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

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

and

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

by Proposition Appendix A.2.7(6).

Proof. The composite

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

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

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

is an equivalence. By [Reference Schwede53, Theorem 7.12] (note that $i^*=\Phi ^e$ ), it remains to show that the induced map

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

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

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

which is equivalent to $0$ too.

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

$$\begin{align*}A^{\wedge {\mathbb{Z}/2}}\wedge_{R^{\wedge {\mathbb{Z}/2}}}B^{\wedge {\mathbb{Z}/2}} \simeq (A\wedge_R B)^{\wedge {\mathbb{Z}/2}}. \end{align*}$$

On the other hand, there are canonical equivalences

$$ \begin{align*} \mathrm{THR}(A)\wedge_{\mathrm{THR}(R)}\mathrm{THR}(B) &\simeq (A\wedge_{A^{\wedge {\mathbb{Z}/2}}} A) \wedge_{R\wedge_{R^{\wedge {\mathbb{Z}/2}}} R} (B\wedge_{B^{\wedge {\mathbb{Z}/2}}} B) \\ &\simeq (A\wedge_R B) \wedge_{A^{\wedge {\mathbb{Z}/2}}\wedge_{R^{\wedge {\mathbb{Z}/2}}}B^{\wedge {\mathbb{Z}/2}}} (A\wedge_R B). \end{align*} $$

Combine the two equations to obtain the desired equivalence.

Proof. Let $f\colon M(G/e)\to L(G/e)$ be 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

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

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

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

Proof. There is an equivalence

(2.9) $$ \begin{align} i^* \mathrm{H}(M^{\oplus {\mathbb{Z}/2}})\simeq \mathrm{H} (M\oplus M)\\[-7pt]\nonumber \end{align} $$

by equation (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 equation (2.8) by adjunction. We need to show that the induced map

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

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

is an isomorphism.

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

$$\begin{align*}M\oplus M \to M\oplus M\oplus M\oplus M \to M\oplus M \end{align*}$$

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 equation (2.10) can be written as the homomorphisms

$$\begin{align*}M\to M\oplus M \to M \end{align*}$$

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 equation (2.9) is an equivalence in $\mathrm {NAlg}_{{\mathbb {Z}/2}}$ . Hence, we obtain equation (2.8) as an equivalence in $\mathrm {NAlg}_{{\mathbb {Z}/2}}$ .

Proof. By [Reference Dotto, Moi, Patchkoria and Reeh16, Theorem 5.1], the morphism (2.11) can be described as a diagram

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 equation (2.11) is an isomorphism.

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 [Reference Lurie42, Proposition 4.6.2.17], to construct equation (2.12), it suffices to construct a map of $N^{{\mathbb {Z}/2}}\mathrm {H} A$ -modules

(2.13) $$ \begin{align} N^{{\mathbb{Z}/2}} \mathrm{H} M \to \mathrm{H} (N_A^{{\mathbb{Z}/2}} M). \end{align} $$

By Proposition Appendix 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 $\mathrm {CAlg}_{\mathbb {Z}/2}$

$$\begin{align*}N^{{\mathbb{Z}/2}}\mathrm{H} A \to \mathrm{H} \underline{\pi}_0(N^{{\mathbb{Z}/2}}\mathrm{H} A) \end{align*}$$

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

$$\begin{align*}N^{{\mathbb{Z}/2}}\mathrm{H} M \to \mathrm{H} \underline{\pi}_0(N^{{\mathbb{Z}/2}}\mathrm{H} M). \end{align*}$$

Hence, to construct equation (2.13), it suffices to construct a map of $\underline {\pi }_0(N^{{\mathbb {Z}/2}} \mathrm {H} A)$ -modules

$$\begin{align*}\underline{\pi}_0(N^{{\mathbb{Z}/2}} \mathrm{H} M) \to N_A^{{\mathbb{Z}/2}}M. \end{align*}$$

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) $$ \begin{align} \pi_0(i^*N^{{\mathbb{Z}/2}}\mathrm{H} M) \to M\otimes_A M, \end{align} $$

that is, 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 Appendix A.2.7(6). The canonical assignment $x\otimes y\to x\otimes y$ finishes the construction.

