\section{Introduction}
In Bhatt--Morrow--Scholze \cite{BMS18}, they compute $p$-complete topological periodic cyclic homology of the ring of integers $\mathcal{O}_L$ in a finite extension $L$ of $\mathbb{Q}_p$ over the base $E_{\infty}$ ring $\bS[z]:=\Sigma^{\infty}\mathbb{N}_+$, denoted $\TP(\mathcal{O}_L/\bS[z];\mathbb{Z}_p^{\wedge})$. They prove that $\TP_*(\mathcal{O}_L/\bS[z];\mathbb{Z}_p^{\wedge})$ is isomorphic to a periodized Frobenius twist of the Breul--Kisin module.\footnotemark \footnotetext{As a ring, the Frobenius twist of a Breul-Kisin module $\mathfrak{S}^{-1}$ is simply $\mathfrak{S}=W(\mathbb{F}_q)[[z]]$, but it is regarded as a $\mathfrak{S}$-algebra via the canonical Frobenius map $\varphi$, which lifts the Frobenius map on $W(\mathbb{F}_q)$ and sends $z$ to $z^p$.} The starting point of this note is the observation that the spectrum $\TP(\mathcal{O}_L/\bS[z];\mathbb{Z}_p^{\wedge})$ is complex oriented. I show that it carries a height one formal group law that is compatible with the cyclotomic structure map of $E_{\infty}$ rings 
\[\varphi \colon \thinspace \TC^{-}(\mathcal{O}_L/S[z];\mathbb{Z}_p^{\wedge})\to \TP(\mathcal{O}_L/\bS[z];\mathbb{Z}_p^{\wedge}),\]
from \cite[Const. 11.5]{BMS18} (cf. \cite[Prop. 2.11]{LW20}).  
This gives evidence that topological periodic cyclic homology shift chromatic complexity up by one, consistent with the red-shift philosophy of Ausoni--Rognes \cite{AR08}, where chromatic complexity is measured using complex orientations. Since algebraic K-theory is rarely complex oriented, it is less common to measure red-shift behavior using complex orientations, but here we make the case that it can be used as a method for studying red-shift behavior in $\TC^{-}$ and $\TP$ over certain choices of base $E_{\infty}$-ring. 

Throughout, $L$ will denote a finite extension of $\mathbb{Q}_p$
with ring of integers $\mathcal{O}_L$, which is a complete discrete valuation ring of mixed characteristic $(0,p)$. We fix a uniformizer $\varpi_L$ in $\mathcal{O}_L$ and write $\mathbb{F}_q=\mathcal{O}_L/(\varpi_L)$ for the residue field, which is a finite field of characteristic $p$. We note that $\varpi_L$ satisfies $E_L(\varpi_L)=0$ where $E_L(z)=z^{e_L}+p\theta(z)$
is the associated Eisenstein polynomial of degree $e_L$ and $e_L$ is the ramification index. We note that there is an isomorphism of $W(\mathbb{F}_q)$-algebras $\mathcal{O}_L\cong W(\mathbb{F}_q)[z]/(E_L(z))$ by the classification of local fields of mixed characteristic $(0,p)$ \cite[II. \S 5 Thm. 4]{Ser79}. We write $\bS_{W(\mathbb{F}_q)}$ for the spherical Witt vectors, which is constructed as an $E_{\infty}$ ring in \cite{Lur21}. The choice of uniformizer $\varpi_L$ determines a map 
$\mathbb{S}_{W(\mathbb{F}_q)}[z]\to \mathcal{O}_L,$
of $E_{\infty}$ rings sending $z$ to $\varpi_L$. We define
$THH(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z]))$
by replacing all tensors over $\mathbb{S}$ in the standard cyclic bar construction with tensors over $\bS_{W(\mathbb{F}_q)}[z]$.  
By \cite[Rem. 2.12]{LW20}, this is simply the $p$-completion of $THH(\mathcal{O}_L/\mathbb{S}[z])$. We define 
\begin{align*}
\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z]):=&\THH(\mathcal{O}_L/\mathbb{S}_{\mathbb{W}(\mathbb{F}_q)}[z])^{h\mathbb{T}} \text{ and } \\
\TP(\mathcal{O}_L/\mathbb{S}_{\mathbb{W}(\mathbb{F}_q)}[z]):=&\THH(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])^{t\mathbb{T}}
\end{align*}
where $\mathbb{T}\subset \mathbb{C}^{\times}$ is the subgroup of unit vectors. As we recall in Lemma \ref{main lemma}, it is shown in \cite[Prop. 11.10]{BMS18} (cf. Theorem 2.15 \cite{LW20}) that 
\[ \TC_0^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z]) \cong \TP_0(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])\cong \mathbb{W}(\mathbb{F}_q)[[z]]\] 
and the Tate valued Frobenius map induces a map of $W(\mathbb{F}_q)$-algebras 
\[\varphi \colon \thinspace \TC_0^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z]) \to \TP_0(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])\]
extending the Frobenius map on $W(\mathbb{F}_q)$ by sending $z$ to $z^p$.

