# Real topological Hochschild homology of schemes

Jens Hornbostel  Bergische Universität Wuppertal, Fakultät für Mathematik und
Naturwissenschaften
Gaußstrasse 20, 42119 Wuppertal, Germany<EMAIL_ADDRESS>and
Doosung Park  Bergische Universität Wuppertal, Fakultät für Mathematik und
Naturwissenschaften
Gaußstrasse 20, 42119 Wuppertal, Germany<EMAIL_ADDRESS>
###### Abstract.

We prove that real topological Hochschild homology $\mathrm{THR}$ for schemes
with involution satisfies base change and descent for the
${\mathbb{Z}/2}$-isovariant étale topology. As an application, we provide
computations for the projective line (with and without involution) and the
higher dimensional projective spaces.

###### Key words and phrases:

real topological Hochschild homology, equivariant homotopy theory, isovariant
étale descent

###### 1991 Mathematics Subject Classification:

Primary 19D55; Secondary 11E70, 16E40, 55P91

##  1\. Introduction

Hochschild and cyclic homology and their refinements $\mathrm{THH}$ and
$\mathrm{TC}$ have been extensively studied for many decades, both for their
own sake and for their deep connections with algebraic $K$-theory via traces,
see e.g. [10]. Standard textbook references include [35] and [17]. More
recently, Bhatt-Morrow-Scholze [6] have introduced a filtration on
$\mathrm{THH}$ and $\mathrm{TC}$ that is strongly related to integral $p$-adic
Hodge theory.

Both topological and algebraic $K$-theory have real or hermitian refinements.
This is also true for $\mathrm{THH}$ and $\mathrm{TC}$, but the serious study
of their real variants $\mathrm{THR}$ and $\mathrm{TCR}$ has just started, see
e.g. [15], [14], and [43]. These theories apply to very different branches in
algebra and geometry: commutative rings, group rings, schemes, ring spectra, …
all with or without a non-trivial involution. Concerning the
${\mathbb{Z}/2}$-equivariant cyclotomic trace, we understand that there is
work in progress by Harpaz, Nikolaus, and Shah studying this map in the very
general setting of stable Poincaré $\infty$-categories. (When studying traces,
beware of the difference between rings, which is what most algebraic
$K$-theorists and algebraic geometries look at, and ring spectra, which are
the input of $\mathrm{THH}$ and $\mathrm{TC}$.)

The current article contributes to a better understanding of $\mathrm{THR}$ in
algebraic geometry, although some of our results also apply to other settings.
Our first main result is the base change result in Theorem 3.2.3 for the
isovariant étale topology. This then is one of the main ingredients in the
proof of the following isovariant étale descent theorem for $\mathrm{THR}$.

######  Theorem 3.4.3.

The presheaf

$\mathrm{THR}\in\mathcal{P}\mathrm{sh}(\mathbf{Aff}_{{\mathbb{Z}/2}},\mathrm{Sp}_{{\mathbb{Z}/2}})$

satisfies isovariant étale descent, where $\mathbf{Aff}_{{\mathbb{Z}/2}}$
denotes the category of affine schemes with involutions, and
$\mathrm{Sp}_{{\mathbb{Z}/2}}$ denotes the $\infty$-category of
${\mathbb{Z}/2}$-spectra.

In the non-equivariant case, similar results have been established for
$\mathrm{THH}$ and $\mathrm{TC}$ by Weibel-Geller [54] and Geisser-Hesselholt
[19].

This descent theorem implies in particular that $\mathrm{THR}$ satisfies the
equivariant Zariski-Mayer-Vietoris property for affine schemes with
involutions. This can be used to extend $\mathrm{THR}$ to non-affine schemes
with involutions, see Definition 3.4.6, in a way that is compatible with
existing definitions of $\mathrm{THH}$, see Proposition 3.4.7.

Using this Zariski-Mayer-Vietoris theorem and various explicit computations of
$\mathrm{THR}$ of products of monoid rings and maps between them, we are able
to compute $\mathrm{THR}(X)$ for $X=\mathbb{P}^{1},\mathbb{P}^{\sigma}$ and
more generally $\mathbb{P}^{n}$ with trivial involution, see Theorems 5.1.2,
5.1.4 and 5.2.6. Here $\mathbb{P}^{\sigma}$ denotes the projective line with
the involution switching the homogeneous coordinates. We recall that Blumberg
and Mandell [9] compute $\mathrm{THH}(\mathbb{P}^{n})$. For $\mathbb{P}^{n}$
with the trivial involution, we obtain the following:

######  Theorem 5.2.6.

For any separated scheme with involution $X$ and integer $n\geq 0$, there is
an equivalence of ${\mathbb{Z}/2}$-spectra

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

For a ${\mathbb{Z}/2}$-spectra $E$, $i_{*}E:=E\oplus E$ with the obvious
involution, see (A.15). We refer to Remarks 2.0.1, 5.1.1, and 5.2.7 for a
comparison with hermitian and real algebraic $K$-theory.

Recall that unlike algebraic and hermitian $K$-theory, $\mathrm{THH}$ and
$\mathrm{THR}$ are not $\mathbb{A}^{1}$-invariant even on regular schemes. On
the other hand, $\mathrm{THH}$ and $\mathrm{TC}$ do extend to log schemes, and
using descent, trivial $\mathbb{P}^{1}$-bundle formula, and computations on
the log schemes $(\mathbb{P}^{n},\mathbb{P}^{n-1})$ can be shown to be
representable in the log-variant of the $\mathbb{P}^{1}$-stable motivic Morel-
Voevodsky homotopy category. This is done in the very recent joint work [7] of
the second author with Federico Binda and Paul Arne Østvær. We refer to [8]
for an overview of their work. The results of this article are expected to
provide most of the necessary ingredients for showing stable representability
of $\mathrm{THR}$ or at least its fixed points $\mathrm{THO}$ in the
corresponding equivariant log-homotopy categories. Furthermore, the work of
Quigley-Shah [43] allows us to extend many results of this article including
Theorems 3.4.3 and 5.2.6 to $\mathrm{TCR}$. The second author hopes to carry
out motivic representability of $\mathrm{THR}$ and $\mathrm{TCR}$ in
forthcoming work.

We conclude with a short overview of the different sections. Section 2
discusses some generalities about commutative ring spectra with involutions
and their $\mathrm{THR}$. This includes the study of several functors between
${\mathbb{Z}/2}$-equivariant and non-equivariant categories, including the
norm and equivariant notion of flatness. The necessary background on
$G$-stable equivariant homotopy theory for finite groups $G$, both
${\infty}$\- and model categorical, is provided in Appendix A. Section 3
reviews and extends several definitions for Grothendieck topologies on schemes
with involutions, notably Thomason’s isovariant étale topology. Base change
and descent for $\mathrm{THR}$ are established in subsections 3.2 and 3.4.
Section 4 recollects and extends material on real and dihedral nerves, which
is crucial when computing $\mathrm{THR}$ of spherical monoid rings and of
monoid rings over Eilenberg MacLane spectra, and where the monoids may have
involutions. These monoid ring computations together with isovariant Zariski
descent are then used in the computations for projective spaces in section 5.

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

##  2\. Real topological Hochschild homology of rings with involutions

######  Remark 2.0.1.

$\mathrm{THR}$ is to real algebraic $K$-theory $\mathbf{KR}$ what
$\mathrm{THH}$ is to algebraic $K$-theory. Hence we recall some recent results
on real algebraic $K$-theory $\mathbf{KR}$. This is a
${\mathbb{Z}/2}$-equivariant motivic spectrum constructed in [29] and [11]. We
recover hermitian $K$-theory when restricting $\mathbf{KR}$ to schemes with
trivial ${\mathbb{Z}/2}$-action. (This is one incarnation of the philosophy
“the fixed points of $\mathbf{KR}$ are hermitian $K$-theory” in the world of
presheaves on ${\mathbb{Z}/2}$-schemes). Forgetting the action, we recover
Voevodsky’s algebraic $K$-theory spectrum $KGL$. As observed in [55, section
7.3], see also [11], Schlichting’s techniques generalize to show that
$\mathbf{KR}$ is representable in $\mathcal{SH}^{{\mathbb{Z}/2}}(k)$. In
particular, $\mathbf{KR}$ satisfies equivariant Nisnevich descent. Here and
below, following [11] and [29], we consider $\mathbf{KR}$ as a motivic
spectrum with respect to the circle
$T^{\rho}\simeq\mathbb{P}^{1}\wedge\mathbb{P}^{\sigma}$, where $\rho$ is the
regular representation of ${\mathbb{Z}/2}$, and $\mathbb{P}^{\sigma}$ denotes
$\mathbb{P}^{1}$ with involution switching the homogeneous coordinates. It
might be more consistent with the notations for $\mathrm{THH}$ to denote the
(motivic) hermitian $K$-theory spectrum on schemes with involution by
$\mathbf{KO}$, and to reserve the notation $\mathbf{KR}$ for a (motivic)
spectrum with involution whose fixed points are $\mathbf{KO}$. For further
results on $\mathbf{KR}$ and projective spaces we refer to Remarks 5.1.1 and
5.2.7 below.

Throughout this section, we fix morphisms of finite groupoids

(2.1)
$\mathrm{pt}\xrightarrow{i}\mathrm{B}(\mathbb{Z}/2)\xrightarrow{p}\mathrm{pt},$

where $\mathrm{B}G$ denotes the finite groupoid consisting of a single set $*$
with $\operatorname{Hom}_{\mathrm{B}G}(*,*):=G$. According to section A.2, we
often use the alternative notation

$N^{{\mathbb{Z}/2}}=i_{\otimes},\;\Phi^{{\mathbb{Z}/2}}=p_{\otimes},\;\iota=p^{*},\text{
and }(-)^{G}=p_{*}$

instead of the notation in section A.1. In particular, all functors in the
sequel are $\infty$-functors, which admit lifts to Quillen functors between
model categories.

We then have adjoint pairs

(2.2)
$i^{*}:\mathrm{Sp}_{{\mathbb{Z}/2}}\rightleftarrows\mathrm{Sp}:i_{*}\text{ and
}\iota:\mathrm{Sp}\rightleftarrows\mathrm{Sp}_{{\mathbb{Z}/2}}:(-)^{G}$

and functors

$N^{{\mathbb{Z}/2}}:\mathrm{Sp}\to\mathrm{Sp}_{{\mathbb{Z}/2}}\text{ and
}\Phi^{{\mathbb{Z}/2}}:\mathrm{Sp}_{{\mathbb{Z}/2}}\to\mathrm{Sp}.$

By [48, Theorem 7.12], the pair of functors $(i^{*},\Phi^{{\mathbb{Z}/2}})$ is
conservative. We will use this fact frequently. We also have adjoint pairs

$\displaystyle
N^{{\mathbb{Z}/2}}:\mathrm{CAlg}\rightleftarrows\mathrm{NAlg}_{{\mathbb{Z}/2}}:i^{*},\text{
}i^{*}:\mathrm{NAlg}_{{\mathbb{Z}/2}}\rightleftarrows\mathrm{CAlg}:i_{*},$
$\displaystyle\text{ and
}\iota:\mathrm{CAlg}\rightleftarrows\mathrm{CAlg}_{{\mathbb{Z}/2}}:(-)^{{\mathbb{Z}/2}}$

and a colimit preserving functor

$\Phi^{{\mathbb{Z}/2}}:\mathrm{CAlg}_{{\mathbb{Z}/2}}\to\mathrm{CAlg}.$

We refer sections A.1 and A.2 for further properties of all these functors.

We now define real topological Hochschild homology for commutative ring
spectra with involution.

###  2.1. Definition of $\mathrm{THR}$

######  Definition 2.1.1.

Suppose $A\in\mathrm{NAlg}_{{\mathbb{Z}/2}}$. For abbreviation, we set

$A^{\wedge{\mathbb{Z}/2}}:=N^{{\mathbb{Z}/2}}i^{*}A.$

Since $N^{{\mathbb{Z}/2}}$ is left adjoint to $i^{*}$, we have the counit map
$A^{\wedge{\mathbb{Z}/2}}\to A$. We use this map to define

$\mathrm{THR}(A):=A\wedge_{A^{\wedge{\mathbb{Z}/2}}}A\in\mathrm{NAlg}_{{\mathbb{Z}/2}},$

which is called the _real topological Hochschild homology of $A$_. The first
$\wedge$ in the formulation of $\mathrm{THR}$ is the pushout in
$\mathrm{NAlg}_{{\mathbb{Z}/2}}$. Under a certain flatness condition, this is
equivalent to the Bökstedt model of the real topological Hochschild homology,
see [14, Theorem, p. 65].

The map to the second smash factor in $A\wedge_{A^{\wedge{\mathbb{Z}/2}}}A$
gives a canonical map

(2.3) $A\to\mathrm{THR}(A).$

In analogy with hermitian $K$-theory $\mathbf{KO}$ and real algebraic
$K$-theory $\mathbf{KR}$, we define

$\mathrm{THO}(A)=(\mathrm{THR}(A))^{{\mathbb{Z}/2}}\simeq(A\wedge_{N^{{\mathbb{Z}/2}}i^{*}A}A)^{{\mathbb{Z}/2}}\in\mathrm{CAlg}$

for $A\in\mathrm{NAlg}_{{\mathbb{Z}/2}}$. For
$B\in\mathrm{NAlg}=\mathrm{CAlg}$, recall that the _topological Hochschild
homology of $B$_ is defined to be

$\mathrm{THH}(B):=B\wedge_{B\wedge B}B\in\mathrm{CAlg}.$

which in turn generalizes the classical [35, Proposition 1.1.13] from rings to
ring spectra.

######  Definition 2.1.2.

Suppose $\mathcal{C}$ be a category. An object $X$ of
$\mathrm{Fun}(\mathrm{B}({\mathbb{Z}/2}),\mathcal{C})$ is called an _object of
$\mathcal{C}$ with involution_, where $\mathrm{B}({\mathbb{Z}/2})$ is the
finite groupoid associated with ${\mathbb{Z}/2}$. Explicitly, $X$ is an object
of $\mathcal{C}$ equipped with an automorphism $w\colon X\to X$ such that
$w\circ w=\mathrm{id}$.

In particular, we have the notions of commutative rings with involutions,
commutative monoids with involutions, etc.

We do not discuss definitions of $\mathrm{HR}(A)$ refining $\mathrm{HH}(A)$
and comparison results between $\mathrm{THR}(A)$ and $\mathrm{HR}(A)$,
similarly to e.g. [41, Proposition IV.4.2], but see Remark 4.2.3 below. The
adjoint functors $\underline{\pi}_{0}$ and $\mathrm{H}$, which are studied in
the appendix, map some of the adjunctions before the Proposition to the
adjunctions, see e.g. Definition 2.3.2.

######  Proposition 2.1.3.

For $A\in\mathrm{NAlg}_{{\mathbb{Z}/2}}$ and $B\in\mathrm{CAlg}$, there exist
canonical equivalences

$\mathrm{THH}(i^{*}A)\simeq i^{*}\mathrm{THR}(A)\text{ and
}\mathrm{THR}(N^{{\mathbb{Z}/2}}B)\simeq N^{{\mathbb{Z}/2}}\mathrm{THH}(B).$

Hence there exists a canonical equivalence

$\mathrm{THR}(A^{\wedge{\mathbb{Z}/2}})\simeq\mathrm{THR}(A)^{\wedge{\mathbb{Z}/2}}.$

In particular, the first equivalence implies $i^{*}\mathrm{THR}(\iota
B)\simeq\mathrm{THH}(B)$, using (2) of Proposition A.2.7.

###### Proof.

Since $N^{{\mathbb{Z}/2}}$ and $i^{*}$ preserve colimits, we have equivalences

$\mathrm{THH}(i^{*}A)\simeq i^{*}A\wedge_{i^{*}A\wedge i^{*}A}i^{*}A\simeq
i^{*}A\wedge_{i^{*}N^{{\mathbb{Z}/2}}i^{*}A}i^{*}A\simeq i^{*}\mathrm{THR}(A)$

and

$\mathrm{THR}(N^{{\mathbb{Z}/2}}B)\simeq
N^{{\mathbb{Z}/2}}B\wedge_{N^{{\mathbb{Z}/2}}i^{*}N^{{\mathbb{Z}/2}}B}N^{{\mathbb{Z}/2}}B\simeq
N^{{\mathbb{Z}/2}}(B\wedge_{B\wedge B}B)\simeq
N^{{\mathbb{Z}/2}}\mathrm{THH}(B)$

by Proposition A.2.7(6). ∎

######  Proposition 2.1.4.

For $B\in\mathrm{CAlg}$, there exists a canonical equivalence

(2.4) $\mathrm{THR}(i_{*}B)\xrightarrow{\simeq}i_{*}\mathrm{THH}(B).$

###### Proof.

The composite

(2.5) $\mathrm{THR}(i_{*}B)\to
i_{*}i^{*}\mathrm{THR}(i_{*}B)\xrightarrow{\simeq}i_{*}\mathrm{THH}(i^{*}i_{*}B)\to
i_{*}\mathrm{THH}(B)$

defines (2.4), where the first (resp. third) map is induced by the unit (resp.
counit). Proposition 2.1.3 shows that the induced map

$i^{*}\mathrm{THR}(i_{*}B)\to i^{*}i_{*}\mathrm{THH}(B)$

is an equivalence. By [48, Theorem 7.12] (note that $i^{*}=\Phi^{e}$), it
remains to show that the induced map

$\Phi^{{\mathbb{Z}/2}}\mathrm{THR}(i_{*}B)\to\Phi^{{\mathbb{Z}/2}}i_{*}\mathrm{THH}(B)$

is an equivalence. The right hand side is equivalent to $0$ by Proposition
A.2.7(3),(5). On the other hand, we have equivalences

$\Phi^{{\mathbb{Z}/2}}\mathrm{THR}(i_{*}B)\simeq\Phi^{{\mathbb{Z}/2}}(i_{*}B)\wedge_{i_{*}B}\Phi^{{\mathbb{Z}/2}}(i_{*}B)\simeq
0\wedge_{i_{*}B}0,$

which is equivalent to $0$ too. ∎

Applying $i^{*}$ and Proposition A.2.7(2), we obtain
$\mathrm{THO}(i_{*}B)\simeq\mathrm{THH}(B)$, which compares nicely with the
well-known $\mathbf{KO}(X\amalg X)\simeq K(X)$ for schemes $X$, where the
involution on $X\amalg X$ switches the components.

