\dR^\ast_{R[t]/R} \simeq \Lambda_{R[t]}(dt)$, $\dR^\ast_{R/x / R} \simeq R\pdb{x'}/x$. Similarly, there is an equivalence $\HH(R[t]/x / R) \simeq R[t][S^1 \times \CP^\infty]/x$.

Let $i_{n-1}: \cZ(v_{n-1}) \hookrightarrow \cM_{T(n)}$ denote the closed substack cut out by the global section $v_{n-1}\in \H^0(\cM_{T(n)}; \co_{\cM_{T(n)}})$. If $f: X \to \cM_{T(n)}$ is a stack over $\cM_{T(n)}$, let $X^{v_{n-1}=0}$ denote the pullback of $X$ along $i_{n-1}$, and let $f: X^{v_{n-1}=0} \to \cZ(v_{n-1})$ denote the structure morphism. Then $i_{n-1}^\ast \HH(X/\cM_{T(n)}) = \HH(X^{v_{n-1}=0}/\cZ(v_{n-1}))$. In the case $X = \spec(\BP{n}_\ast)/\GG_m$, there is an isomorphism $X^{v_{n-1}=0} = \spec(\BP{n-1}_\ast)/\GG_m$. We will now relate $\HH(X^{v_{n-1}=0}/\cZ(v_{n-1}))$ to $\HH(X^{v_{n-1}=0}/\cM_{T(n-1)})$ by calculating $\HH(\cZ(v_{n-1})/\cM_{T(n-1)})$.

Recall from <ref> that there is a $\GG_a$-bundle $\cM_{T(n)} \to \cM_{T(n-1)}$. Note that $L_{\cM_{T(n)}/\cM_{T(n-1)}}$ is a free $\co_{\cM_{T(n)}}$-module of rank $1$ generated by a class $d(t_n)$,

and that $L_{\cZ(v_{n-1})/\cM_{T(n)}}$ is a free $\co_{\cZ(v_{n-1})}$-module of rank $1$ generated by a class $\sigma^2(v_{n-1})$.

Applying <ref>, we find that

\begin{equation}\label{hh-z-vn1}
\pi_\ast \HH(\cZ(v_{n-1})/\cM_{T(n-1)}) \cong \co_{\cZ(v_{n-1})}\pdb{\sigma^2(v_{n-1})} \otimes_{\co_{\cZ(v_{n-1})}} \Lambda_{\co_{\cZ(v_{n-1})}}(dt_n).
\end{equation}

We therefore see that $\HH(X^{v_{n-1}=0}/\cM_{T(n-1)})$ a subquotient of the tensor product of $\HH(X^{v_{n-1}=0}/\cZ(v_{n-1}))$ and $f^\ast \HH(\cZ(v_{n-1})/\cM_{T(n-1)}) \cong \co_{X^{v_{n-1}=0}}\pdb{\sigma(v_{n-1})}[dt_n]/(dt_n)^2$.
Let us now take $f$ to be the morphism $\spec(\BP{n-1}_\ast)/\GG_m \to \cM_{T(n)}$. The $E^2$-page of the Adams-Novikov-Bökstedt spectral sequence for $\THH(\BP{n-2}/T(n-1))$ is given by

$$E^2_{\ast,\ast} = \BP{n-2}_\ast\pdb{\sigma^2 v_j | j\geq n-1} \otimes_{\BP{n-1}_\ast} \Lambda_{\BP{n-1}_\ast}(dt_j | j \geq n),$$

and the extensions on the $E^\infty$-page are given by $(\sigma^2 v_j)^{p^{n-j}} = \sigma^2 v_n$. The above discussion therefore shows that $f^\ast \HH(\cZ(v_{n-1})/\cM_{T(n-1)})$ precisely detects the “bottom piece” of this $E^2$-page, i.e., the subalgebra $\BP{n-2}_\ast\pdb{\sigma^2 v_{n-1}} \otimes_{\BP{n-2}_\ast} \Lambda_{\BP{n-2}_\ast}(dt_n)$. Therefore, the preceding calculation of $\HH(\cZ(v_{n-1})/\cM_{T(n-1)})$ gives one explanation for why $\pi_\ast(\THH(\BP{n-1}/T(n)) \otimes_{\BP{n-1}} \BP{n-2})$ is (additively) equivalent to the “subalgebra” $\BP{n-2}_\ast[\theta_{n-2}^p]$ of $\pi_\ast \THH(\BP{n-2}/T(n-1)) \cong \BP{n-2}_\ast[\theta_{n-2}]$.

We can extend the analysis of <ref> further.
Let $0\leq j\leq n-1$, and let $i_{j,\cdots,n-1}: \cZ(v_{[j,n)}) \hookrightarrow \cM_{T(n)}$ denote the closed substack cut out by the global sections $v_j,\cdots,v_{n-1} \in \H^0(\cM_{T(n)}; \co_{\cM_{T(n)}})$. If $f: X \to \cM_{T(n)}$ is a stack over $\cM_{T(n)}$, let $X^{v_j,\cdots,v_{n-1}=0}$ denote the pullback of $X$ along $i$, and let $f: X^{v_j,\cdots,v_{n-1}=0} \to \cZ(v_{[j,n)})$ denote the structure morphism. Then $i_{j,\cdots,n-1}^\ast \HH(X/\cM_{T(n)})$ is equivalent to $\HH(X^{v_j,\cdots,v_{n-1}=0}/\cZ(v_{[j,n)}))$. In the case $X = \spec(\BP{n-1}_\ast)/\GG_m$, there is an isomorphism $X^{v_j,\cdots,v_{n-1}=0} = \spec(\BP{j-1}_\ast)/\GG_m$. We can now relate $\HH(X^{v_j,\cdots,v_{n-1}=0}/\cZ(v_{[j,n)}))$ to $\HH(X^{v_j,\cdots,v_{n-1}=0}/\cM_{T(j)})$ by calculating $\HH(\cZ(v_{[j,n)})/\cM_{T(j)})$.

We claim that there is an isomorphism

$$\HH(\cZ(v_{[j,n)})/\cM_{T(j)}) \cong \co_{\cZ(v_{[j,n)})}\pdb{\sigma(v_i) | j\leq i\leq n-1} \otimes_{\co_{\cZ(v_{[j,n)})}} \Lambda_{\co_{\cZ(v_{[j,n)})}}(dt_i | j+1\leq i \leq n)$$

To prove this, we will use descending induction on $j$; the base case $j=n-1$ was studied in <ref>. For the inductive step, suppose we know the result for $j+1$. Let $i_j: \cZ(v_{[j,n)}) \to \cZ(v_{j+1}, \cdots, v_{n-1})$ denote the closed substack cut out by $v_j$. Then there are isomorphisms

\begin{align*}
\HH(\cZ(v_{[j,n)})/\cM_{T(j+1)}^{v_j=0}) & \cong i_j^\ast \HH(\cZ(v_{j+1}, \cdots, v_{n-1})/\cM_{T(j+1)})\\
& \cong \co_{\cZ(v_{[j,n)})}\pdb{\sigma^2(v_i) | j+1\leq i\leq n-1} \otimes_{\co_{\cZ(v_{[j,n)})}} \Lambda_{\co_{\cZ(v_{[j,n)})}}(dt_i | j+2\leq i \leq n)
%[\sigma(v_{j+1}), \cdots, \sigma(v_{n-1})]/(\sigma(v_{j+1})^p, \cdots, \sigma(v_{n-1})^p).
\end{align*}