We say a formal group law $G$ over a commutative $\mathbb{F}_p$-algebra $R$ has $p$-height $n$ if the $p$-series satisfies 
\[ [p]_G(X)=aX^{p^n}+O(X^{p^n+1})\in R[[x]]\]
where $a\ne 0$ in $R$.\footnote{This choice is different than the choice made by some authors. See \cite[Rem. 3.1.7]{Smi11} for a discussion of the benefits of each choice.}

\begin{thm}\label{main thm}
The complex orientation  
\[ \MU\to \TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q}[z]))\]
maps $x_{p-1}$ to $E_L(z)t^{1-p}$. The elements $x_{p^n-1}$ map trivially for $n>1$. Consequently, the complex oriented spectrum $\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])/p$ carries a $p$-height one formal group law $F$ with $p$-series 
\[[p]_{F}(X)=z^eX^{p}+O(X^{p+1})\]	
and the complex oriented spectrum $\TP(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])/p$ carries a $p$-height one formal group law $G$ with $p$-series 
\[[p]_{F}(X)=z^{pe}X^{p}+O(X^{p+1}).\]
\end{thm}
\begin{rem2}
By \cite[Prop. 11.10]{BMS18} and \cite[Thm 2.15]{LW20}, we know that $E_L(z)$ maps to $E_L(\varpi_L)=0\in \pi_0\THH(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)})$, so the multiplicative formal group law of Theorem \ref{main thm} reduces to the additive formal group law on $\THH(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)})/p$. This is consistent with the red-shift philosophy. Similarly, $E_L(z)\in \TP_0(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)})$ maps to $E_L(\varpi_L)=p\in\TP_0(\mathbb{F}_q)$ so this formal group law also reduces to the additive formal group law in $\TP_*(\mathbb{F}_q)/p$, which is also consistent with the red-shift philosophy.  
\end{rem2}

\subsection{Conventions}
To fix models, we work in the $\infty$-category of spectra throughout and write $\otimes$ for the symmetric monoidal product and $\mathbb{S}$ for the symmetric monoidal unit. When we make statements in the category of abelian groups however, we also write $\otimes$ for the symmetric monoidal product in abelian groups and not the derived category and hope the distinction is clear from context. We write $\mathcal{A}_*$ for the dual Steenrod algebra defined as $\mathcal{A}_*=(\mathbb{F}_p)_*(\mathbb{F}_p)$ using our conventions. By an $E_n$ ring, I mean an algebra over the little $n$ disks operad $E_n$ in the $\infty$-category of spectra. 