######  Proposition 2.1.5.

Let $R\to A,B$ be maps in $\mathrm{NAlg}_{{\mathbb{Z}/2}}$. Then there exists
a canonical equivalence

$\mathrm{THR}(A)\wedge_{\mathrm{THR}(R)}\mathrm{THR}(B)\simeq\mathrm{THR}(A\wedge_{R}B).$

###### Proof.

Both $N^{{\mathbb{Z}/2}}$ and $i^{*}$ preserve colimits. Hence we obtain a
canonical equivalence

$A^{\wedge{\mathbb{Z}/2}}\wedge_{R^{\wedge{\mathbb{Z}/2}}}B^{\wedge{\mathbb{Z}/2}}\simeq(A\wedge_{R}B)^{\wedge{\mathbb{Z}/2}}.$

On the other hand, there are canonical equivalences

$\displaystyle\mathrm{THR}(A)\wedge_{\mathrm{THR}(R)}\mathrm{THR}(B)$
$\displaystyle\simeq(A\wedge_{A^{\wedge{\mathbb{Z}/2}}}A)\wedge_{R\wedge_{R^{\wedge{\mathbb{Z}/2}}}R}(B\wedge_{B^{\wedge{\mathbb{Z}/2}}}B)$
$\displaystyle\simeq(A\wedge_{R}B)\wedge_{A^{\wedge{\mathbb{Z}/2}}\wedge_{R^{\wedge{\mathbb{Z}/2}}}B^{\wedge{\mathbb{Z}/2}}}(A\wedge_{R}B).$

Combine the two equations to obtain the desired equivalence. ∎

###  2.2. Mackey functors for ${\mathbb{Z}/2}$

We refer to the appendix for a general discussion of Mackey and Green functors
for finite groups $G$. We now restrict to the case $G={\mathbb{Z}/2}$.

######  Example 2.2.1.

According to [48, Example 4.38], a Mackey functor $C$ for $G:={\mathbb{Z}/2}$
can be described as a diagram

${C(G/e)}$${C(G/G),}$$\scriptstyle{w}$$\scriptstyle{\mathrm{tran}}$$\scriptstyle{\mathrm{res}}$

where $C(G/e)$ is an abelian group with an involution $w$, $C(G/G)$ is an
abelian group with the trivial involution, $\mathrm{res}$ and $\mathrm{tran}$
are homomorphisms of abelian groups with involutions, and the equality

$\mathrm{res}\circ\mathrm{tran}=\mathrm{id}+w$

is satisfied (i.e. the _double coset formula_ holds). A _morphism of Mackey
functors $C\to D$_ is a diagram of abelian groups with involutions

${C(G/e)}$${D(G/e)}$${C(G/G)}$${D(G/G)}$$\scriptstyle{\mathrm{tran}}$$\scriptstyle{\mathrm{res}}$$\scriptstyle{\mathrm{tran}}$$\scriptstyle{\mathrm{res}}$

such that the horizontal homomorphisms commute with $\mathrm{tran}$ and
$\mathrm{res}$.

######  Example 2.2.2.

If $M$ is an abelian group with involution, then we can associate the Mackey
functor

${M}$${M^{{\mathbb{Z}/2}},}$$\scriptstyle{\mathrm{tran}}$$\scriptstyle{\mathrm{res}}$

where $\mathrm{tran}$ maps $x\in M$ to $x+w(x)$, and $\mathrm{res}$ is the
inclusion. In this way, we obtain a fully faithful functor from the category
of abelian groups with involutions to the category of Mackey functors
$\mathrm{Mack}_{{\mathbb{Z}/2}}$. We often regard an abelian group with
involution as a Mackey functor if no confusion seems likely to arise.

######  Lemma 2.2.3.

Let $A$ be a Green functor for $G:={\mathbb{Z}/2}$, and let $M$ and $L$ are
$A$-modules. If $L$ is associated (in the sense of the previous example) with
an abelian group with involution, then the induced map

(2.6)
$\operatorname{Hom}_{\mathrm{Mod}_{A}}(M,L)\to\operatorname{Hom}_{\mathrm{Mod}_{A(G/e)}}(M(G/e),L(G/e))$

is an isomorphism.

###### Proof.

Suppose $f\colon M(G/e)\to L(G/e)$ is a homomorphism of $A(G/e)$-modules. Then
the image of $f\circ\mathrm{res}$ is in $L(G/e)^{{\mathbb{Z}/2}}$. Hence there
exists a unique homomorphism of $A(G/G)$-modules $g\colon M(G/G)\to L(G/G)$
such that in the diagram

${M(G/e)}$${L(G/e)}$${M(G/G)}$${L(G/G)}$$\scriptstyle{f}$$\scriptstyle{\mathrm{tran}}$$\scriptstyle{\mathrm{res}}$$\scriptstyle{\mathrm{tran}}$$\scriptstyle{\mathrm{res}}$$\scriptstyle{g}$

the pair $(f,g)$ commutes with $\mathrm{res}$. We have

$\mathrm{res}\circ
g\circ\mathrm{tran}=f\circ\mathrm{res}\circ\mathrm{tran}=f\circ(\mathrm{id}+w)=(\mathrm{id}+w)\circ
f=\mathrm{res}\circ\mathrm{tran}\circ f.$

Since $\mathrm{res}$ for $L$ is injective, we deduce that the pair $(f,g)$
commutes with $\mathrm{tran}$. This constructs an inverse of (2.6). ∎

###  2.3. Equivariant Eilenberg-MacLane spectra

In this subsection, we explain basic properties of Equivariant Eilenberg-
MacLane spectra. We also explain how to define $\mathrm{THR}$ of commutative
rings.

######  Definition 2.3.1.

Suppose $\mathcal{C}$ is a category (not an $\infty$-category). Let

$\mathcal{C}\xrightarrow{\iota}\mathrm{Fun}(\mathrm{B}({\mathbb{Z}/2}),\mathcal{C})\xrightarrow{i^{*}}\mathcal{C}$

be the functors induced by (2.1). Let $(-)^{{\mathbb{Z}/2}}$ denote the right
adjoint of $\iota$ if exists.

Let us give some examples. For a commutative ring $A$, $\iota A$ is the
commutative ring $A$ with the trivial involution. For an $A$-module $M$,
$\iota M$ is the $\iota A$-module $M$ with the trivial involution. By abuse of
notation we sometimes denote the constant Mackey functors by $\iota A$ and
$\iota M$ as well.

For a commutative ring $B$ with involution, $i^{*}B$ is the commutative ring
obtained by forgetting the involution, and $B^{{\mathbb{Z}/2}}$ is the
${\mathbb{Z}/2}$-fixed point ring. For an $B$-module $L$, $i^{*}L$ is the
$i^{*}B$-module obtained by forgetting the involution.

######  Definition 2.3.2.

For a commutative monoid $M$ with involution, we regard $M$ as a Mackey
functor, and take the functor (A.25) to obtain the equivariant Eilenberg-
MacLane spectrum $\mathrm{H}M$. There are canonical equivalences

(2.7) $i^{*}\mathrm{H}M\simeq\mathrm{H}i^{*}M\text{ and
}(\mathrm{H}M)^{{\mathbb{Z}/2}}\simeq\mathrm{H}(M^{\mathbb{Z}/2}).$

For a commutative ring $A$ with involution, we can regard $\mathrm{H}A$ as an
object of $\mathrm{NAlg}_{{\mathbb{Z}/2}}$ as explained in [48, Example
11.12].

Note that for a given commutative ring $B$ the commutative ring spectra
$\mathrm{H}\iota B$ and $\iota\mathrm{H}B$ are quite different. E.g., applying
$\underline{\pi}_{0}$ to the first one yields the constant Mackey functor
associated with $B$ whereas for the second one a tensor product with the
Burnside ring Mackey functor of ${\mathbb{Z}/2}$ appears.

######  Proposition 2.3.3.

Let $M$ be an abelian group. Then there is a canonical equivalence

(2.8) $\mathrm{H}(M^{\oplus{\mathbb{Z}/2}})\simeq i_{*}\mathrm{H}M,$

where $M^{\oplus{\mathbb{Z}/2}}$ denotes the abelian group $M\oplus M$ with
the involution given by $(x,y)\mapsto(y,x)$. The equivalence (2.8) can be
promoted to an equivalence in $\mathrm{NAlg}_{{\mathbb{Z}/2}}$ if $M$ is a
commutative ring.

###### Proof.

There is an equivalence

(2.9) $i^{*}\mathrm{H}(M^{{\mathbb{Z}/2}})\simeq\mathrm{H}(M\oplus M)$

by (2.7). Compose this with the map $\mathrm{H}(M\oplus M)\to\mathrm{H}M$
induced by the summation homomorphism $M\oplus M\to M$, and then we construct
(2.8) by adjunction. We need to show that the induced map

$\operatorname{Hom}_{\mathrm{Ho}(\mathrm{Sp}_{{\mathbb{Z}/2}})}(\Sigma^{n}\Sigma^{\infty}X_{+},\mathrm{H}(M^{\oplus{\mathbb{Z}/2}}))\to\operatorname{Hom}_{\mathrm{Ho}(\mathrm{Sp}_{{\mathbb{Z}/2}})}(\Sigma^{n}\Sigma^{\infty}X_{+},i_{*}\mathrm{H}M)$

is an isomorphism for $X={\mathbb{Z}/2},e$ and integer $n\in\mathbb{Z}$. If
$n\neq 0$, then both sides are vanishing. Assume $n=0$. More concretely, it
remains to show that the composite of the induced maps

(2.10)
$\begin{split}\operatorname{Hom}_{\mathrm{Ho}(\mathrm{Sp}_{{\mathbb{Z}/2}})}(\Sigma^{\infty}X_{+},\mathrm{H}(M^{\oplus{\mathbb{Z}/2}}))\to&\operatorname{Hom}_{\mathrm{Ho}(\mathrm{Sp})}(i^{*}\Sigma^{\infty}X_{+},i^{*}\mathrm{H}(M^{\oplus{\mathbb{Z}/2}}))\\\
\to&\operatorname{Hom}_{\mathrm{Ho}(\mathrm{Sp})}(i^{*}\Sigma^{\infty}X_{+},\mathrm{H}M)\end{split}$

is an isomorphism.

If $X={\mathbb{Z}/2}$, then (2.10) can be written as the homomorphisms

$M\oplus M\to M\oplus M\oplus M\oplus M\to M\oplus M$

given by $(x,y)\mapsto(x,0,y,0)$ and $(x,y,z,w)\mapsto(x+y,z+w)$. The
composite is an isomorphism. If $X=e$, then (2.10) can be written as the
homomorphisms

$M\to M\oplus M\to M$

given by $x\mapsto(x,0)$ and $(x,y)\mapsto x+y$. The composite is also an
isomorphism.

If $M$ is a commutative ring, then (2.9) is an equivalence in
$\mathrm{NAlg}_{{\mathbb{Z}/2}}$. Hence we obtain (2.8) as an equivalence in
$\mathrm{NAlg}_{{\mathbb{Z}/2}}$. ∎

######  Definition 2.3.4.

For a commutative ring $A$ with involution, we set

$\mathrm{THR}(A):=\mathrm{THR}(\mathrm{H}A).$

From the map (2.3), we see that $\mathrm{THR}(A)$ is an $\mathrm{H}A$-module.

Recall from (A.18) that $\underline{\pi}_{0}$ of an equivariant spectrum is a
Mackey functor.

######  Proposition 2.3.5.

For every commutative ring $A$, the morphism of Mackey functors

(2.11) $\iota A\to\underline{\pi}_{0}(\mathrm{THR}(\iota A))$

induced by (2.3) is an isomorphism.

###### Proof.

By [14, Corollary 5.2], the morphism (2.11) can be described as a diagram

${A}$${A}$${A}$${(A\otimes
A)/T,}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\mathrm{tran}}$$\scriptstyle{\mathrm{res}}$$\scriptstyle{\mathrm{tran}}$$\scriptstyle{\mathrm{res}}$$\scriptstyle{\alpha}$

where $\alpha(x)=1\otimes x$ for $x\in A$, and $T$ is the subgroup generated
by $ax\otimes b-a\otimes xb$ for $a,b,x\in A$. The description of $T$ means
that $\alpha$ is an isomorphism. It follows that (2.11) is an isomorphism. ∎

###  2.4. Norm functors and flat modules

The purpose of this subsection is to prove Proposition 2.4.2, which is one
ingredient of the proof of Theorem 3.2.3.

######  Definition 2.4.1.

Suppose $A$ is a commutative ring and $M$ is an $A$-module. Let
$N_{A}^{{\mathbb{Z}/2}}M$ denote the $\iota A$-module $M\otimes_{A}M$ whose
${\mathbb{Z}/2}$-action is given by $a\otimes b\mapsto b\otimes a$. If
$A=\mathbb{Z}$, we simply write $N^{{\mathbb{Z}/2}}M$ instead of
$N_{\mathbb{Z}}^{{\mathbb{Z}/2}}M$.

Observe that there is an isomorphism $N_{A}^{{\mathbb{Z}/2}}A\cong\iota A$ of
commutative rings with involutions. We prefer to use the notation $\iota A$
instead of $N_{A}^{{\mathbb{Z}/2}}A$ for brevity.

######  Proposition 2.4.2.

Let $M$ be an $A$-module, where $A$ is a commutative ring. Then there exists a
canonical map of $\mathrm{H}\iota A$-modules

(2.12)
$N^{{\mathbb{Z}/2}}\mathrm{H}M\wedge_{N^{{\mathbb{Z}/2}}\mathrm{H}A}\mathrm{H}\iota
A\to\mathrm{H}(N_{A}^{{\mathbb{Z}/2}}M).$

If $M$ is flat, then this map is an equivalence.

###### Proof.

First, we have a map $N^{{\mathbb{Z}/2}}\mathrm{H}A\to\mathrm{H}\iota A$ using
the adjunction $(N^{{\mathbb{Z}/2}},i^{*})$, that $i^{*}$ commutes with
$\mathrm{H}$, and that $i^{*}\iota\simeq id$. We have the $\iota A$-module
structure on $N_{A}^{{\mathbb{Z}/2}}M$ given by $a(x\otimes y)=ax\otimes y$
for all $a\in A$ and $x\otimes y\in N_{A}^{{\mathbb{Z}/2}}M$. Hence we can
regard $\mathrm{H}(N_{A}^{{\mathbb{Z}/2}}M)$ as an $\mathrm{H}\iota A$-module.
By [37, Proposition 4.6.2.17], to construct (2.12), it suffices to construct a
map of $N^{{\mathbb{Z}/2}}\mathrm{H}A$-modules

(2.13) $N^{{\mathbb{Z}/2}}\mathrm{H}M\to\mathrm{H}(N_{A}^{{\mathbb{Z}/2}}M).$

By Proposition A.3.7, $N^{{\mathbb{Z}/2}}\mathrm{H}A$ and
$N^{{\mathbb{Z}/2}}\mathrm{H}M$ are $(-1)$-connected. From the adjoint pairs
(A.27) and (A.29), we obtain a map in $\mathrm{CAlg}_{G}$

$N^{{\mathbb{Z}/2}}\mathrm{H}A\to\mathrm{H}\underline{\pi}_{0}(N^{{\mathbb{Z}/2}}\mathrm{H}A)$

and a map of $N^{{\mathbb{Z}/2}}\mathrm{H}A$-modules

$N^{{\mathbb{Z}/2}}\mathrm{H}M\to\mathrm{H}\underline{\pi}_{0}(N^{{\mathbb{Z}/2}}\mathrm{H}M).$

Hence to construct (2.13), it suffices to construct a map of
$\underline{\pi}_{0}(N^{{\mathbb{Z}/2}}\mathrm{H}A)$-modules

$\underline{\pi}_{0}(N^{{\mathbb{Z}/2}}\mathrm{H}M)\to
N_{A}^{{\mathbb{Z}/2}}M.$

By Lemma 2.2.3 and Definition 2.4.1, this is equivalent to constructing a
morphism of $\pi_{0}(i^{*}N^{{\mathbb{Z}/2}}\mathrm{H}A)$-modules

(2.14) $\pi_{0}(i^{*}N^{{\mathbb{Z}/2}}\mathrm{H}M)\to M\otimes_{A}M,$

i.e., a morphism of $A\otimes A$-modules $M\otimes M\to M\otimes_{A}M$ since
$i^{*}N^{{\mathbb{Z}/2}}\simeq(-)^{\wedge 2}$ by Proposition A.2.7(6). The
canonical assignment $x\otimes y\to x\otimes y$ finishes the construction.

The class of $M$-modules such that (2.12) is an equivalence is closed under
filtered colimits. By Lazard’s theorem, every flat $A$-module is a filtered
colimit of finitely generated free $A$-modules. Hence to show that (2.12) is
an equivalence if $M$ is flat, we may assume $M=A^{n}$. In this case, there is
an equivalence

$N^{{\mathbb{Z}/2}}\mathrm{H}M\simeq V_{n}\wedge
N^{{\mathbb{Z}/2}}\mathrm{H}A,$

where $V_{n}$ is the set $[n]\times[n]$ with the ${\mathbb{Z}/2}$-action given
by $(a,b)\mapsto(b,a)$. Hence we obtain equivalences

$N^{{\mathbb{Z}/2}}\mathrm{H}M\wedge_{N^{{\mathbb{Z}/2}}\mathrm{H}A}\mathrm{H}\iota
A\simeq V_{n}\wedge\mathrm{H}\iota A\simeq\mathrm{H}(V_{n}\times\iota A).$

Combining this with (2.12) we obtain a map

(2.15) $f\colon\mathrm{H}(V_{n}\times\iota
A)\to\mathrm{H}(N_{A}^{{\mathbb{Z}/2}}A^{n}).$

To show that $f$ is an equivalence, it suffices to show that
$\underline{\pi}_{0}(f)$ is an equivalence. Since $V_{n}\times\iota A$ and
$N_{A}^{{\mathbb{Z}/2}}A^{n}$ are rings with involutions, by Lemma 2.2.3 it
suffices to show that $\pi_{0}(f)$ is an equivalence of modules over rings
with involution. Hence to show that (2.12) is an equivalence, it suffices to
show that it is an equivalence after applying $\pi_{0}$, i.e., the induced map

$g\colon(M\otimes M)\otimes_{A\otimes A}A\to M\otimes_{A}M$

is an isomorphism. From the description of (2.14), we see that $g$ is given by

$g((x\otimes y)\otimes a)=ax\otimes y$

for $x,y\in M$ and $a\in A$. One can readily check that this $g$ is an
isomorphism. ∎

##  3\. Descent properties of $\mathrm{THR}$

###  3.1. Some equivariant topologies

We refer to [22] for the definition of the stable ${\mathbb{Z}/2}$-equivariant
motivic homotopy category $\mathcal{SH}^{{\mathbb{Z}/2}}(k)$, which is
compatible with the later work of Hoyois [28], but differs from Hermann’s. See
his Corollary 2.13 and Example 3.1, as well as [22, Example 2.16 and section
6.1], for a comparison. The following definitions are taken from [22].
_Throughout this section, we assume that $G$ is an abstract finite group,
which we will identify with its associated finite group scheme over a fixed
base scheme._ We are mostly interested in the case $G={\mathbb{Z}/2}=C_{2}$.