The class of A-modules such that equation (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 equation (2.12) is an equivalence if M is flat, we may assume $M=A^n$ . In this case, there is an equivalence

$$\begin{align*}N^{{\mathbb{Z}/2}} \mathrm{H} M \simeq V_n \wedge N^{{\mathbb{Z}/2}} \mathrm{H} A, \end{align*}$$

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

$$\begin{align*}N^{{\mathbb{Z}/2}}\mathrm{H} M\wedge_{N^{{\mathbb{Z}/2}}\mathrm{H} A} \mathrm{H} \iota A \simeq V_n \wedge \mathrm{H} \iota A \simeq \mathrm{H}(V_n \times \iota A). \end{align*}$$

Combining this with equation (2.12), we obtain a map

(2.15) $$ \begin{align} f \colon \mathrm{H}(V_n\times \iota A) \to \mathrm{H}(N_A^{{\mathbb{Z}/2}} A^n). \end{align} $$

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 equation (2.12) is an equivalence, it suffices to show that it is an equivalence after applying $\pi _0$ , that is, the induced map

$$\begin{align*}g \colon (M\otimes M)\otimes_{A\otimes A} A \to M\otimes_A M \end{align*}$$

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

$$\begin{align*}g((x\otimes y)\otimes a)=ax\otimes y \end{align*}$$

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

Proof. This appears on [Reference Heller, Krishna and Østvær26, p. 1225].

Proof. We refer to [Reference Heller, Krishna and Østvær26, Proposition 6.11].

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 diagonal morphism of schemes $\operatorname {\mathrm {Spec}}(B)\to \operatorname {\mathrm {Spec}}(B)\times _{\operatorname {\mathrm {Spec}}(A)}\operatorname {\mathrm {Spec}}(B)$ is an open and closed immersion by the implication a) $\Rightarrow $ b) in [Reference Dieudonné and Grothendieck15, Corollaire IV.17.4.2]. This implies that 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

$$\begin{align*}\mathop{\mathrm{colim}}(N_A^{{\mathbb{Z}/2}}B \xrightarrow{\cdot e}N_A^{{\mathbb{Z}/2}}B \xrightarrow{\cdot e} \cdots) \cong \iota B. \end{align*}$$

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

Proof. Consider the commutative square

(3.3)

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 equation (2.7). The same equivalence is used when identifying the upper left entry in the square.) This implies that equation (3.2) is equivalent to the upper horizontal map of equation (3.3). Proposition Appendix A.5.5 and Lemma 3.2.1 show that the lower horizontal map of equation (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 Appendix A.5.6. By Proposition Appendix 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 Appendix A.4.4, and are reduced to showing that the induced morphism

$$\begin{align*}\underline{\pi}_0(\mathrm{THR}(\iota A)) \Box_{\iota A} \iota B \to \underline{\pi}_0(\mathrm{THR}(\iota B)) \end{align*}$$

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

Proof. We have isomorphisms $\operatorname {\mathrm {Spec}}(A^{{\mathbb {Z}/2}})\simeq \operatorname {\mathrm {Spec}}(A)/({\mathbb {Z}/2})$ and $\operatorname {\mathrm {Spec}}(B^{{\mathbb {Z}/2}})\simeq \operatorname {\mathrm {Spec}}(B)/({\mathbb {Z}/2})$ . 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) $$ \begin{align} B\cong A\otimes_{\iota (A^{{\mathbb{Z}/2}})}\iota (B^{{\mathbb{Z}/2}}). \end{align} $$

Since $A^{{\mathbb {Z}/2}}\to B^{{\mathbb {Z}/2}}$ is flat, $B^{{\mathbb {Z}/2}}$ is a filtered colimit of finite free $A^{{\mathbb {Z}/2}}$ -modules. It follows that $\iota A^{{\mathbb {Z}/2}}\to \iota B^{{\mathbb {Z}/2}}$ is flat by Proposition Appendix 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 Appendix A.5.7. Then Proposition 2.1.5 yields an equivalence

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

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.