\subsection{Acknowledgements}
The author would like to thank Mona Merling, Thomas Nikolaus, and Holger Reich for comments on a draft of this note, and Jack Morava for inspiration. The author was informed that similar calculations where done by Bhatt--Morrow--Scholze, Hesselholt, and Krause--Nikolaus independently in unpublished work, so the author acknowledges these other computations. 
\section{Spectral sequences}
Let $E$ be a homotopy associative ring spectrum, $A$ a $E_{\infty}$ ring and $B$ an $E_1$ algebra over $A$. Then topological Hochschild homology $\THH(B/A)$, defined as the realization of the cyclic bar construction in $A$-modules,
\[\THH(B/A)=|\xymatrix{A \ar[r]& \ar@<1ex>[l]\ar@<-1ex>[l]A\otimes_{B}A\ar@<.5ex>[r] \ar@<-.5ex>[r]& \ar@<-1ex>[l]\ar@<1ex>[l]\ar[l] A\otimes_{B}A\otimes_{B}A \ar@<1ex>[r]\ar@<-1ex>[r] \ar[r] & \ar@<.5ex>[l]\ar@<1.5ex>[l]\ar@<-1.5ex>[l]\ar@<-.5ex>[l]\dots }|\]  
has an action of the circle group $\mathbb{T}$. There are associated multiplicative conditionally convergent spectral sequences with signature
\begin{align}
\label{S^1 homotopy} E_{2}^{*,*}=H^*(\mathbb{T};E_*\THH(B/A))\Rightarrow E_*^c\TC^{-}(B/A),\\
\label{S^1 Tate} \widehat{E}_{2}^{*,*}=\hat{H}^*(\mathbb{T};E_*\THH(B/A))\Rightarrow E_*^c\TP(B/A),
\end{align}
which are multiplicative when $A$ is an $E_{\infty}$ ring spectrum over $B$ by  \cite[Sec. 4]{BR05} and \cite[Thm. 6.47]{HR20} respectively. Here we write 
\begin{align}
E_*^c(\TC^{-}(B/A))=&\lim_kE_*(\TC^{-}(B/A)[k]) \text{ and }\\
E_*^c(\TP(B/A))=&\lim_kE_*(\TP(B/A)[k])
\end{align}
where 
\[\TC^{-}(B/A)[k]:=F (E\mathbb{T}_+^{(k)},THH(B/A))^{\mathbb{T}}\text{ and }\TP(B/A)[k]:= (\widetilde{E}\mathbb{T}/\widetilde{E}\mathbb{T}_k\wedge F (E\mathbb{T},THH(B/A)) )^{\mathbb{T}}\]
where $E\mathbb{T}^{(k)}$ is the $k$-skeleton of standard CW decomposition of $E\mathbb{T}$, $\widetilde{E}\mathbb{T}_k$ is the cofiber of the canonical map $E\mathbb{T}_+^{(k)}\to \mathbb{S}$ when $k\ge 0$ and 
$\widetilde{E}\mathbb{T}_k=F(\widetilde{E}\mathbb{T}_{-k+1},\mathbb{S})$ when $k<0$. 
\begin{rem2}
It is not always the case that $\THH(B/A)$ is a cyclotomic spectrum, but it will be a cyclotomic spectrum in our main cases of interest. 	
\end{rem2}