######  Definition 3.1.1.

Let $x$ be a point of a $G$-scheme $X$. The _set-theoretic stabilizer of $X$
at $x$_ is defined to be

$S_{x}:=\\{g\in G:gx=x\\}.$

The _scheme-theoretic stabilizer of $X$ at $x$_ is defined to be

$G_{x}:=\ker(S_{x}\to\mathrm{Aut}(k(x))).$

Let $f\colon Y\to X$ be an equivariant morphism of $G$-schemes. We say that

  1. (i)

$f$ is _(equivariant) étale_ if its underlying morphism of schemes is étale,

  2. (ii)

$f$ is an _equivariant étale cover_ if $f$ is étale and surjective,

  3. (iii)

$f$ is _isovariant_ if for every point $y\in Y$, the induced homomorphism
$G_{y}\to G_{f(y)}$ is an isomorphism,

  4. (iv)

$f$ is an _isovariant étale cover_ if $f$ is isovariant and an equivariant
étale cover,

  5. (v)

$f$ is a _fixed point étale cover_ if it is an étale cover and for every point
$x\in X$, there exists a point $y\in f^{-1}(x)$ such that $G_{x}\simeq G_{y}$.

  6. (vi)

$f$ is a _equivariant Nisnevich cover_ if $f$ is an étale cover and for every
point $x\in X$, there exists a point $y\in f^{-1}(x)$ such that $k(x)\simeq
k(y)$ and $S_{x}\simeq S_{y}$.

These covers define _equivariant étale, isovariant étale, fixed point étale,
and equivariant Nisnevich topologies_ on the category of $G$-schemes.

The isovariant étale topology was first studied by Thomason [50], see e.g.
[22, section 6.1]. The equivariant Nisnevich topology is due to Voevodsky
[12].

The discussion in [22, p. 1223] shows that the equivariant Nisnevich topology
is coarser than the fixed point étale topology. As observed in the proof of
[22, Corollary 6.6]. the fixed point étale topology is equivalent to the
isovariant étale topology. Hence we have the following inclusions of
topologies:

$\displaystyle\text{(equivariant Nisnevich)}\subset$ (fixed point étale)
$\displaystyle\simeq$ $\displaystyle\text{(isovariant
\'{e}tale)}\subset\text{(equivariant \'{e}tale).}$

For a $G$-scheme $X$, let $X/G$ denote the geometric quotient, which is an
algebraic space. For the existence, see e.g. [46, Corollary 5.4]. If $S$ is a
locally noetherian scheme with the trivial $G$-action and $X\to S$ is a quasi-
projective $G$-equivariant morphism, then $X/G$ is representable by an
$S$-scheme according to [13, Théorème V.7.1]. If $A$ is a commutative ring
with $G$-action, then there is a canonical isomorphism
$\operatorname{Spec}(A)/G\simeq\operatorname{Spec}(A^{G})$, see e.g. [46,
Theorem 4.1].

######  Proposition 3.1.2.

Let $f\colon Y\to X$ be a separated isovariant étale morphism of $G$-schemes.
Then the quotient morphism $Y/G\to X/G$ of algebraic spaces is étale, and the
induced square

${Y}$${X}$${Y/G}$${X/G}$$\scriptstyle{f}$

is cartesian.

###### Proof.

This is a special case of [46, Proposition 6.7] by [46, Corollary 2.17]. ∎

######  Definition 3.1.3.

For a $G$-scheme $X$, let $X_{iso{\acute{e}t}}$ denote the small isovariant
étale site with the isovariant étale coverings $Y\to X$ of $G$-schemes.

######  Proposition 3.1.4 (Thomason).

Let $X$ be a $G$-scheme. Then there exists an equivalence of sites

$(X/G)_{{\acute{e}t}}\xrightarrow{\simeq}X_{iso{\acute{e}t}}$

sending any $X/G$-scheme $Y$ to $Y\times_{X/G}X$.

###### Proof.

We refer to [22, Proposition 6.11]. ∎

######  Definition 3.1.5.

Let $X$ be a separated $G$-scheme. The presheaf $X^{G}$ on the category of
separated schemes $\mathbf{Sch}$ is defined to be

$X^{G}(Y):=\operatorname{Hom}_{\mathbf{Sch}}(Y,X)^{G}$

for $Y\in\mathbf{Sch}$. By [13, Proposition XII.9.2], $X^{G}$ is representable
by a closed subscheme of $X$. Furthermore, the points of the topological space
underlying the scheme $X^{G}$ are in canonical bijection with the points of
the topological space of fixed points (recall $G$ is finite).

###  3.2. Isovariant étale base change

######  Lemma 3.2.1.

Let $A\to B$ be an étale homomorphism of commutative rings. Then the map
$\mathrm{H}(m)\colon\mathrm{H}(N_{A}^{{\mathbb{Z}/2}}B)\to\mathrm{H}(\iota B)$
induced by the multiplication map

$m\colon N_{A}^{{\mathbb{Z}/2}}B\to\iota B,\;x\otimes y\mapsto xy$

is flat.

###### Proof.

Let $C$ be the kernel of $m$, which is an ideal of $B\otimes_{A}B$ with the
induced involution. Since $A\to B$ is étale, the ring structure on
$B\otimes_{A}B$ makes a ring structure on $C$. Hence we have an isomorphism of
commutative rings with involutions $N_{A}^{{\mathbb{Z}/2}}B\cong\iota B\times
C$. We set $e:=(1.0)\in\iota B\times C$. There is an isomorphism
$N_{A}^{{\mathbb{Z}/2}}B[1/e]\cong\iota B$, which gives an isomorphism

$\mathop{\mathrm{colim}}(N_{A}^{{\mathbb{Z}/2}}B\xrightarrow{\cdot
e}N_{A}^{{\mathbb{Z}/2}}B\xrightarrow{\cdot e}\cdots)\cong\iota B.$

Hence $\iota B$ is a filtered colimit of free
$N_{A}^{{\mathbb{Z}/2}}B$-modules. By Proposition A.5.11, $m$ is flat. It
follows that $\mathrm{H}(m)$ is flat too. ∎

Let $A\to B$ be a map in $\mathrm{NAlg}_{{\mathbb{Z}/2}}$. Then we have the
commutative square in $\mathrm{NAlg}_{{\mathbb{Z}/2}}$

${A}$${\mathrm{THR}(A)}$${B}$${\mathrm{THR}(B),}$

where the horizontal maps are given by (2.3). Hence we obtain an induced map

(3.1) $\mathrm{THR}(A)\wedge_{A}B\to\mathrm{THR}(B)$

in $\mathrm{NAlg}_{{\mathbb{Z}/2}}$. We will now study this map for
$\mathrm{H}\iota A\to\mathrm{H}\iota B$.

Note that the following result does not directly follow from the étale base
change result [19, Proposition 3.2.1] for $\mathrm{THH}$ as neither
$\mathrm{H}$ nor $\mathrm{THR}$ commute with $\iota$. This result is crucial
for the descent results further below.

######  Proposition 3.2.2.

Let $A\to B$ be an étale homomorphism of commutative rings. Then we have an
induced equivalence

(3.2) $\mathrm{THR}(\iota A)\wedge_{\mathrm{H}\iota A}\mathrm{H}\iota
B\stackrel{{\scriptstyle\simeq}}{{\to}}\mathrm{THR}(\iota B).$

###### Proof.

Consider the commutative square

(3.3) ${\mathrm{H}\iota A\wedge_{N^{{\mathbb{Z}/2}}\mathrm{H}A}\mathrm{H}\iota
B}$${\mathrm{H}\iota B\wedge_{N^{{\mathbb{Z}/2}}\mathrm{H}B}\mathrm{H}\iota
B}$${\mathrm{H}(N_{A}^{{\mathbb{Z}/2}}B)\wedge_{N^{{\mathbb{Z}/2}}\mathrm{H}B}\mathrm{H}\iota
B}$${\mathrm{H}(N_{B}^{{\mathbb{Z}/2}}B)\wedge_{N^{{\mathbb{Z}/2}}\mathrm{H}B}\mathrm{H}\iota
B,}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

where the horizontal maps are induced by the map $A\to B$, the vertical
equivalences are obtained by Proposition 2.4.2, and the right hand one even
comes from the algebraic isomorphism mentioned after Definition 2.4.1. We have
an equivalence $H\iota B\wedge_{N^{{\mathbb{Z}/2}}\mathrm{H}B}\mathrm{H}\iota
B\simeq\mathrm{THR}(\iota B)$. (This equivalence involves a computation for
the index under the $\wedge$, namely $i^{*}\mathrm{H}\iota
B\simeq\mathrm{H}B$, which follows from (2.7). The same equivalence is used
when identifying the upper left entry in the square.) This implies that (3.2)
is equivalent to the upper horizontal map of (3.3). Proposition A.5.5 and
Lemma 3.2.1 show that the lower horizontal map of (3.3) is flat. Putting
everything together, we deduce that the map (3.2) is flat.

We will show that this map is an equivalence as claimed by applying
Proposition A.5.6. By Proposition A.3.7, $N^{{\mathbb{Z}/2}}\mathrm{H}A$ and
$N^{{\mathbb{Z}/2}}\mathrm{H}B$ are $(-1)$-connected. Hence we may use
Proposition A.4.4, and are reduced to showing that the induced morphism

$\underline{\pi}_{0}(\mathrm{THR}(\iota A))\Box_{\iota A}\iota
B\to\underline{\pi}_{0}(\mathrm{THR}(\iota B))$

is an isomorphism. This follows easily from applying Proposition 2.3.5 to $A$
and $B$. ∎

We now establish isovariant étale base change for commutative rings with
possibly non-trivial involution.

######  Theorem 3.2.3.

Let $A\to B$ be an isovariant étale homomorphism of commutative rings with
involutions. Then there are canonical equivalences in
$\mathrm{NAlg}_{{\mathbb{Z}/2}}$

$\mathrm{THR}(B)\simeq\mathrm{THR}(A)\wedge_{\mathrm{H}\iota(A^{{\mathbb{Z}/2}})}\mathrm{H}\iota(B^{{\mathbb{Z}/2}})\simeq\mathrm{THR}(A)\wedge_{\mathrm{H}A}\mathrm{H}B.$

###### Proof.

We have isomorphisms
$\operatorname{Spec}(A^{{\mathbb{Z}/2}})\simeq\operatorname{Spec}(A)/({\mathbb{Z}/2})$
and
$\operatorname{Spec}(B^{{\mathbb{Z}/2}})\simeq\operatorname{Spec}(B)/({\mathbb{Z}/2})$.
Hence by Proposition 3.1.2, the induced homomorphism $A^{{\mathbb{Z}/2}}\to
B^{{\mathbb{Z}/2}}$ is étale, and there is an isomorphism of commutative rings
with involution

(3.4) $B\cong A\otimes_{\iota(A^{{\mathbb{Z}/2}})}\iota(B^{{\mathbb{Z}/2}}).$

Since $A^{{\mathbb{Z}/2}}\to B^{{\mathbb{Z}/2}}$ is flat, $B^{{\mathbb{Z}/2}}$
is a filtered colimit of finite free $A^{{\mathbb{Z}/2}}$-module. It follows
that $\iota A^{{\mathbb{Z}/2}}\to\iota B^{{\mathbb{Z}/2}}$ is flat by
Proposition A.5.11. Hence the isomorphism (3.4) induces an equivalence
$\mathrm{H}B\simeq\mathrm{H}A\wedge_{\mathrm{H}(\iota
A^{{\mathbb{Z}/2}})}\mathrm{H}(\iota B^{{\mathbb{Z}/2}})$ by Proposition
A.5.7. Then Proposition 2.1.5 yields an equivalence

$\mathrm{THR}(B)\simeq\mathrm{THR}(A)\wedge_{\mathrm{THR}(\iota(A^{{\mathbb{Z}/2}}))}\mathrm{THR}(\iota(B^{{\mathbb{Z}/2}})).$

Now Proposition 3.2.2 implies the left equivalence of the Proposition after
canceling out one smash factor $\mathrm{THR}(\iota(A^{{\mathbb{Z}/2}}))$.
Applying the isovariance condition once more gives the right hand side
equivalence. ∎

###  3.3. Presheaves of equivariant spectra

Zariski and other sheaves and completely determined by their behaviors on
affine schemes. This is known to be true in some homotopical settings as well.
The purpose of this section is to establish a rather general result, namely
Proposition 3.3.12, which applies to our setting, that is the isovariant étale
site and ${\mathbb{Z}/2}$-equivariant spectra. For this, a result of [2] will
be very useful.

######  Definition 3.3.1.

Let $\mathcal{C}$ be a category with a Grothendieck topology $t$, and let
$\mathcal{V}$ be a presentable $\infty$-category. Let

$\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V}):=\mathrm{Fun}(\mathrm{N}(\mathcal{C})^{op},\mathcal{V})$

denote the $\infty$-category of presheaves on $\mathcal{C}$ with values in
$\mathcal{V}$. We say that a presheaf
$\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V})$ satisfies _t-
descent_ if the induced map

(3.5) $\mathcal{F}(X)\to\lim_{i\in\Delta}\mathcal{F}(\mathscr{X}_{i})$

is an equivalence for every $t$-hypercover $\mathscr{X}\to X$. Let
$\mathcal{S}\mathrm{hv}_{t}(\mathcal{C},\mathcal{V})$ denote the full
subcategory of $\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V})$ consisting of
presheaves satisfying $t$-descent. We often omit $t$ in the above notation if
it is clear from the context.

The above condition is sometimes called _t-hyperdescent_ , in order to
distinguish it from the weaker descent condition only for covering sieves
rather than all hypercovers. We refer to [36, section 6.5] and [16] for a
careful comparison.

If $\mathcal{V}\to\mathcal{V}^{\prime}$ is a functor of $\infty$-categories,
then this induces a functor
$\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V})\to\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V}^{\prime})$.
In this way, we obtain functors $i^{*}$, $(-)^{{\mathbb{Z}/2}}$, etc for the
presheaf categories.

######  Proposition 3.3.2.

Suppose
$\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}})$,
where $\mathcal{C}$ is a site. Then
$\mathcal{F}\in\mathcal{S}\mathrm{hv}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}})$
if and only if
$i^{*}\mathcal{F},\mathcal{F}^{{\mathbb{Z}/2}}\in\mathcal{S}\mathrm{hv}(\mathcal{C},\mathrm{Sp})$.

###### Proof.

Let $\mathscr{X}\to X$ be a hypercover. Since $i^{*}$ and
$(-)^{{\mathbb{Z}/2}}$ preserve limits, (3.5) is an equivalence if and only if

$i^{*}\mathcal{F}(X)\to\lim_{i\in\Delta}i^{*}\mathcal{F}(\mathscr{X}_{i})\text{
and
}\mathcal{F}(X)^{{\mathbb{Z}/2}}\to\lim_{i\in\Delta}\mathcal{F}(\mathscr{X}_{i})^{{\mathbb{Z}/2}}$

are equivalences. Equivalently,
$i^{*}\mathcal{F},\mathcal{F}^{{\mathbb{Z}/2}}\in\mathcal{S}\mathrm{hv}(\mathcal{C},\mathrm{Sp})$.
∎

######  Definition 3.3.3.

Suppose $\mathcal{C}$ is a site. As a consequence of Proposition 3.3.2, we see
that $\mathcal{S}\mathrm{hv}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}})$ is the
full subcategory of local objects of [36, Definition 5.5.4.1] with respect to
the class of maps consisting of

$\iota\Sigma^{n}\Sigma^{\infty}\mathscr{X}_{+}\to\iota\Sigma^{n}\Sigma^{\infty}X_{+}\text{
and }i_{\sharp}\Sigma^{n}\Sigma^{\infty}\mathscr{X}_{+}\to
i_{\sharp}\Sigma^{n}\Sigma^{\infty}X_{+}$

for all hypercover $\mathscr{X}\to X$ and integer $n$. In particular, there is
an adjoint pair

(3.6)
$L:\mathcal{P}\mathrm{sh}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}})\rightleftarrows\mathcal{S}\mathrm{hv}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}}):\eta$

by [36, Proposition 5.5.4.15(3)], where $\eta$ is the inclusion functor. A map
$\mathcal{F}\to\mathcal{G}$ in
$\mathcal{P}\mathrm{sh}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}})$ is called a
_local equivalence_ if $L\mathcal{F}\to L\mathcal{G}$ is an equivalence.

######  Proposition 3.3.4.

Let $\mathcal{C}$ be a site. Then the functor

$i^{*}\colon\mathcal{P}\mathrm{sh}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}})\to\mathcal{P}\mathrm{sh}(\mathcal{C},\mathrm{Sp})$

preserves local equivalences.

###### Proof.

We need to show that the maps

$i^{*}\iota\Sigma^{n}\Sigma^{\infty}\mathscr{X}_{+}\to
i^{*}\iota\Sigma^{n}\Sigma^{\infty}X_{+}\text{ and
}i^{*}i_{\sharp}\Sigma^{n}\Sigma^{\infty}\mathscr{X}_{+}\to
i^{*}i_{\sharp}\Sigma^{n}\Sigma^{\infty}X_{+}$

are local equivalences for all hypercover $\mathscr{X}\to X$ and integer $n$,
which follow from Proposition A.2.7(2),(7). ∎

######  Proposition 3.3.5.

Let $\mathcal{C}$ be a site. Then the functor

$(-)^{{\mathbb{Z}/2}}\colon\mathcal{P}\mathrm{sh}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}})\to\mathcal{P}\mathrm{sh}(\mathcal{C},\mathrm{Sp})$

preserves local equivalences.

