Doosung Park
Bergische Universität Wuppertal,
Fakultät Mathematik und Naturwissenschaften

Gaußstrasse 20, 42119 Wuppertal, Germany

Motivic real topological Hochschild spectrum
Primary 19D55; Secondary 11E70, 14A21, 16E40, 55P91

We define real topological Hochschild homology of noetherian separated fs log schemes with involutions.
We show that real topological Hochschild homology is $(\P^n,\P^{n-1})$-invariant,
which leads to the definition of the motivic real topological Hochschild spectrum living in a certain $\Ztwo$-equivariant logarithmic motivic category.
We also explore properties of real topological Hochschild homology that can be deduced from the logarithmic motivic homotopy theory.

§ INTRODUCTION

Topological Hochschild homology $\THH$ and its cousin $\TC$ have a deep connection with algebraic $K$-theory via the cyclotomic trace of Bökstedt-Hsiang-Madsen [8].
The Dundas-Goodwillie-McCarthy theorem [11] provides a computational tool for algebraic $K$-theory.
It is also discovered that $\THH$ contains arithmetic data.
Bhatt-Morrow-Scholze [3] study filtrations on the $S^1$-homotopy fixed point spectrum $\TC^-$ of $\THH$,
whose graded pieces are closely related to the prismatic cohomology of Bhatt-Scholze according to <cit.>.

Hesselholt-Madsen [15] define real algebraic $K$-theory,
which refines both algebraic and hermitian $K$-theories in a uniform manner via equivariant homotopy theory.
They also define real topological Hochschild homology $\THR$ of rings.
For further developments on $\THR$,
we refer to [20], [10], and [24].
A forthcoming work of Harpaz-Nikolaus-Shah shows a real refinement of the Dundas-Goodwillie-McCarthy theorem,
which would justify the real trace method as a computational tool for real or hermitian $K$-theory.

One feature of $\THH$ different from algebraic $K$-theory is that $\THH$ is not $\A^1$-invariant even for regular schemes,
the induced map $\THH(X)\to \THH(X\times \A^1)$ is not an equivalence of spectra for every nonempty scheme $X$.
The definition of $\THH$ of schemes is due to Geisser-Hesselholt [13].
Hence the motivic methods in $\A^1$-homotopy theory initiated by Morel-Voevodsky [22] are not directly applicable to $\THH$.

Non $\A^1$-invariance of $\THH$ is related to the fact that the sequence of spectra

\[
\THH(Z)\xrightarrow{i_*} \THH(X)
\xrightarrow{j^*} \THH(X-Z)
\]

is not a fiber sequence when $i\colon Z\to X$ is a closed immersion of regular schemes and $j\colon X-Z\to X$ is its open complement.
On the other hand,
the localization sequence in algebraic $K$-theory can be read as the fiber sequence of spectra

\[
\Kth(Z)\xrightarrow{i_*} \Kth(X)\xrightarrow{j^*} \Kth(X-Z).
\]

Hesselholt-Madsen [16] and Rognes-Sagave-Schlichtkrull [26] study localization sequences for $\THH$ in the logarithmic setting.

Author's joint work with Binda-Østvær [6] introduces the $\infty$-category of logarithmic motivic spectra $\logSH(S)$ for fs log schemes $S$,
which aims to incorporate various non $\A^1$-invariant cohomology theories into the motivic framework using logarithmic geometry.
We refer to Ogus's book [23] for logarithmic geometry.
For a closed immersion of scheme $Z\to X$,
let $(X,Z)$ denote the log scheme with the underlying scheme $X$ and with the compactifying log structure associated with the open immersion $X-Z\to X$.
In the construction of $\logSH(S)$ when $S$ is a scheme (with the trivial log structure),
the interval $\A^1$ is replaced with the set of log schemes $(\P^n,\P^{n-1})$ for all integers $n\geq 1$,
and the Nisnevich topology is replaced with the strict Nisnevich topology in [7].
To construct a motivic spectrum in $\logSH(S)$ that represents an existing cohomology theory of schemes,
one can take the following two steps:

Step 1. Extend the cohomology theory to fs log schemes.

Step 2. Show that the extension is $(\P^n,\P^{n-1})$-invariant for all integers $n\geq 1$ and satisfies strict Nisnevich descent.

Rognes [25] defines $\THH$ of log rings.
It is possible to define $\THH$ of noetherian fs log schemes based on Rognes' definition and to construct a $\P^1$-spectrum $\bTHH$ (=$\mathbf{logTHH}$) representing $\THH$,
see <cit.>.
Author's joint work with Binda-Lundemo-Østvær <cit.> constructs a $\P^1$-spectrum representing Hochschild homology.

In this article,
we employ this strategy for $\THR$.
We define $\THR$ of log rings with involutions,
where an involution means an automorphism $\sigma$ such that $\sigma\circ \sigma=\id$.
With the aid of author's joint work with Hornbostel [18],
we define $\THR$ of noetherian separated fs log schemes with involutions.
Then we prove that $\THR$ is $(\P^n,\P^{n-1})$-invariant for all integers $n\geq 1$.
This allows us to apply motivic methods in logarithmic motivic homotopy theory to $\THR$.
The following two results are examples of this.

Let $Z\to X$ be a closed immersion of smooth schemes over a noetherian separated scheme $S$.
Then there exists a fiber sequence of $\Ztwo$-spectra

\[
\THR(\Th(\Normal_Z X))
\to
\THR(X)
\to
\THR(\Blow_Z X,E),
\]

where $E$ is the exceptional divisor.

We refer to (<ref>) for the notation $\THR(\Th(\Normal_Z X))$.
Proposition <ref> implies that
$\THR(\Th(\Normal_Z X))$ can be written in terms of $\THR$ of schemes.
This sequence can be considered as the localization sequence for $\THR$.
The map $\THR(\Th(\Normal_Z X)) \to \THR(X)$ is called the Gysin map.

Let $Z\to X$ be a closed immersion of smooth schemes over a noetherian separated scheme $S$.
Then the induced square of $\Ztwo$-spectra

\[
\begin{tikzcd}
\THR(X)\ar[d]\ar[r]&
\THR(Z)\ar[d]
\\
\THR(Z\times_X \Blow_Z X)\ar[r]&
\THR(\Blow_Z X)
\end{tikzcd}
\]

is cartesian,
where $\Blow_Z X$ denotes the blow-up of $X$ along $Z$.

The statement in this theorem involves no log schemes,
but the proof uses log schemes.

Hu-Kriz-Ormsby [19] consider $\P^1\wedge \P^\sigma$-spectra instead of $\P^1$-spectra to define a motivic $\Ztwo$-spectrum,
where $\P^\sigma$ is the scheme $\P^1$ with the involution given by $[x:y]\mapsto [y:x]$.
In Definition <ref>,
we analogously introduce the $\infty$-category of prelogarithmic motivic $\Ztwo$-spectra $\prelogSH_{\Ztwo}(S)$ for any scheme $S$,
which is helpful for constructing logarithmic motivic spectra using the fixed point functor

\[
\colon
\prelogSH_{\Ztwo}(S)
\to
\logSH(S)
\]

in Definition <ref>.
We also construct the “forgetful” functor

\[
\colon
\prelogSH_{\Ztwo}(S)
\to
\logSH(S)
\]

in Definition <ref>.
The reason why we add “pre” in the notation $\prelogSH_{\Ztwo}(S)$ is that a further localization is desired to obtain a better behaved $\infty$-category of logarithmic motivic $\Ztwo$-spectra,
see Remark <ref> for an explanation.

Together with the $\P^1\wedge \P^\sigma$-periodicity of $\THR$ in <cit.>,
we can define the periodic  $\P^1\wedge \P^\sigma$-spectrum

\[
\bTHR
\in
\prelogSH_{\Ztwo}(S)
\]

for every noetherian separated scheme $S$,
where $\THO(X):=\THR(X)^{\Ztwo}$ for every noetherian separated fs log scheme $X$ with involution.
The construction of $\bTHR$ resembles the construction of the motivic real $K$-theory $\Ztwo$-spectrum <cit.>.
We obtain the new $\P^1$-spectrum $\bTHO:=\bTHR^{\Ztwo}$,
while we show $i^*\bTHR\simeq \bTHH$ in Proposition <ref>.

§.§ Organization of the article

To define the $\P^1\wedge \P^\sigma$-spectrum $\bTHR$,
we need the following ingredients:

The definition of $\THR$ of log rings with involutions.

Isovariant étale descent property for $\THR$.

Invariance of $\THR$ under passing to the associated log structure.

$\P^1\wedge \P^\sigma$-periodicity of $\THR$.

$(\P^n,\P^{n-1})$-invariance of $\THR$.

Due to [18],
we have (2) and (4).
In <ref>,
we deal with (1).
This requires dihedral replete bar constructions in <ref>,
which is an equivariant analogue of Rognes' replete bar constructions <cit.>.
In <ref>,
we show (3) for noetherian separated fs log schemes.
The strategy is to work in a sufficiently local situation.
In <ref>,
we show (5) by providing an explicit computation of $\THR(\P^n,\P^{n-1})$ using cubes in $\infty$-categories.
Then we discuss properties of $\THR$ that can be deduced from [6],
and we construct the $\P^1\wedge \P^\sigma$-spectrum $\bTHR$.

§.§ Acknowledgement

This research was conducted in the framework of the DFG-funded research training group GRK 2240: Algebro-
Geometric Methods in Algebra, Arithmetic and Topology.

§ DIHEDRAL REPLETE BAR CONSTRUCTIONS

Rognes <cit.> defines the replete bar construction of a commutative monoid,
which is a key input to define topological Hochschild homology of log rings.
In this section,
we discuss a dihedral refinement of this construction.

We refer to <cit.> for a review of real simplicial sets and dihedral sets,
which are defined using crossed simplicial groups of Fiedorowicz-Loday [12].
A real simplicial set $X$ is a simplicial set equipped with involutions $w_q\colon X_q \to X_q$ (i.e., an automorphism such that $w_q\circ w_q=\id$) for all simplicial degrees $q$ satisfying certain relations.
A dihedral simplicial set $X$ is a real simplicial set equipped with automorphisms $t_q\colon X_q\to X_q$ for all simplicial degrees $q$ satisfying certain relations.

Let $\oplus$ denote the coproduct in the category of commutative monoids.
For commutative monoids $P$ and $Q$,
the coproduct $P\oplus Q$ is naturally isomorphic to the product $P\times Q$.