Recall that <ref> gives an isomorphism between $\HH(\cM_{T(j+1)}^{v_j=0}/\cM_{T(j)})$ and $\co_{\cM_{T(j+1)}^{v_j=0}}\pdb{\sigma^2(v_j)} \otimes_{\co_{\cM_{T(j+1)}^{v_j=0}}} \Lambda_{\co_{\cM_{T(j+1)}^{v_j=0}}}(dt_{j+1})$. The desired calculation of $\HH(\cZ(v_{[j,n)})/\cM_{T(j)})$ is now a simple computation with the transitivity sequence for the composite

$$\cZ(v_{[j,n)}) \to \cM_{T(j+1)}^{v_j=0} \to \cM_{T(j)}.$$

Let $X = \spec(\BP{j-1}_\ast)/\GG_m$, and let $f: X \to \cZ(v_{[j,n)})$ be the structure map. Then the above discussion implies that $\HH(\spec(\BP{j-1}_\ast)/\GG_m/\cM_{T(j)})$ is isomorphic to the tensor product of $f^\ast \HH(\cZ(v_{[j,n)})/\cM_{T(j)})$ and $\HH(\spec(\BP{j-1}_\ast)/\GG_m/\cZ(v_{[j,n)}))$. This gives the $E^2$-page of the Adams-Novikov-Bökstedt spectral sequence computing $\pi_\ast \THH(\BP{j-1}/T(j))$ (see <ref>), and one can run this spectral sequence as in <ref>. If $\BP{j-1}$ admits the structure of an $\E{3}$-algebra, there are extensions $\sigma^2(v_i)^p = \sigma^2(v_{i+1})$ modulo decomposables on the $E^\infty$-page of this spectral sequence.

Let $T(n)/v_{[j,n)}$ denote $T(n)/(v_j, \cdots, v_{n-1})$. Since $\theta_j\in \pi_\ast \THH(\BP{j-1}/T(j))$ is represented by $\sigma^2(v_j)$, we find that $\THH(T(n)/v_{[j,n)}/T(j))$ is (additively) equivalent to as $T(n)[\theta_j]/(v_{[j,n)}, \theta_j^{p^{n-j}})$. (See <ref> for a more topological perspective on this observation.) This discussion provides an algebraic perspective on why $\pi_\ast \THH(\BP{n-1}/T(n))/v_{[j,n)}$ is (additively) equivalent to as the “subalgebra” of $\pi_\ast \THH(\BP{j-1}/T(j))$ generated by $\theta_j^{p^{n-j}}$.

In topology, <ref> plays out as follows, if we assume[There is an unconditional variant of the following discussion, obtained by replacing $T(n)$ with $X(p^{n+1}-1)$. However, this comes at the cost of adding the spaces $\Delta_n$ into the mix.] <ref>. Let $n\geq 1$. We begin by observing that $T(n)/v_{n-1}$ is the Thom spectrum of an $\E{1}$ map $\mu: \Omega J_{p-1}(S^{2p^{n-1}}) \to \BGL_1(T(n-1))$; in particular, $T(n)/v_{n-1}$ admits the structure of an $\E{1}$-ring. To see this, we first define the map $\mu$ as follows. There is a map $S^{2p^{n-1}} \to \B^2\GL_1(X(p^n-1))$ which detects the class $v_{n-1}\in \pi_{2p^{n-1}-2} X(p^n-1)$, which naturally extends to a map $J_{p-1}(S^{2p^{n-1}}) \to \B^2\GL_1(X(p^n-1))$ since we are working $p$-locally. Therefore, we obtain an $\E{1}$-map $\Omega J_{p-1}(S^{2p^{n-1}}) \to \BGL_1(X(p^n-1))$. The projection $X(p^n-1) \to T(n-1)$ is a map of $\E{2}$-rings by <ref>, and therefore induces an $\E{1}$-map $\BGL_1(X(p^n-1))\to \BGL_1(T(n-1))$. Composition with
the $\E{1}$-map $\Omega J_{p-1}(S^{2p^{n-1}}) \to \BGL_1(X(p^n-1))$ produces the desired map $\mu$.
The fact that the Thom spectrum of $\mu$ can be identified with $T(n)/v_{n-1}$ can be proved directly using <ref> below.

It follows from this discussion that there is an equivalence

$$\THH(T(n)/v_{n-1}/T(n-1)) \simeq T(n)[J_{p-1}(S^{2p^{n-1}})]/v_{n-1}.$$

Moreover, under the equivalence $\THH(\BP{n-2}/T(n-1)) \simeq \BP{n-2}[\Omega S^{2p^{n-1} + 1}]$ of <ref>(a), the map $\THH(T(n)/v_{n-1}/T(n-1)) \to \THH(\BP{n-2}/T(n-1))$ induced by the map $T(n)/v_{n-1} \to \BP{n-2}$ is given by the skeletal inclusion of $J_{p-1}(S^{2p^{n-1}}) \to \Omega S^{2p^{n-1}+1}$. The projection $\THH(\BP{n-2}/T(n-1)) \to \THH(\BP{n-1}/T(n))/v_{n-1}$ can be identified with the effect on $\BP{n-2}$-chains of the James-Hopf map $\Omega S^{2p^{n-1}+1} \to \Omega S^{2p^n+1}$. Therefore, the EHP sequence

$$J_{p-1}(S^{2p^{n-1}}) \to \Omega S^{2p^{n-1}+1} \to \Omega S^{2p^n+1}$$

shows that $\THH(\BP{n-1}/T(n))/v_{n-1}$ is (additively) equivalent to precisely as the “subalgebra” of $\THH(\BP{n-2}/T(n-1))$ generated by $\theta_{n-1}^p$.
The above calculation of $\THH(T(n)/v_{n-1}/T(n-1))$ is a topological incarnation of the calculation of $\HH(\cZ(v_{n-1})/\cM_{T(n-1)})$ in <ref>. Indeed, the Adams-Novikov-Bökstedt spectral sequence (see <ref>) runs

\begin{equation}\label{anss-z-vn1}
E^2_{\ast,\ast} = \pi_\ast \HH(\cZ(v_{n-1})/\cM_{T(n-1)}) \Rightarrow \pi_\ast \gr^\bull_\ev \THH(T(n)/v_{n-1}/T(n-1)),
\end{equation}

and the $E^2$-page is given by <ref>. Again, one can establish analogues of the Bökstedt differentials <ref> in the Adams-Novikov-Bökstedt spectral sequence, and thereby obtain an alternative approach to the above calculation of $\THH(T(n)/v_{n-1}/T(n-1))$.

Let us continue to assume <ref>, and let $0\leq j \leq n-1$. Recall that $T(n)/v_{[j,n)}$ denote $T(n)/(v_j, \cdots, v_{n-1})$. Recall that

\begin{equation}\label{tn-vj-vn1-homology}
\H_\ast(T(n)/v_{[j,n)}; \FF_p) =
\begin{cases}
\FF_2[\zeta_1^2, \cdots, \zeta_j^2, \zeta_{j+1}, \cdots, \zeta_n] & p=2, \\
\FF_p[\zeta_1, \cdots, \zeta_n] \otimes \Lambda_{\FF_p}(\tau_j, \cdots, \tau_{n-1}) & p>2.
\end{cases}
\end{equation}

It is natural to ask if the discussion in <ref> extends to a description of $\THH(T(n)/v_{[j,n)}/T(j))$, paralleling <ref>. This is an ill-posed question, since it is not clear that $T(n)/v_{[j,n)}$ admits the structure of an $\E{1}$-algebra. Nevertheless, if $T(n)/v_{[j,n)}$ did admit the structure of an $\E{1}$-$T(j)$-algebra, then an analysis similar to <ref> shows that

$$\THH(T(n)/v_{[j,n)} / T(j)) \simeq T(n)[J_{p^{n-j}-1}(S^{2p^j})]/v_{[j,n)}.$$

This is the topological analogue of the calculation of <ref>. Under the equivalence $\THH(\BP{j-1}/T(j)) \simeq \BP{j-1}[\Omega S^{2p^j + 1}]$ of <ref>(a), the map $\THH(T(n)/v_{[j,n)}/T(j)) \to \THH(\BP{j-1}/T(j))$ induced by the map $T(n)/v_{[j,n)} \to \BP{j-1}$ is given by the skeletal inclusion of $J_{p^{n-j}-1}(S^{2p^j}) \to \Omega S^{2p^j+1}$. The projection

$$\THH(\BP{j-1}/T(j)) \to \THH(\BP{n-1}/T(n))/v_{[j,n)}$$

can be identified with the effect on $\BP{j-1}$-chains of the James-Hopf map $\Omega S^{2p^j+1} \to \Omega S^{2p^n+1}$. Therefore, the EHP sequence

$$J_{p^{n-j}-1}(S^{2p^j}) \to \Omega S^{2p^j+1} \to \Omega S^{2p^n+1}$$

shows that $\pi_\ast \THH(\BP{n-1}/T(n))/v_{[j,n)}$ is (additively) equivalent to precisely the “subalgebra” of $\pi_\ast \THH(\BP{j-1}/T(j))$ generated by $\theta_j^{p^{n-j}}$.