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

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

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

Proof. We need to show that the maps

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

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

Proof. We need to show that the maps

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

are local equivalences. Since $(-)^{{\mathbb {Z}/2}} i_{\sharp }\simeq (-)^{{\mathbb {Z}/2}} i_*\simeq \mathrm {id}$ by Proposition Appendix A.2.7(2),(3), the second map in equation (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 [Reference Greenlees and May23, Theorem 3.10], we have an equivalence

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

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

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

are local equivalences in $\mathcal {P}\mathrm {sh}(\mathcal {C},\mathrm {Sp})$ , the first map in equation (3.7) 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 $\mathcal {F}\to \mathcal {F}'$ with $\mathcal {F}'\in \mathcal {S}\mathrm {hv}(\mathcal {C},\mathrm {Sp}_{{\mathbb {Z}/2}})$ . By Propositions 3.3.4 and 3.3.5, $i^*\mathcal {F}'$ and $\mathcal {F}^{\prime {\mathbb {Z}/2}}$ are local equivalent to $0$ . Since $i^*\mathcal {F}',\mathcal {F}^{\prime {\mathbb {Z}/2}}\in \mathcal {S}\mathrm {hv}(\mathcal {C},\mathrm {Sp})$ by Proposition 3.3.2, it follows that $i^*\mathcal {F}'$ and $\mathcal {F}^{\prime {\mathbb {Z}/2}}$ are equivalent to $0$ . Hence, $\mathcal {F}'$ is equivalent to $0$ , that is, $\mathcal {F}$ is local equivalent to $0$ .

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

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

becomes an isomorphism after sheafification for $H=e,{\mathbb {Z}/2}$ and integer n. By [Reference Dugger, Hollander and Isaksen18, Theorem 1.3], this is equivalent to saying that $i^*g$ and $g^{{\mathbb {Z}/2}}$ are local equivalences. Proposition 3.3.7 finishes the proof.

Proof. Apply [Reference Ayoub3, Proposition 4.4.56] to the canonical (compose Yoneda and sheafification) functors $\mathcal {C}\to \mathbf {Shv}(\mathcal {C})$ and $\mathcal {C}'\to \mathbf {Shv}(\mathcal {C}')$ , where the right-hand side categories are equipped with the topology described in [Reference Ayoub3, after Théorème 4.4.51]. Hence, we obtain left Quillen equivalences

$$\begin{align*}\mathbf{Psh}(\mathcal{C},\mathrm{Sp}^O_{{\mathbb{Z}/2}}) \to \mathbf{Psh}(\mathbf{Shv}(\mathcal{C}),\mathrm{Sp}^O_{{\mathbb{Z}/2}}) \text{ and } \mathbf{Psh}(\mathcal{C}',\mathrm{Sp}^O_{{\mathbb{Z}/2}}) \to \mathbf{Psh}(\mathbf{Shv}(\mathcal{C}'),\mathrm{Sp}^O_{{\mathbb{Z}/2}}) \end{align*}$$

with respect to the local model structures of Definition 3.3.11. Owing to Proposition 3.3.12 and [Reference Lurie42, Lemma 1.3.4.21], we obtain equivalences of $\infty $ -categories

$$\begin{align*}\mathcal{S}\mathrm{hv}(\mathcal{C},\mathrm{Sp}_{{\mathbb{Z}/2}}) \simeq \mathcal{S}\mathrm{hv}(\mathbf{Shv}(\mathcal{C}),\mathrm{Sp}_{{\mathbb{Z}/2}}) \text{ and } \mathcal{S}\mathrm{hv}(\mathcal{C}',\mathrm{Sp}_{{\mathbb{Z}/2}}) \simeq \mathcal{S}\mathrm{hv}(\mathbf{Shv}(\mathcal{C}'),\mathrm{Sp}_{{\mathbb{Z}/2}}). \end{align*}$$

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

Proof. By Proposition Appendix 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.

Proof. Since $\mathrm {H} A\to \mathrm {H} B$ is flat in the sense of [Reference Lurie42, Definition 7.2.2.10], we have an isomorphism

$$\begin{align*}\pi_q \mathcal{F}(\operatorname{\mathrm{Spec}}(B))) \simeq \pi_q(M)\otimes_A B \end{align*}$$