\section{The homotopical Tate spectral sequence for relative $\TP$ of $\mathbb{Z}_p$}
In this section, we recall the computation of 
\[ \TC_*^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z]) \text{ and }\TP_*(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])\]
by \cite{BMS18,KN19,LW20}. In \cite[Thm. 3.1]{KN19}, they compute 
\[ \THH_*(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])\cong \mathcal{O}_L[x]\]
where $\mathcal{O}_L[x]$ is a polynomial algebra over $\mathcal{O}_L$ with $|x|=2$. At the outset, we fix a preferred choice of generator $x=\sigma^2E_L(z)$ where 
\[ \sigma^2\colon \thinspace \Sigma^2\overline{\mathcal{O}}_L\to \THH(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])\]
	is the map constructed in \cite[Const. 3.2.3.]{HW20}, where $\overline{\mathcal{O}}_L$ is the fiber of the unit map $S_{W(\mathbb{F}_q)}[z]\to \mathcal{O}_{L}$. Note that $\THH(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])$ is an $E_\infty$ cyclotomic spectrum by \cite[Const. 11.5]{BMS18} (cf.  \cite[Prop. .11]{LW20}). 
Consequently, we can compute $\TC_*^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])$ and $\TP_*(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])$ groups the conditionally convergent $\mathbb{T}$ homotopy fixed point and Tate spectral sequences, as well as the Frobenius map. 
When $E=\mathbb{S}$, $A=\mathbb{S}_{W(\mathbb{F}_q)}[z]$ and $B=\mathcal{O}_L$, each of the spectral sequences \eqref{S^1 homotopy} and \eqref{S^1 Tate} collapse at the $E_2$-pages, however there are also hidden multiplicative extensions. We simply record the computation, as it appears in \cite[Prop. 11.10]{BMS18} (cf. \cite[Thm. 2.15]{LW20}). 
\begin{lem}[Proposition 11.10 \cite{BMS18} (cf. Theorem 2.15 \cite{LW20})]\label{main lemma}
There are isomorphisms 
\begin{align*}
\TC_*^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])\cong & W(\mathbb{F}_q)[[z]][x,t]/(tx-E_L(z))\text{ and }\\
\TP_*(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])\cong & W(\mathbb{F}_q)[[z]][t,t^{-1}] 
\end{align*}
of graded $\mathfrak{S}=W(\mathbb{F}_q)[[z]]$-algebras with $|x|=2$, $|t|=-2$, and $|z|=0$. The Frobenius map 
\[\varphi \colon \thinspace \TC^{-}_0(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)})\to \TP_0(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)})\]
is the lift of Frobenius on $W(\mathbb{F}_q)$ and sends $z$ to $z^p$. More generally, the map 
\begin{align}\label{frob}
\varphi \colon \thinspace \TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)})\to \TP^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)})
\end{align}
induces a map of graded $\mathfrak{S}$-algebras on homotopy groups $\pi_*$ by sending $x$ to $t^{-1}$, $t$ to $\varphi(E_L(z))t$ 
\end{lem}
\begin{rem2}
The relation $tx=E_L(z)$ in $\TC_*^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q}[z])$ follows by \cite[Prop. 3.2.7]{HW20} and our choice of generator
\[ x=\sigma^2E_L(z).\]
\end{rem2}

\section{Complex orientations and $\TP$ of complete DVRS}
Most of this section will be devoted to proving the following result. 
\begin{thm}\label{main thm}
The complex orientation  
\[ \MU\to \TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q}[z]))\]
maps $x_{p-1}$ to $E_L(z)x^{p-1}$ and the elements $x_{p^n-1}$ map trivially for $n>1$. Consequently, the complex oriented spectrum $\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])/p$ carries a $p$-typical $p$-height one formal group law $F$ with $p$-series 
\[[p]_{F}(X)=z^{e_L}X^{p}+O(X^{p+1})\]	
up to multiplication by a unit in $W(\mathbb{F}_q)/(p)[[z]]$. 
\end{thm}