###### Proof.

We need to show that the maps

(3.7)
$(\iota\Sigma^{\infty}\mathscr{X}_{+})^{{\mathbb{Z}/2}}\to(\iota\Sigma^{\infty}X_{+})^{{\mathbb{Z}/2}}\text{
and
}(i_{\sharp}\Sigma^{\infty}\mathscr{X}_{+})^{{\mathbb{Z}/2}}\to(i_{\sharp}\Sigma^{\infty}X_{+})^{{\mathbb{Z}/2}}$

are local equivalences. Since
$(-)^{{\mathbb{Z}/2}}i_{\sharp}\simeq(-)^{{\mathbb{Z}/2}}i_{*}\simeq\mathrm{id}$
by Proposition A.2.7(2),(3), the second map in (3.7) is an equivalence.

The formulation (A.4) means that $\iota$ commutes with $\Sigma^{\infty}$, and
there is an equivalence $(\iota X)^{{\mathbb{Z}/2}}\simeq X$. Together with
the tom Dieck splitting [20, Theorem 3.10], we have an equivalence

$(\iota\Sigma^{\infty}X_{+})^{{\mathbb{Z}/2}}\simeq\Sigma^{\infty}X_{+}\oplus\Sigma^{\infty}(\mathrm{B}({\mathbb{Z}/2})\times
X)_{+}.$

We have a similar equivalence for $\mathscr{X}$ too. Since

$\Sigma^{\infty}\mathscr{X}_{+}\to\Sigma^{\infty}X_{+}\text{ and
}\Sigma^{\infty}(\mathrm{B}({\mathbb{Z}/2})\times\mathscr{X})_{+}\to\Sigma^{\infty}(\mathrm{B}({\mathbb{Z}/2})\times
X)_{+}$

are local equivalences in $\mathcal{P}\mathrm{sh}(\mathcal{C},\mathrm{Sp})$,
the first map in (3.7) is a local equivalence. ∎

######  Proposition 3.3.6.

Let $f\colon\mathcal{F}\to\mathcal{G}$ be a morphism in
$\mathcal{P}\mathrm{sh}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}})$, where
$\mathcal{C}$ is a site. If $i^{*}f$ and $f^{{\mathbb{Z}/2}}$ are local
equivalences, then $f$ is a local equivalence.

###### Proof.

By considering the fiber of $f$, we reduce to the case when $\mathcal{G}=0$.
This means that $i^{*}\mathcal{F}$ and $\mathcal{F}^{{\mathbb{Z}/2}}$ are
local equivalent to $0$. The adjoint pair (2.4) gives a local equivalence
$\mathcal{F}\to\mathcal{F}^{\prime}$ with
$\mathcal{F}^{\prime}\in\mathcal{S}\mathrm{hv}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}})$.
By Propositions 3.3.4 and 3.3.5, $i^{*}\mathcal{F}^{\prime}$ and
$\mathcal{F}^{\prime{\mathbb{Z}/2}}$ are local equivalent to $0$. Since
$i^{*}\mathcal{F}^{\prime},\mathcal{F}^{\prime{\mathbb{Z}/2}}\in\mathcal{S}\mathrm{hv}(\mathcal{C},\mathrm{Sp})$
by Proposition 3.3.2, it follows that $i^{*}\mathcal{F}^{\prime}$ and
$\mathcal{F}^{\prime{\mathbb{Z}/2}}$ are equivalent to $0$. Hence
$\mathcal{F}^{\prime}$ is equivalent to $0$, i.e., $\mathcal{F}$ is local
equivalent to $0$. ∎

######  Definition 3.3.7.

Let $\mathfrak{M}$ be a combinatorial model category. For a category
$\mathcal{C}$, we set

$\mathbf{Psh}(\mathcal{C},\mathfrak{M}):=\mathrm{Fun}(\mathcal{C}^{op},\mathfrak{M}).$

A morphism $\mathcal{F}\to\mathcal{G}$ in
$\mathbf{Psh}(\mathcal{C},\mathfrak{M})$ is a weak equivalence (resp.
fibration) if $\mathcal{F}(X)\to\mathcal{G}(X)$ is a weak equivalence (resp.
fibration) for all $X\in\mathcal{C}$. One can form a projective model
structure based on these, see [2, Definition 4.4.18].

If $\mathcal{V}$ is the underlying $\infty$-category of $\mathfrak{M}$, then
the underlying $\infty$-category of $\mathbf{Psh}(\mathcal{C},\mathfrak{M})$
with respect to the projective model structure is equivalent to
$\mathcal{P}\mathrm{sh}(\mathcal{C},\mathcal{V})$ by [37, Proposition
1.3.4.25].

######  Definition 3.3.8 ([2, Definition 4.4.23]).

A _category of coefficients_ is a left proper cofibrantly generated stable
model category $\mathfrak{M}$ satisfying the following conditions:

  1. (i)

Finite coproducts of weak equivalences are weak equivalences.

  2. (ii)

there exists a set $\mathcal{E}$ of compact objects of $\mathfrak{M}$ that
generates the homotopy category $\mathrm{Ho}(\mathfrak{M})$.

The set $\mathcal{E}$ is considered as a part of the data.

######  Example 3.3.9.

According to [24, Proposition B.63], $\mathrm{Sp}^{O}_{G}$ is a cofibrantly
generated stable model category. By [24, Example B.10, Remark B.64],
$\mathrm{Sp}^{O}_{G}$ is left proper. A consequence of [24, Corollary B.43] is
that finite coproducts of weak equivalences are weak equivalences. Let
$\mathcal{E}$ be the set of $\Sigma^{n}\Sigma^{\infty}(G/H)_{+}$ for all
subgroup $H$ of $G$ and integer $n$, which consists of compact objects and
generates $\mathrm{Ho}(\mathrm{Sp}^{O}_{G})$ by Proposition A.1.6. In
conclusion, $\mathrm{Sp}^{O}_{G}$ is a category of coefficients.

We also note that $\mathrm{Sp}^{O}_{G}$ is a combinatorial model category.

######  Definition 3.3.10.

Suppose $\mathcal{C}$ is a site. According to [2, Definition 4.4.28], a
morphism $\mathcal{F}\to\mathcal{G}$ in
$\mathbf{Psh}(\mathcal{C},\mathrm{Sp}^{O}_{G})$ is called a _local weak
equivalence_ if the induced morphism of presheaves

$\displaystyle(X\in\mathcal{C}\mapsto\operatorname{Hom}_{\mathrm{Ho}(\mathbf{Psh}(\mathcal{C},\mathrm{Sp}^{O}_{G}))}(\Sigma^{n}\Sigma^{\infty}(G/H\times
X)_{+},\mathcal{F})$ $\displaystyle\to$
$\displaystyle(X\in\mathcal{C}\mapsto\operatorname{Hom}_{\mathrm{Ho}(\mathbf{Psh}(\mathcal{C},\mathrm{Sp}^{O}_{G}))}(\Sigma^{n}\Sigma^{\infty}(G/H\times
X)_{+},\mathcal{G})$

becomes an isomorphism after sheafification for all subgroup $H$ of $G$ and
integer $n$. There is a local projective model structure, see [2, Definition
4.4.34]. This is a Bousfield localization of the projective model structure
with respect to local weak equivalences.

######  Proposition 3.3.11.

Let $\mathcal{C}$ be a site. The underlying $\infty$-category of
$\mathbf{Psh}(\mathcal{C},\mathrm{Sp}^{O}_{{\mathbb{Z}/2}})$ with respect to
the local projective model structure is equivalent to
$\mathcal{S}\mathrm{hv}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}})$.

###### Proof.

Let $f\colon\mathcal{F}\to\mathcal{G}$ be a morphism of fibrant objects in
$\mathbf{Psh}(\mathcal{C},\mathrm{Sp}^{O}_{G})$ with respect to the projective
model structure. We need to show that $f$ is a local weak equivalence if and
only if the corresponding map $g$ in
$\mathcal{P}\mathrm{sh}(\mathcal{C},\mathrm{Sp}_{G})$ is a local equivalence.

By adjunction, $f$ is a local weak equivalence if and only if the induced
morphism of presheaves

$(X\in\mathcal{C}\mapsto\operatorname{Hom}_{\mathrm{Ho}(\mathrm{Sp}^{O})}(\Sigma^{n},\mathcal{F}(X)^{H}))\to(X\in\mathcal{C}\mapsto\operatorname{Hom}_{\mathrm{Ho}(\mathrm{Sp}^{O})}(\Sigma^{n},\mathcal{G}(X)^{H}))$

becomes an isomorphism after sheafification for $H=e,{\mathbb{Z}/2}$ and
integer $n$. By [16, Theorem 1.3], this is equivalent to saying that $i^{*}g$
and $g^{{\mathbb{Z}/2}}$ are local equivalences. Proposition 3.3.6 finishes
the proof. ∎

######  Proposition 3.3.12.

Let $\mathcal{C}$ and $\mathcal{C}^{\prime}$ be sites. If there is an
equivalence of topoi
$\mathbf{Shv}(\mathcal{C})\simeq\mathbf{Shv}(\mathcal{C}^{\prime})$, then
there is an equivalence of $\infty$-categories

$\mathcal{S}\mathrm{hv}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}})\simeq\mathcal{S}\mathrm{hv}(\mathcal{C}^{\prime},\mathrm{Sp}_{{\mathbb{Z}/2}}).$

###### Proof.

Apply [2, Proposition 4.4.56] to the canonical (compose Yoneda and
sheafification) functors $\mathcal{C}\to\mathbf{Shv}(\mathcal{C})$ and
$\mathcal{C}^{\prime}\to\mathbf{Shv}(\mathcal{C}^{\prime})$, where the right
hand side categories are equipped with the topology described in [2, after
Théorème 4.4.51]. Hence we obtain left Quillen equivalences

$\mathbf{Psh}(\mathcal{C},\mathrm{Sp}^{O}_{{\mathbb{Z}/2}})\to\mathbf{Psh}(\mathbf{Shv}(\mathcal{C}),\mathrm{Sp}^{O}_{{\mathbb{Z}/2}})\text{
and
}\mathbf{Psh}(\mathcal{C}^{\prime},\mathrm{Sp}^{O}_{{\mathbb{Z}/2}})\to\mathbf{Psh}(\mathbf{Shv}(\mathcal{C}^{\prime}),\mathrm{Sp}^{O}_{{\mathbb{Z}/2}})$

with respect to the local model structures of Definition 3.3.10. Owing to
Proposition 3.3.11 and [37, Lemma 1.3.4.21], we obtain equivalences of
$\infty$-categories

$\mathcal{S}\mathrm{hv}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}})\simeq\mathcal{S}\mathrm{hv}(\mathbf{Shv}(\mathcal{C}),\mathrm{Sp}_{{\mathbb{Z}/2}})\text{
and
}\mathcal{S}\mathrm{hv}(\mathcal{C}^{\prime},\mathrm{Sp}_{{\mathbb{Z}/2}})\simeq\mathcal{S}\mathrm{hv}(\mathbf{Shv}(\mathcal{C}^{\prime}),\mathrm{Sp}_{{\mathbb{Z}/2}}).$

We obtain the desired equivalence of $\infty$-categories thanks to the
equivalence of topoi (and hence sites)
$\mathbf{Shv}(\mathcal{C})\simeq\mathbf{Shv}(\mathcal{C}^{\prime})$. ∎

###  3.4. Isovariant étale descent

For an abelian group $M$, (2.7) gives a canonical equivalence

$\mathrm{H}M\xrightarrow{\simeq}(\mathrm{H}\iota M)^{{\mathbb{Z}/2}}.$

The last adjunction in (2.2) then yields a maps
$\iota\mathrm{H}M\to\mathrm{H}\iota M$. We have a similar map for a
commutative ring $A$.

######  Lemma 3.4.1.

Let $A$ be a commutative ring and $M$ a flat $A$-module. Then there is an
equivalence

$\mathrm{H}\iota M\simeq\mathrm{H}\iota
A\wedge_{\iota\mathrm{H}A}\iota\mathrm{H}M.$

###### Proof.

By Proposition A.3.6, $\mathrm{H}$ preserves filtered colimits. The two
functors $\iota$ also preserve filtered colimits since they are left adjoints.
Every flat $A$-module is a filtered colimit of finitely generated free
$A$-modules by Lazard’s theorem, so we reduce to the case when $M$ is a
finitely generated free $A$-module. In this case, the claim is clear. ∎

For a scheme $X$, let $\mathbf{\acute{E}t}\mathbf{Aff}/X$ denote the category
of affine schemes étale over $X$. We start by considering the affine case
$X=\operatorname{Spec}(A)$.

######  Lemma 3.4.2.

Let $A$ be a commutative ring, and let $M$ be an $\mathrm{H}A$-module. The
presheaf $\mathcal{F}$ of spectra on
$\mathbf{\acute{E}t}\mathbf{Aff}/\operatorname{Spec}(A)$ given by

$\mathcal{F}(\operatorname{Spec}(B)):=M\wedge_{\mathrm{H}A}\mathrm{H}B$

for étale homomorphisms $A\to B$ satisfies étale descent.

###### Proof.

Since $\mathrm{H}A\to\mathrm{H}B$ is flat in the sense of [36, Definition
7.2.2.10], we have an isomorphism

$\pi_{q}\mathcal{F}(\operatorname{Spec}(B)))\simeq\pi_{q}(M)\otimes_{A}B$

for all integers $q$. In particular, $\pi_{q}\mathcal{F}$ is a quasi-coherent
sheaf on $\mathbf{\acute{E}t}\mathbf{Aff}/\operatorname{Spec}(A)$.

According to the paragraph preceding [19, Proposition 3.1.2], there is a
conditionally convergent spectral sequence

$E_{2}^{st}:=H_{{\acute{e}t}}^{s}(X,a_{{\acute{e}t}}\pi_{-t}\mathcal{F})\Rightarrow\pi_{-s-t}L_{{\acute{e}t}}\mathcal{F}.$

Since $\pi_{-t}\mathcal{F}$ is a quasi-coherent sheaf for all integer $t$, the
cohomology $H^{s}(X,a_{{\acute{e}t}}\pi_{-t}\mathcal{F})$ vanishes for all
integer $s\neq 0$. It follows that
$\pi_{-t}\mathcal{F}\to\pi_{-t}L_{{\acute{e}t}}\mathcal{F}$ is an isomorphism
for all integer $t$, i.e., $\mathcal{F}$ satisfies étale descent. ∎

Suppose $A$ is a commutative ring with involution. Let
$\mathbf{iso\acute{E}tAff}/\operatorname{Spec}(A)$ denote the category of
affine schemes with involutions isovariant étale over
$\operatorname{Spec}(A)$.

######  Theorem 3.4.3.

The presheaf

$\mathrm{THR}\in\mathcal{P}\mathrm{sh}(\mathbf{Aff}_{{\mathbb{Z}/2}},\mathrm{Sp}_{{\mathbb{Z}/2}})$

satisfies isovariant étale descent, where $\mathbf{Aff}_{{\mathbb{Z}/2}}$
denotes the category of affine schemes with involutions.

###### Proof.

We only need to show that the restriction of $\mathrm{THR}$ to
$\mathbf{iso\acute{E}tAff}/\operatorname{Spec}(A)$ satisfies isovariant étale
descent. The functor

(3.8)
$\mathbf{\acute{E}t}\mathbf{Aff}/\operatorname{Spec}(A^{{\mathbb{Z}/2}})\to\mathbf{iso\acute{E}tAff}/\operatorname{Spec}(A)$

sending $\operatorname{Spec}(B)$ to $\operatorname{Spec}(A\otimes_{\iota
A^{{\mathbb{Z}/2}}}\iota B)$ is an equivalence of sites by Proposition 3.1.4.
Let

$\mathcal{F}\in\mathcal{P}\mathrm{sh}(\mathbf{\acute{E}t}\mathbf{Aff}/\operatorname{Spec}(A^{{\mathbb{Z}/2}}),\mathrm{Sp}_{{\mathbb{Z}/2}})$

be the presheaf given by
$\mathcal{F}(\operatorname{Spec}(B)):=\mathrm{THR}(A)\wedge_{\mathrm{H}\iota
A^{{\mathbb{Z}/2}}}\mathrm{H}\iota B$ for étale homomorphisms
$A^{{\mathbb{Z}/2}}\to B$. If we show that $\mathcal{F}$ satisfies étale
descent, then the presheaf

$\mathcal{G}\in\mathcal{P}\mathrm{sh}(\mathbf{iso\acute{E}tAff}/\operatorname{Spec}(A),\mathrm{Sp}_{{\mathbb{Z}/2}})$

obtained from $\mathcal{F}$ and (3.8) satisfies isovariant étale descent.
Theorem 3.2.3 gives the third equivalence in

$\mathcal{G}(\operatorname{Spec}(\iota B\otimes_{\iota
A^{{\mathbb{Z}/2}}}A))=\mathcal{F}(\operatorname{Spec}(B))=\mathrm{THR}(A)\wedge_{\mathrm{H}\iota
A^{{\mathbb{Z}/2}}}\mathrm{H}\iota B\simeq\mathrm{THR}(A\otimes_{\iota
A^{{\mathbb{Z}/2}}}\iota B),$

i.e., $\mathcal{G}\simeq\mathrm{THR}$. This means that $\mathrm{THR}$
satisfies isovariant étale descent.

Hence it remains to check that $\mathcal{F}$ satisfies étale descent. There is
an equivalence

$i^{*}\mathcal{F}(\operatorname{Spec}(B))\simeq
i^{*}\mathrm{THR}(A)\wedge_{\mathrm{H}A^{{\mathbb{Z}/2}}}\mathrm{H}B.$

By Lemmas A.2.9 and 3.4.1, there are equivalences

$\mathcal{F}(\operatorname{Spec}(B))^{{\mathbb{Z}/2}}\simeq(\mathrm{THR}(A)\wedge_{\iota\mathrm{H}A^{{\mathbb{Z}/2}}}\iota\mathrm{H}B)^{{\mathbb{Z}/2}}\simeq\mathrm{THR}(A)^{{\mathbb{Z}/2}}\wedge_{\mathrm{H}A^{{\mathbb{Z}/2}}}\mathrm{H}B.$

Lemma 3.4.2 implies that $i^{*}\mathcal{F}$ and $\mathcal{F}^{{\mathbb{Z}/2}}$
satisfy étale descent, which means that $\mathcal{F}$ satisfies étale descent.
∎

######  Proposition 3.4.4.

For every separated scheme $X$ with involution, there exists an isovariant
étale covering $\\{U_{i}\to X\\}_{i\in I}$ such that each $U_{i}$ is an affine
scheme with involution.