Since $\THH(T(n)/v_{[j,n)} / T(j)) \simeq T(n)[J_{p^{n-j}-1}(S^{2p^j})]/v_{[j,n)}$, one expects $T(n)/v_{[j,n)}$ to have an $\E{1}$-cell structure over $T(j)$ described by the cell structure of $J_{p^{n-j}-1}(S^{2p^j})$. Although we do not know how to prove this unconditionally, it is not difficult to show if we further assume <cit.>. In this case, <cit.> says that there is a map $f: \Omega^2 S^{2p^j+1} \to \BGL_1(T(j))$ which detects $v_j\in \pi_{2p^j-2} T(j)$ on the bottom cell of the source, such that the Thom spectrum of $f$ is a form of $\BP{j-1}$. Let[In <ref>, we described the map $f_{n,n-1}$ without assuming <cit.>. It is generally not possible to describe $f_{n,j}$ similarly if $j<n-1$: although there is a map $S^{2p^j} \to \B^2\GL_1(T(j))$ which detects $v_j\in \pi_{2p^j-2} T(j)$, there are $p$-local obstructions to extending along $S^{2p^j} \to J_{p^{n-j}-1}(S^{2p^j})$ if $n-j>1$. These obstructions can be viewed as the $\E{2}$-Browder brackets on $v_j$; <cit.> implies that these Browder brackets can be compatibly trivialized.] $f_{n,j}: \Omega J_{p^{n-j}-1}(S^{2p^j}) \to \BGL_1(T(j))$ denote the composite of $f$ with the $\E{1}$-map $\Omega J_{p^{n-j}-1}(S^{2p^j}) \to \Omega^2 S^{2p^j+1}$. Then the Thom spectrum of $f_{n,j}$ is equivalent to $T(n)/v_{[j,n)}$ as a $T(j)$-module. This is not quite an “$\E{1}$-cell structure” for $T(n)/v_{[j,n)}$, since $f_{n,j}$ is not an $\E{1}$-map; nevertheless, this construction of $T(n)/v_{[j,n)}$ suffices to calculate $\THH(T(n)/v_{[j,n)} / T(j))$.

If $R$ is an $\E{2}$-algebra which is an $\E{1}$-$T(n)$-algebra, one can loosely interpret the above discussion as saying that the square

\begin{equation}\label{pushout-ehp}
\xymatrix{
R[J_{p^{n-j}-1}(S^{2p^j})] \simeq R \otimes_{T(n)} \THH(T(n)/v_{[j,n)}/T(j)) \ar[r] \ar[d] & \THH(R/v_{[j,n)}/T(j)) \ar[d] \\
R \ar[r] & \THH(R/T(n))/v_{[j,n)}
\end{equation}

exhibits the top-right corner as the “tensor product of the top-left and bottom-right corners”. Note that the homotopy of the top-left corner is $R[\theta_j]/\theta_j^{p^{n-j}}$.
The bottom-right corner should be thought of as $\THH(R/v_{[j,n)}/T(n)/v_{[j,n)})$, although it is difficult to make this picture precise (since $T(n)/v_{[j,n)}$ does not admit the structure of an $\E{2}$-algebra).

For instance, if $R = \BP{n-1}$, then the square <ref> says that the square

\begin{equation}
\xymatrix{
\BP{n-1}[J_{p^{n-j}-1}(S^{2p^j})] \ar[r] \ar[d] & \THH(\BP{j-1}/T(j)) \simeq \BP{j-1}[\Omega S^{2p^j+1}] \ar[d] \\
R \ar[r] & \THH(\BP{n-1}/T(n))/v_{[j,n)} \simeq \BP{j-1}[\Omega S^{2p^n+1}]
\end{equation}

exhibits the top-right corner as the tensor product of the top-left and bottom-right corners. This is essentially the observation that the map $\THH(\BP{j-1}/T(j)) \to \THH(\BP{n-1}/T(n))/v_{[j,n)}$ sends $\theta_j^m\mapsto 0$ unless $p^{n-j}|m$, in which case $\theta_j^m \mapsto \theta_n^{m/p^{n-j}}$. This therefore explains the similarity between $\THH(\BP{n-1}/T(n))$ and $\THH(\BP{j-1}/T(j))$ given by <ref>(a).

As mentioned before, the lack of structure on the objects involved above make it difficult to use the above picture to understand the multiplicative structure on $\THH(\BP{n-1})$; but it does point to a plan of attack. Namely, one can attempt to understand the even filtration on $\THH(\BP{n-1})/v_{[0,n)}$ by considering the natural map $\THH(\BP{n-1})/v_{[0,n)} \to \THH(\FF_p)$. It is not hard to see that this map is an eff cover, so that the stack associated to the even filtration on $\THH(\BP{n-1})/v_{[0,n)}$ is the quotient of the scheme associated to the even filtration on $\THH(\FF_p)$ by a certain group scheme. The scheme associated to the even filtration on $\THH(\FF_p)$ is precisely $\GG_a$, and the above discussion suggests that the stack associated to the even filtration on $\THH(\BP{n-1})/v_{[0,n)}$ is isomorphic to $\GG_a/W[F^n]$; this is also suggested by work of Lee in [65]. We hope to study this in future work joint with Jeremy Hahn and Arpon Raksit. To this end, we set up some groundwork for future investigation of this stack in <ref>, where we study some basic properties of $W[F^n]$.

The calculation of $\THH(T(n)/v_{[j,n)} / T(j))$ in <ref> shows that more is true: if $n\geq k-1$, the structure of $\BP{n}$ as an $\E{1}$-$X(p^k)$-algebra (i.e., $\THH(\BP{n}/T(k))$) mirrors the structure of $\BP{n-k}$ as an $\E{1}$-algebra over the sphere (i.e., $\THH(\BP{n-k})$).

Note that the Thom spectrum of the map $f_{n,0}$ has been studied in [84], where it was denoted $y(n)$. Just as the $y(n)$ describe a filtration of $y(\infty) = \FF_p$ by $\E{1}$-algebras, the spectra $T(n)/v_{[j,n)}$ describe a filtration of $\BP{j-1}$. For instance, it is not difficult to show that for $j\leq k \leq n-1$, the spectrum $T(n)/v_{[j,n)}$ is $\mathrm{Tel}(k)$-acyclic. Therefore, if $T(n)/v_{[j,n)}$ admits the structure of an $\E{1}$-ring, the same argument as <cit.> implies that the map $K(T(n)/v_{[j,n)}) \to K(\BP{j-1})$ is an $\mathrm{Tel}(k)$-equivalence for $j\leq k \leq n-1$. Since $K(\BP{j-1})$ is $\mathrm{Tel}(k)$-locally contractible for $k\geq j+1$, the only interesting case is $k = j$; in this case, we find that the maps

$$K(T(j+1)/v_j) \to K(T(j+2)/(v_j, v_{j+1})) \to \cdots \to K(\BP{j-1})$$

are all $\mathrm{Tel}(j)$-equivalences.