\begin{proof}[Proof of Theorem \ref{main thm}]
We consider the spectral sequence \eqref{S^1 homotopy} for $E=\mathbb{F}_p$, $A=\mathbb{S}_{W(\mathbb{F}_q})[z]$ and $B=\mathcal{O}_L$
\[ H^*(\mathbb{T};(\mathbb{F}_p)_*(\THH(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q})[z])))\Rightarrow (\mathbb{F}_p)^c_*(\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z];\mathbb{F}_p)\]
which has input 
\[ \mathbb{F}_q\otimes \mathcal{A}//E(0)_* \otimes \mathbb{F}_p[x]\otimes \mathbb{F}_p[t]\]
regarded as a $\mathcal{A}_*$ comodule algebra with the evident algebra structure, the $\mathcal{A}_*$-coaction on $(\mathbb{F}_p)_*(\mathcal{O}_L)$ is given by restriction along the inclusion 
$(\mathbb{F}_p)_*(\mathcal{O}_L)\rightarrow (\mathbb{F}_p)_*(\mathbb{F}_q)$ where $(\mathbb{F}_p)_*(\mathbb{F}_q)\cong \mathcal{A}_*\otimes \mathbb{F}_q$ has coaction induced by the coproduct, $x$ is a $\mathcal{A}_*$-comodule primitive, and $P(t)$ has the standard $\mathcal{A}_*$-coaction computed in \cite{Mil58}. 
We claim that the first nontrivial differentials are 
\[ d_{2}(\bar{\tau}_k)=tx^{p^k}\]
for $k\ge 1$. To see this, we first note that, for any $y\in H_*(\THH(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z]))$,
\[ d_{2}(y)=t\sigma y \]
by \cite[Prop. 3.2]{BR05}. We then recall that the map $\THH(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])\to \THH(\mathbb{F}_q)$ is a map of $E_{\infty}$ ring spectra and therefore the map in mod $p$ homology is compatible with Dyer-Lashof operations. Consequently, we can use the formula $Q^{p^k}\sigma=\sigma Q^{p^k}$ by \cite{Bok87a} and the computation
\[\sigma \overline{\tau}_k=\sigma Q^{p^k}\overline{\tau}_0=Q^{p^k}(\sigma \tau_0)=(\mu_0)^{p^k} \]
by \cite{BMMS86} for $k\ge 1$, to show that $\sigma(\overline{\tau}_k)=x^{p^k}$ in $(\mathbb{F}_p)_*(\THH(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z]))$. This implies the claimed differential for $k\ge 1$. By \cite[Thm. 1.2]{BR05}, we know that 
$\beta Q^{p^k}(\overline{\tau}_k)$ and 
\[Q^{p^k}(\overline{\tau}_k)-\overline{\tau}_k(\sigma \overline{\tau}_k)^{p-1}\]
also survive to the $E_{\infty}$-page and they are infinite cycles in the $\mathbb{T}$ homotopy fixed point spectra sequence (and consequently permanent cycles in the $\mathbb{T}$ homotopy fixed point spectral sequence). Where 
\[ \beta Q^{p^k}(\overline{\tau}_{k})=\overline{\xi}_{k+1} \text{ and }\tau^{\prime}_{k+1}:=Q^{p^k}(\overline{\tau}_k)-\overline{\tau}_k(\sigma \overline{\tau}_k)^{p-1}=\overline{\tau}_{k+1}-\overline{\tau}_k x^{p^k(p-1)}\]
for $k\ge 1$ by \cite{BMMS86}. Therefore, all of the generators are permanent cycles in the homological $\mathbb{T}$ homotopy fixed point spectral sequence except possibly $\overline{\xi}_1$ and $x$. In particular, the classes $\tau_k^{\prime}$ are permanent cycles in the homological $\mathbb{T}$ homotopy fixed point spectral sequence. So all products of these classes with other permanent cycles are also permanent cycles.

A priori, there is a possible differential of length $r\ge 2$ on $x$ and a possible differential of length $r\ge 2$ on $\overline{\xi}_1$, but these are the only remaining possibilities. Since $x$ is in the Hurewicz image it must also be a permanent cycle; i.e. the Hurewicz map 
\[ \TC_*^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])\to (\mathbb{F}_p)_*^c\TC^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])\]
maps the class $x$ to the class with the same name. Therefore, the only remaining possibility is a differential on $\xi_1$ 