###### Proof.

Let $X^{{\mathbb{Z}/2}}$ denote the closed subscheme of $X$ obtained by
Definition 3.1.5. We set $Y:=X-X^{{\mathbb{Z}/2}}$, which is an open subscheme
of $X$.

If $x$ is a point of $X^{{\mathbb{Z}/2}}$, choose an affine open neighborhood
$U_{x}$ of $x$ in $X$. Then $V_{x}:=U_{x}\cap w(U_{x})$ is again an affine
open neighborhood of $x$ since $X$ is separated, and $w$ can be restricted to
$V_{x}$. It follows that

$\\{V_{x}\to X\\}_{x\in X}\cup\\{Y\to X\\}$

is a Zariski covering.

The ${\mathbb{Z}/2}$-action on $Y$ is free, so the quotient morphism $Y\to
Y/({\mathbb{Z}/2})$ is étale, and there is an isomorphism
$Y\times_{Y/({\mathbb{Z}/2})}Y\cong Y\times{\mathbb{Z}/2}$. It follows that
the morphism $Y\times{\mathbb{Z}/2}\to Y$ obtained by the first projection
$Y\times_{Y/({\mathbb{Z}/2})}Y\to Y$ is isovariant étale. Choose a Zariski
covering $\\{W_{j}\to Y\\}_{j\in J}$ after forgetting involution such that
each $W_{j}$ is an affine scheme, and then

(3.9) $\\{V_{x}\to X\\}_{x\in X}\cup\\{W_{j}\times{\mathbb{Z}/2}\to X\\}_{j\in
J}$

is an isovariant étale covering. ∎

Let $\mathbf{Sch}_{{\mathbb{Z}/2}}$ denote the category of separated schemes
with involutions.

######  Proposition 3.4.5.

There is an equivalence of topoi

(3.10)
$\mathbf{Shv}_{iso{\acute{e}t}}(\mathbf{Aff}_{{\mathbb{Z}/2}})\simeq\mathbf{Shv}_{iso{\acute{e}t}}(\mathbf{Sch}_{{\mathbb{Z}/2}}).$

Hence there is an equivalence of $\infty$-categories

(3.11)
$\mathcal{S}\mathrm{hv}_{iso{\acute{e}t}}(\mathbf{Aff}_{{\mathbb{Z}/2}},\mathrm{Sp}_{{\mathbb{Z}/2}})\simeq\mathcal{S}\mathrm{hv}_{iso{\acute{e}t}}(\mathbf{Sch}_{{\mathbb{Z}/2}},\mathrm{Sp}_{{\mathbb{Z}/2}}).$

###### Proof.

Combine [1, Théorème III.4.1] and Proposition 3.4.4 to obtain (3.10). Use
Proposition 3.3.12 for (3.11). ∎

######  Definition 3.4.6.

By Theorem 3.4.3 and Proposition 3.4.5, we obtain

$\mathrm{THR}\in\mathcal{S}\mathrm{hv}_{iso{\acute{e}t}}(\mathbf{Sch}_{{\mathbb{Z}/2}},\mathrm{Sp}_{{\mathbb{Z}/2}}).$

This definition immediately implies that $\mathrm{THR}(X)$ that satisfies
isovariant étale descent for all $X\in\mathbf{Sch}_{{\mathbb{Z}/2}}$.

Below, we carry out computations of $\mathrm{THR}(X)$ for projective spaces
$X=\mathbb{P}^{n}$, even with involution for $n=1$. Let us first check that
extending Proposition 2.1.3 to schemes yields a definition of $\mathrm{THH}$
for schemes equivalent to the one of [19] as follows. See e.g. [9, p. 1055 and
chapter 3] for a discussion of other equivalent definitions of
$\mathrm{THH}(X)$.

######  Proposition 3.4.7.

For $X\in\mathbf{Sch}_{{\mathbb{Z}/2}}$ and $Y\in\mathbf{Sch}$, there are
canonical equivalences

$\mathrm{THH}(i^{*}X)\simeq i^{*}\mathrm{THR}(X)\text{ and
}\mathrm{THR}(Y\amalg Y)\simeq i_{*}\mathrm{THH}(Y),$

where the involution on $Y\amalg Y$ switches the components.

###### Proof.

The question is isovariant étale local on $X$ and étale local on $Y$, so we
reduce to the case when $X=\operatorname{Spec}(A)$ for some commutative ring
$A$ with involution and $Y=\operatorname{Spec}(B)$ for some commutative ring
$B$. We obtain equivalences

$\mathrm{THR}(i^{*}A)=\mathrm{THR}(\mathrm{H}(i^{*}A))\simeq\mathrm{THR}(i^{*}\mathrm{H}A)\simeq
i^{*}\mathrm{THR}(\mathrm{H}A)=i^{*}\mathrm{THR}(A)$

using Proposition 2.1.3 and (2.7). We similarly obtain the remaining desired
equivalence using Propositions 2.1.4 and 2.3.3. ∎

######  Definition 3.4.8.

Recall from [22, section 2.1] that an _equivariant Nisnevich distinguished
square_ is a cartesian square in $\mathbf{Sch}_{{\mathbb{Z}/2}}$ of the form

(3.12) $Q:=\leavevmode\hbox to58.35pt{\vbox
to46.41pt{\pgfpicture\makeatletter\hbox{\hskip 29.17355pt\lower-23.2526pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-29.17355pt}{-23.15276pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
8.33331pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${V}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{f}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-12.37569pt}{-20.65276pt}\pgfsys@lineto{10.89308pt}{-20.65276pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{j}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

such that $j$ is an open immersion, $f$ is equivariant étale, and the morphism
of schemes $(Y-V)_{\mathrm{red}}\to(X-U)_{\mathrm{red}}$ is an isomorphism.
The collection of such squares forms a cd-structure, which is bounded,
complete, and regular in the sense of [52] by [22, Theorem 2.3]. Furthermore,
the associated topology is the equivariant Nisnevich topology in Definition
3.1.1 by [22, Proposition 2.17].

######  Corollary 3.4.9.

For every equivariant Nisnevich distinguished square $Q$ of the form (3.12),
the induced square

$\mathrm{THR}(Q)=\leavevmode\hbox to118.07pt{\vbox
to52.74pt{\pgfpicture\makeatletter\hbox{\hskip
59.03471pt\lower-26.41928pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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is cartesian.

###### Proof.

Since $\mathrm{THR}$ satisfies isovariant étale descent, it satisfies
equivariant Nisnevich descent. By [52, Proposition 3.8(2)],
$\Omega^{\infty}\Sigma^{n}i^{*}\mathrm{THR}(Q)$ and
$\Omega^{\infty}\Sigma^{n}\Phi^{{\mathbb{Z}/2}}\mathrm{THR}(Q)$ are cartesian
for all integer $n$, which implies the claim. ∎

For the computations about $\mathrm{THR}(\mathbb{P}^{n})$ we provide in the
next sections, we may simply use appropriate equivariant Nisnevich covers by
affine schemes and Corollary 3.4.9, which e.g. exhibits
$\mathrm{THR}(\mathbb{P}^{1})$ as part of a homotopy cocartesian square in
which all other entries are of the form $\mathrm{THR}(Y)$ for
$Y=\operatorname{Spec}(A[M])$ corresponding to some commutative monoid rings
for $M=\mathbb{N}$ or $M=\mathbb{Z}$ over the commutative base ring $A$. We
could even go one step further and define
$\mathrm{THR}(\mathbb{P}^{n}_{\mathbb{S}},\sigma)$ of THR for projective
spaces, possibly with non-trivial involution $\sigma$, over the sphere
spectrum $\mathbb{S}$ rather than over $A$ or $\mathrm{H}A$, using the
homotopy pushout of the appropriate diagram of THR of the corresponding
spherical monoid rings $\mathbb{S}[M]$, compare the proofs in section 5.1.

##  4\. The dihedral replete bar construction

###  4.1. Crossed simplicial groups

Hochschild and cyclic homology and their real refinements are closely related
to cyclic, real and dihedral nerves. We now present a uniform treatment of
these constructions. The original references are [18] and [35], parts of this
are also explained e.g. in [15] and [14].

######  Definition 4.1.1.

(see [18, Definition 1.1],[35, chapter 6.3]) A _crossed simplicial group_ $G$
is a sequence of groups $\\{G_{n}\\}_{n\geq 0}$ together with a category
$\Delta G$ satisfying the following conditions:

  1. (i)

$\Delta G$ contains $\Delta$ as a subcategory with the same objects.

  2. (ii)

$\mathrm{Aut}([n])$ is the opposite group $G_{n}^{op}$.

  3. (iii)

A morphism in $\Delta G$ can be uniquely written as the composite $\alpha\circ
g$ for some $\alpha\in\operatorname{Hom}_{\Delta}([m],[n])$ and $g\in
G_{m}^{op}$.

Observe that every crossed simplicial group is a (or rather has an underlying)
simplicial set, see [18, Lemma 1.3]. A _$G$ -set_ is a functor $\Delta
G^{op}\to\mathbf{Set}$, which by restriction has an underlying simplicial set.

######  Construction 4.1.2.

Let $X$ be a simplicial set. For a crossed simplicial group $G$, the $G$-set
$F_{G}(X)$ is defined in [18, Definition 4.3]. In simplicial degree $q$, we
have

$F_{G}(X)_{q}:=G_{q}\times X_{q}.$

The $G_{q}$ action on $F_{G}(X)_{q}$ is the left multiplication on $G_{q}$.
The faces and degeneracy maps are given by

(4.1) $d_{i}(g,x):=(d_{i}(g),d_{g^{-1}(i)}(x))\text{ and
}s_{i}(g,x):=(s_{i}(g),s_{g^{-1}(i)}(x)).$

According to [18, Proposition 5.1], we have the projection $p_{1}\colon
F_{G}(X)\to G$ given by $p_{1}(g,x):=g$. We also have the projection
$p_{2}\colon\lvert F_{G}(X)\rvert\to\lvert X\rvert$ given by

$p_{2}[(g,x),u]:=[x,gu]$

for $u\in\Delta_{top}^{q}$. Furthermore, the two projections define a
homeomorphism

$\lvert F_{G}(X)\rvert\cong\lvert G\rvert\times\lvert X\rvert.$

If $X$ is a $G$-set, then we have the evaluation map $ev\colon F_{G}(X)\to X$
given by

$ev(g,x):=gx.$

According to [18, Theorem 5.3], the composite map

$\lvert G\rvert\times\lvert X\rvert\xrightarrow{\cong}\lvert
F_{G}(X)\rvert\xrightarrow{ev}\lvert X\rvert$

defines a $\lvert G\rvert$-action on $\lvert X\rvert$. The _$G$ -geometric
realization of $X$_ is $\lvert X\rvert$ with this $\lvert G\rvert$-action.

######  Example 4.1.3.

We have the following three fundamental examples of crossed simplicial groups.
(1) [18, Example 2, section 1.5] introduces a crossed simplicial group $G$
with $G_{n}:={\mathbb{Z}/2}$. For this $G$, a $G$-set is called a _real
simplicial set_. The $G$-geometric realization is called the _real geometric
realization_.

Explicitly, a real simplicial set $X$ is a simplicial set equipped with
isomorphisms $w_{n}\colon X_{n}\to X_{n}$ with $w_{n}^{2}=\mathrm{id}$ for all
integer $n\geq 1$ satisfying the relations

$d_{i}w_{n}=w_{n-1}d_{n-i}\text{ and }s_{i}w_{n}=w_{n+1}s_{n-i}$

for $0\leq i\leq n$.
(2) [18, Example 4, section 1.5] introduces a crossed simplicial group $G$
with $G_{n}:=C_{n+1}$. For this $G$, a $G$-set is called a _cyclic set_ ,
which was defined by Connes. The $G$-geometric realization is called the
_cyclic geometric realization_ , and comes with an action of $S^{1}=SO(2)$
(see also [35, Theorem 7.1.4]).

Explicitly, a cyclic set $X$ is a simplicial set equipped with isomorphisms
$t_{n}\colon X_{n}\to X_{n}$ with $(t_{n})^{n+1}=\mathrm{id}$ for all integers
$n\geq 1$ satisfying the relations

$d_{i}t_{n}=t_{n-1}d_{i-1}\text{ and }s_{i}t_{n}=t_{n+1}s_{i-1}$

for $1\leq i\leq n$.
(3) [18, Example 5, section 1.5] introduces a crossed simplicial group $G$
with $G_{n}:=D_{n+1}$, where $D_{n}$ denotes the dihedral group of order $2n$.
For this $G$, a $G$-set is called a _dihedral set_. The $G$-geometric
realization is called the _dihedral geometric realization_.

Explicitly, a dihedral set $X$ is a simplicial set equipped with $t_{n}$ and
$w_{n}$ for all integer $n\geq 1$ satisfying $w_{n}t_{n}=t_{n}^{-1}w_{n}$ and
all the above conditions for $t_{n}$ and $w_{n}$.

There are obvious forgetful functors from dihedral to real and to cyclic sets.

For $\lvert G\rvert$, we obtain ${\mathbb{Z}/2}$, $S^{1}=U(1)=SO(2)$, and
$O(2)$ in the three examples above.

######  Example 4.1.4.

Let $G$ be the crossed simplicial set in Example 4.1.3(1). For a simplicial
set $X$, the bijection $G\times X_{q}=X_{q}\amalg X_{q}$ induces a canonical
isomorphism of $G$-sets

$F_{G}(X)\cong X\amalg X^{op},$

where $w_{n}\colon X_{n}\amalg X_{n}^{op}\to X_{n}\amalg X_{n}^{op}$ is the
switching map. If $X$ is a real simplicial set, then the evaluation map
$ev\colon F_{G}(X)\to X$ sends $x\in X_{n}^{op}$ to $w_{n}(x)$. It follows
that the ${\mathbb{Z}/2}$-action on $\lvert X\rvert$ is given by the composite
map

$\lvert X\rvert\xrightarrow{\cong}\lvert X^{op}\rvert\xrightarrow{w}\lvert
X\rvert,$

where the first map is the canonical identity.

######  Example 4.1.5.

Let $G$ be the crossed simplicial set in Example 4.1.3(2). For every integer
$n\geq 0$, we set

$\Lambda^{n}:=F_{G}(\Delta^{n}).$

This is Connes’ cyclic $n$-complex. We warn the reader that $F_{G}(X)$ for a
simplicial set is different from the cyclic bar construction.

###  4.2. Real and dihedral nerves

We now recall how commutative monoids with involution give rise to real and
dihedral simplicial sets, refining nerves and cyclic nerves for commutative
monoids without involutions.

######  Definition 4.2.1.

Suppose $M$ is a commutative monoid with involution $\sigma$. The _real nerve
of $M$_, denoted $\mathrm{N}^{\sigma}M$, is the real simplicial set whose
underlying simplicial set is the nerve of $M$ with
$(\mathrm{N}^{\sigma}M)_{q}:=N^{\times q}$ and $w$ is given by

$w(x_{1},\ldots,x_{q}):=(\sigma(x_{q}),\ldots,\sigma(x_{1}))$

in simplicial degree $q$. The _real bar construction_ , denoted
$\mathrm{B}^{\sigma}M$, is the real geometric realization of
$\mathrm{N}^{\sigma}M$. We refer to [15, Example 2.1.1] a similar account.

######  Definition 4.2.2.

Suppose $M$ is a commutative monoid with involution $\sigma$. The _dihedral
nerve of $M$_, denoted $\mathrm{N}^{\mathrm{di}}M$, is the dihedral set
defined as follows. In simplicial degree $q$, we have
$(\mathrm{N}^{\mathrm{di}}M)_{q}:=M^{\times q+1}$. The face maps are

$d_{i}(x_{0},\ldots,x_{q}):=\left\\{\begin{array}[]{ll}(x_{0},\ldots,x_{i-1},x_{i}+x_{i+1},x_{i+2},\ldots,x_{q})&\text{if
}i=0,\ldots,q-1\\\ (x_{q}+x_{0},x_{1},\ldots,x_{q-1})&\text{if
}i=q.\end{array}\right.$

The degeneracy maps are

$s_{i}(x_{0},\ldots,x_{q}):=(x_{0},\ldots,x_{i},0,x_{i+1},\ldots,x_{q})\text{
for }i=0,\ldots,q.$

The rotation maps are

$t(x_{0},\ldots,x_{q}):=(x_{0},x_{i},x_{i+1},\ldots,x_{q},x_{1},\ldots,x_{i-1})\text{
for }i=0,\ldots,q.$

The involution is

$w(x_{0},\ldots,x_{q}):=(\sigma(x_{0}),\sigma(x_{q}),\ldots,\sigma(x_{1})),$

compare also with [15, Example 2.1.2].

######  Remark 4.2.3.

This involution looks similar to the involution on the Hochschild complex as
studied in [35, 5.2.1]. In [35, section 5.2] Loday investigates several “real”
versions of Hochschild and cyclic homology for rings. The decompositions of
[35, Proposition 5.2.3, (5.2.7.1)] should be compared to the Bott sequence
[47, section 6] relating algebraic to hermitian $K$-theory, which also splits
after inverting $2$ or rather $1-\epsilon$.

Note that we obtain the _cyclic nerve of $M$_ [53, section 2.3], denoted
$\mathrm{N}^{\mathrm{cy}}M$, if we forget the involution structure.

The _dihedral bar construction_ , denoted $\mathrm{B}^{\mathrm{di}}M$, is the
dihedral geometric realization of $\mathrm{N}^{\mathrm{di}}M$. By Construction
4.1.2, $\mathrm{B}^{\mathrm{di}}M$ admits a canonical $O(2)$-action. The
underlying real simplicial set of $\mathrm{N}^{\mathrm{di}}M$ is different
from $\mathrm{N}^{\sigma}M$.