Since $T(n)/v_{[j,n)}$ is closely related to $T(j)$ by <ref>, it is natural to wonder if there is a relationship between $T(n)/v_{[j,n)}$ and $T(n+k)/v_{[j,n+k)}$, in a manner compatible with their relationship to $T(j)$. By <ref>, $T(n+k)/v_{[n,n+k)}$ is the Thom spectrum of a map $\Omega J_{p^k-1}(S^{2p^n}) \to \BGL_1(T(n))$. It follows that if $T(n)/v_{[j,n)}$ admits the structure of an $\E{1}$-ring, then $T(n+k)/v_{[j,n+k)}$ is the Thom spectrum of a map $\Omega J_{p^k-1}(S^{2p^n}) \to \BGL_1(T(n)/v_{[j,n)})$. As mentioned in <ref>, if we further assume <cit.>, the spectrum $T(n)/v_{[j,n)}$ (resp. $T(n+k)/v_{[j,n+k)}$) is the Thom spectrum of a map $\Omega J_{p^{n-j}-1}(S^{2p^j}) \to \BGL_1(T(j))$ (resp. $\Omega J_{p^{n+k-j}-1}(S^{2p^j}) \to \BGL_1(T(j))$).

The relationship between the two presentations of $T(n+k)/v_{[j,n+k)}$ (as a Thom spectrum over $T(n)/v_{[j,n)}$ and over $T(j)$) is explained by the following observation in unstable homotopy theory: there is a fibration[To construct the fibration <ref>, recall that there is an EHP sequence

$$J_{p^m-1}(S^{2d}) \to \Omega S^{2d+1} \xar{H} \Omega S^{2dp^m+1}.$$

By dimension considerations, the canonical map $J_{p^{m+k}-1}(S^{2d}) \to \Omega S^{2d+1}$ factors through $J_{p^{m+k}-1}(S^{2d}) \to J_{p^k-1}(S^{2dp^m}) \times_{\Omega S^{2dp^m+1}} \Omega S^{2d+1}$, and one can easily check that this map is an equivalence. This implies the desired fiber sequence <ref>.]

\begin{equation}\label{partial-ehp}
J_{p^m-1}(S^{2d}) \to J_{p^{m+k}-1}(S^{2d}) \xar{H} J_{p^k-1}(S^{2dp^m}).
\end{equation}

Indeed, applying <ref> when $m = {n-j}$ and $d=p^j$, we obtain a fibration of $\E{1}$-spaces:

$$\Omega J_{p^{n-j}-1}(S^{2p^j}) \to \Omega J_{p^{n+k-j}-1}(S^{2p^j}) \xar{H} \Omega J_{p^k-1}(S^{2p^n}).$$

The composite of $f_{n+k,j}: \Omega J_{p^{n+k-j}-1}(S^{2p^j}) \to \BGL_1(T(j))$ with the map $\Omega J_{p^{n-j}-1}(S^{2p^j}) \to \Omega J_{p^{n+k-j}-1}(S^{2p^j})$ is $f_{n,j}: \Omega J_{p^{n-j}-1}(S^{2p^j}) \to \BGL_1(T(j))$. Therefore, <cit.> implies that there is a map $f_{n+k,j}: \Omega J_{p^k-1}(S^{2p^n}) \to \BGL_1(T(n)/v_{[j,n)})$ whose Thom spectrum is $T(n+k)/v_{[j,n+k)}$. This is the desired relationship between the various presentations of $T(n+k)/v_{[j,n+k)}$.

Observe that the preceding discussion implies, in particular, that there is a map $q_k: \Omega J_{p^k-1}(S^{2p^n}) \to \BGL_1(y(n))$ such that the composite $S^{2p^n-1} \to \Omega J_{p^k-1}(S^{2p^n}) \to \BGL_1(y(n))$ detects $v_n\in \pi_{2p^n-2} y(n)$, and such that the Thom spectrum of the map $q_k$ is $y(n+k)$. Taking $k \to \infty$, this implies that there is a map $q_\infty: \Omega^2 S^{2p^n+1} \to \BGL_1(y(n))$ whose Thom spectrum is $y(\infty) = \FF_p$. The map $q_\infty$ is adjoint to the $\E{1}$-map $\Omega^3 S^{2p^n+1}_+ \to y(n)$ from <cit.> which detects $v_n$ on the bottom cell of the source.

§ ANALOGUES FOR $\KO$ AND $\TMF$

Many of the results from the body of this article extend to the case of $\ko$ and $\tmf$. In this section, we will state these results; since the proofs are essentially the same, we will not give arguments unless the situation is substantially different. We will specialize to the case $p=2$ for simplicity. One of the main observations in <ref> is that the structure of $\BP{n}$ as an $\E{1}$-algebra over $X(p^n)$ (or rather, $T(n)$) mirrors the structure of $\Z_p$ as an $\E{1}$-algebra over $S^0$. For $\ko$ and $\tmf$, there are analogues of $X(p^n)$, which we studied in [34].

Let $A$ denote the free $\E{1}$-algebra $S\mmod \nu$ with a nullhomotopy of $\nu$, i.e., the Thom spectrum of the $\E{1}$-map $\Omega S^5 \to \BGL_1(S)$ which detects $\nu\in \pi_3(S)$ on the bottom cell[The spectrum $A$ has been studied before by Mahowald and his coauthors in [73, 42, 75, 74, 76, 87], where it is often denoted $X_5$.]. This spectrum has the property that $\H_\ast(A; \FF_2) \cong \FF_2[\zeta_1^4]$ (in fact, $\BPP_\ast(A) \cong \BPP_\ast[\frac{\eta_R(v_1^2) - v_1^2}{4}] \cong \BPP_\ast[t_1^2 + v_1 t_1]$). There is an $\E{1}$-map $i: A \to \ko$ such that under the isomorphism $\H_\ast(\ko; \FF_2) \cong \FF_2[\zeta_1^4, \zeta_2^2, \zeta_3, \cdots]$, the map $i$ corresponds to the inclusion of $\FF_2[\zeta_1^4]$. In particular, the map $A \to \ko$ is an equivalence in dimensions $\leq 4$. There is in fact an $\E{1}$-map $A \to \MSpin$, induced from an $\E{1}$-map $\Omega S^5 \to \BSpin$. There is also an $\E{1}$-map $A \to X(2) = T(1)$, such that $T(1) \simeq A \otimes C\eta$. We note that the “$Q_0$-Margolis homology” of $\H_\ast(\ko; \FF_2)$ (i.e., the homology of $\Sq^1$ viewed as a differential acting on $\H_\ast(\ko; \FF_2)$) is precisely $\H_\ast(A; \FF_2)$.

Similarly, let $B$ denote the $\E{1}$-algebra of <cit.>[The $\E{1}$-ring $B$ has been briefly studied under the name $\ol{X}$ in [50].], so that there is an $\E{1}$-space $N$ such that $B$ is the Thom spectrum of an $\E{1}$-map $N \to \BString$. We will not recall the construction of $N$ here; we only say that $B$ is obtained from the $\E{1}$-quotient $S\mmod\sigma$ by further taking an “$\E{1}$-quotient” by the class in $\pi_{11}(S\mmod\sigma)$ constructed from a nullhomotopy of $\nu\sigma\in \pi_{10}(S)$. This spectrum has the property that $\H_\ast(B; \FF_2) \cong \FF_2[\zeta_1^8, \zeta_2^4]$ (in fact, $\BPP_\ast(B) \cong \BPP_\ast[b_4, y_6]$, where $b_4 \equiv t_1^4$ and $y_6 \equiv t_2^2$ modulo $(2,v_1, \cdots)$)[For the sake of illustration, we remark that if $p=2$, then $b_4$ can be taken to be the following cobar representative for $\sigma = \alpha_{4/4}$, where the $v_i$s are Hazewinkel's generators:

\begin{align*}
b_4 & = \frac{1}{2}\left(\frac{\eta_R(v_1^4) - v_1^4}{8} - (\eta_R(v_1 v_2) - v_1 v_2)\right)\\
& = 5 t_1^4 + 9 t_1^3 v_1 + 7 t_1^2 v_1^2 - 2 t_1 t_2 + 2 t_1 v_1^3 - t_1 v_2 - t_2 v_1.
\end{align*}