We consider the map of homological Tate spectral sequences 
\[ 
	\xymatrix{
	H^*(\mathbb{T}, (\mathbb{F}_p)_*(\THH_*(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)})) \ar[r] \ar@{=>}[d] &H^*(\mathbb{T} ,(\mathbb{F}_p)_*(\THH(k))) \ar@{=>}[d] \\
	(\mathbb{F}_p)_*^c(\TC^{-}_*(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}) \ar[r] & (\mathbb{F}_p)_*(\TC^{-}_*(k))) 
	}
\]
induced by the map
\[\THH(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])\to \THH(k).\]
This map of Tate spectral sequences sends $x$ to $\mu_0$ and $\overline{\xi}_1$ to $\overline{\xi}_1$ by direct inspection. 
In the target spectral sequence, we know that $\mu_0$ and $\overline{\xi}_1$ are infinite cycles. 
This implies that if there is a differential on $\overline{\xi}_1$ in the source of the map of spectral sequences, then its target must map to zero at the same page as the length of this differential and similarly for $\mu_0$. Therefore, the only remaining possibilities are the differentials
\begin{align*}
d_r(\overline{\xi}_1)=t^rw_1x\text{ and }d_r(\overline{\xi}_1)=t^rw_2\xi_1 \\
\end{align*}
for some classes $w_1,w_2\in (\mathbb{F}_p)_*(\THH(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])$ such that $w_1x$ and $x_2\xi_i$ map to zero in 
\[(\mathbb{F}_p)_*(\THH(\mathbb{F}_q/\bS_{W(\mathbb{F}_q)}[z]).\] 
We now rule out these possibilities. Suppose there is a differential of the form
\[ d_r(\overline{\xi}_1)=t^rxw_1\text{ or }d_r(\overline{\xi}_1)=t^rw_2\xi_1 \]
for some nontrivial classes $w_1$, $w_2$. Then $\overline{\xi}_1$ does not survive the spectral sequence. 
Consequently, when computing the Adams spectral sequence
with signature
\[ \Cotor_{\mathcal{A}_*}^{*,*}\left (\mathbb{F}_p;(\mathbb{F}_p)_*\left (\TC^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])[2]\right )\right )\Rightarrow \pi_*\left (\TC^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])[2] \right)\]
then the class 
\[\overline{\xi}_1\otimes 1\in \mathcal{A}_*\otimes (\mathbb{F}_p)_*\left (\TC^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])[2]\right )\]
in the cobar complex would survive to the $E_2$-page giving a nontrivial element 
\[\overline{\xi}_1\otimes 1\in  \Cotor_{\mathcal{A}_*}^{*,*}(\mathbb{F}_p;(\mathbb{F}_p)_*(\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])[2]).\]
because it would not be killed by a $d_1$ differential on $\overline{\xi}_1\in (\mathbb{F}_p)_*(\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])[2])$ in the Cobar complex and there are no further elements in 
$(\mathbb{F}_p)_{2p-2}(\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])[1])$ that could be the source of a differential killing $\overline{\xi}_1\otimes 1$. This would in turn imply that the map of Adams spectral sequences 
\[ \Cotor_{\mathcal{A}_*}^{*,*}(\mathbb{F}_p,\mathbb{F}_p)\to \Cotor_{\mathcal{A}_*}^{*,*}(\mathbb{F}_p,(\mathbb{F}_p)_*(\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])[2])),\]
sends $\overline{\xi}_1\otimes 1$ to the class of the same name. Since $\overline{\xi}_1\otimes 1$ detects $\alpha_1$ in $\pi_{2p-3}(S_p^{\wedge})$, it is a permanent cycle and since $\overline{\xi}_1\otimes 1$ is on the $1$-line of the $E^2$-page of the Adams spectral sequence and we have shown that it is an infinite cycle, it must also be a permanent cycle. Thus, $\alpha_1$ maps to a nontrivial element in $\pi_{2p-3}(\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])[2])$. This of course cannot be the case because $\pi_*(\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])[2]))$ is necessarily concentrated in even degrees since the input of the truncated homotopical homotopy fixed point spectral sequence 
\[ E(t)\otimes \pi_*\left (\THH(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z]\right )\]
computing it is concentrated in even total degrees. We therefore reach a contradiction, so there cannot be a differential on $\overline{\xi}_1$.  There are therefore no further differentials, the $E_{\infty}$-page is isomorphic to  
\[ \mathbb{F}_q\otimes \mathcal{A}//E(1)_*\otimes \mathbb{F}_p[x]/x^p\otimes \mathbb{F}_p[t,t^{-1}]\]
as $\mathcal{A}_*$-comodule algebras where $\mathcal{A}//E(1)_*$ is  the homology of the Adams summand $\ell$. 