Let $\Delta_\sigma^1$ be the real simplicial set whose underlying simplicial set is $\Delta^1$ and whose involution $w\colon (\Delta^1)_q\to (\Delta^1)_q$ in simplicial degree $q$ sends the $q$-simplex $a_0 \cdots a_q$ to $(1-a_0)\cdots (1-a_q)$.
For a commutative monoid $P$ with involution $\sigma$,
let $P\otimes \Delta_\sigma^1$ be the real simplicial set whose underlying simplicial set is the tensor product $P\otimes \Delta^1$ defined by the formula

\[
(P\otimes \Delta^1)_q
\bigoplus_{i\in (\Delta^1)_q}P
\]

and whose involution is given by $(x_0,\ldots,x_{q+1})\to (w(x_{q+1}),\ldots,w(x_0))$ in simplicial degree $q$,
where the indices $0,1,\ldots,q+1$ correspond to the $q$-simplicies $0\cdots 00,0\cdots 01,\ldots,1\cdots 11$ in $\Delta^1$.

Let $P$ be a commutative monoid with involution.
Then the map of real simplicial sets

\[
P\otimes \Delta_\sigma^1
\to
\]

given by $(x_0,\ldots,x_q)\mapsto x_0+\cdots+x_q$ in simplicial degree $q$ is a $\Ztwo$-weak equivalence.

Let $\sd_\sigma$ be the Segal subdivision functor [27].
We have an isomorphism of simplicial sets with involutions

\[
\sd_\sigma(\Delta_\sigma^1)\simeq \Delta^1\amalg_{\{1\}} \Delta^1
\]

where the involution on the right-hand side switches the factors.
This induces an isomorphism of simplicial sets with involutions

\[
\sd_\sigma(P\otimes \Delta_\sigma^1)
\simeq
(P\otimes \Delta^1) \oplus_P (P\otimes \Delta^1),
\]

where the involution on the right-hand side is obtained by the formula $(x,y)\mapsto (w(y),w(x))$ with $w$ in Definition <ref>.
Since $\Delta^1$ is contractible,
the map $P\to P\otimes \Delta^1$ induced by $\{1\}\to \Delta^1$ is a homotopy equivalence.
This yields a $\Ztwo$-homotopy equivalence

\[
\xrightarrow{\simeq}
(P\otimes \Delta^1) \oplus_P (P\otimes \Delta^1),
\]

which implies the claim.

We review the $\infty$-categorical formulation of equivariant homotopy theory due to Bachmann-Hoyois <cit.>.
See <cit.> for a more detailed review.
Let $\FinGpd$ denote the $2$-category of finite groupoids.
Bachmann-Hoyois construct the functor

\[
\SH
\colon
\Span(\FinGpd)
\to
\CAlg(\Cat_\infty),
\;
(X\xleftarrow{f} Y\xrightarrow{p} Z)
\mapsto
p_\otimes f^*,
\]

see <cit.> for the notation $\Span$.
For a morphism $f$ in $\FinGpd$,
$f^*$ admits a right adjoint $f_*$.
If $f$ is a finite covering,
then $f^*$ admits a left adjoint $f_\sharp$.

For a finite groupoid $X$,
let $\NAlg(\SH(X))$ be the $\infty$-category of normed $X$-spectra <cit.>,
and let $\CAlg(\SH(X))$ be the $\infty$-category of $\E_\infty$-rings in $\SH(X)$.
We have the forgetful functor $\NAlg(\SH(X))\to \CAlg(\SH(X))$.
We use the notation $\wedge$ for the coproduct in $\NAlg(\SH(X))$ and $\CAlg(\SH(X))$.
For a morphism of finite groupoids $f\colon X\to S$,
we have the induced adjoint functors

\[
\begin{tikzcd}
\CAlg(\SH(S))
\ar[r,shift left=0.75ex,"f^*"]
\ar[r,shift right=0.75ex,"f_*"',leftarrow]
\CAlg(\SH(X)).
\end{tikzcd}
\]

If $f\colon X\to S$ is a finite covering of finite groupoids,
then we have the induced sequence of adjoint functors

\[
\begin{tikzcd}[column sep=large]
\NAlg(\SH(X))
\ar[r,shift left=1.5ex,"f_\sharp"]
\ar[r,"f^*" description,leftarrow]
\ar[r,"f_*"',shift right=1.5ex]
\NAlg(\SH(S)).
\end{tikzcd}
\]

On the other hand,
if $f$ has connected fibers,
then the induced functor

\[
f_\otimes \colon \CAlg(\SH(X))\to \CAlg(\SH(S))
\]

preserves colimits.

For a finite group $G$,
the $\infty$-category of $G$-spectra is

\[
\Sp_G
\SH(\rB G).
\]

We also set $\NAlg_G:=\NAlg(\SH(\rB G))$ and $\CAlg_G:=\CAlg(\SH(\rB G))$.
We omit the subscripts $G$ in this notation if $G$ is trivial.
There is an equivalence of $\infty$-categories $\CAlg\simeq \NAlg$.

Consider the obvious functors $\pt \xrightarrow{i} \rB (\Ztwo) \xrightarrow{p} \pt$.
Observe that $i$ is a finite covering and $p$ has connected fibers.
We often use the alternative notation

\[
\;
\;
\Phi^{\Ztwo}
\]

The functor $N^{\Ztwo}$ is the norm functor of Hill-Hopkins-Ravenel [17],
and the functor $\Phi^{\Ztwo}$ is the geometric fixed point functor.
We have the induced functors

\begin{gather*}
i^*\colon \Sp_{\Ztwo} \to \Sp,
\;
N^{\Ztwo},i_\sharp,i_* \colon \Sp\to \Sp_{\Ztwo},
\\
i^*\colon \NAlg_{\Ztwo} \to \CAlg,
\;
N^{\Ztwo},i_\sharp,i_* \colon \CAlg\to \NAlg_{\Ztwo},
\\
p^*\colon \Sp\to \Sp_{\Ztwo},
\;
\Phi^{\Ztwo},(-)^{\Ztwo} \colon \Sp_{\Ztwo}\to \Sp
\\
p^*\colon \CAlg\to \CAlg_{\Ztwo},
\;
\Phi^{\Ztwo},(-)^{\Ztwo} \colon \CAlg_{\Ztwo}\to \CAlg,
\;
\end{gather*}

We refer to <cit.> for fundamental relations among these functors.
We note that the pair of functors $(i^*,\Phi^{\Ztwo})$ is conservative.

Let $f\colon X\to S$ be a morphism of finite groupoids.
For every map $R\to A$ in $\NAlg(\SH(S))$ and map $f^*R\to B$ in $\NAlg(\SH(X))$,
we have the natural map

\[
A\wedge_R f_*B
\to
\]

given by the composite

\[
A\wedge_R f_*B
\xrightarrow{ad}
f_*f^*(A\wedge_R f_*B)
\xrightarrow{\simeq}
f_*(f^*A \wedge_{f^*R} f^*f_*B)
\xrightarrow{ad'}
f_*(f^*A \wedge_{f^*R} B),
\]

where $ad$ (resp. $ad'$) denotes the map obtained by the unit (resp. counit).

Let $q\colon \Ztwo\to \pt$ be the obvious morphism of finite groupoids.
Then the natural map

\[
A\wedge_R q_*B
\to
q_*(q^*A\wedge_{q^*R} B)
\]

is an equivalence for every map $R\to A$ in $\CAlg$ and map $q^*R\to B$ in $\NAlg(\SH(\Ztwo))$.

The functor $q^*\colon \CAlg\to \NAlg(\SH(\Ztwo))$ can be identified with the diagonal functor $\CAlg \to \CAlg\times \CAlg$, and the functor $q_*\colon \NAlg(\SH(\Ztwo))\to \CAlg$ can be identified with the direct sum functor $\oplus \colon \CAlg\times \CAlg\to \CAlg$.
Use these explicit descriptions to show the claim.

For $R,A\in \NAlg_{\Ztwo}$ and $B\in \CAlg$,
the natural map

\begin{equation}
\label{thrlog.6.1}
A\wedge_R i_*B
\to
\end{equation}

is an equivalence.

It suffices to show that (<ref>) becomes equivalences after applying $i^*$ and $\Phi^{\Ztwo}$.
With the aid of the forgetful functor $\NAlg_{\Ztwo}\to \CAlg_{\Ztwo}$, we obtain an equivalence $\Phi^{\Ztwo}(A\wedge_R i_*B)\simeq \Phi^{\Ztwo}A \wedge_{\Phi^{\Ztwo}R}\Phi^{\Ztwo}i_*B$.
By <cit.>,
we have $\Phi^{\Ztwo}i_*\simeq 0$.
Hence both sides of (<ref>) vanish after applying $\Phi^{\Ztwo}$.

Apply <cit.> to the cartesian square

\[
\begin{tikzcd}
\Ztwo\ar[d,"q"']\ar[r,"q"]&
\pt\ar[d,"i"]&
\\
\pt\ar[r,"i"]&
\rB (\Ztwo)
\end{tikzcd}
\]

to obtain a natural equivalence $q_*q^*\simeq i^*i_*$.
We have natural equivalences

\[
q_*q^*((-)\wedge_{(-)} (-))
\simeq
\simeq
\]

where the second one is due to Lemma <ref>.
We have the induced commutative diagram

\[
\begin{tikzcd}[column sep=tiny]
i^*(A\wedge_R i_*B)\ar[r,"ad"]\ar[d,"\simeq"']&
i^*i_*i^*(A\wedge_R i_*B)\ar[r,"\simeq"]&
i^*i_*(i^*A\wedge_{i^*R} i^*i_*B)\ar[r,"ad'"]\ar[d,"\simeq"]&
i^*i_*(i^*A\wedge_{i^*R} B)\ar[d,"\simeq"]
\\
\\
\end{tikzcd}
\]

Hence to show that (<ref>) becomes an equivalence after applying $i^*$, it suffices to show that the composite of the upper vertical maps in the commutative diagram

\[
\begin{tikzcd}
\\
\end{tikzcd}
\]

is an equivalence.
This is true since the composite of the lower horizontal maps is an equivalence by the counit-unit identity.

For a simplicial set (resp. real simplicial set) $X$,
we use the notation $\Sphere[X]:=\Sigma^\infty X_+$,
which is a spectra (resp. $\Ztwo$-spectra).
If $P$ is a commutative monoid,
then $\Sphere[P]$ is an object of $\CAlg$.
If $P$ is a commutative monoid with involution,
then $\Sphere[P]$ is a commutative monoid in the model category of orthogonal $\Ztwo$-spectrum in the sense of <cit.>.
Due to <cit.>,
we can regard $\Sphere[P]$ as an object of $\NAlg_{\Ztwo}$.