Here, we used the formula

$$\eta_R(v_2) = v_2 - 5v_1 t_1^2 - 3v_1^2 t_1+ 2t_2 - 4t_1^3.$$

]. There is an $\E{1}$-map $i: B \to \tmf$ such that under the isomorphism $\H_\ast(\tmf; \FF_2) \cong \FF_2[\zeta_1^8, \zeta_2^4, \zeta_3^2, \zeta_4, \cdots]$, the map $i$ corresponds to the inclusion of $\FF_2[\zeta_1^8, \zeta_2^4]$. In particular, the map $B \to \tmf$ is an equivalence in dimensions $\leq 12$. There is in fact an $\E{1}$-map $B \to \MString$. There is also an $\E{1}$-map $B \to T(2)$ such that $T(2) \simeq B \otimes DA_1$, where $DA_1$ is an $8$-cell complex whose mod $2$ cohomology is isomorphic to the subalgebra of the Steenrod algebra generated by $\Sq^2$ and $\Sq^4$. We note that the “$Q_0$-Margolis homology” of $\H_\ast(\tmf; \FF_2)$ (i.e., the homology of $\Sq^1$ viewed as a differential acting on $\H_\ast(\tmf; \FF_2)$) is precisely $\H_\ast(B; \FF_2)$.

The following was implicitly stated in [34], but we make it explicit here:

The $\E{1}$-algebra structures on $A$ and $B$ admit extensions to $\Efr{2}$-algebra structures such that the maps $A \to X(2)$, $B \to X(4)_{(2)}$, $A\to \ko$, and $B \to \tmf$ admit the structure of $\Efr{2}$-maps.

A calculation paralleling <ref> shows:

Assume <ref>. There are isomorphisms

\begin{align*}
\H_\ast(\THH(\ko/A); \FF_2) & \cong \H_\ast(\ko; \FF_2)[\sigma(\zeta_3)] \otimes_{\FF_2} \Lambda_{\FF_2}(\sigma(\zeta_2^2)), \\
\H_\ast(\THH(\tmf/B); \FF_2) & \cong \H_\ast(\tmf; \FF_2)[\sigma(\zeta_4)] \otimes_{\FF_2} \Lambda_{\FF_2}(\sigma(\zeta_3^2)).
\end{align*}

Here, $|\sigma(\zeta_3)| = 8$, $|\sigma(\zeta_2^2)|=7$, $|\sigma(\zeta_4)| = 16$, and $|\sigma(\zeta_3^2)| = 15$.

Using the Adams spectral sequence for $\pi_\ast \THH(\ko/A)$ and $\pi_\ast \THH(\tmf/B)$ as in <ref>(b) (and using $\ko$- and $\tmf$-linearity), one finds:

Assume <ref>. Upon $2$-completion, there are equivalences

\begin{align*}
\THH(\ko/A) & \simeq \ko \oplus \bigoplus_{j\geq 1} \Sigma^{8j - 1} \ko/2j, \\
\THH(\tmf/B) & \simeq \tmf \oplus \bigoplus_{j\geq 1} \Sigma^{16j - 1} \tmf/2j.
\end{align*}

Since $\ko \otimes C\eta \simeq \ku$, <ref> implies that $\THH(\ko/A) \otimes C\eta \simeq \THH(\ku/T(1))$. Relatedly, there is an equivalence $\ko \otimes T(1) \simeq \ku[\Omega S^5]$ of $\E{1}$-$T(1)$-algebras, which implies that

$$\THH(\ko) \otimes T(1) \simeq \THH(\ko \otimes T(1)/T(1)) \simeq \ku[S^5] \oplus \bigoplus_{j\geq 1} \Sigma^{8j - 1} \ku[S^5]/2j.$$

Along similar lines, <ref> implies that $\THH(\tmf/B) \otimes DA_1 \simeq \THH(\BP{2}/T(2))$. There is also a $2$-local equivalence $\tmf \otimes T(2) \simeq \BP{2}[N]$ of $\E{1}$-$T(2)$-algebras, so that

$$\THH(\tmf) \otimes T(2) \simeq \BP{2}[\B N] \oplus \bigoplus_{j\geq 1} \Sigma^{16j - 1} \BP{2}[\B N]/2j.$$

Note that $\BP{2}[N] \simeq \BP{2}[\Omega S^9 \times \Omega S^{13}]$, so that $\pi_\ast(\tmf \otimes T(2)) \cong \mathbf{Z}_{(2)}[v_1, v_2, x_8, y_{12}]$, where $|v_1| = 2$, $|v_2| = 6$, $|x_8|=8$, and $|y_{12}|=12$.

This gives a potential approach to calculating $\THH(\ko)$ (resp. $\THH(\tmf)$) via the $T(1)$-based (resp. $T(2)$-based) Adams-Novikov spectral sequence. Describing this spectral sequence is essentially equivalent to calculating the analogue of the topological Sen operator for $\THH(\ko/A)$, whose construction is described below in <ref>.

Recall from <ref> that the structure of $\ko$ over $A$ mirrors the structure of $\tmf$ over $B$, which in turn mirrors the structure of $\BP{n}$ over $T(n)$; in other words, the calculation of <ref> is along the diagonal line $(n,n)$ in <ref>. It is natural to wonder whether there is an $\E{1}$-ring $\tilde{A}$ equipped with an $\E{1}$-map $A \to \tilde{A}$ and an $\E{1}$-map $\tilde{A} \to \ko$ such that the structure of $\ko$ over $\tilde{A}$ mirrors the structure of $\BP{n-1}$ over $T(n)$. (This is the “off-diagonal line” $(n,n-1)$ in <ref>.) This question is only interesting when $p=2$, since $\ko_{(p)}$ splits as a direct sum of even shifts of $\BP{1}$ if $p>2$. Let us localize at $2$ for the remainder of this discussion. Examining the argument establishing <ref> when $p=2$, one finds that the mod $2$ homology of $\tilde{A}$ must be $\H_\ast(\tilde{A}; \FF_2) \cong \FF_2[\zeta_1^4, \zeta_2^2]$.

If $A$ admits the structure of an $\Efr{2}$-ring, then $\tilde{A}$ can be constructed as follows. The class $\sigma_1\in \pi_5(A)$ determined by a nullhomotopy of $\eta\nu$ (see <cit.>) defines a map $S^6 \to \BGL_1(A)$, which, thanks to our assumption on $A$, extends to an $\E{1}$-map $\Omega S^7 \to \BGL_1(A)$. The desired $\E{1}$-$A$-algebra $\tilde{A}$ can be defined as Thom spectrum of this map.
(According to <cit.>, one should not expect $\tilde{A}$ to admit a natural construction as a Thom spectrum over the sphere.) Note that $\tilde{A} \otimes C\eta \simeq T(2)$.

The same argument as <ref>(a) shows:

If both $A$ and $\tilde{A}$ admit the structure of $\Efr{2}$-rings and $\ko$ admits the structure of an $\E{1}$-$\tilde{A}$-algebra, then there is a $2$-complete equivalence

$$\THH(\ko/\tilde{A}) \simeq \ko[\Omega S^9],$$

where the generator in $\pi_8 \THH(\ko/\tilde{A})$ is $\sigma^2(v_2)$. Moreover,

\begin{equation}\label{tilde-A-tp}
\H^c_\ast(\TP(\ko/\tilde{A}); \FF_2) \cong \FF_2[\zeta_1^4, \zeta_2^2, \zeta_3^2, \zeta_4, \cdots]\ls{\hbar}.
\end{equation}

Similarly, if $B$ admits the structure of an $\Efr{2}$-ring, the class $\sigma_2\in \pi_{13}(B)$ from <cit.> defines a map $S^{14} \to \BGL_1(B)$. Thanks to our assumption on $B$, this extends to an $\E{1}$-map $\Omega S^{15} \to \BGL_1(B)$. Define $\tilde{B}$ to be Thom spectrum of this map, so that $\H_\ast(\tilde{B}; \FF_2) \cong \FF_2[\zeta_1^8, \zeta_2^4, \zeta_3^2]$. Note that $\tilde{B} \otimes DA_1 \simeq T(3)$.