We have the map of dihedral sets

(4.2) $\mathrm{N}^{\mathrm{di}}M\to M$

sending $(x_{0},\ldots,x_{q})$ to $x_{0}+\cdots+x_{q}$ in simplicial degree
$q$, where we regard $M$ as the constant dihedral set whose rotation maps are
the identities and whose involutions are given by $\sigma$. For every
$\sigma$-orbit $I$ in $M$, we set

(4.3) $\mathrm{N}^{\mathrm{di}}(M;I):=\mathrm{N}^{\mathrm{di}}M\times_{M}I.$

Let $\mathrm{B}^{\mathrm{di}}(M;I)$ be its dihedral geometric realization. We
have an isomorphism of dihedral sets

(4.4)
$\mathrm{N}^{\mathrm{di}}M\cong\coprod_{I}\mathrm{N}^{\mathrm{di}}(M;I),$

where the coproduct runs over the $\sigma$-orbits in $M$.

######  Proposition 4.2.4.

Let $M$ and $L$ are commutative monoids with involutions. Then there is an
isomorphism of dihedral sets

(4.5) $\mathrm{N}^{\mathrm{di}}(M\times
L)\cong\mathrm{N}^{\mathrm{di}}M\times\mathrm{N}^{\mathrm{di}}L.$

###### Proof.

The two projections $M\times L\rightrightarrows M,L$ induce (4.5). To show
that it is an isomorphism, observe that it is given by the shuffle
homomorphism

$(M\times L)^{\times(q+1)}\to M^{\times(q+1)}\times L^{\times(q+1)}$

in simplicial degree $q$. ∎

######  Proposition 4.2.5.

Let $M$ and $L$ are commutative monoids with the involutions. Then there is an
isomorphism of dihedral sets

$\mathrm{N}^{\mathrm{di}}(M\times
L;(x,y))\cong\mathrm{N}^{\mathrm{di}}(M;x)\times\mathrm{N}^{\mathrm{di}}(L;y)$

for all $x\in M$ and $y\in L$.

###### Proof.

Follows from the commutativity of the square

${(M\times L)^{\times(q+1)}}$${M^{\times(q+1)}\times
L^{\times(q+1)}}$${M\times L}$${M\times L,}$$\scriptstyle{\mathrm{id}}$

where the upper horizontal homomorphism is the shuffle homomorphism, and the
vertical homomorphisms are the summation homomorphisms. ∎

######  Proposition 4.2.6.

Let $G$ be an abelian group with an involution $w$, and let $j$ be an element
of $G$. If $j=w(j)$, then there is an isomorphism of real simplicial sets

(4.6) $\mathrm{N}^{\mathrm{di}}(G;j)\xrightarrow{\cong}\mathrm{N}^{\sigma}G.$

If $j\neq w(j)$, then there is an isomorphism of real simplicial sets

(4.7)
$\mathrm{N}^{\mathrm{di}}(G;\\{j,w(j)\\})\xrightarrow{\cong}{\mathbb{Z}/2}\times\mathrm{N}^{\sigma}G,$

As a consequence, there is an isomorphism of real simplicial sets

(4.8) $\mathrm{N}^{\mathrm{di}}G\simeq G\times\mathrm{N}^{\sigma}G.$

###### Proof.

If $j=w(j)$, the assignment

$(x_{0},\ldots,x_{q})\mapsto(x_{1},\ldots,x_{q})$

in simplicial degree $q$ constructs the isomorphism (4.6). If $j\neq w(j)$,
the assignment

$(x_{0},\ldots,x_{q})\mapsto(a,x_{1},\ldots,x_{q})$

in simplicial degree $q$ constructs the isomorphism (4.7), where $a:=0$ (resp.
$a:=1$) if $x_{0}+\cdots+x_{q}=j$ (resp. $x_{0}+\cdots+x_{q}=-j$). ∎

######  Definition 4.2.7.

Let $S^{\sigma}$ be the ${\mathbb{Z}/2}$-space whose underlying space is
$S^{1}$ and whose ${\mathbb{Z}/2}$-action is given by the reflection $(x,y)\in
S^{1}\subset\mathbb{R}^{2}\mapsto(x,-y)$. This is the real geometric
realization of the real simplicial set whose underlying simplicial set is the
simplicial circle $\Delta^{1}/\partial\Delta^{1}$ and whose involution on the
non-degenerate simplicies is the identity. One easily checks that the
relations in Example 4.1.3 uniquely determines the higher $w_{n}$ on
$\Delta^{1}/\partial\Delta^{1}$icies is the identity. To understand the
involution on the real realization, the reader is advised to look at the
explanations in Example 4.1.4.

######  Proposition 4.2.8.

For the additive monoid $\mathbb{Z}$ with the trivial involution, there is a
${\mathbb{Z}/2}$-homotopy equivalence

$\mathrm{B}^{\sigma}\mathbb{Z}\simeq S^{\sigma}.$

###### Proof.

Consider the element $1\in(\mathrm{N}^{\sigma}\mathbb{Z})_{1}\simeq\mathbb{Z}$
in simplicial degree $1$. This is fixed by the involution on
$\mathrm{N}^{\sigma}\mathbb{Z}$, so the real geometric realization of the real
simplicial subset of $\mathrm{N}^{\sigma}\mathbb{Z}$ whose only non-degenerate
simplicies are $1$ and the base point is $S^{\sigma}$. Hence we obtain a map
of ${\mathbb{Z}/2}$-spaces $S^{\sigma}\to\mathrm{B}^{\sigma}\mathbb{Z}$. This
is the usual homotopy equivalence after forgetting the involutions. It remains
to check that the induced map

(4.9)
$(S^{\sigma})^{{\mathbb{Z}/2}}\to(\mathrm{B}^{\sigma}\mathbb{Z})^{{\mathbb{Z}/2}}$

is a homotopy equivalence as well.

By [15, Proposition 2.1.6], $(\mathrm{B}^{\sigma}\mathbb{Z})^{\mathbb{Z}/2}$
is the classifying space of the category $\mathop{\mathrm{Sym}}\mathbb{Z}$,
whose objects are integers, and whose morphisms are given by

$\operatorname{Hom}_{\mathop{\mathrm{Sym}}\mathbb{Z}}(a,b):=\left\\{\begin{array}[]{ll}*&\text{if
}2\mid a-b,\\\ \emptyset&\text{if }2\nmid a-b.\end{array}\right.$

The classifying space of $\mathop{\mathrm{Sym}}\mathbb{Z}$ is obviously
homotopy equivalent to $S^{0}$, and hence we deduce a homotopy equivalence

(4.10) $S^{0}\simeq(\mathrm{B}^{\sigma}\mathbb{Z})^{\mathbb{Z}/2}.$

This implies that $\pi_{n}((\mathrm{B}^{\sigma}\mathbb{Z})^{\mathbb{Z}/2})$ is
trivial for all integer $n>0$ and
$\pi_{0}((\mathrm{B}^{\sigma}\mathbb{Z})^{\mathbb{Z}/2})$ consists of two
points.

Hence it suffices to show that the map

(4.11)
$\pi_{0}((S^{\sigma})^{{\mathbb{Z}/2}})\to\pi_{0}((\mathrm{B}^{\sigma}\mathbb{Z})^{{\mathbb{Z}/2}})$

induced by (4.9) is injective. Let $\mathrm{sd}_{\sigma}$ denote Segal’s
edgewise subdivision functor in [49]. In simplicial degree $1$, the two
degeneracy maps in $\mathrm{sd}_{\sigma}(\mathrm{N}^{\sigma}\mathbb{Z})$ are
given by

$(x_{1},x_{2},x_{3})\mapsto x_{2},x_{1}+x_{2}+x_{3}.$

Then edges $(x_{1},x_{2},x_{1})$ in the ${\mathbb{Z}/2}$-fixed point space
connects $x_{2}$ and $2x_{1}+x_{2}$. In simplicial degree $0$,
$\mathrm{sd}_{\sigma}(S^{\sigma})$ consists of two vertices $0$ and $1$, whose
images in $\mathrm{B}^{\sigma}\mathbb{Z}$ are not connected. Hence (4.11) is
injective as claimed. ∎

Together with (4.6), we obtain an equivalence

(4.12) $\mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)\simeq S^{\sigma}$

for all $j\in\mathbb{Z}$.

######  Definition 4.2.9.

Let $\mathbb{Z}^{\sigma}$ denote the monoid $\mathbb{Z}$ with the involution
$x\mapsto-x$ for $x\in\mathbb{Z}$.

######  Proposition 4.2.10.

There is a ${\mathbb{Z}/2}$-homotopy equivalence

$\mathrm{B}^{\sigma}\mathbb{Z}^{\sigma}\simeq S^{1}.$

###### Proof.

See [14, Example 5.13]. ∎

Combine with Proposition 4.2.6 to obtain an equivalence in
$\mathrm{Sp}_{{\mathbb{Z}/2}}$

(4.13)
$\mathbb{S}[\mathrm{B}^{\mathrm{di}}\mathbb{Z}^{\sigma}]\simeq\mathbb{S}[S^{1}]\oplus\bigoplus_{j\in\mathbb{Z}}i_{*}i^{*}\mathbb{S}[S^{1}].$

For every integer $j$, let $S^{\sigma}(j)$ denote the space $S^{1}$ with the
$O(2)$-action, whose $SO(2)$-action is given by $(t,x)\mapsto t^{j}x$ for
$t\in SO(2)$ and $x\in S^{1}$, and whose ${\mathbb{Z}/2}$-action is given by
the complex conjugate $x\mapsto\overline{x}$. Here, we regard $S^{1}$ as the
unit circle in $\mathbb{C}$. For $j\geq 0$, let $\Delta^{j}_{\sigma}$ be the
${\mathbb{Z}/2}$-space whose involution is the reflection mapping the vertex
$i$ to $j-i$ for all $0\leq i\leq j$.

The following computations refine [44, Proposition 3.21].

######  Proposition 4.2.11.

We have $\mathrm{B}^{\mathrm{di}}(\mathbb{N};0)\cong*$. For every integer
$j\geq 1$, there is an $O(2)$-equivariant homeomorphism

(4.14)
$\mathrm{B}^{\mathrm{di}}(\mathbb{N};j)\simeq(S^{\sigma}\times\Delta^{j-1}_{\sigma})/C_{j},$

where the $C_{j}$-action on $\Delta^{j-1}_{\sigma}$ is the cyclic permutation.
Hence there is an $O(2)$-equivariant deformation retract

(4.15) $\mathrm{B}^{\mathrm{di}}\mathbb{N}\simeq*\amalg\coprod_{j\geq
1}S^{\sigma}(j).$

###### Proof.

The $(j-1)$-simplex $(1,\ldots,1)$ generates
$\mathrm{N}^{\mathrm{cy}}(\mathbb{N};j)$ as a cyclic set. Use this to
construct a surjective map of cyclic sets

(4.16) $\Lambda^{j-1}\to\mathrm{N}^{\mathrm{cy}}(\mathbb{N};j).$

In simplicial degree $q$, this can be written as the map

$C_{q+1}\times\operatorname{Hom}([q],[j-1])\to\\{(x_{0},\ldots,x_{q})\in\mathbb{N}^{q+1}:x_{0}+\cdots+x_{q}=j\\}.$

sending $(t,f)$ to

$(f(t(0))-f(t(q)),f(t(1))-f(t(0)),\ldots,f(t(q))-f(t(q-1))).$

In this formulation, the values are calculated modulo $j$. Let
$\Lambda_{\sigma}^{j-1}$ be the dihedral set whose underlying cyclic set is
$\Lambda^{j-1}$ and whose involution is given by

$(t,f)\mapsto(q+1-t,\rho\circ f),$

where $\rho\colon[j-1]\to[j-1]$ is the map sending $x\in[j-1]$ to $j-1-x$.
Then (4.16) becomes a morphism of dihedral sets

(4.17) $\Lambda_{\sigma}^{j-1}\to\mathrm{N}^{\mathrm{di}}(\mathbb{N};j).$

The cyclic geometric realization of (4.16) is $S^{1}$-homeomorphic to the
quotient map

$S^{1}\times\Delta^{j-1}\to(S^{1}\times\Delta^{j-1})/C_{j},$

see the proof of [23, Lemma 2.2.3] or [44, Proposition 3.20]. Use the
observation in Example 4.1.4 to show that the real geometric realization of
$\lvert\Lambda_{\sigma}^{j-1}\rvert$ is ${\mathbb{Z}/2}$-homeomorphic to
$S^{\sigma}\times\Delta^{j-1}_{\sigma}$. Combine these two facts to deduce
that the dihedral geometric realization of (4.17) is $O(2)$-homeomorphic to
the quotient map

$S^{\sigma}\times\Delta^{j-1}_{\sigma}\to(S^{\sigma}\times\Delta^{j-1}_{\sigma})/C_{j}$

In particular, we obtain (4.14). Since $\Delta_{\sigma}^{j-1}$ is
$D_{j}$-contractible to its barycenter, we obtain (4.15), ∎

Let us review what the proof of [44, Proposition 3.21] contains. For every
integer $r\geq 1$, let $\mathrm{sd}_{r}$ denote the $r$-fold edgewise
subdivision functor in [10]. For every integer $j$, the $r$-fold power map
$\mathrm{N}^{\mathrm{di}}(\mathbb{Z};j)\to\mathrm{sd}_{r}\mathrm{N}^{\mathrm{di}}(\mathbb{Z};j)$
given by

(4.18)
$(m_{0},\ldots,m_{q})\mapsto(m_{0},\ldots,m_{q},\ldots,m_{0},\ldots,m_{q})$

induces a homeomorphism

(4.19)
$\mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)\xrightarrow{\cong}\mathrm{B}^{\mathrm{di}}(\mathbb{Z};rj)^{C_{r}}.$

If $r\nmid j$, then we have

(4.20) $\mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)^{C_{j}}=\emptyset.$

Suppose $j\geq 0$. We similarly have a homeomorphism

(4.21)
$\mathrm{B}^{\mathrm{di}}(\mathbb{N};j)\xrightarrow{\cong}\mathrm{B}^{\mathrm{di}}(\mathbb{N};rj)^{C_{r}}$

If $r\nmid j$, then we have we have

(4.22) $\mathrm{B}^{\mathrm{di}}(\mathbb{N};j)^{C_{r}}=\emptyset.$

If $H$ is a closed subgroup of $SO(2)$, then the induced map

(4.23)
$\mathrm{B}^{\mathrm{di}}(\mathbb{N};j)^{H}\to\mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)^{H}$

is a homotopy equivalence.

######  Proposition 4.2.12.

For every integer $j\geq 1$, the induced map

$\mathrm{B}^{\mathrm{di}}(\mathbb{N};j)\to\mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)$

is an $O(2)$-equivariant homotopy equivalence.

###### Proof.

We need to show that (4.23) is a homotopy equivalence for all closed subgroups
$H$ of $O(2)$.

If $H=O(2)$, then
$\mathrm{B}^{\mathrm{di}}(\mathbb{N};j)^{O(2)}=\mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)^{O(2)}=\emptyset$
by (4.20) and (4.22). Hence the remaining case is $H=D_{r}$ for every integer
$r\geq 1$. Since (4.18) commutes with $w$, (4.21) is
${\mathbb{Z}/2}$-equivariant. Similarly, (4.19) is
${\mathbb{Z}/2}$-equivariant. Hence we have the induced homeomorphisms

$\mathrm{B}^{\mathrm{di}}(\mathbb{N};j)^{{\mathbb{Z}/2}}\xrightarrow{\cong}\mathrm{B}^{\mathrm{di}}(\mathbb{N};rj)^{D_{r}}\text{
and
}\mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)^{{\mathbb{Z}/2}}\xrightarrow{\cong}\mathrm{B}^{\mathrm{di}}(\mathbb{Z};rj)^{D_{r}}.$

Combine with (4.22) to reduce to the case when $H={\mathbb{Z}/2}$.