Let $\Ndi P$ be the dihedral nerve of $P$ in <cit.>, and let $\Bdi P$ be its dihedral geometric realization.
In simplicial degree $q$,
we have $(\Ndi P)_q:=P^{\times (q+1)}$.
We obtain the cyclic nerve $\Ncy i^* P$ <cit.> of $i^*P$ if we forget the involution structure on $\Ndi P$.
There is a map of dihedral sets

\begin{equation}
\label{drep.2.1}
\Ndi P
\to
\end{equation}

sending $(x_0,\ldots,x_q)$ to $x_0+\cdots+x_q$ in simplicial degree $q$.

Let $\cC$ be an ordinary category.
An object of $\cC$ with involution is an object of $\cC_{\Ztwo}:=\Fun(\rB (\Ztwo),\cC)$.

\[
i^*\colon \Fun(\rB (\Ztwo),\cC)
\to
\cC
\]

be the forgetful functor,
and let $i_\sharp$ (resp. $i_*$) be its left adjoint (resp. right adjoint) if it exists.

If $P$ is a commutative monoid,
then $i_*P$ is the commutative monoid $P\times P$ with the involution $w$ given by $w(x,y):=(w(y),w(x))$.
there is a natural isomorphism $i_\sharp P\simeq i_*P$.
We have a similar description of $i_*A$ for any commutative ring $A$,
but $i_\sharp A \not\simeq i_*A$.

If $X$ is a scheme,
then $i_\sharp X$ is the scheme $X\amalg X$ with the involution switching the two components.

Let $P\to Q$ be maps of commutative monoids with involutions.
Then there is a natural equivalence of $\Ztwo$-spectra

\[
\Sphere[P\amalg P]\wedge_{\Sphere[P]}\Sphere[Q]
\simeq
\Sphere[(P\amalg P)\oplus_P Q],
\]

where the involution on $P\amalg P$ in the formulation switches the components.

There is a natural isomorphism of commutative monoids with involutions

\[
(P\amalg P) \oplus_P Q
\simeq
Q\amalg Q,
\]

where the involution on the right-hand side switches the components.
By the explicit description of the functors $i^*$ and $i_*$ in terms of orthogonal spectra in <cit.>,
we have natural equivalences

\[
\Sphere[P\amalg P]
\simeq
\text{ and }
\Sphere[Q\amalg Q]
\simeq
\]

Lemma <ref> yields a natural equivalence

\[
\wedge_{\Sphere[P]}
\Sphere[Q]
\simeq
i_*(i^*\Sphere[P]\wedge_{i^*\Sphere[P]} i^*\Sphere[Q]).
\]

Combine what we have discussed above to obtain the desired equivalence.

Let $P$ be a commutative monoid with involution.
Then there is a natural equivalence of $\Ztwo$-spectra

\[
\Sphere[\Bdi P]
\simeq
\Sphere[P]\wedge_{\Sphere[i_\sharp i^*P]} \Sphere[P],
\]

where the homomorphism $i_\sharp i^*P\to P$ in the formulation is obtained by the counit of the adjunction pair $(i_\sharp,i^*)$.

There is an isomorphism of real simplicial sets

\begin{equation}
\label{drep.2.2}
(P\otimes \Delta_\sigma^1)\oplus_{i_\sharp i^*P} P
\simeq
\Ndi P,
\end{equation}

where the map $i_\sharp i^*P\to P\otimes \Delta_\sigma^1$ is given by $(x,y)\mapsto (x,0,\ldots,0,y)$ in simplicial degree $q$.
The composite

\[
i_\sharp i^*P \to P\otimes \Delta_\sigma^1 \to P
\]

coincides with the counit homomorphism.
Since $P\otimes \Delta_\sigma^1$ is degreewise the disjoint union of finite copies of $i_\sharp i^* P$ and $i_\sharp i^*P \amalg i_\sharp i^*P$ with the switching involution as an $i_\sharp i^*P$-set,
Lemma <ref> yields a natural equivalence

\[
\Sphere[P\otimes \Delta_\sigma^1]
\wedge_{\Sphere[i_\sharp i^* P]}
\Sphere[P]
\simeq
\Sphere[(P\otimes \Delta_\sigma^1)\oplus_{i_\sharp i^* P} P].
\]

Use Proposition <ref> and (<ref>) to finish the proof.

Let $P$ be a commutative monoid with involution.
The dihedral replete nerve of $P$ is the dihedral set

\begin{equation}
\label{dih.13.1}
\Ndrep P
\Ndi P^\gp \times_{P^\gp} P,
\end{equation}

where the map $\Ndi P^\gp\to P^\gp$ in this formulation is given by (<ref>) for $P^\gp$.
The dihedral replete bar construction of $P$, denoted $\Bdrep P$,
is the dihedral geometric realization of $\Ndrep P$.

We obtain the replete bar construction $\Bdi i^*P$ <cit.> of $i^*P$ if we forget the involution structure on $\Bdrep P$.

Let $P$ and $Q$ be commutative monoids with involutions.
Then there is a natural isomorphism of dihedral sets

\[
\Ndrep (P\times Q)
\simeq
\Ndrep P \times \Ndrep Q.
\]

By <cit.>,
we have a natural isomorphism of dihedral sets

\[
\Ndi (P^\gp \times Q^\gp) \times_{P^\gp \times Q^\gp} (P\times Q)
\simeq
(\Ndi P^\gp \times \Ndi Q^\gp) \times_{P^\gp \times Q^\gp} (P\times Q).
\]

The left-hand side is isomorphic to $\Ndrep (P\times Q)$,
and the right-hand side is isomorphic to $\Ndrep P \times \Ndrep Q$.

For a commutative monoid $P$ with involution,
let $N^\sigma P$ denote the real nerve of $P$,
see <cit.>.

Let $P$ be a commutative monoid with involution.
Then there is a natural isomorphism of dihedral sets

\[
\Ndrep P
\simeq
P\times \Nsigma P^\gp.
\]

Immediate from <cit.> and (<ref>).

For a commutative monoid $P$ with involution $w$,
let $(i_\sharp i^*P)^{ex}$ denote the commutative monoid $P\oplus P^\gp$ with the involution $w$ given by

\[
\]

We have the commutative triangle

\[
\begin{tikzcd}
& (i_\sharp i^*P)^{ex}\ar[rd,"\theta^{ex}"]
\\
i_\sharp i^*P\ar[ru,"\eta"]\ar[rr,"\theta"]&
\end{tikzcd}
\]

such that $\theta(x,y):=x+y$, $\theta^{ex}(x,y):=x$, and $\eta(x,y):=(x+y,y)$ for $x,y\in P$.
We note that $\theta$ is the counit homomorphism.
The construction of $(i_\sharp i^*P)^{ex}$ is an equivariant analogue of the exactification in <cit.>.

Let $P$ be a commutative monoid with involution.
Then there is a natural equivalence of $\Ztwo$-spectra

\[
\Sphere[\Bdrep P]
\simeq
\Sphere[P]\wedge_{\Sphere[(i_\sharp i^*P)^{ex}]} \Sphere[P].
\]

Let $\rE P^\gp$ denote the total simplicial set of $P^\gp$,
and let $Q$ be the real simplicial set whose underlying simplicial set is $P\times \rE P^\gp$,
and whose involution is given by

\[
(x,g_0,\ldots,g_q)\in P\times (P^\gp)^{\times(q+1)}
\mapsto
\]

in simplicial degree $q$ for every integer $q\geq 0$.
Consider the maps of real simplicial sets

\[
(i_\sharp i^* P)^{ex} \to Q \to P\times \Nsigma P^\gp
\]

given by
\mapsto
\mapsto
in simplicial degree $q$.

There is a real simplicial isomorphism

\[
Q \oplus_{(i_\sharp i^*P)^{ex}} P
\simeq
P\times \Nsigma P^\gp.
\]

Since $Q$ is degreewise the disjoint union of finite copies of $(i_\sharp i^*P)^{ex}$ and $(i_\sharp i^*P)^{ex}\amalg (i_\sharp i^*P)^{ex}$ with the switching involution as an $(i_\sharp i^*P)^{ex}$-set, Lemma <ref> yields a natural equivalence of $\Ztwo$-spectra

\begin{equation}
\label{dih.15.1}
\Sphere[Q] \wedge_{\Sphere[(i_\sharp i^*P)^{ex}]} \Sphere[P]
\simeq
\Sphere[P\times \Nsigma P^\gp].
\end{equation}

Let us show that the map $Q\to P$ given by

\[
(x,g_0,\ldots,g_q)\mapsto x
\]

in simplicial degree $q$ is a $\Ztwo$-homotopy equivalence. Its underlying map of simplicial sets is the projection $P\times \rE P^\gp \to P$, which is a homotopy equivalence.
The $\Ztwo$-fixed point of the Segal subdivision $\sd_{\sigma} Q$ is in simplicial degree $q$ the set

\[
\{
x\in P^{\Ztwo},g_0,\ldots,g_q\in P^\gp
\}.
\]

From this description,
we see that the induced map

\[
(\sd_{\sigma} Q)^{\Ztwo}
\to
(\sd_{\sigma} P)^{\Ztwo}
\]

can be identified with the projection $P\times \rE P^\gp \to P$.
Hence the map $Q\to P$ is a $\Ztwo$-homotopy equivalence.
Combine this with Proposition <ref> and (<ref>) to obtain the desired equivalence.

For a commutative monoid $P$ with involution,
we can regard $\Sphere[\Bdi P]$ and $\Sphere[\Bdrep P]$ as objects of $\NAlg_{\Ztwo}$ by Propositions <ref> and <ref>.

Let $P$ be a commutative monoid with involution.
Then there are natural equivalences of $\Ztwo$-spectra

\[
i^*\Sphere[\Bdi P]
\simeq
\Sphere[\Bcy i^*P]
\text{ and }
i^*\Sphere[\Bdrep P]
\simeq
\Sphere[\Brep i^*P].
\]

Observe that we obtain $\Bcy i^*P$ (resp. $\Brep i^*P$) by forgetting the involution structure on $\Bdi P$ (resp. $\Bdrep P$).