If $\tilde{B}$ admits the structure of an $\Efr{2}$-ring and $\tmf$ admits the structure of an $\E{1}$-$\tilde{B}$-algebra, then the same argument as in <ref>(a) shows that there is a $2$-complete equivalence

$$\THH(\tmf/\tilde{B}) \simeq \tmf[\Omega S^{17}],$$

where the generator in $\pi_{16} \THH(\tmf/\tilde{B})$ is $\sigma^2(v_3)$. Moreover,

\begin{equation}\label{tilde-B-tp}
\H^c_\ast(\TP(\tmf/\tilde{B}); \FF_2) \cong \FF_2[\zeta_1^8, \zeta_2^4, \zeta_3^2, \zeta_4^2, \zeta_5, \cdots]\ls{\hbar}.
\end{equation}

[Topological Sen operator for $\THH$ relative to $A$]
By <cit.>, $\THH(A)$ is equivalent to the Thom spectrum of the composite

$$\cL S^5 \xar{\cL \nu} \cL \B^2\GL_1(S) \simeq \B^2 \GL_1(S) \times \BGL_1(S) \xar{\id\times \eta} \BGL_1(S).$$

There is a nonsplit fiber sequence

\begin{equation}\label{free-loops-S5}
\Omega S^5 \to \cL S^5 \to S^5,
\end{equation}

and the restriction of the above composite along the map $\Omega S^5 \to \cL S^5$ is the map $\Omega S^5 \to \BGL_1(S)$ which defines $A$. It follows from the fiber sequence <ref> that $\THH(A)$ is the Thom spectrum of a map $S^5 \to \BGL_1(A)$ which detects a class $x\in \pi_4(A) \cong \pi_4(\ko)$. In particular, $\THH(A)$ is an $A$-module with two cells. Now assume <ref>; then <cit.> gives a splitting $\THH(A) \to A$, which implies that the class $x\in \pi_4(A)$ must be trivial. In other words, $\THH(A) \simeq A[S^5]$. If $\cC$ is an $A$-linear $\infty$-category, this implies the existence of a cofiber sequence

\begin{equation}\label{A-sen}
\THH(\cC) \to \THH(\cC/A) \xar{\Theta'} \Sigma^6 \THH(\cC/A).
\end{equation}

One should be able to recover the calculation of $\THH(\ko)$ from [2] using <ref> in the case $\cC = \Mod_\ko$ and the calculation of <ref>.

Similarly to <ref>, <ref> and <ref> imply that

\begin{align*}
\THH(\ko/A)/2 & \simeq \ko[S^7 \times \Omega S^9]/2, \\
\THH(\ko) \otimes_\ko \FF_2 & \simeq \FF_2[S^5 \times S^7 \times \Omega S^9].
\end{align*}

The latter of these has been proven by Angeltveit-Rognes in <cit.>.

Recall from <cit.> that Mahowald-Rezk duality gives an equivalence $W \ko \simeq \Sigma^6 \ko$ (resp. $W \BP{1} \simeq \Sigma^{2p} \BP{1}$); the shift of $6$ (resp. $2p$) in this equivalence arises for the same reason as in <ref> (resp. <ref> with $n=1$): both correspond to the class $\sigma^2(t_1^2)$ (resp. $\sigma^2(t_1)$).
We hope to explore this further in future work.

There is also an analogue of the topological Sen operator for $B$. To describe it, one observes using an argument similar to <ref> that $\THH(B/S\mmod\sigma) \simeq B[S^{13}]$ and that $\THH(S\mmod\sigma) \simeq (S\mmod\sigma)[S^9]$. This implies that if $\cC$ is a $B$-linear $\infty$-category, there are cofiber sequences

\begin{align*}
\THH(\cC/S\mmod\sigma) & \to \THH(\cC/B) \xar{\Theta_B} \Sigma^{14} \THH(\cC/B),\\
\THH(\cC) & \to \THH(\cC/S\mmod\sigma) \xar{\Theta_B'} \Sigma^{10} \THH(\cC/S\mmod\sigma).
\end{align*}

However, it is significantly more complicated to describe these cofiber sequences in almost any nontrivial example, so we omit further discussion. Nevertheless, one can use <ref> to show the following equivalences analogous to <ref>:

\begin{align*}
\THH(\tmf/B)/2 & \simeq \tmf[S^{15} \times \Omega S^{17}]/2, \\
\THH(\tmf) \otimes_\tmf \FF_2 & \simeq \FF_2[S^9 \times S^{13} \times S^{15} \times \Omega S^{17}];
\end{align*}

note that $\FF_2[S^9 \times S^{13}] \simeq \FF_2[BN]$. The latter of these has been proven by Angeltveit-Rognes in <cit.>.

Assume <ref>, and let $p=2$. Then there is a map $\cM_{T(1)} \to \cM_A$ of stacks over $\Mfg$, which exhibits $\cM_{T(1)}$ as a $2$-fold fppf cover of $\cM_A$. Recall that $\cM_{T(1)}$ is isomorphic to the moduli stack of graded formal groups equipped with a coordinate up to order $\leq 2$ (equivalently, order $\leq 3$ for $2$-typical formal groups). Similarly, we have:

The stack $\cM_A$ is isomorphic to the moduli stack of graded formal groups equipped with an even coordinate up to order $\leq 5$.

Recall that there is a fiber sequence

$$\SU(2)/\U(1) \cong S^2 \to \BU(1) \simeq \CP^\infty \to \BSU(2) \simeq \HHP^\infty.$$

Let $n\geq 1$. There is a homotopy equivalence $\HHP^n\times_{\HHP^\infty} \CP^\infty \simeq \CP^{2n+1}$ (since $S^{4n+3}/\SU(2) = \HHP^n$ and $S^{4n+3}/\U(1) = \CP^{2n+1}$), which produces the “twistor fibration”, i.e., the fiber sequence

\begin{equation}\label{s2-bundle-hpn}
S^2 \to \CP^{2n+1} \to \HHP^n.
\end{equation}

The map $\CP^{2n+1} \to \HHP^n$ is given in coordinates by the map $[z_1: \cdots: z_{2n+2}] \mapsto [z_1 + z_2 \mathbf{j} : \cdots: z_{2n+1} + z_{2n+2} \mathbf{j}]$.
Note that $\SU(2)/\U(1) = \CP^1$ is the unit sphere $S(\fr{su}(2))$ in the adjoint representation of $\SU(2)$, so <ref> equivalently says that $\CP^{2n+1}$ is the sphere bundle of the adjoint bundle of rank $3$ over $\HHP^n$.

Let $R$ be a complex-oriented homotopy commutative ring with associated formal group $\GG$ over $\pi_\ast(R)$; we will assume for simplicity that $2$ is not a zero-divisor in $\pi_\ast R$. Then $R^\ast(\CP^5)$ is isomorphic to the ring of functions on $\GG$ which vanish to order $\geq 6$. The Serre spectral sequence associated to the fiber sequence <ref> implies that $R^\ast(\HHP^2)$ is isomorphic to $R^\ast(\CP^5)^{\Z/2}$, where $\Z/2$ acts by inversion on the formal group. This implies the desired claim, since there is an equivalence $\HHP^2 \simeq \Sigma^4 C\nu$ of spectra, and $A$ is the free $\E{1}$-ring whose unit factors through the inclusion $S^0 \to C\nu$.