We apply the strongly convergent inverse-limit Adams spectral sequence of \cite[Prop. 2.2]{LNR10} with signature
\[ \lim_k \Cotor_{\mathcal{A}_*}^{*,*}(\mathbb{F}_p;(\mathbb{F}_p)_*(\TC^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])[k])\Rightarrow \pi_*(\TC^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])\]
to compute $\pi_*(\TC^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)})$. By a change of rings isomorphism the input is isomorphic to 
\[ \Cotor_{E(\bar{\tau}_0,\bar{\tau}_1)}^{*,*}(\mathbb{F}_p,\mathbb{F}_q[x]/x^p\otimes_{\mathbb{F}_p} \mathbb{F}_p[t])\] 
and this is isomorphic to
\[ P(v_0,v_1)\otimes \mathbb{F}_q[x]/x^p\otimes_{\mathbb{F}_p} \mathbb{F}_p[t].\]
The spectral sequence then collapses for bidegree reasons. We argue, however, there are hidden extensions so that the $v_0$-tower on $\mathbb{F}_q\{1\}$ forms a copy of $\mathcal{O}_L$ and then there is a further hidden extension $tx=E_L(z)$ so in fact the $v_0$ tower on $1$ and $tx$ together form a copy of $W(\mathbb{F}_q)[[z]]$. We also must have a hidden extension $v_0tx^p=v_1$. This is the only way to produce the known abutment 
\[ \pi_*(\TC^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])\cong  W(\mathbb{F}_q)[[z]][x,t]/(tx-E_L(z)).\]
of Lemma \ref{main lemma}.

We also note that $\TC^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])$ is an $E_{\infty}$ ring and it is complex oriented because $\TC_*^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])$ is concentrated in even degrees. Consequently, there is a map of ring spectra
\[\MU \to \TC^{-}(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z]).\]
By Chadwick-Mandell \cite[Theorem 1.2.]{CM15}, this lifts to a map of $E_2$ rings. There is a multiplicative map of Adams spectral sequences
\begin{align*}
\Cotor_{\mathcal{A}_*}^{*,*}(\mathbb{F}_p,(\mathbb{F}_p)_*(\MU))\to \Cotor_{\mathcal{A}_*}^{*,*}(\mathbb{F}_p,(\mathbb{F}_p)_*(\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q}[z])))
\end{align*}
which factors the map of Adams spectral sequences 
\begin{align}
\Cotor_{\mathcal{A}_*}^{*,*}(\mathbb{F}_p,(\mathbb{F}_p)_*(\MU))\to \Cotor_{\mathcal{A}_*}^{*,*}(\mathbb{F}_p,(\mathbb{F}_p)_*(\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])[2]))
\end{align}
where 
\[(\mathbb{F}_p)_*(\TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q}[z])[2])\cong H\left (\mathbb{F}_q\otimes\mathcal{A}//E(0))_*\otimes P(x)\otimes P(t^{-1})\otimes E(t),d_2 \right)\]
and $d_2(w)=t\sigma w$ for all $w\in \mathbb{F}_q\otimes\mathcal{A}//E(0))_*\otimes P(x)\otimes P(t^{-1})$. In particular, $d_2(\overline{\tau}_1)=tx^p$. Consequently,  
the element $\overline{\tau}_1\otimes 1$ 
maps to $\bar{\tau}_1\otimes 1$ and there is no differential in the target spectral sequence hitting this element. This element is on the 1-line and it is an infinite cycle since it is a permanent cycle on the source, surviving to become $x_{p-1}\in \pi_{2p-2}(\MU)$ in the abutment. 
We therefore conclude that $x_{p-1}\in \pi_{2p-2}(\MU)$ maps to some element in $\TC^{-}_{2p-2}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])[2]$. Since $\THH(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])$ is complex oriented and it is a $\mathcal{O}_L$-module it must be equipped with the additive formal group law, so we know that $x_{p-1}$ must map to a $t$-divisible element. The only element in the correct degree in $\TC^{-}_{2p-2}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])[2]$ is $tx^p$ up to multiplication by a unit in $\mathcal{O}_L$. Since $\mathcal{O}_L\cong W(\mathbb{F}_q)[z]/E_L(z)$ we know that this unit must map to a unit in $W(\mathbb{F})_q$ modulo $z$. Consequently, $v_1$ maps to $f(z)x^p$ in $\TC_*^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}[z])$ where $f(z)\ne 0$ in $W(\mathbb{F}_q)[[z]]$, $f(z)\in (E_L(z))$ and $f(z)\equiv E_L(z)$ modulo $(z^{e_L+1})$. This implies $f(z)\dot{=}E_L(z)$ where $\dot{=}$ means that the two are equal up to a unit in $W(\mathbb{F}_q)[[z]]$. Since $E_L(z)=z^{e_L}+p\theta(z)$, we observe that $E_L(z)\equiv z^{e_L}$ modulo $p$ and consequently, writing $\bar{f}(z)$ for $f(z)$ modulo $p$, then $\bar{f}(z)\dot{=}z^{e_L}$ where now $\dot{=}$ means up to multiplication by a unit in $W(\mathbb{F}_q)/(p)[[z]]$. 