§ THR OF LOG RINGS WITH INVOLUTIONS

For $R\in \NAlg_{\Ztwo}$,
the real topological Hochschild homology of $R$ is defined to be

\[
\THR(R)
\]

where the maps $N^{\Ztwo}i^*R \to R$ in the formulation are the counit of the adjunction pair $(N^{\Ztwo},i^*)$.
If $A$ is a commutative ring with involution,
then we set $\THR(A):=\THR(\EM A)$.
For a commutative monoid $P$ with involution,
there is a natural equivalence of $\Ztwo$-spectra

\begin{equation}
\THR(\Sphere[P])
\simeq
\Sphere[\Bdi P],
\end{equation}

see [20] and also  <cit.>.
One can also show this using Proposition <ref> and the natural equivalence of $\Ztwo$-spectra $N^{\Ztwo} i^*\Sphere[P]\simeq \Sphere[i_\sharp i^*P]$.

Recall from <cit.> that a log ring[A log ring is often called a pre-log ring in the literature, e.g.,  <cit.>.]  $(A,P)$ is a commutative ring $A$ equipped with a homomorphism $P\to A$ of commutative monoids,
where the monoid operation on $A$ is the multiplication.
A homomorphism of log rings $(A,P)\to (B,Q)$ is a pair of homomorphisms $A\to B$ and $P\to Q$ such that the square

\[
\begin{tikzcd}
\\
\end{tikzcd}
\]

We regard a commutative ring $A$ as a log ring $(A,\{1\})$.

Rognes <cit.>
defines the topological Hochschild homology of a log ring $(A,P)$ as the coproduct in $\CAlg$

\begin{equation}
\THH(A,P)
\THH(A)\wedge_{\Sphere[\Bcy P]}\Sphere[\Brep P],
\end{equation}

where the map $\Sphere[\Bcy P]\simeq \THH(\Sphere[P])\to \THH(A)$ in the formulation is obtained by applying $\THH$ to the induced map $\Sphere[P]\to \EM A$.

The real topological Hochschild homology of a log ring $(A,P)$ with involution is the coproduct in $\NAlg_{\Ztwo}$

\begin{equation}
\label{dih.22.1}
\THR(A,P)
\THR(A)\wedge_{\Sphere[\Bdi P]}\Sphere[\Bdrep P],
\end{equation}

where the map $\Sphere[\Bdi P]\simeq \THR(\Sphere[P])\to \THR(A)$ in this formulation is obtained by applying $\THR$ to the induced map $\Sphere[P]\to \EM A$.

We obviously have $\THR(A,\{1\})\simeq\THR(A)$ for every commutative ring $A$.

Let $P\to M$ be a homomorphism of commutative monoids with involutions.
For notational convenience,
we set

\[
\THR(\Sphere[M],P)
\Sphere[\Bdi M]\wedge_{\Sphere[\Bdi P]}\Sphere[\Bdrep P].
\]

Let $Q\to M$ be a homomorphism of commutative monoids with involutions.
Then there is a natural equivalence of $\Ztwo$-spectra

\[
\THR(A[M],P\oplus Q)
\simeq
\THR(A,P) \wedge \THR(\Sphere[M],Q).
\]

By <cit.> and Proposition <ref>,
there are natural isomorphisms of dihedral sets

\[
\Ndi (P\oplus Q)
\simeq
\Ndi P \times \Ndi Q
\text{ and }
\Ndrep (P\oplus Q)
\simeq
\Ndrep P \times \Ndrep Q.
\]

Apply $\Sphere[-]$ to these,
and use (<ref>) to obtain the desired equivalence.

Let $S^\sigma$ be $S^1$ with the involution given by $e^{i\theta}\in S^1 \mapsto e^{-i\theta}$.

By <cit.>,
we have an equivalence of $\Ztwo$-spectra

\[
\THR(\Sphere[\N],\N)
\simeq
\bigoplus_{d=0}^\infty \Sphere[S^\sigma].
\]

<cit.> implies that the induced map

\[
\THR(\Sphere[\N])
\to
\THR(\Sphere[\N],\N)
\]

can be written as the componentwise induced map

\[
\Sphere\oplus \bigoplus_{d=1}^\infty \Sphere[S^\sigma]
\to
\bigoplus_{d=0}^\infty \Sphere[S^\sigma].
\]

In the remaining part of this section,
we investigate how $\THH$ and $\THR$ interact with the functors $i^*$ and $i_*$.

Let $(A,P)$ be a log ring with involution.
Then there is a natural equivalence of spectra

\[
\simeq
\THH(i^*A,i^*P).
\]

We have a natural equivalence $i^*\THR(A)\simeq \THH(i^*A)$
by <cit.>.
Since $i^*\colon \NAlg_{\Ztwo}\to \CAlg$ preserves colimits,
we have a natural equivalence

\[
\simeq
i^*\THR(A)\wedge_{i^*\Sphere[\Bdi P]}i^*\Sphere[\Bdrep P].
\]

Together with Proposition <ref>,
we obtain the desired equivalence.

For a commutative monoid $P$,
let $\iota P$ denote the commutative monoid $P$ with the trivial involution.

Let $(A,P)$ be a log ring.
Then there is a natural equivalence of $\Ztwo$-spectra

\[
\simeq
\THR(i_*A,\iota P),
\]

where the structure homomorphism $\iota P\to i_*A$ sends $p\in P$ to $(p,p)$.

Lemma <ref> and Proposition <ref>
yield natural equivalences of $\Ztwo$-spectra

\begin{align*}
\simeq &
i_*(\THH(A)\wedge_{i^*\Sphere[\Bdi \iota P]}i^*\Sphere[\Bdrep \iota P])
\\
\simeq &
i_*\THH(A)\wedge_{\Sphere[\Bdi \iota P]}\Sphere[\Bdrep \iota P].
\end{align*}

By <cit.>,
we have a natural equivalence of $\Ztwo$-spectra

\[
\simeq
\THR(i_*A).
\]

Combine these with the definition of $\THR(i_*A,\iota P)$ to conclude.

§ THR OF FS LOG SCHEMES WITH INVOLUTIONS

So far,
we have discussed $\THR$ of log rings $(A,P)$ with involutions.
it will be hasty to claim that

\begin{equation}
\label{descent.0.1}
\THR(\Spec(A,P))
\THR(A,P)
\end{equation}

is the correct definition unless we show that there is an equivalence $\THR(A,P)\simeq \THR(A,Q)$ whenever $(A,Q)$ is another log ring with involution such that $\Spec(A,P)$ and $\Spec(A,Q)$ are isomorphic as log schemes.

Our approach for this problem is to work on fs log affine schemes $X$ in the sense of Definition <ref> below.
The main result in this section is Theorem <ref>,
which shows that (<ref>) is reasonable for the fs log affine case.
Together with the isovariant étale descent result <cit.>,
we obtain the definition of $\THR$ of noetherian fs log schemes.

Let us briefly review basic notation and terminology in log geometry.
We refer to Ogus's book [23] for the details.

* For a log scheme $X$,
let $\ul{X}$ be its underlying scheme,
and let $\cM_X$ be its structure sheaf of monoids.

* A morphism of log schemes $f\colon Y\to X$ is strict if the induced morphism $Y\to X\times_{\ul{X}}\ul{Y}$ is an isomorphism.

* For a log ring $(A,P)$,
let $\Spec(A,P)$ denote its associated log scheme in the sense of <cit.>.

* For a commutative monoid $P$,
we set $\A_P:=\Spec(\Z[P],P)$,
whose structure homomorphism $P\to \Z[P]$ sends $p\in P$ to $p$.

* A chart $P$ of a log scheme $X$ is a commutative monoid $P$ together with a strict morphism $X\to \A_P$ of log schemes.

* A log scheme $X$ is fine saturated (or simply fs) if $X$ admits strict étale locally a chart $P$ such that $P$ is a fine saturated monoid.

Let $X$ be an fs log scheme such that its underlying scheme is affine.
We say that $X$ is an fs log affine scheme if the following condition is satisfied:

The monoid $\Gamma(X,\cM_X)/\Gamma(X,\cO_X^*)$ is finitely generated,
the identity homomorphism $\Gamma(X,\cM_X)\to \Gamma(X,\cM_X)$ exhibits $\Gamma(X,\cM_X)$ as a chart of $X$,
and the induced homomorphism

\[
\Gamma(X,\cM_X)/\Gamma(X,\cM_X^*)
\to
\Gamma(U,\cM_U)/\Gamma(U,\cO_U^*)
\]

is surjective for every open subscheme $U$ of $X$.

By <cit.>,
this condition is equivalent to the following condition:

There exists a chart $P$ of $X$ such that $P$ is finitely generated and the induced homomorphism

\[
P\to \Gamma(U,\cM_U)/\Gamma(U,\cO_U^*)
\]

is surjective for every open subscheme $U$ of $X$.

Observe that every open subscheme of an fs log affine scheme whose underlying scheme is affine is an fs log affine scheme.

Let $X$ be a noetherian fs log affine scheme with involution.
The real topological Hochschild homology of $X$ is

\[
\THR(X)
\THR(\Gamma(X,\cO_X),\Gamma(X,\cM_X)).
\]

Let $X$ be a noetherian fs log scheme.
Then there exists a strict étale covering $\{U_i\to X\}_{i\in I}$ such that each $U_i$ is an fs log affine scheme.

Immediate from <cit.>.

Let $G$ be a finite group.
For a point $x$ of a $G$-scheme $X$,
the scheme-theoretic stabilizer of $X$ at $x$ is

\[
\ker(
\{
g\in G:gx=x
\}\to \mathrm{Aut}(k(x))).
\]

A morphism of $G$-schemes $Y\to X$ is an isovariant étale cover if the induced homomorphism $G_y\to G_{f(x)}$ is an isomorphism for every point $y\in Y$
and $f$ is étale and surjective after forgetting the $G$-action.

Let $(A,P)$ be a log ring with involution.
Then the presheaf $\THR(-,P)$ on the opposite category of $A$-algebras is an isovariant étale hypersheaf.

Let $B\to C$ be an isovariant étale homomorphism of commutative $A$-algebras with involutions.
By <cit.>,
there are natural equivalences of $\Ztwo$-spectra

\[
\THR(B,P)
\wedge_{\EM B}
\EM C
\simeq
\THR(B,P)
\wedge_{\EM \iota (B^{\Ztwo})}
\EM \iota (C^{\Ztwo})
\simeq
\THR(C,P).
\]

Argue as in the proof of <cit.> to show the claim.

For a commutative monoid $P$,
let $P^*$ denote its submonoid of units,
and let $P^\gp$ denote the group completion of $P$.
We set $\ol{P}:=P/P^*$.
We say that $P$ is integral if the induced homomorphism $P\to P^\gp$ is injective.