The description of $\cM_A$ in <ref> has concrete applications; for instance, in [33], we show that $\cM_{\tmf \otimes A} = \cM_A \times_{\Mfg} \Mell$ can be identified with the moduli stack of elliptic curves $\ce$ equipped with a splitting of the Hodge filtration on $\H^1_\dR(\ce)$, and use this to describe an topological analogue of the integral ring of quasimodular forms.

As explained in <cit.>, there is a $\Z/2$-equivariant $\E{1}$-algebra $A_{\Z/2}$ whose underlying $\E{1}$-algebra is $A$, and such that $\Phi^{\Z/2} A_{\Z/2} = X(2)_{(2)}$ as $\E{1}$-algebras. This is a topological interpretation of the following observation suggested by <ref>: $\cM_A$ is “half” of $\cM_{X(2)}$; more precisely, there is a two-fold fppf cover $\cM_{X(2)} \to \cM_A$. This is an algebraic analogue of the equivalence $A \otimes C\eta \simeq X(2)$.

We also note that there is a $\Z/2$-equivariant analogue  of the fiber sequence <ref>: namely, there is a $\Z/2$-equivariant twistor fibration

S^\rho \ar[d]^-\cong \ar[r] & \CP^{2n-1}_\RR \ar[d]^-\cong \ar[r] & \HHP^{n-1}_\RR \ar[d]^-\cong \\
S^{2\rho-1}/S^\sigma \ar[r] & S^{2n\rho-1}/S^\sigma \ar[r] & S^{2n\rho-1}/S^{2\rho-1}.

where $\Z/2$ acts on $\HHP^n$ via the action of $\Z/2\subseteq S^1\subseteq \SO(3)$ on $\bH$. The underlying fibration is <ref>, while the $\Z/2$-fixed points gives the fibration

$$S^1 \to \RP^{2n-1} \to \CP^{n-1}$$

which exhibits $\RP^{2n-1}$ as the sphere bundle of the complex line bundle $\co(2)$ on $\CP^{n-1}$.

One consequence of the identification of $\cM_A$ in <ref> is that $\cM_A \to \Mfg$ is an affine bundle, so that the pullback of the cotangent complex $L_{\cM_A/\Mfg}$ to $\spec(\BPP_\ast(A))/\GG_m \cong \spec(\BPP_\ast[t_1^2 + v_1 t_1])/\GG_m$ can be identified with a free $\BPP_\ast[t_1^2 + v_1 t_1]$-module of rank $1$ generated by the class $d(t_1^2 + v_1 t_1)$ in weight $2$. Using <ref>, we obtain the algebraic analogue of <ref>: if $X$ is a stack over $\cM_A$, there is a cofiber sequence

\begin{equation*}
\HH(X/\Mfg) \to \HH(X/\cM_A) \xar{\Theta_\mot} \Sigma^{6,3} \HH(X/\cM_A).
\end{equation*}

The stack $\cM_{\ko}$ can be identified with the moduli stack of curves of the form $y = x^2 + bx + c$ with change of coordinate $x\mapsto x+r$, and $\HH(\cM_{\ko}/\Mfg)$ describes the $E_1$-page of the Adams-Novikov-Bökstedt spectral sequence calculating $\THH(\ko)$ (see <ref>). It would be interesting to explicitly describe $\HH(\cM_{\ko}/\Mfg)$; note that

$$\pi_\ast \HH(\cM_\ko/\cM_A) \cong \co_{\cM_\ko}\pdb{\sigma^2 v_j | j\geq 2} \otimes_{\co_{\cM_\ko}} \Lambda_{\co_{\cM_\ko}}(dt_i | i\geq 2),$$

where $\sigma^2(v_j)$ lives in degree $2^{j+1}$ and weight $2^j$, and $dt_i$ lives in degree $2^{i+1}-1$ and weight $2^i$.
This can be proved exactly as in <ref>; weight considerations presumably allow one to fully describe $\Theta_\mot: \HH(X/\cM_A) \to \Sigma^{6,3} \HH(X/\cM_A)$, and hence $\HH(\cM_\ko/\Mfg)$.

Assume <ref>, and let $p=2$. It is trickier to describe the stack $\cM_B$ in a manner analogous to <ref>. As a first approximation, if we assume that $S\mmod \sigma$ admits the structure of a homotopy commutative ring, one can attempt to describe the moduli stack $\cM_{S\mmod \sigma}$. However, it is provably impossible to construct a Hurewicz fibration

$$S^4 \to \HHP^5 \to \OP^2$$

in the point-set category. This is a consequence of <cit.>, which states more generally that if $F \to E \to X$ is a Hurewicz fibration where $E$ is homotopy equivalent to $\HHP^{2n+1}$ and $F$ and $X$ are homotopy equivalent to finite CW-complexes, then either $F$ or $X$ must be contractible. Note that this result implies that there cannot even be a Hurewicz fibration

$$S^4 \to \HHP^3 \to S^8.$$

Similarly, there cannot be Hurewicz fibrations

\begin{align*}
\CP^3 & \to \CP^7 \to S^8, \\
\CP^3 & \to \CP^{11} \to \OP^2;
\end{align*}

see [71] for the impossibility of the first Hurewicz fibration (which implies the impossibility of the second Hurewicz fibration). These no-go results make it difficult to give a formal group-theoretic description of $R^\ast(\OP^2)$ (and hence of $\cM_{S\mmod \sigma}$, since $\OP^2 \simeq \Sigma^8 C\sigma$) where $R$ is a complex-oriented homotopy commutative ring.

The story for $\ko$ admits a slightly different generalization to higher heights.

Observe that $S^5 = \SU(4)/\Sp(2)$, and that the map $\Omega S^5 \to \BU$ (whose Thom spectrum is $A$) can be viewed as the composite

$$\Omega(\SU(4)/\Sp(2)) \to \Omega(\SU/\Sp) \simeq \BSp \to \BU.$$

The equivalence $\Omega(\SU/\Sp) \simeq \BSp$ is given by Bott periodicity, and the map $\BSp \to \BU$ takes a symplectic bundle to its underlying unitary bundle.

Motivated by <ref>, we are led to the following definition:

Define an $\E{1}$-algebra $X_\bH(n)$ via the Thom spectrum of the composite

$$\Omega(\SU(2n)/\Sp(n)) \to \Omega(\SU/\Sp) \simeq \BSp \to \BU.$$

There is a canonical $\E{1}$-map $X_\bH(n) \to X_\bH(\infty) = \MSp$.

The spectrum $X_\bH(n)$ has been studied by Andy Baker.

See [29] for a detailed study of the space $\Omega(\SU(2n)/\Sp(n))$. Let us note that if $\SU(2n)_\bH$ denotes the $\Z/2$-equivariant loop space with the $\Z/2$-action given by the symplectic involution

$$A \mapsto J\ol{A} J^{-1}, \ J = \begin{pmatrix}
0 & -1\\
1 & 0
\end{pmatrix}^{\oplus n},$$

then $\Omega(\SU(2n)/\Sp(n)) \simeq (\Omega^\sigma \SU(2n)_\bH)^{\Z/2}$. Indeed, the fixed points of $\SU(2n)_\bH$ is $\Sp(n)$ (by definition), so we can apply the first sentence of <ref> to conclude.

As the notation indicates, $X_\bH(n)$ should be viewed as a quaternionic analogue of the $X(n)$ spectra from [96]; see <ref>.