Propositions 4.2.6 and 4.2.8 give a homotopy equivalence

(4.24) $\mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)^{{\mathbb{Z}/2}}\simeq S^{0}.$

By Proposition 4.2.11, we also have a homotopy equivalence
$\mathrm{B}^{\mathrm{di}}(\mathbb{N};j)^{{\mathbb{Z}/2}}\simeq S^{0}$. Hence
it remains to check that the induced map

(4.25)
$\pi_{0}(\mathrm{B}^{\mathrm{di}}(\mathbb{N};j)^{{\mathbb{Z}/2}})\to\pi_{0}(\mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)^{{\mathbb{Z}/2}})$

is a bijection. Recall that $\mathrm{sd}_{\sigma}$ denotes Segal’s edgewise
subdivision functor. In simplicial degree $1$, the two degeneracy maps in
$\mathrm{sd}_{\sigma}(\mathrm{N}^{\mathrm{di}}(\mathbb{N};j))$ and
$\mathrm{sd}_{\sigma}(\mathrm{N}^{\mathrm{di}}(\mathbb{Z};j))$ are given by

$(x_{0},x_{1},x_{2},x_{3})\mapsto(x_{3}+x_{0}+x_{1},x_{2}),(x_{0},x_{1}+x_{2}+x_{3}).$

The edge $(x_{0},x_{1},x_{2},x_{1})$ in the ${\mathbb{Z}/2}$-fixed point
spaces connects $(x_{0},2x_{1}+x_{2})$ and $(x_{0}+2x_{1},x_{2})$. Hence we
have

$\pi_{0}(\mathrm{B}^{\mathrm{di}}(\mathbb{N};j))\cong\\{(x_{0},x_{1})\in\mathbb{N}^{2}:x_{0}+x_{1}=j\\}/\sim$

and

$\pi_{0}(\mathrm{B}^{\mathrm{di}}(\mathbb{Z};j))\cong\\{(x_{0},x_{1})\in\mathbb{Z}^{2}:x_{0}+x_{1}=j\\}/\sim,$

where $(x_{0},x_{1})\sim(x_{0}^{\prime},x_{1}^{\prime})$ if and only if
$x_{0}-x_{0}^{\prime}$ is even. This shows that (4.25) is a bijection. ∎

######  Proposition 4.2.13.

For a commutative monoid $M$ with involution, there is an equivalence in
$\mathrm{Sp}_{{\mathbb{Z}/2}}$

$\mathrm{THR}(\mathbb{S}[M])\simeq\mathbb{S}[\mathrm{B}^{\mathrm{di}}M].$

###### Proof.

See [30], and also [14, Proposition 5.9]. ∎

######  Proposition 4.2.14.

Let $A$ be a commutative ring with involution, and let $M$ be a commutative
monoid with involution. Then there is an equivalence in
$\mathrm{Sp}_{{\mathbb{Z}/2}}$

$\mathrm{H}(A[M])\simeq\mathrm{H}A\wedge\mathbb{S}[M].$

###### Proof.

If we regard $M$ as a ${\mathbb{Z}/2}$-set, then $M$ is a coproduct of copies
${\mathbb{Z}/2}$ and $e$. If $M=e$, then the claim is clear. Hence it suffices
to show that there is an equivalence

(4.26)
$\mathrm{H}((i^{*}A)^{\oplus{\mathbb{Z}/2}})\simeq\mathrm{H}A\wedge\Sigma^{\infty}({\mathbb{Z}/2})_{+},$

where $(i^{*}A)^{\oplus{\mathbb{Z}/2}}$ is the commutative monoid $A\oplus A$
with the involution given by $(x,y)\mapsto(y,x)$. By (2.7) and Propositions
A.1.10, A.2.7(3), and 2.3.3, we obtain equivalences

$\mathrm{H}A\wedge i_{\sharp}i^{*}\mathbb{S}\simeq
i_{\sharp}(i^{*}\mathrm{H}A\wedge i^{*}\mathbb{S})\simeq
i_{\sharp}\mathrm{H}i^{*}A\simeq
i_{*}\mathrm{H}i^{*}A\simeq\mathrm{H}((i^{*}A)^{\oplus{\mathbb{Z}/2}}).$

Together with the equivalence
$i_{\sharp}i^{*}\mathbb{S}\simeq\Sigma^{\infty}({\mathbb{Z}/2})_{+}$, we
obtain (4.26). ∎

######  Proposition 4.2.15.

Let $A$ be a commutative ring with involution, and let $M$ be a commutative
monoid with involution. Then there is a canonical equivalence in
$\mathrm{Sp}_{{\mathbb{Z}/2}}$

$\mathrm{THR}(A[M])\simeq\mathrm{THR}(A)\wedge\mathbb{S}[\mathrm{B}^{\mathrm{di}}M].$

###### Proof.

By Propositions 2.1.5 and 4.2.13, we have equivalences

$\mathrm{THR}(\mathrm{H}A\wedge\mathbb{S}[M])\simeq\mathrm{THR}(\mathrm{H}A)\wedge\mathrm{THR}(\mathbb{S}[M])\simeq\mathrm{THR}(A)\wedge\mathbb{S}[\mathrm{B}^{\mathrm{di}}M].$

Proposition 4.2.14 finishes the proof. ∎

##  5\. Properties of real topological Hochschild homology

###  5.1. THR of the projective line

We now establish the first computations of $\mathrm{THR}$ for non-affine
schemes, namely $\mathbb{P}^{1}$ and $\mathbb{P}^{\sigma}$.

######  Remark 5.1.1.

uite qHermitian $K$-theory $\mathbf{KO}$ resp. $\mathbf{KR}$ is not an
orientable theory, that is the usual projective bundle formula as for Chow
groups and algebraic $K$-theory does not hold. The computation of
$\mathbb{P}^{1}$ over regular rings is [26, Proposition 6.1], where a
$(8,4)$-motivic periodic spectrum $\mathbf{KO}$ is constructed. This
computation is extended by [47, Theorem 9.10] to rather general base schemes
(still with 2 invertible). For the projective line with involution, a
variation of Schlichting’s proof leads to the computation of
$\mathbf{KR}(\mathbb{P}^{\sigma})$, see [11, Theorem 5.1] and compare [55,
Theorem 7.1] and [29] for different proofs. In particular, this leads to an
equivariant motivic spectrum $\mathbf{KR}$ which is
$\mathbb{P}^{1}\wedge\mathbb{P}^{\sigma}$-periodic. For further periodicities
of $\mathbf{KR}$ see [29, Theorem 10]. In their notation, we have
$\mathbb{P}^{1}\simeq S^{1}\wedge S^{\alpha}$ and $\mathbb{P}^{\sigma}\simeq
S^{\gamma}\wedge S^{\gamma\alpha}\simeq\mathbb{P}^{1}_{-}$. For the definition
of $S^{\gamma}=S^{\sigma}$ in the motivic setting and a proof of the last
equivalence, we refer to [11, section 2.5]. Although $\mathrm{THR}$ is not
$\mathbb{A}^{1}$-invariant, it seems reasonable to expect that the formulas
for $\mathrm{THR}(\mathbb{P}^{n})$ are similar to those for $\mathbf{KR}$. In
the cases considered below this is indeed the case: the following Proposition
implies that
$\Omega^{1+\alpha}\mathrm{THR}\simeq\Sigma^{\gamma-1}\mathrm{THR}$. Smashing
with $S^{1}$, we obtain the same periodicity as (35) in [29]. Similarly, the
next proposition corresponds to (36) of loc. cit.

The following computations rely on Proposition 4.2.13 and the computations for
dihedral nerves in the previous section.

######  Theorem 5.1.2.

For any $X\in\mathbf{Sch}_{{\mathbb{Z}/2}}$, there is an equivalence of
${\mathbb{Z}/2}$-spectra

$\mathrm{THR}(X\times\mathbb{P}^{1})\simeq\mathrm{THR}(X)\oplus\Sigma^{\sigma-1}\mathrm{THR}(X).$

###### Proof.

For notational convenience, we will write the proof as if everything takes
place over $\mathbb{S}$ rather than $X$. Using the description of
$\mathrm{THR}$ for spherical groups rings from Proposition 4.2.13 and its
extension to log schemes with involution from Proposition 4.2.15, we are
reduced to consider the following homotopy cocartesian square, corresponding
to the standard Zariski cover of $\mathbb{P}^{1}$ by two copies of
$\mathbb{A}^{1}$ and using Corollary 3.4.9 and the description of
$\mathrm{THR}(\mathbb{S}[M])$ from Proposition 4.2.13:

${\mathrm{THR}(\mathbb{P}^{1}_{\mathbb{S}})}$${\mathbb{S}[\mathrm{B}^{\mathrm{di}}\mathbb{N}]}$${\mathbb{S}[\mathrm{B}^{\mathrm{di}}(-\mathbb{N})]}$${\mathbb{S}[\mathrm{B}^{\mathrm{di}}\mathbb{Z}]}$

Here the notation $\mathbb{N}$ and $-\mathbb{N}$ indicates the two different
embeddings of $\mathbb{A}^{1}$ in $\mathbb{P}^{1}$. It is crucial to notice
that even as $\mathbb{G}_{m}$ has trivial involution, the involution given by
the dihedral nerve (see Definition 4.2.2 above) yields nontrivial involutions.
Using the decomposition of (4.4) and Propositions 4.2.6, 4.2.8, and 4.2.11, we
obtain the following ${\mathbb{Z}/2}$-equivariant (homotopy) cocartesian
square:

${\mathrm{THR}(\mathbb{P}^{1}_{\mathbb{S}})}$${\mathbb{S}[*\amalg\coprod_{j\geq
1}S^{\sigma}]}$${\mathbb{S}[*\amalg\coprod_{j\leq-1}S^{\sigma}]}$${\mathbb{S}[\coprod_{j\in\mathbb{Z}}S^{\sigma}]}$

An obvious cancellation, using Proposition 4.2.12 on the relevant maps, yields
the following ${\mathbb{Z}/2}$-equivariant (homotopy) cocartesian square

${\mathrm{THR}(\mathbb{P}^{1}_{\mathbb{S}})}$${\mathbb{S}[*]}$${\mathbb{S}[*]}$${\mathbb{S}[S^{\sigma}]}$

and the result follows. ∎

We now turn to the slightly more subtle computation of the projective line
$\mathbb{P}^{\sigma}$ with involution. Recall that $(i_{\sharp},i^{*},i_{*})$
denotes the free-forgetful-cofree adjunction for the map
$i\colon\mathrm{pt}\to\mathrm{B}({\mathbb{Z}/2})$. Let
$\Sigma^{\sigma}\colon\mathrm{Sp}_{{\mathbb{Z}/2}}\to\mathrm{Sp}_{{\mathbb{Z}/2}}$
be the functor $\Sigma^{\infty}S^{\sigma}\wedge(-)$, which has an inverse
functor $\Sigma^{-\sigma}$ since the sphere $\Sigma^{\infty}S^{\sigma}$ is
$\wedge$-invertible in $\mathrm{Sp}_{{\mathbb{Z}/2}}$. For integers $m$ and
$n$, we set $\Sigma^{m+n\sigma}:=\Sigma^{m}(\Sigma^{\sigma})^{\wedge n}$,
which is a functor
$\mathrm{Sp}_{{\mathbb{Z}/2}}\to\mathrm{Sp}_{{\mathbb{Z}/2}}$. We also set
$\Sigma^{m+n\sigma}:=\Sigma^{m+n\sigma}\mathbb{S}\in\mathrm{Sp}_{{\mathbb{Z}/2}}$
for abbreviation.

For an adjoint pair of $\infty$-categories
$F:\mathcal{C}\rightleftarrows\mathcal{D}:G$, let $ad\colon\mathrm{id}\to GF$
(resp. $ad^{\prime}\colon FG\to\mathrm{id}$) denote the unit (resp. counit).

######  Lemma 5.1.3.

There is a natural equivalence of functors

$\Sigma^{-\sigma}\simeq\mathrm{fib}(\mathrm{id}\xrightarrow{ad}i_{*}i^{*}).$

###### Proof.

Consider the cocartesian square of ${\mathbb{Z}/2}$-spaces

${S^{\sigma}-\\{(1,0),(-1,0)\\}}$${S^{\sigma}-\\{(1,0)\\}}$${S^{\sigma}-\\{(-1,0)\\}}$${S^{\sigma}.}$

There are ${\mathbb{Z}/2}$-homotopy equivalences
$S^{\sigma}-\\{(1,0),(-1,0)\\}\simeq{\mathbb{Z}/2}$ and
$S^{\sigma}-\\{(1,0)\\}\simeq S^{\sigma}-\\{(-1,0)\\}\simeq\mathrm{pt}$.
Together with the explicit descriptions of $i_{\sharp}$ and $i^{*}$ in
Construction A.2.4, we obtain a natural equivalence

(5.1)
$\Sigma^{\sigma}\simeq\mathrm{cofib}(i_{\sharp}i^{*}\xrightarrow{ad^{\prime}}\mathrm{id}).$

By adjunction, we obtain the desired natural equivalence. ∎

######  Theorem 5.1.4.

For any $X\in\mathbf{Sch}_{{\mathbb{Z}/2}}$, there is an equivalence of
${\mathbb{Z}/2}$-spectra

$\mathrm{THR}(X\times\mathbb{P}^{\sigma})\simeq\mathrm{THR}(X)\oplus\Sigma^{1-\sigma}\mathrm{THR}(X).$

###### Proof.

As before, we will write the proof as if everything takes place over
$\mathbb{S}$. Consider the cartesian equivariant Nisnevich square

(5.2)
${{\mathbb{Z}/2}\times\mathbb{G}_{m,\mathbb{S}}^{\sigma}}$${(\mathbb{P}_{\mathbb{S}}^{1}-\infty)\amalg(\mathbb{P}_{\mathbb{S}}^{1}-0)}$${\mathbb{G}_{m,\mathbb{S}}^{\sigma}}$${\mathbb{P}_{\mathbb{S}}^{\sigma}.}$

as in [11, Lemma 2.23], where the ${\mathbb{Z}/2}$-action on the upper right
corner is induced by the action on $\mathbb{P}_{\mathbb{S}}^{\sigma}$.

Propositions 2.1.3 and 3.4.7 give equivalences

$\mathrm{THR}((\mathbb{P}_{\mathbb{S}}^{1}-\infty)\amalg(\mathbb{P}_{S}^{1}-0))\simeq
i_{*}\mathrm{THR}(i^{*}\mathbb{A}_{\mathbb{S}}^{1})\simeq
i_{*}i^{*}\mathrm{THR}(\mathbb{A}_{S}^{1}).$

We similarly have an equivalence
$\mathrm{THR}({\mathbb{Z}/2}\times\mathbb{G}_{m,\mathbb{S}}^{\sigma})\simeq
i_{*}i^{*}\mathrm{THR}(\mathbb{G}_{m,\mathbb{S}}^{\sigma})$ since there are
isomorphisms
${\mathbb{Z}/2}\times\mathbb{G}_{m,\mathbb{S}}^{\sigma}\cong{\mathbb{Z}/2}\times\mathbb{G}_{m,\mathbb{S}}$
and $i^{*}\mathbb{G}_{m,\mathbb{S}}^{\sigma}\cong
i^{*}\mathbb{G}_{m,\mathbb{S}}$. Applying $\mathrm{THR}$ to (5.2),
identifications similar to those in the proof of Theorem 5.1.2 – but using
(4.13) rather than Proposition 4.2.8 – yield the following homotopy
cocartesian square:

(5.3)
${\mathrm{THR}(\mathbb{P}_{\mathbb{S}}^{\sigma})}$${i_{*}i^{*}(\mathbb{S}\oplus\bigoplus_{j>0}\Sigma^{\infty}_{+}S^{\sigma})}$${i_{*}i^{*}(\mathbb{S}\oplus\bigoplus_{j>0}\Sigma^{\infty}_{+}S^{1})}$${\Sigma_{+}^{\infty}S^{1}\oplus\bigoplus_{j>0}i_{*}i^{*}\Sigma_{+}^{\infty}S^{1}}$${i_{*}i^{*}(\Sigma_{+}^{\infty}S^{1}\oplus\bigoplus_{j>0}i_{*}i^{*}\Sigma_{+}^{\infty}S^{1})}$$\scriptstyle{\simeq}$

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extracted from (5.3), where the right vertical map (resp. lower horizontal
map) is obtained by applying $i_{*}i^{*}$ to the left (resp. right) of the
unit map $\mathrm{id}\to i_{*}i^{*}$. Since $\Phi^{{\mathbb{Z}/2}}i_{*}\simeq
0$ by Proposition A.2.7(3),(5), $\Phi^{{\mathbb{Z}/2}}Q$ is cartesian. The
objects in the square $i^{*}Q$ are direct sums of
$\bigoplus_{j>0}\Sigma_{+}^{\infty}S^{1}$, and the right vertical and lower
horizontal maps are the matrix multiplications given by

$\left(\begin{array}[]{cc}1&0\\\ 1&0\\\ 0&1\\\
0&1\end{array}\right)\quad\left(\begin{array}[]{cc}1&0\\\ 0&1\\\ 0&1\\\
1&0\end{array}\right)$

for certain choices of bases. From this, one can check that $i^{*}Q$ is
cartesian too. It follows that $Q$ is cartesian.

Hence the other direct summand of the square (5.3)

${\mathrm{THR}(\mathbb{P}_{\mathbb{S}}^{\sigma})}$${i_{*}i^{*}\Sigma^{0}}$${\Sigma^{0}\oplus\Sigma^{1}}$${i_{*}i^{*}\Sigma^{0}\oplus
i_{*}i^{*}\Sigma^{1}}$

is also cartesian. It follows that
$\mathrm{THR}(\mathbb{P}_{\mathbb{S}}^{\sigma})$ is equivalent to the direct
sum

$\lim(\Sigma^{0}\to i_{*}i^{*}\Sigma^{0}\leftarrow
i_{*}i^{*}\Sigma^{0})\oplus\lim(\Sigma^{1}\to i_{*}i^{*}\Sigma^{1}\leftarrow
0).$

Together with Lemma 5.1.3, we obtain the desired equivalence. ∎

######  Proposition 5.1.5.

There is an equivalence of ${\mathbb{Z}/2}$-spectra

$\mathrm{THR}(X)\simeq\Omega_{\mathbb{P}^{1}\wedge\mathbb{P}^{\sigma}}\mathrm{THR}(X).$

###### Proof.

Combine Theorems 5.1.2 and 5.1.4. ∎

###  5.2. THR of projective spaces

######  Definition 5.2.1.

As usual, for any integer $n\geq 1$, we consider the $n$-cube
$(\Delta^{1})^{n}$ as a partially ordered set, and use the same symbol for the
associated category. For an $\infty$-category $\mathcal{C}$, an _$n$ -cube in
$\mathcal{C}$_ is a functor

$Q\colon\mathrm{N}(\Delta^{1})^{n}\to\mathcal{C},$

compare [37, Definition 6.1.1.2]. If $\mathcal{C}$ admits limits, the _total
fiber of $Q$_ is defined to be

$\mathrm{tfib}(Q):=\mathrm{fib}(Q(0,\ldots,0)\to\lim
Q|_{(\Delta^{1})^{n}-\\{(0,\ldots,0)\\}}).$

The following $\infty$-categorical result can be shown by dualizing the
arguments from [7, Appendix A].

######  Proposition 5.2.2.

Let $Q$ be an $n$-cube in an $\infty$-category $\mathcal{C}$ with small
limits, where $n$ is a nonnegative integer. Then for every integer $1\leq
i\leq n$, there exists a fiber sequence

$\mathrm{tfib}(Q)\to\mathrm{tfib}(Q|_{(\Delta^{1})^{i-1}\times\\{0\\}\times(\Delta^{1})^{n-i-1}})\to\mathrm{tfib}(Q|_{(\Delta^{1})^{i-1}\times\\{1\\}\times(\Delta^{1})^{n-i-1}}).$

######  Proposition 5.2.3.

Let $\mathcal{C}$ be a symmetric monoidal stable $\infty$-category with small
limits such that the tensor product operation on $\mathcal{C}$ preserves fiber
sequences in each variable. If $Q$ is an $n$-cube and $f\colon X_{0}\to X_{1}$
is a map in $\mathcal{C}$, then there is a canonical equivalence

$\mathrm{tfib}(Q)\otimes\mathrm{fib}(f)\simeq\mathrm{tfib}(Q\otimes f),$

where $Q\otimes f$ is the associated $(n+1)$-cube sending
$(a_{1},\ldots,a_{n+1})\in(\Delta^{1})^{n+1}$ to $Q(a_{1},\ldots,a_{n})\otimes
X_{a_{n+1}}$.