Let $\theta\colon P\to Q$ be a homomorphism of integral monoids.
If $\ol{\theta}\colon \ol{P}\to \ol{Q}$ is an isomorphism, then the induced homomorphism of monoids

\begin{equation}
\label{strict.2.1}
\eta
\colon
\to
\end{equation}

is an isomorphism.

Since $\ol{\theta}$ is an isomorphism,
$\eta$ is surjective.
To show that $\eta$ is injective,
assume $\eta(p,v)=\eta(p',v')$ with $p,p'\in P$ and $v,v'\in Q^*$.
This implies $\theta(p)+v=\theta(p')+v'$.
Since $\ol{\theta}$ is an isomorphism,
there exists $u\in P^*$ satisfying $p=p'+u$.
Together with the assumption that $Q$ is integral,
we have $v'=v+u$.
Use <cit.> to see that $(p,v)=(p',v')$ in $P\oplus_{P^*} Q^*$.
Hence $\eta$ is an isomorphism.

Let $\theta\colon P\to Q$ be a homomorphism of integral monoids with involutions.
If $\ol{\theta}\colon \ol{P}\to \ol{Q}$ is surjective,
then the induced map

\[
\THR(\Sphere[Q],P)
\to
\THR(\Sphere[Q],Q)
\]

is an equivalence of $\Ztwo$-spectra.

Due to Propositions <ref> and <ref>,
it suffices to show that the induced map

\[
(\Sphere[Q] \wedge_{\Sphere[i_\sharp i^*Q]} \Sphere[Q])
\wedge_{(\Sphere[P] \wedge_{\Sphere[i_\sharp i^*P]} \Sphere[P])}
(\Sphere[P] \wedge_{\Sphere[(i_\sharp i^*P)^{ex}]} \Sphere[P])
\to
\Sphere[Q] \wedge_{\Sphere[(i_\sharp i^*Q)^{ex}]} \Sphere[Q]
\]

is an equivalence of $\Ztwo$-spectra.
By canceling some terms,
it suffices to show that the induced map

\begin{equation}
\label{strict.3.1}
\Sphere[i_\sharp i^*Q]\wedge_{\Sphere[i_\sharp i^*P]}\Sphere[(i_\sharp i^*P)^{ex}]
\to
\Sphere[(i_\sharp i^*Q)^{ex}]
\end{equation}

is an equivalence of $\Ztwo$-spectra.

Consider the localization $P_F$ with respect to the face $F:=\theta^{-1}(Q^*)$ of $P$.
By <cit.>,
$i^*P_F$ is a filtered colimit of free $i^*P$-sets.
This implies that $\Sphere[i_\sharp i^*P_F]$ is a filtered colimit of free $\Sphere[i_\sharp i^*P]$-modules.
It follows that the induced map

\[
\Sphere[i_\sharp i^*P_F]\wedge_{\Sphere[i_\sharp i^*P]}\Sphere[(i_\sharp i^*P)^{ex}]
\to
\Sphere[(i_\sharp i^*P_F)^{ex}]
\]

is an equivalence.
Replace $P\to Q$ with $P_F\to Q$ to reduce to the case when $\ol{\theta}$ is an isomorphism.

We have the induced commutative diagram

\begin{equation}
\label{strict.3.2}
\begin{tikzcd}
i^*P^*\oplus i^*P^*\ar[d]\ar[r]&
i^*P\oplus i^*P\ar[d]\ar[r,"\eta"]&
i^*P\oplus i^*P^\gp\ar[d]
\\
i^*Q^*\oplus i^*Q^*\ar[r]&
i^*Q\oplus i^*Q\ar[r]&
i^*Q\oplus i^*Q^\gp,
\end{tikzcd}
\end{equation}

where $\eta$ sends $(p,p')$ to $(p+p',p')$.
By Lemma <ref> and the description of the pushout in the category of integral monoids in <cit.>,
we see that the left and outer squares in (<ref>) are cocartesian.
Observe that $i^*P\oplus i^*P$ and $i^*P\oplus i^*P^\gp$ are free $i^*P^*\oplus i^*P^*$-sets.
Apply $\Sphere[-]$ to the left and outer squares in (<ref>) and use the pasting law to deduce that the induced map

\[
\Sphere[i^*Q \oplus i^*Q]
\wedge_{\Sphere[i^*P\oplus i^*P]}
\Sphere[i^*P\oplus i^*P^\gp]
\to
\Sphere[i^*Q\oplus i^*Q^\gp].
\]

is an equivalence of spectra,
(<ref>) becomes an equivalence after applying $i^*$.

Observe that the $\Ztwo$-fixed point monoids of $i_\sharp i^* P$ and $i_\sharp i^*Q$ are isomorphic to $i^*P$ and $i^*Q$.
Let $w$ denote the involutions on $P$ and $Q$,
and let $L$ and $M$ be the $\Ztwo$-fixed point monoids of $(i_\sharp i^*P)^{ex}$ and $(i_\sharp i^*Q)^{ex}$.
Observe that $M$ is a submonoid of $Q^\gp$ consisting of $y$ such that $y+w(y)\in Q$.
We have a similar description for $L$ too.
For such an element $y$,
there exist $x\in P^\gp$ and $u\in Q^*$ such that $\theta(x)+u=y$.
This implies $\theta(x+w(x))\in Q$.
Since $\theta$ is exact by <cit.>,
we have $x+w(x)\in P$.
It follows that the induced homomorphism

\[
L\oplus_{i^*P} {i^*Q}\to M
\]

is an isomorphism of commutative monoids by the description of the pushout in the category of integral monoids.
Hence we have the commutative diagram with cocartesian squares

\begin{equation}
\label{strict.3.3}
\begin{tikzcd}
\\
\end{tikzcd}
\end{equation}

Observe that $i^*P$ and $L$ are free $i^*P^*$-sets.
Apply $\Sphere[-]$ to the left and outer squares in (<ref>) and use the pasting law to deduce that the induced map

\[
\Sphere[i^*Q]\wedge_{\Sphere[i^*P]}\Sphere[L]
\to
\Sphere[M]
\]

is an equivalence,
(<ref>) becomes an equivalence after applying $\Phi^{\Ztwo}$.

Let $(A,P)$ be an integral log ring with involution.
Assume that $X:=\Spec(A,P)$ is a noetherian fs log affine scheme with involution.
Then the induced map

\[
\THR(A,P)
\to
\THR(X)
\]

is an equivalence.

If $U:=\Spec(B,P)$ is an open subscheme of $X$,
then the induced homomorphism $\Gamma(X,\cM_X)\to \Gamma(U,\cM_U)/\Gamma(U,\cO_U^*)$ is surjective since $X$ is an fs log affine scheme.
By Proposition <ref> and Lemma <ref>,
the induced map

\[
\THR(B,\Gamma(X,\cM_X))
\to
\THR(U)
\]

is an equivalence.
Together with Proposition <ref>,
we see that the presheaf $\THR$ on the category of open fs log affine subschemes of $X$ is an isovariant étale hypersheaf.
Hence showing that the induced map $\THR(A,P)\to \THR(X)$ an equivalence is isovariant étale local on $\ul{X}$.
Using <cit.>,
we reduce to the case when $P\to \Gamma(X,\cM_X)/\Gamma(X,\cO_X^*)$ is surjective.
To finish the proof,
use Proposition <ref> and Lemma <ref> again.

A morphism of fs log schemes with involutions $f\colon Y\to X$ is a strict isovariant étale cover if $f$ is strict and its underlying morphism of schemes with involutions $\ul{f}\colon \ul{Y}\to \ul{X}$ is an isovariant étale cover.
The strict isovariant étale topology is the topology generated by strict isovariant étale covers.
Let $siso\et$ be the shorthand for this topology.

The presheaf $\THR$ on the category of noetherian fs log affine schemes with involutions is a strict isovariant étale hypersheaf.

Let $X$ be a noetherian fs log affine scheme.
Choose any chart $P$ of $X$ such that $P$ is fine and saturated,
and use Proposition <ref> and Theorem <ref>.

Recall from Definition <ref> that the forgetful functor from the category of fs log schemes with involutions to the category of fs log schemes is denoted by $i^*$.
We have its left adjoint $i_\sharp$.

Let $X$ be an fs log affine scheme.
Then $i_\sharp X$ is an fs log affine scheme with involution.

We set $P:=\Gamma(X,\cM_X)$.
Consider the induced strict morphism $f\colon X\to \A_P$.
We also have the morphism $u\colon \Spec(\Z)\simeq \A_0 \to \A_P$ induced by the homomorphism $P\to 0$.
Then we have the strict morphism $g\colon X\amalg X\to \A_P\times \A_P$ given by $(f,u)$ (resp. $(u,f)$) in the first (resp. second) component.
It follows that $P\oplus P \simeq \Gamma(X\amalg X,\cM_{X\amalg X})$ is a chart of $X\amalg X$.

For all open subschemes $U$ and $V$ of $X$,
we have a natural isomorphism

\[
\Gamma(U\amalg V,\cM_{U\amalg V})/ \Gamma(U\amalg V,\cO_{U\amalg V}^*)
\simeq
\Gamma(U,\cM_U)/\Gamma(U,\cO_U^*)
\oplus
\Gamma(V,\cM_V)/\Gamma(V,\cO_V^*).
\]

Use this to show that $i_\sharp X$ satisfies the condition (i) in Definition <ref>.

A morphism of fs log schemes with involutions is a strict equivariant Nisnevich cover if it is strict and its underlying morphism of schemes with involutions is an equivariant Nisnevich cover in the sense of Voevodsky <cit.>.
See also <cit.>.

The strict equivariant Nisnevich topology is the topology generated by strict equivariant Nisnevich covers.
Let $seNis$ be the shorthand for this topology.
Observe that the strict equivariant Nisnevich topology is coarser than the strict isovariant étale topology.

Let $X$ be a noetherian separated fs log scheme with involution.
Then there exists a strict isovariant Nisnevich covering $\{U_i\to X\}_{i\in I}$ such that each $U_i$ is a noetherian fs log affine scheme with involution.

Let $w$ be the involution on $X$.
Choose an open neighborhood $U_x$ of $x$ in $X$ such that $U_x$ is an fs log affine scheme.
If $x=w(x)$,
then $V_x:=U_x\cap w(U_x)$ is an fs log affine scheme with involution.
If $x\neq w(x)$,
then $V_x:=U_x\amalg w(U_x)$ is an fs log affine scheme with involution by Proposition <ref>.
We finish the proof using <cit.>.