Normed division algebra     $\RR$     $\cc$     $\bH$

$\E{1}$-ring     $y(n) \subseteq \Phi^{\Z/2} X(2^n)_\RR$     $T(n) \subseteq X(2^n)$     $X_\bH(2^n)$

Limit as $n\to \infty$     $\FF_2 \subseteq \Phi^{\Z/2} \MU_\RR = \MO$     $\BPP \subseteq \MU$     $\MSp$

Mod $2$ homology     $\FF_2[\zeta_1, \cdots, \zeta_n]$     $\FF_2[\zeta_1^2, \cdots, \zeta_n^2]$     $\FF_2[\zeta_1^4, \cdots, \zeta_n^4]$

The analogies between $y(n)$, $T(n)$, and $X_\bH(n)$, where the implicit prime is $p=2$. The inclusion $y(n) \subseteq \Phi^{\Z/2} X(2^n)_\RR$ is discussed in <ref>; see [104]. The final row is to be interpreted as follows: $\H_\ast(\Phi^{\Z/2} X(2^n)_\RR; \FF_2)$ is a direct sum of even shifts of $\FF_2[\zeta_1, \cdots, \zeta_n]$; similarly for $\H_\ast(X(2^n); \FF_2)$ and $\H_\ast(X_\bH(2^n); \FF_2)$.

Note that there are isomorphisms of algebras

\begin{align*}
\H_\ast(\Phi^{\Z/2} X(n)_\RR; \FF_2) & \cong \H_\ast(\Omega(\SU(n)/\SO(n)); \FF_2) \cong \FF_2[x_1, \cdots, x_{n-1}], \\
\H_\ast(X_\bH(n); \Z) & \cong \H_\ast(\Omega(\SU(2n)/\Sp(n)); \Z) \cong \Z[y_1, \cdots, y_{n-1}],
\end{align*}

where $|x_j| = j$ and $|y_j| = 4j$.

By construction, $X_\bH(2) \simeq A = S\mmod \nu$.

Suppose that $X_\bH(n)$ admits the structure of a homotopy commutative ring. One can then also ask for an interpretation of the stack $\cM_{X_\bH(n)}$ analogous to <ref>. It turns out that the difficulties of <ref> are no longer an issue for $X_\bH(n)$. Indeed, the analogue of the map $S^4 \to \Omega S^5$ (whose Thomification is the map $C\nu \to A$ used in the proof of <ref>) is given by a map $\iota: \HHP^{n-1} \to \Omega(\SU(2n)/\Sp(n))$ which exhibits $\HHP^{n-1}$ as the generating complex of $\Omega(\SU(2n)/\Sp(n))$. (See, e.g., <cit.>.)
Moreover, the composite

$$\HHP^{n-1} \xar{\iota} \Omega(\SU(2n)/\Sp(n)) \to \Omega(\SU/\Sp) \simeq \BSp$$

factors as $\HHP^{n-1} \hookrightarrow \HHP^\infty \simeq \BSp(1) \to \BSp$. Since the Thom spectrum of the tautological quaternionic line bundle over $\HHP^{n-1}$ is $\Sigma^{-4} \HHP^n$, the map $\iota$ Thomifies to a map $\Sigma^{-4} \HHP^n \to X_\bH(n)$.

Using the twistor fibration <ref> and the map $\Sigma^{-4} \HHP^n \to X_\bH(n)$ of <ref>, one can argue as in <ref> to show:

The stack $\cM_{X_\bH(n)}$ is isomorphic to the moduli stack of graded formal groups equipped with an even coordinate up to order $\leq 2n+1$.

Suppose that $X_\bH(n)$ admits the structure of an $\Efr{2}$-ring.
There is also a canonical map $\cM_{X_\bH(n-1)} \to \cM_{X_\bH(n)}$ which exhibits $\cM_{X_\bH(n)}$ as the quotient of $\cM_{X_\bH(n-1)}$ by the group scheme $\GG_a^{(2n-2)}/\GG_m$ over $B\GG_m$, where $\GG_a^{(2n-2)}$ denotes the affine line with $\GG_m$-action of weight $2n-2$. This is the algebraic analogue of the following:

The spectrum $X_\bH(n)$ is equivalent to the Thom spectrum of a map $\Omega S^{4n-3} \to \BGL_1(X_\bH(n-1))$.

By <cit.> (see also [13]), it suffices to establish that there is a fiber sequence of $\E{1}$-spaces

\begin{equation*}%\label{e1-su2n}
\Omega(\SU(2n-2)/\Sp(n-1)) \to \Omega(\SU(2n)/\Sp(n)) \to \Omega S^{4n-3}.
\end{equation*}

To see this, observe that there is a diffeomorphism $\SU(2n)/\Sp(n) \cong \SU(2n-1)/\Sp(n-1)$, and hence a fibration

\SU(2n-2)/\Sp(n-1) \ar@{=}[d] \ar[r] & \SU(2n-1)/\Sp(n-1) \ar[d]^-\cong \ar[r] & \SU(2n-1)/\SU(2n-2) \ar[d]^-\cong \\
\SU(2n-2)/\Sp(n-1) \ar[r] & \SU(2n)/\Sp(n) \ar[r] & S^{4n-3}.

The desired fiber sequence is obtained by looping the bottom row.

On the bottom cell of the source, the map $\Omega S^{4n-3} \to \BGL_1(X_\bH(n-1))$ defines a class $\chi_n^\bH\in \pi_{4n-5} X_\bH(n-1)$, and $\chi_{2^n}^\bH$ is detected in the $E_2$-page of the Adams-Novikov spectral sequence for $X_\bH(2^n-1)$ by $[t_n^2]$. Moreover, if $X_\bH(n-1)$ admits the structure of an $\Efr{2}$-ring and $X_\bH(n)$ admits the structure of an $\E{1}$-$X_\bH(n-1)$-algebra, then $\THH(X_\bH(n)/X_\bH(n-1)) \simeq X_\bH(n)[\Omega S^{4n-3}]$.

We can then conclude (as in <ref> and <ref>) that if $\cC$ is an $X_\bH(n)$-linear $\infty$-category and $X$ is a stack over $\cM_{X_\bH(n)}$, then there are cofiber sequences

\begin{align*}
\THH(\cC/X_\bH(n-1)) & \to \THH(\cC/X_\bH(n)) \xar{\Theta'} \Sigma^{4n-2} \THH(\cC/X_\bH(n)), \\
\HH(X/\cM_{X_\bH(n-1)}) & \to \HH(X/\cM_{X_\bH(n)}) \xar{\Theta_\mot} \Sigma^{4n-2,2n-1} \HH(X/\cM_{X_\bH(n)}).
\end{align*}

Only the first cofiber sequence requires that $X_\bH(n-1)$ and $X_\bH(n)$ admit the structure of $\Efr{2}$-rings, and that $X_\bH(n)$ admits the structure of an $\E{1}$-$X_\bH(n-1)$-algebra; the second cofiber sequence only requires that $X_\bH(n)$ admit the structure of a homotopy commutative ring.

§ ALTERNATIVE CALCULATION OF $\DIFFR_{\Z/P^N}$

In this brief section, we give an alternative algebraic argument for <ref> following <cit.>. I am very grateful to Sasha Petrov for an illuminating discussion about this entire appendix; see also <cit.>.

Let $R$ be a (discrete) commutative $\Z/p^n$-algebra. Then <cit.> implies that

\begin{align*}
\spec(\Z/p^n)^{\slashed{D}}(R) & \simeq \Map_\CAlg(\Z/p^n, W(R)/V(1))\\
& \simeq \{z\in W(R) | zV(1) = p^n\} = \{x\in W(R) | V(Fz) = p^n\}.
\end{align*}

Since $V$ is injective, this is a torsor for $\W[F](R) = \GG_a^\sharp(R)$. Moreover, this torsor is trivializable, i.e., $p^n$ is in the image of $VF$. In fact, we claim that

\begin{equation}\label{pn V}
p^n = V(p^{n-1}) = VF(V(p^{n-2}))\in W(\Z/p^n).
\end{equation}

To see this, let us compute in ghost coordinates. Recall that if $w(x) = (w_0(x), w_1(x), \cdots)$ are the ghost coordinates of $x\in W(R)$, then $w_{n+1}(Vx) = pw_n(x)$. Since $w(p^n) = (p^n, p^n, \cdots)$ and $w(V(p^{n-1})) = (0, p^n, p^n, \cdots)$, we see that

$$w(p^n - V(p^{n-1})) = (p^n, 0, 0, \cdots).$$