In sum, the class $x_{p-1}\in \pi_{2p-2}\MU$ maps to $E_L(z)x^{1-p}$ in $\pi_{2p-2}TC^{-}(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)})$ up to multiplication by a unit in $W(\mathbb{F}_q)$ and consequently $v_1\in \pi_{2p-2}\BP$ maps to $z^{e_L}t^{1-p}$ in $(\mathbb{S}/p)_{2p-2}\TP(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)})$ up to multiplication by a unit in $\mathbb{F}_p$. 

Suppose $x_{p^n-1}\in \pi_{2p^n-2}\MU_{(p)}$ is mapped nontrivially via the map 
\[ \pi_*(\MU)\to \bS/p\wedge \TC^{-}_*(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)})\]
for $n>1$. Then $x_{p^n-1}^k$ also maps nontrivially for all $k\ge 1$ because this is a map of rings and by Lemma \ref{main lemma}.  Consequently, this implies that $\pi_*(L_{T(n)}\TP(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)}))$ is nonzero where $T(n)$ is telescope of a $v_n$ power self map of a type $n$ spectrum $V$. This also implies that $L_{T(n)}K(\mathcal{O}_L)$ is nontrivial. However, that contradicts Mitchell's theorem \cite{Mit90} and we reach a contradiction. Consequently, the $p$-typical formal group law of $\bS/p\wedge \TP(\mathcal{O}_L/\mathbb{S}_{W(\mathbb{F}_q)})$ has height exactly one. 
\end{proof}
\begin{cor}
There $E_{\infty}$ ring complex orientation 
\[ \MU\to \TP(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])\]
that factors through the Tate valued Frobenius map $\varphi$ of \eqref{frob} associates to $\TP(\mathcal{O}_L/\bS_{W(\mathbb{F}_q)}[z])/p$ a $p$-typical formal group law $G$ of $p$-height $1$ with $p$-series 
\[ [p]_G(X)=z^{pe_L}X^p+O(X^{p+1})\]
up to multiplication by a unit in $W(\mathbb{F}_q)/(p)[[z]]$. 
\end{cor}
\begin{proof}
This follows by combining Theorem \ref{main thm} with the description of the Tate valued Frobenius map \eqref{frob} in Lemma \ref{main lemma}. 	
\end{proof}

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