Let $\lSch$ (resp. $\lAff$) denote the category of noetherian separated fs log schemes (resp. noetherian fs log affine schemes).
Following Definition <ref>,
we obtain the category
$\lSch_{\Ztwo}$ (resp. $\lAff_{\Ztwo}$).

If $\cC$ is a category with a topology $t$ and $\cV$ is an $\infty$-category,
let $\Sh_t^\wedge(\cC,\cV)$ denote the $\infty$-category of $t$-hypercomplete sheaves with values in $\cV$ on $\cC$.

There is an equivalence of $\infty$-categories

\[
\Sh_{siso\et}^{\wedge}(\lAff_{\Ztwo},\Sp_{\Ztwo})
\simeq
\Sh_{siso\et}^{\wedge}(\lSch_{\Ztwo},\Sp_{\Ztwo}).
\]

By <cit.>,
it suffices to show that the direct image functor

\[
\Sh_{siso\et}(\lSch_{\Ztwo})
\to
\Sh_{siso\et}(\lAff_{\Ztwo})
\]

is an equivalence of categories.
This is a consequence of Proposition <ref> and <cit.>.

As a consequence,
we obtain the hypersheaf $\THR$ on $\lSch_{\Ztwo}$.
For a noetherian separated fs log scheme $X$,
the real topological Hochschild homology of $X$ is defined to be $\THR(X)$.

Let $(A,P)$ be a noetherian fs log ring with involution.
Then there is a natural equivalence of $\Ztwo$-spectra

\[
\THR(A,P)
\simeq
\THR(\Spec(A,P)).
\]

The functors $\THR(-,P)$ and $\THR(\Spec(-,P))$ are strict isovariant étale hypersheaves on the opposite category of noetherian $A$-algebras.
Together with Proposition <ref>,
we reduce to the case when $\Spec(A,P)$ is a noetherian fs log affine scheme with involution.
Theorem <ref> finishes the proof.

In the next results,
we explain how $\THR$ is related with $\THH$ under the functors $i^*$ and $i_*$.

Let $X$ be a noetherian separated fs log scheme with involution.
Then there is a natural equivalence of spectra

\[
\THH(i^*X)
\simeq
\]

By Proposition <ref>,
it suffices to construct a natural equivalence

\[
\THH(i^*(-))
\simeq
\]

in $\Sh^{\wedge}_{siso\et}(\lAff_{\Ztwo},\Sp_{\Ztwo})$.
Together with Proposition <ref> and Theorem <ref>,
we finish the proof.

Let $X$ be a noetherian separated fs log scheme with involution.
Then there is a natural equivalence of $\Ztwo$-spectra

\[
\THR(i_\sharp X)
\simeq
\]

As above,
we reduce to the case when $X$ is a noetherian fs log affine scheme.
In this case,
there is a natural isomorphism

\[
i_\sharp X
\simeq
\Spec(i_*\Gamma(X,\cO_X),\iota \Gamma(X,\cM_X)).
\]

Together with Proposition <ref> and Theorem <ref>,
we finish the proof.

Let $X$ be a noetherian separated fs log scheme with involution.
The orthogonal topological Hochschild homology of $X$ is defined to be the $\Ztwo$-fixed point spectrum

\[
\THO(X)
\THR(X)^{\Ztwo}.
\]

Let $X$ be a noetherian separated fs log scheme with involution.
Then there is a natural equivalence of spectra

\[
\THO(i_\sharp X)
\simeq
\THH(X).
\]

We have a natural equivalence of spectra $\THO(i_\sharp X)\simeq (i_*\THH(X))^{\Ztwo}$ by Proposition <ref>.
Together with <cit.>,
we finish the proof.

§ MOTIVIC REPRESENTABILITY

Throughout this section,
we fix a finite dimensional noetherian separated scheme $S$ (with the trivial log structure and involution).
The purpose of this section is to represent orthogonal topological Hochschild homology in $\logSH(S)$ and real topological Hochschild homology in a $\Ztwo$-equivariant analogue of $\logSH(S)$.

We begin with recalling the ingredients for defining $\logSH(S)$.
Let $\SmlSm/S$ denote the category of fs log schemes $Y$ of finite type over $S$ such that $Y\to S$ is log smooth and $\ul{Y}\to S$ is smooth.
A morphism in $\SmlSm/S$ is a strict Nisnevich cover if it is strict and its underlying morphism of schemes is a Nisnevich cover.
We have the $\infty$-category of strict Nisnevich sheaves of spectra $\Sh_{sNis}(\SmlSm/S,\Sp)$.

For a smooth scheme $X$ of finite type over $S$ with a strict normal crossing divisor $D$,
let $(X,D)$ denote the fs log scheme with the underlying scheme $X$ and the compactifying log structure associated with the open immersion $X-D\to X$.
By <cit.>,
every object of $\SmlSm/S$ arises as this form.
We often regard $\P^{n-1}$ as a closed complement of $\A^n$ in $\P^n$,
and we form $(\P^n,\P^{n-1})$.
We set $\square:=(\P^1,\infty)$.

Now, the $\infty$-category of logarithmic motivic $S^1$-spectra is defined to be the localization

\[
\logSH_{S^1}(S)
\]

where $(\P^\bullet,\P^{\bullet-1})^{-1}$ denotes the class of projections $(\P^n,\P^{n-1})\times X\to X$ for all $X\in \SmlSm/S$ and integers $n\geq 1$.
This is one of the various models of $\logSH_{S^1}(S)$ in <cit.>.
The $\infty$-category of logarithmic motivic spectra is defined to be the $\infty$-category of $\P^1$-spectra in $\logSH_{S^1}(S)$:

\[
\logSH(S)
\Sp_{\P^1}(\logSH_{S^1}(S)).
\]

In this section,
we often compute $\THR(X\times Y)$ in terms of $\THR(X)$ for noetherian separated fs log schemes $X$ and $Y$ such that there is a Zariski covering $\{\Spec(\Z[M_i],P_i)\to Y\}_{i\in I}$ for some homomorphisms of monoids $P_i\to M_i$.
In view of Proposition <ref>,
the base $X$ is irrelevant for the computation.
We will often argue as if everything takes place on $\Sphere$ instead of $X$ for notational convenience.

Recall that $S^\sigma$ is $S^1$ with the involution given by $e^{i\theta}\in S^1 \mapsto e^{-i\theta}$.
We have the functor

\[
\Sigma^\sigma
\Sigma^\infty S^\sigma \wedge (-)
\colon
\Sp_{\Ztwo}\to \Sp_{\Ztwo}.
\]

Let us recall the computation of $\THR$ for the projective spaces with the trivial involutions as follows.

Let $X$ be a noetherian separated fs log scheme.
Then there exists a natural equivalence of $\Ztwo$-spectra

\[
\THR(X\times \P^n)
\simeq
\left\{
\begin{array}{ll}
\THR(X) \oplus \bigoplus_{j=1}^{\lfloor n/2 \rfloor} i_*\THH(X) & \text{if $n$ is even},
\\
\THR(X) \oplus \bigoplus_{j=1}^{\lfloor n/2 \rfloor} i_*\THH(X) \oplus \Sigma^{n(\sigma-1)}\THR(X) & \text{if $n$ is odd}.
\end{array}
\right.
\]

Following Remark <ref>,
we can argue as if everything takes place on $\Sphere$ instead of $X$.
Then the claim is due to <cit.>.

Our next goal is to show that $\THO$ is representable in $\logSH_{S^1}(S)$.
For this,
we need the notion of cubes <cit.>,
which we recall as follows.

For a set $I$,
let $\bP(I)$ denote the set of subsets of $I$.
We impose the partially ordered set structure on $\bP(I)$ with respect to the inclusion,
and we regard $\bP(I)$ as the associated category.
For an $\infty$-category $\cC$,
an $I$-cube in $\cC$ is a functor

\[
\colon
\bP(I)
\to
\cC.
\]

If the cardinal of $I$ is an integer $n$,
then an $I$-cube is called an $n$-cube.
The total cofiber of $Q$ is

\[
\tcofib(Q)
\cofib(\colim(Q|_{\bP(I)-\{I\}}) \to Q(I))
\]

whenever the colimit and cofiber exist.
The total fiber of $Q$ is

\[
\tfib(Q)
\fib(Q(\emptyset)\to \lim(Q|_{\bP(I)-\{\emptyset\}}))
\]

whenever the limit and fiber exist.

Let $X$ be a noetherian separated fs log scheme with involution.
Then the induced map

\[
\THR(X)
\to
\THR(X\times (\P^n,\P^{n-1}))
\]

is an equivalence of $\Ztwo$-spectra,
where we impose the trivial involution on $(\P^n,\P^{n-1})$.

Following Remark <ref>,
we will argue for $\Sphere$ instead of $X$.
Let $\ul{U_0},\ldots,\ul{U_n}$ be the standard cover of $\P_{\Sphere}^n$,
and we set $U_i:=(\P_{\Sphere}^n,\P_{\Sphere}^{n-1})\times_{\P_{\Sphere}^n} \ul{U_i}$.
For every nonempty subset $I\subset [n]$,
we set $U(I):=U_I:=\bigcap_{i\in I} U_i$.
We also set $U_\emptyset:=\Sphere$.
We regard $U$ as an $(n+1)$-cube.
We need to show $\tcofib(\THR(U))\simeq 0$.

For notational convenience,
we consider the commutative monoids

\begin{gather*}
P_j:=\{(x_1,\ldots,x_n)\in \Z^n : x_j\geq 0\}
\text{ for $j=1,\ldots,n$},
\\
P_0:=\{(x_1,\ldots,x_n)\in \Z^n : x_1+\ldots+x_n \leq 0\},
\\
F_0:=\{(x_1,\ldots,x_n)\in \Z^n : x_1+\ldots+x_n = 0\}.
\end{gather*}

For every subset $J\subset [n]$,
we set $P_J:=\bigcap_{j\in J}P_j$.
By convention,
we have
There is a canonical equivalence

\[
\THR(U_I)
\simeq
\left\{
\begin{array}{ll}
\THR(\Sphere[P_{[n]-I}])& \text{if $0\in I$},
\\
\THR(\Sphere[P_{[n]-I}],P_{[n]-I}\cap F_0)
&\text{if $0\notin I$}.
\end{array}
\right.
\]

Each $P_J$ is isomorphic to products of finite copies of $\Z$ and $\N$.
The computations of $\Bdi \N$ and $\Bdi \Z$ in <cit.> and the fact that $\Bdi$ preserves finite products <cit.> yield a natural equivalence of $\Ztwo$-spectra

\[
\THR(U_I)
\simeq
\Sphere[\Phi(I)]
\]

\begin{gather*}
\Phi(I)
\coprod_{(x_1,\ldots,x_n)\in \Z^n} \Phi(I;(x_1,\ldots,x_n)),
\\
\Phi(I;(x_1,\ldots,x_n))
\left\{
\begin{array}{ll}
\Phi_1(I;x_1) \times \cdots \times \Phi_n(I;x_n)
\text{if $(x_1,\ldots,x_n)\in P_{[n]-I}$},
\\
\emptyset
\text{otherwise},
\end{array}
\right.
\\
\Phi_i(I;x_i)
\left\{
\begin{array}{ll}
* & \text{if $x_i=0$ and $i\in I$},
\\
S^\sigma & \text{otherwise}.
\end{array}
\right.
\end{gather*}

Since there is a natural decomposition

\[
\tcofib(\Phi)
\simeq
\coprod_{x\in \Z^n}\tcofib(\Phi(-;x)),
\]

it suffices to show $\tcofib(\Phi(-;x))\simeq 0$ for every $x\in \Z^n$.