###### Proof.

This is again obtained by dualizing the arguments for the corresponding
Proposition in [7]. An intermediate step is to show $\mathrm{tfib}(Q)\otimes
X\simeq\mathrm{tfib}(Q\otimes X)$ for any $X\in\mathcal{C}$, where $Q\otimes
X$ is the associated $n$-cube sending
$(a_{1},\ldots,a_{n})\in(\Delta^{1})^{n}$ to $Q(a_{1},\ldots,a_{n})\otimes X$.
Then one can use Proposition 5.2.2. ∎

######  Proposition 5.2.4.

Let $\mathcal{C}$ be a symmetric monoidal stable $\infty$-category with small
limits such that the tensor product operation on $\mathcal{C}$ preserves fiber
sequences in each variable. If $i_{1}\colon X_{0,1}\to X_{1,1}$, $\ldots$,
$i_{n}\colon X_{0,n}\to X_{1,n}$ are maps in $\mathcal{C}$, then there is a
canonical equivalence

$\mathrm{fib}(i_{1})\otimes\cdots\otimes\mathrm{fib}(i_{n})\simeq\mathrm{tfib}(i_{1}\otimes\cdots\otimes
i_{n}),$

where $i_{1}\otimes\cdots\otimes i_{n}$ is the associated $n$-cube sending
$(a_{1},\ldots,a_{n})\in(\Delta^{1})^{n}$ to $X_{a_{1},1}\otimes\cdots\otimes
X_{a_{n},n}$ and arrows given by tensor products of $i_{j}$s and identities.

###### Proof.

Use Proposition 5.2.3 repeatedly. ∎

######  Proposition 5.2.5.

Suppose $X\in\mathbf{Sch}_{{\mathbb{Z}/2}}$ and $\\{U_{1},\ldots,U_{n}\\}$ is
a Zariski covering of $X$ with the induced involutions. Let $Q$ be the
$S$-cube given by

$Q(\\{i_{1},\ldots,i_{r}\\}):=U_{i_{1}}\cap\cdots\cap U_{i_{r}}$

for nonempty $\\{i_{1},\ldots,i_{r}\\}\subset S:=\\{1,\ldots,n\\}$ and
$Q(\emptyset):=X$. Then there is an equivalence

$\mathrm{tfib}(\mathrm{THR}(Q))\simeq 0.$

###### Proof.

Let us use the equivalence $\mathrm{N}(\mathbf{P}(S))\simeq(\Delta^{1})^{n}$.
We proceed by induction on $n$. The claim is clear if $n=1$. Assume $n>1$. By
induction, we fiber sequences

$\mathrm{tfib}(\mathrm{THR}(Q|_{\\{0\\}\times(\Delta^{1})^{n-1}}))\to\mathrm{THR}(X)\to\mathrm{THR}(U_{2}\cup\cdots\cup
U_{n})$

and

$\mathrm{tfib}(\mathrm{THR}(Q|_{\\{1\\}\times(\Delta^{1})^{n-1}}))\to\mathrm{THR}(U_{1})\to\mathrm{THR}(U_{1}\cap(U_{2}\cup\cdots\cup
U_{n})).$

Together with Proposition 5.2.2, we reduce to showing that the induced square

${\mathrm{THR}(X)}$${\mathrm{THR}(U_{2}\cup\cdots\cup
U_{n})}$${\mathrm{THR}(U_{1})}$${\mathrm{THR}(U_{1}\cap(U_{2}\cup\cdots\cup
U_{n}))}$

is cartesian. This follows from Corollary 3.4.9. ∎

######  Theorem 5.2.6.

For any $X\in\mathbf{Sch}_{{\mathbb{Z}/2}}$ and integer $n\geq 0$, there is an
equivalence of ${\mathbb{Z}/2}$-spectra

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

###### Proof.

Proposition A.2.7(3) allows us to replace $i_{*}$ by $i_{\sharp}$ in the
claim. We set

$M_{j}:=\\{(x_{1},\ldots,x_{n})\in\mathbb{Z}^{n}:x_{j}\geq 0\\}$

for $j=1,\ldots,n$ and

$M_{n+1}:=\\{(x_{1},\ldots,x_{n})\in\mathbb{Z}^{n}:x_{1}+\cdots+x_{n}\leq
0\\}.$

Observe that there is an isomorphism of commutative monoids
$M_{j}\cong\mathbb{Z}^{n-1}\oplus\mathbb{N}$ for all $j=1,\ldots,n+1$. We set
$M_{I}:=M_{i_{1}}\cap\cdots\cap M_{i_{q}}$ for all nonempty subset
$I:=\\{i_{1},\ldots,i_{q}\\}$ of $\\{1,\ldots,n+1\\}$, and consequently
$M_{\emptyset}:=\mathbb{Z}^{n}$. Together with the obvious maps, we obtain an
$(n+1)$-cube $M$ in commutative monoids associated with $M_{I}$ for any subset
$I\subset\\{1,\ldots,n+1\\}$. Here we set $M(b_{1},...,b_{n+1})=M_{I}$, where
$I$ is the set of indices $i$ such that $b_{i}=0$. By (4.4), we have a
canonical decomposition

$\mathrm{B}^{\mathrm{di}}M_{I}\simeq\coprod_{v\in\mathbb{Z}^{n}}\mathrm{B}^{\mathrm{di}}(M_{I};v),$

where $\mathrm{B}^{\mathrm{di}}(M_{I};v):=\emptyset$ if $v\notin M_{I}$. Hence
for every $v$ the above $\mathrm{B}^{\mathrm{di}}(M_{I};v)$ assemble to an
$(n+1)$-cube in ${\mathbb{Z}/2}$-spaces, and a decomposition of the
$(n+1)$-cube $\mathrm{B}^{\mathrm{di}}M$ into these smaller cubes as $v$
varies in $\mathbb{Z}^{n}$. Combine Propositions 4.2.4 and 4.2.12 to show that
the induced map in this cube

$\mathrm{B}^{\mathrm{di}}(\mathbb{N}\oplus\mathbb{N}^{s}\oplus\mathbb{Z}^{n-s-1};v)\to\mathrm{B}^{\mathrm{di}}(\mathbb{Z}\oplus\mathbb{N}^{s}\oplus\mathbb{Z}^{n-s-1};v)$

is an equivalence for all integer $0\leq s\leq n-1$ if the first coordinate of
$v$ is greater than $0$. By a change of coordinates in the target, we see that
the induced map

$\mathrm{B}^{\mathrm{di}}(M_{I\cup\\{j\\}};v)\to\mathrm{B}^{\mathrm{di}}(M_{I};v)$

is an equivalence for all $j=1,\ldots,n+1$, subsets $I$ of
$\\{1,\ldots,n+1\\}-\\{j\\}$, and $v\in M_{j}^{+}$, where $M_{j}^{+}$ denotes
the set of non-units of $M_{j}$. Use Proposition 5.2.2 repeatedly to show

(5.5) $\mathrm{tfib}(\mathbb{S}[\mathrm{B}^{\mathrm{di}}(M;v)])\simeq 0$

for all $v\in M_{j}^{+}$. This means
$\mathrm{tfib}(\mathbb{S}[\mathrm{B}^{\mathrm{di}}(M;v)])\simeq 0$ whenever
$v\neq O$ since $M_{\emptyset}-(M_{1}^{+}\cup\cdots\cup M_{n+1}^{+})=\\{O\\}$,
where $O$ denotes the origin in $\mathbb{Z}^{n}$.

Now consider the standard cover of $\mathbb{P}^{n}_{\mathbb{S}}$ by $(n+1)$
copies of $\mathbb{A}^{n}_{\mathbb{S}}$, and recall that
$\mathbb{A}^{n}_{\mathbb{S}}$ is the spherical monoid ring of
$\mathbb{N}^{n}$. (We continue switching between spherical monoid rings and
honest schemes over rings as before. Also, note that by Proposition 4.2.15
products of affine schemes correspond to products of monoids when computing
$\mathrm{THR}$.) Choosing suitable coordinates, the intersections of $s$
elements of the cover with $0<s\leq n+1$ are given by (the spherical monoid
rings of) $M_{I}$ above such that $\lvert I\rvert=n+1-s$. By Proposition
5.2.5, we have an equivalence

$\mathrm{THR}(\mathbb{P}_{\mathbb{S}}^{n})\simeq\lim\mathbb{S}[\mathrm{B}^{\mathrm{di}}M|_{(\Delta^{1})^{n}-\\{(0,\ldots,0)\\}}].$

Since $M_{\\{1,\ldots,n\\}}=0$, we also have an equivalence

$\mathrm{fib}(0\to\lim\mathbb{S}[\mathrm{B}^{\mathrm{di}}M|_{(\Delta^{1})^{n}-\\{(0,\ldots,0)\\}}])\simeq\mathrm{tfib}(\mathbb{S}[\mathrm{B}^{\mathrm{di}}M]).$

Combine these two to obtain an equivalence

$\Sigma^{-1}\mathrm{THR}(\mathbb{P}_{\mathbb{S}}^{n})\simeq\mathrm{tfib}(\mathbb{S}[\mathrm{B}^{\mathrm{di}}M]).$

Together with (5.5), we have an equivalence

$\Sigma^{-1}\mathrm{THR}(\mathbb{P}_{\mathbb{S}}^{n})\simeq\mathrm{tfib}(\mathbb{S}[\mathrm{B}^{\mathrm{di}}(M;O)]).$

Consider the canonical decomposition of $(n+1)$-cubes

$\mathbb{S}[\mathrm{B}^{\mathrm{di}}(M;O)]\simeq Q\oplus Q^{\prime}$

such that $Q^{\prime}(0,\ldots,0)=0$ and the other entries of $Q^{\prime}$ are
obtained by $\mathbb{S}[-]$ of the base point of
$\mathrm{B}^{\mathrm{di}}(M;O)$. Use Proposition 5.2.2 repeatedly to have an
equivalence $\mathrm{tfib}(Q^{\prime})\simeq\Sigma^{-1}$, and hence we have an
equivalence

(5.6)
$\mathrm{THR}(\mathbb{P}_{\mathbb{S}}^{n})\simeq\mathbb{S}\oplus\Sigma^{1}\mathrm{tfib}(Q).$

We claim that

$\mathrm{tfib}(Q|_{(\Delta^{1})^{d+1}\times\\{0\\}^{n-d}})\simeq\left\\{\begin{array}[]{ll}\bigoplus_{j=1}^{\lfloor
d/2\rfloor}i_{\sharp}\Sigma^{-1}&\text{if $d$ is even},\\\
\bigoplus_{j=1}^{\lfloor
d/2\rfloor}i_{\sharp}\Sigma^{-1}\oplus\Sigma^{d\sigma-d-1}&\text{if $d$ is
odd}.\end{array}\right.$

Let us proceed by induction on $d$. The claim is clear if $d=0$ by the
definition of the total fiber. Assume $0<d\leq n$. Let
$\\{e_{1},\ldots,e_{n}\\}$ be the standard basis in $\mathbb{Z}^{n}$. The
$d$-cube $M|_{(\Delta^{1})^{d}\times\\{1\\}\times\\{0\\}^{n-d}}$ is isomorphic
to the naturally associated $d$-cube sending
$(a_{1},\ldots,a_{d})\in(\Delta^{1})^{d}$ to

$P_{a_{1}}(e_{1}-e_{d+1})\oplus\cdots\oplus
P_{a_{d}}(e_{d}-e_{d+1})\oplus\mathbb{N}(-e_{d+1})\oplus\mathbb{N}(e_{d+2}-e_{d+1})\oplus\cdots\oplus\mathbb{N}(e_{n}-e_{d+1}),$

where $P_{0}:=\mathbb{N}$, $P_{1}:=\mathbb{Z}$, and $e_{n+1}:=0$. Combine
Propositions 4.2.5 and 5.2.4 to obtain an equivalence

$\displaystyle\mathrm{tfib}(\mathbb{S}[\mathrm{B}^{\mathrm{di}}(M|_{(\Delta^{1})^{d}\times\\{1\\}\times\\{0\\}^{n-d}};O)])$
$\displaystyle\simeq$
$\displaystyle\mathrm{fib}(\mathbb{S}[\mathrm{B}^{\mathrm{di}}(\mathbb{N};0)]\to\mathbb{S}[\mathrm{B}^{\mathrm{di}}(\mathbb{Z};0)])^{\wedge
d}\wedge\mathbb{S}[\mathrm{B}^{\mathrm{di}}(\mathbb{N};0)]^{\wedge n-d}.$

Use Proposition 5.2.2 repeatedly to deduce an equivalence

$\mathrm{tfib}(Q^{\prime}|_{(\Delta^{1})^{d}\times\\{1\\}\times\\{0\\}^{n-d}})\simeq
0.$

Together with Proposition 4.2.11 and (4.12), we obtain equivalences

$\displaystyle\mathrm{tfib}(Q|_{(\Delta^{1})^{d}\times\\{1\\}\times\\{0\\}^{n-d}})\simeq$
$\displaystyle\mathrm{fib}(\mathbb{S}\to\mathbb{S}[S^{\sigma}])^{\wedge
d}\wedge\mathbb{S}^{\wedge n-d}$ $\displaystyle\simeq$
$\displaystyle\Sigma^{-d}(\Sigma^{\infty}S^{\sigma}/(1,0))^{\wedge
d}\simeq\Sigma^{d\sigma-d},$

where $(1,0)$ is the base point of $S^{\sigma}$. Proposition 5.2.2 gives a
fiber sequence

$\mathrm{tfib}(Q|_{(\Delta^{1})^{d+1}\times\\{0\\}^{n-d}})\to\mathrm{tfib}(Q|_{(\Delta^{1})^{d}\times\\{0\\}\times\\{0\\}^{n-d}})\to\mathrm{tfib}(Q|_{(\Delta^{1})^{d}\times\\{1\\}\times\\{0\\}^{n-d}}).$

If $d$ is odd, then we obtain a fiber sequence

(5.7)
$\mathrm{tfib}(Q|_{(\Delta^{1})^{d+1}\times\\{0\\}^{n-d}})\to\bigoplus_{j=1}^{\lfloor
d/2\rfloor}i_{\sharp}\Sigma^{-1}\xrightarrow{f}\Sigma^{d\sigma-d}$

by induction. Since $\pi_{0}(\Sigma^{-1})=0$, the map $\Sigma^{-1}\to
i^{*}\Sigma^{d\sigma-d}\simeq\Sigma^{0}$ obtained by adjunction is equivalent
to $0$. It follows that $f$ is equivalent to $0$, i.e., (5.7) splits. This
completes the induction argument for odd $d$.

If $d$ is even, we obtain a fiber sequence

$\mathrm{tfib}(Q|_{(\Delta^{1})^{d+1}\times\\{0\\}^{n-d}})\to\bigoplus_{j=1}^{\lfloor
d/2\rfloor-1}i_{\sharp}\Sigma^{-1}\oplus\Sigma^{(d-1)\sigma-d}\to\Sigma^{d\sigma-d}$

by induction. As above, the induced map $\bigoplus_{j=1}^{\lfloor
d/2\rfloor-1}i_{\sharp}\Sigma^{-1}\to\Sigma^{d\sigma-d}$ is equivalent to $0$.
It follows that we have an equivalence

$\mathrm{tfib}(Q|_{(\Delta^{1})^{d+1}\times\\{0\\}^{n-d}})\simeq\bigoplus_{j=1}^{\lfloor
d/2\rfloor-1}i_{\sharp}\Sigma^{-1}\oplus\mathrm{fib}(\Sigma^{(d-1)\sigma-d}\xrightarrow{g}\Sigma^{d\sigma-d}).$

We now analyze the non-trivial map
$g\colon\Sigma^{(d-1)\sigma-d}\to\Sigma^{d\sigma-d}$. On the level of
commutative monoids, this corresponds to the inclusion

$\displaystyle\mathbb{Z}(u_{1}-u_{d})\oplus\cdots\oplus\mathbb{Z}(u_{d-1}-u_{d})\oplus\mathbb{N}(-e_{d})\oplus\mathbb{N}(e_{d+1}-e_{d})\oplus\cdots\oplus\mathbb{N}(e_{n}-e_{d})$
$\displaystyle\to$
$\displaystyle\mathbb{Z}u_{1}\oplus\cdots\oplus\mathbb{Z}u_{d}\oplus\mathbb{N}u_{d+1}\oplus\cdots\oplus\mathbb{N}u_{n},$

where $u_{i}:=e_{1}-e_{d+1}$ for $i\in\\{1,\ldots,n\\}-\\{d+1\\}$ and
$u_{d+1}:=e_{d+1}$. As the $0$-entry for $\mathrm{B}^{\mathrm{di}}\mathbb{N}$
is just a point, we only need to study the homomorphism
$\mathbb{Z}^{d-1}\to\mathbb{Z}^{d}$ given by

$(a_{1},\ldots,a_{d-1})\mapsto(a_{1},\ldots,a_{n-1},-a_{1}-\cdots-a_{d-1}).$

In the degree $(0,\ldots,0)$, this is easily seen to induce via Proposition
4.2.8 the map $(S^{\sigma})^{\times d-1}\to(S^{\sigma})^{\times d}$ given by

$(x_{1},\ldots,x_{d-1})\mapsto(x_{1},\ldots,x_{d-1},x_{1}^{-1}\cdots
x_{d-1}^{-1}).$

After a further cancellation of base points, we are left with studying the map
$h\colon(S^{\sigma})^{\wedge d-1}\to(S^{\sigma})^{\wedge d}$. This is an
equivariant cofibration, as it is the $(S^{\sigma})^{\wedge d-1}$-suspension
of the push-out of the cofibration $G\times S^{0}\to G\times I$ along the
projection $G\times S^{0}\to(G/G)\times S^{0}$. Hence unstably the equivariant
(homotopy) cofiber of $f$ is given by $(S^{\sigma})^{\wedge
d}/h((S^{\sigma})^{\wedge d-1})\simeq S^{d}\vee S^{d}$ where
$G={\mathbb{Z}/2}$ acts on the latter by switching the spheres. Thus the
stable equivariant (homotopy) fiber $\mathrm{fib}(g)$ is given by
$\Sigma^{-d-1}\Sigma^{\infty}(S^{d}\vee S^{d})\simeq i_{\sharp}\Sigma^{-1}$
using (A.15). This completes the induction argument for even $d$.

Together with (5.6), we obtain an equivalence

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