If $x\neq (0,\ldots,0)$,
then there exists $i\in [n]$ such that $x\notin P_i$.
The induced map $\Phi(I;x) \to
\Phi(I\cup \{i\};x)$
is a $\Ztwo$-homeomorphism whenever $i\notin I$.
Together with the categorical result <cit.>,
we have $\tcofib(\Phi(-;x))\simeq 0$.

If $x=(0,\ldots,0)$,
then the induced map $\Phi(I;x)\to \Phi(I\cup \{0\};x)$ is a $\Ztwo$-homeomorphism whenever $0\notin I$.
We have $\tcofib(\Phi(-;x))\simeq 0$ similarly.

The $\THH$ part of the following result is proved in <cit.>.

Let $S$ be a noetherian separated scheme.
Then the sheaves

\[
\THH,
\THO
\in
\Sh_{sNis}(\SmlSm/S,\Sp)
\]

are in $\logSH_{S^1}(S)$.

Immediate from Theorem <ref>.

Since $\THR$ is not a sheaf of spectra but a sheaf of $\Ztwo$-spectra,
we cannot directly apply the results in [7] and [6] to $\THR$.
Instead, we will often use the fact that the pair $(i^*,(-)^{\Ztwo})$ is conservative to reduce to the case of $\THH$ and $\THO$.

For any vector bundle $\cE\to X$ with $X\in \SmlSm/S$,
the real topological Hochschild homology of the Thom space $\Th(\cE)$ is defined to be

\begin{equation}
\label{mot.15.1}
\THR(\Th(\cE))
\fib(\THR(\cE)
\to
\THR(\Blow_{Z}\cE,E)),
\end{equation}

where $Z$ is the zero section of $\cE$,
and $E$ is the exceptional divisor.

With the above notation,
there is a natural equivalence of $\Ztwo$-spectra

\[
\THR(\Th(\cE))
\simeq
\fib(
\THR(\P(\cE\oplus \cO))
\to
\THR(\P(\cE))).
\]

Let $Y$ and $Y'$ be the blow-up of $\cE$ and $\P(\cE\oplus \cO)$ along the zero section,
and let $E$ and $E'$ be the exceptional divisors.
By the proof of <cit.>,
there exists a commutative diagram

\[
\begin{tikzcd}
\P(\cE)
\\
\cE\ar[r]&
\P(\cE\oplus \cO)
\end{tikzcd}
\]

such that the induced maps

\begin{gather*}
\fib(\cF(\P(\cE\oplus \cO))\to \cF(Y',E'))
\to
\fib(\cF(\cE)\to \cF(Y,E)),
\\
\cF(Y',E')
\to
\cF(\P(\cE))
\end{gather*}

are equivalences for $\cF\in \logSH_{S^1}(S)$.
In particular,
these are equivalences for $\THH$ and $\THO$ by Theorem <ref>.
From this,
we obtain the desired equivalence.

Let $\cE_1\to X_1$ and $\cE_2\to X_2$ be vector bundles,
where $X_1,X_2\in \SmlSm/S$.
We regard $T:=\cE_1\times_S \cE_2$ as a vector bundle over $X_1\times_S X_2$.
We set

\[
T_1:=(\Blow_{Z_1} \cE_1,E_1)\times_S \cE_2,
\;
T_2:=\cE_1\times_S (\Blow_{Z_2}\cE_2,E_2),
\;
T_{12}:=T_1\times_S T_2,
\]

where $Z_1$ and $Z_2$ are the zero sections of $\cE_1$ and $\cE_2$.
Then there is a natural equivalence of $\Ztwo$-spectra

\[
\THR(\Th(T))
\simeq
\tfib(\THR(C_3)),
\]

where $C_3$ is the cartesian square

\[
\begin{tikzcd}
\\
\end{tikzcd}
\]

Consider the fs log schemes $T_I\in \SmlSm/S$ for every subset $I$ of $\{1,2,4\}$ in <cit.>,
and consider the induced squares

\[
\begin{tikzcd}
\\
\end{tikzcd}
\;\;\;
\begin{tikzcd}
\\
\end{tikzcd}
\]

We have the induced maps of squares $C_1\leftarrow C_2\to C_3$.
By Theorem <ref> and the proof of <cit.>,
the induced maps

\[
\tfib(\cF(C_1))\to \tfib(\cF(C_2))\leftarrow \tfib(\cF(C_3))
\]

are equivalences for $\cF:=\THH,\THO$.
This implies that we have a natural equivalence

\[
\tfib(\THR(C_1))\simeq \tfib(\THR(C_3)),
\]

which yields the desired equivalence.

For $X\in \SmlSm/S$ and integer $n\geq 0$,
there is a natural equivalence

\[
\THR(\Th(X\times \A^n))
\simeq
\Sigma^{n(\sigma-1)}\THR(X),
\]

where we regard $X\times \A^n$ as the rank $n$ trivial bundle over $X$.

We proceed by induction on $n$.
The claim is trivial if $n=0$.
Assume $n>0$.
We set $X_1:=X$,
$\cE_1:=X\times \A^{n-1}$,
and $\cE_2:=S\times \A^1$.
The square $C_3$ in Lemma <ref> becomes

\[
\begin{tikzcd}
(\Blow_X(X\times \A^{n-1}),E_{n-1})\times (\A^1,0)\ar[r]\ar[d]&
(\Blow_X(X\times \A^{n-1}),E_{n-1})\times \A^1\ar[d]
\\
X\times \A^{n-1}\times (\A^1,0)\ar[r]&
X\times \A^{n-1}\times \A^1,
\end{tikzcd}
\]

where $E_{n-1}$ is the exceptional divisor.
By <cit.>, Proposition <ref>, and Lemma <ref>,
we have a natural equivalence

\[
\THR(\Th(X\times \A^n))
\simeq
\THR(\Th(X\times \A^{n-1}))
\wedge
\fib(\THR(\Sphere[\N])\to \THR(\Sphere[\N],\N)).
\]

Combine this with Example <ref> and use the induction hypothesis to conclude.

Next, we concern about the Gysin cofiber sequence in logarithmic motivic homotopy theory.
For this,
we recall the deformation to the normal cone construction in the logarithmic setting as follows.
Assume that $X\in \SmlSm/S$ has the form $(\ul{X},Z_1+\cdots+Z_n)$,
and let $Z$ be a strict closed subscheme of $X$ such that $\ul{Z}$ is strict normal crossing with $Z_1+\cdots+Z_n$ in the sense of <cit.>.
The blow-up of $X$ along $Z$ is

\[
\Blow_Z X:=(\Blow_{\ul{Z}}\ul{X},\widetilde{Z}_1+\cdots+\widetilde{Z}_n),
\]

where $\widetilde{Z}_i$ is the strict transform of $Z_i$ for $1\leq i\leq n$.
The normal bundle of $Z$ in $X$ is defined to be

\[
\Normal_Z X
\Normal_{\ul{Z}} \ul{X} \times_{\ul{X}} X,
\]

where $\Normal_{\ul{Z}} \ul{X}$ denotes the normal bundle of $\ul{Z}$ in $\ul{X}$.
The deformation space associated with $Z\to X$ is defined to be

\[
\Deform_Z X
\Blow_{Z\times \{0\}}(X\times \square)
\Blow_{Z\times \{0\}}(X\times \{0\}).
\]

With the above notation,
there exists a natural cofiber sequence of $\Ztwo$-spectra

\[
\THR(\Th(\Normal_Z X))
\to
\THR(X)
\to
\THR(\Blow_Z X,E),
\]

where $E$ is the exceptional divisor.

We have the induced maps

\begin{align*}
\fib(\THR(X)\to \THR(\Blow_Z X,E))
\leftarrow &
\fib(
\THR(\Deform_Z X)
\to
\THR(\Blow_{Z\times \square}(\Deform_Z X),E^D)
\\
\to &
\fib(
\THR(\Normal_Z X)
\to
\THR(\Blow_Z (\Normal_Z X),E^N)
\end{align*}

where $E$, $E^D$, and $E^N$ are the exceptional divisors.
It suffices to show that the corresponding maps for $\THH$ and $\THO$ are equivalences of spectra since the pair of functors $(i^*,(-)^{\Ztwo})$ is conservative.
This is a consequence of <cit.> (see also <cit.>) and Theorem <ref>.

Let $X$ be a scheme,
and let $\cE\to X$ be a rank $n$ vector bundle.
There exists a Thom equivalence

\[
\THH(\Th(\cE))
\simeq
\THH(X).
\]

see <cit.>.
This is a consequence of the fact that $\bTHH$ is an orientable logarithmic motivic spectrum <cit.>.
Unlike this, in general,
we do not have an equivalence

\[
\THR(\Th(\cE))
\simeq
\Sigma^{n(\sigma-1)}\THR(X),
\]

i.e., $\THR(\Th(\cE))\not \simeq \THR(\Th(X\times \A^n))$ by Proposition <ref>.

For example,
if $X:=\P_{\Sphere}^n$ and $Z:=\P_{\Sphere}^{n-1}$ with even $n\geq 2$,
then Proposition <ref> and Theorems <ref> and <ref> yield

\[
\THR(\Th(\Normal_Z X))
\simeq
\bigoplus_{j=1}^{\lfloor n/2 \rfloor}
\]

This is not equivalent to

\[
\Sigma^{\sigma-1}\THR(Z)
\simeq
\Sigma^{\sigma-1}\bigoplus_{j=1}^{\lfloor (n-1)/2 \rfloor}
\oplus
\Sigma^{n(\sigma-1)}
\]

that is obtained by Proposition <ref>.
We have a similar conclusion if $n$ is odd too.

We also obtain the following descent result with respect to blow-ups along smooth centers (with the trivial involutions).

Let $Z\to X$ be a closed immersion of smooth schemes over $S$.
Then the induced square of $\Ztwo$-spectra

\[
\begin{tikzcd}
\THR(X)\ar[d]\ar[r]&
\THR(Z)\ar[d]
\\
\THR(Z\times_X \Blow_Z X)\ar[r]&
\THR(\Blow_Z X)
\end{tikzcd}
\]

is cartesian.

The corresponding squares of spectra for $\THH$ and $\THO$ are cartesian by <cit.> (see also <cit.>) and Theorem <ref>.
This implies the claim since the pair of functors $(i^*,(-)^{\Ztwo})$ is conservative.

Let $\P^\sigma$ be the scheme $\P^1$ with the involution given by $[x:y]\mapsto [y:x]$.
Let $1$ be the base points of $\P^1$ and $\P^\sigma$,
and then we form the endofunctors
$\Omega_{\P^1}$, $\Omega_{\P^\sigma}$, and $\Omega_{\P^1\wedge \P^\sigma}\simeq \Omega_{\P^\sigma}\Omega_{\P^1}$ on $\PSh((\SmlSm/S)_{\Ztwo},\Sp)$ and $\PSh((\SmlSm/S)_{\Ztwo},\Sp_{\Ztwo})$.

Let $X$ be a noetherian separated fs log scheme.
Then there is a natural equivalence of $\Ztwo$-spectra

\[
\THR(X)
\simeq
\Omega_{\P^1\wedge \P^\sigma}\THR(X).
\]

Following <ref>,
we reduce to <cit.>.

The $\infty$-category of prelogarithmic motivic $\Ztwo$-spectra over $S$ is defined to be

\[
\prelogSH_{\Ztwo}(S)
\Sp_{\P^1\wedge \P^\sigma}
\]

Let $\A^w$ denote the affine line $\A^1$ with the involution $w$ given by $x\mapsto -x$.
We have the $\A^1$-homotopy

\begin{equation}
\label{mot.9.1}
\A^1\times \A^w
\to
\A^w
\end{equation}

given by $(x,y)\mapsto xy$.
Due to this,
the projections
$X\times \A^w \to X$ are automatically inverted if we invert the projections $X\times \A^1\to X$ for all $X\in (\Sm/S)_{\Ztwo}$.

this phenomenon is not generalized to the logarithmic setting since there exists no morphism of fs log schemes

\[
\square \times \square^w \to \square^w
\]

extending (<ref>),
where $\square^w$ denotes the fs log scheme $\square$ with the involution $[x,y]\mapsto [-x,y]$.
This indicates that inverting the projections $X\times \square^w\to X$ for all $X\in (\SmlSm/S)_{\Ztwo}$ is at least required for a better behaved $\infty$-category of logarithmic motivic $\Ztwo$-spectra.

The prelogarithmic motivic $\Ztwo$-fixed point functor is defined to be the functor

\[
\colon
\prelogSH_{\Ztwo}(S)
\to
\logSH(S)
\]

sending a $\P^1\wedge \P^\sigma$-spectrum $(\cF_0,\cF_1,\cF_2,\ldots)$ to the $\P^1$-spectrum

\[
\Omega_{\P^\sigma}\cF_1,
\Omega_{\P^{2\sigma}} \cF_2,\ldots
\]

restricting to the objects of $\SmlSm/S$ with the trivial involutions,
where the bonding map $\Omega_{\P^{n\sigma}}\cF_n\to \Omega_{\P^1\wedge\P^{(n+1)\sigma}}\cF_{n+1}$ is induced by the bounding map $\cF_n\to \Omega_{\P^1\wedge \P^\sigma}\cF_{n+1}$.

The motivic real topological Hochschild $\Ztwo$-spectrum is defined to be the $\P^1\wedge \P^\sigma$-spectrum

\[
\bTHR
\in
\prelogSH_{\Ztwo}(S)
\]

whose bonding maps are given by Proposition <ref>.

The motivic orthogonal topological Hochschild spectrum is defined to be

\[
\bTHO
\bTHR^{\Ztwo}
\in
\logSH(S).
\]

For a strict Nisnevich sheaf of spectra $\cF$ on $(\SmlSm/S)_{\Ztwo}$,
let $i^*\cF$ be the strict Nisnevich sheaf of spectra on $\SmlSm/S$ given by

\[
\cF(i_\sharp X).
\]

Observe that for every integer $n\geq 1$,
$i^*\cF$ is $(\P^n,\P^{n-1})$-invariant if $\cF$ is $(\P^n,\P^{n-1})$-invariant.
We have the “forgetful” functor

\[
\colon
\prelogSH_{\Ztwo}(S)
\to
\logSH(S)
\]

sending a $\P^1\wedge \P^\sigma$-spectrum $(\cF_0,\cF_1,\cF_2,\ldots)$ to the $\P^1$-spectrum

\[
(i^*\cF_0,\Omega_{\P^\sigma}i^*\cF_1,\Omega_{\P^{2\sigma}} i^*\cF_2,\ldots)
\]

with the induced bonding maps.

There is an equivalence

\[
\simeq
\bTHH.
\]

A consequence of Corollary <ref>.

[1]
M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et
cohomologie étale des schémas, vol. 269, 270, 305 of Lecture Notes in
Mathematics, Springer-Verlag, 1972–1973.
Séminaire de Géométrie Algébrique du Bois–Marie 1963–64.

[2]
T. Bachmann and M. Hoyois, Norms in motivic homotopy theory,
Astérisque, 425 (2021), pp. ix+207.

[3]
B. Bhatt, M. Morrow, and P. Scholze, Topological Hochschild
homology and integral $p$-adic Hodge theory, Publ. Math. Inst. Hautes
Études Sci., 129 (2019), pp. 199–310.

[4]
B. Bhatt and P. Scholze, Prisms and prismatic cohomology, Ann.
Math. (2), 196 (2022), pp. 1135–1275.

[5]
F. Binda, T. Lundemo, D. Park, and P. A. Østvær, A
Hochschild-Kostant-Rosenberg theorem and residue sequences for
logarithmic Hochschild homology.

[6]
F. Binda, D. Park, and P. A. Østvær, Logarithmic motivic
homotopy theory.

[7]
height 2pt depth -1.6pt width 23pt, Triangulated
categories of logarithmic motives over a field, Astérisque, 433 (2022).

[8]
M. Bökstedt, W. C. Hsiang, and I. Madsen, The cyclotomic trace
and algebraic $K$-theory of spaces, Invent. Math., 111 (1993),
pp. 465–539.

[9]
P. Deligne, Voevodsky's lectures on motivic cohomology 2000/2001,
in Algebraic topology. The Abel symposium 2007. Proceedings of the fourth
Abel symposium, Oslo, Norway, August 5–10, 2007, Berlin: Springer, 2009,
pp. 355–409.

[10]
E. Dotto, K. Moi, I. Patchkoria, and S. P. Reeh, Real topological
Hochschild homology, J. Eur. Math. Soc. (JEMS), 23 (2021), pp. 63–152.

[11]
B. I. Dundas, T. G. Goodwillie, and R. McCarthy, The local structure
of algebraic K-theory, vol. 18 of Algebra and Applications,
Springer-Verlag London, Ltd., London, 2013.

[12]
Z. Fiedorowicz and J.-L. Loday, Crossed simplicial groups and their
associated homology, Trans. Amer. Math. Soc., 326 (1991), pp. 57–87.

[13]
T. Geisser and L. Hesselholt, Topological cyclic homology of
schemes, in Algebraic $K$-theory (Seattle, WA, 1997), vol. 67 of Proc.
Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1999, pp. 41–87.

[14]
J. Heller, A. Krishna, and P. A. Østvær, Motivic homotopy
theory of group scheme actions, J. Topol., 8 (2015), pp. 1202–1236.

[15]
L. Hesselholt and I. Madsen, Real algebraic ${K}$-theory.

[16]
height 2pt depth -1.6pt width 23pt, On the $K$-theory
of local fields, Ann. of Math. (2), 158 (2003), pp. 1–113.

[17]
M. A. Hill, M. J. Hopkins, and D. C. Ravenel, On the nonexistence of
elements of Kervaire invariant one, Ann. of Math. (2), 184 (2016),
pp. 1–262.

[18]
J. Hornbostel and D. Park, Real topological Hochschild homology of
To appear in J. Inst. Math. Jussieu.

[19]
P. Hu, I. Kriz, and K. Ormsby, The homotopy limit problem for
Hermitian K-theory, equivariant motivic homotopy theory and motivic
Real cobordism, Adv. Math., 228 (2011), pp. 434–480.

[20]
A. Høgenhaven, Real topological cyclic homology of spherical group

[21]
J. Lurie, Higher algebra.
https://www.math.ias.edu/$\sim$lurie/, 2017.

[22]
F. Morel and V. Voevodsky, ${\bf A}^1$-homotopy theory of
schemes, Inst. Hautes Études Sci. Publ. Math.,  (1999), pp. 45–143

[23]
A. Ogus, Lectures on logarithmic algebraic geometry, vol. 178 of
Cambridge Studies in Advanced Mathematics, Cambridge University Press,
Cambridge, 2018.

[24]
J. D. Quigley and J. Shah, On the equivalence of two theories of
real cyclotomic spectra.

[25]
J. Rognes, Topological logarithmic structures, in New topological
contexts for Galois theory and algebraic geometry (BIRS 2008), vol. 16 of
Geom. Topol. Monogr., Geom. Topol. Publ., Coventry, 2009, pp. 401–544.

[26]
J. Rognes, S. Sagave, and C. Schlichtkrull, Localization sequences
for logarithmic topological Hochschild homology, Math. Ann., 363 (2015),
pp. 1349–1398.

[27]
G. Segal, Configuration-spaces and iterated loop-spaces, Invent.
Math., 21 (1973), pp. 213–221.

[28]
T. Tsuji, \(p\)-adic étale cohomology and crystalline
cohomology in the semi-stable reduction case, Invent. Math., 137 (1999),
pp. 233–411.

[29]
F. Waldhausen, Algebraic K-theory of topological spaces. II.
Algebraic topology, Proc. Symp., Aarhus 1978, Lect. Notes Math. 763,
356-394 (1979)., 1979.