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

According to the paragraph preceding [Reference Geisser and Hesselholt21, Proposition 3.1.2], there is a conditionally convergent spectral sequence

$$\begin{align*}E_2^{st} := H_{{\acute{e}t}}^s(X,a_{{\acute{e}t}}\pi_{-t}\mathcal{F}) \Rightarrow \pi_{-s-t}L_{{\acute{e}t}}\mathcal{F}(X), \end{align*}$$

where $a_{{\acute {e}t}}$ denotes the étale sheafification functor. Since $\pi _{-t}\mathcal {F}$ is a quasi-coherent sheaf for every integer t, the cohomology $H^s(X,a_{{\acute {e}t}} \pi _{-t}\mathcal {F})$ vanishes for every integer $s\neq 0$ . It follows that $\pi _{-t}\mathcal {F}\to \pi _{-t}L_{{\acute {e}t}}\mathcal {F}$ is an isomorphism for every integer t, that is, $\mathcal {F}$ satisfies étale descent.

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

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

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

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

be the presheaf given by $ \mathcal {F}(\operatorname {\mathrm {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

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

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

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

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

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

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

By Lemmas Appendix A.2.9 and 3.4.1, there are equivalences

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

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.

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

$$\begin{align*}\{V_x\to X\}_{x\in X} \cup \{Y\to X\} \end{align*}$$

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 since the algebraic space $Y/({\mathbb {Z}/2})$ is formed by the étale equivalence relation $Y\times {\mathbb {Z}/2} \rightrightarrows Y$ , 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) $$ \begin{align} \{V_x\to X\}_{x\in X} \cup \{W_j\times {\mathbb{Z}/2}\to X\}_{j\in J} \end{align} $$

is an isovariant étale covering.

Proof. Combine [Reference Artin, Grothendieck and Verdier2, Théorème III.4.1] and Proposition 3.4.4 to obtain equation (3.10). Use Proposition 3.3.13 for equation (3.11).

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

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

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

Proof. Let P be the collection of equivariant Nisnevich distinguished squares. Consider the class $G_P$ of morphisms between simplicial presheaves in [Reference Voevodsky58, p. 1392]. By [Reference Voevodsky58, Proposition 3.8(2)], every $G_P$ -local equivalence is an equivariant Nisnevich local equivalence. This implies that if $\mathcal {F}$ is a simplicial presheaf satisfying equivariant Nisnevich descent, then $\mathcal {F}(Q)$ is Cartesian for all $Q\in P$ .

Since $\mathrm {THR}$ satisfies isovariant étale descent, it satisfies equivariant Nisnevich descent. Hence, we can apply the argument in the above paragraph to $\Omega ^{\infty } \Sigma ^n i^* \mathrm {THR}$ and $\Omega ^{\infty } \Sigma ^n \Phi ^{{\mathbb {Z}/2}} \mathrm {THR}$ for all integers n to see that $i^*\mathrm {THR}(Q)$ and $\Phi ^{{\mathbb {Z}/2}}\mathrm {THR}(Q)$ are Cartesian, which implies the claim.

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 [Reference Fiedorowicz and Loday20, Definition 4.3]. In simplicial degree q, we have

$$\begin{align*}F_G(X)_q := G_q\times X_q. \end{align*}$$

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) $$ \begin{align} d_i(g,x):=(d_i(g),d_{g^{-1}(i)}(x)) \text{ and } s_i(g,x):=(s_i(g),s_{g^{-1}(i)}(x)). \end{align} $$

According to [Reference Fiedorowicz and Loday20, 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

$$\begin{align*}p_2[(g,x),u] := [x,gu] \end{align*}$$

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

$$\begin{align*}\lvert F_G(X) \rvert \cong \lvert G \rvert \times \lvert X \rvert. \end{align*}$$

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

$$\begin{align*}ev(g,x) := gx. \end{align*}$$

According to [Reference Fiedorowicz and Loday20, Theorem 5.3], the composite map

$$\begin{align*}\lvert G \rvert \times \lvert X \rvert \xrightarrow{\cong} \lvert F_G(X) \rvert \xrightarrow{ev} \lvert X \rvert \end{align*}$$

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.

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

$$\begin{align*}(M\times L)^{\times (q+1)} \to M^{\times (q+1)} \times L^{\times (q+1)} \end{align*}$$

in simplicial degree q.

Proof. This follows from the commutativity of the square

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

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

$$\begin{align*}(x_0,\ldots,x_q)\mapsto (x_1,\ldots,x_q) \end{align*}$$

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

$$\begin{align*}(x_0,\ldots,x_q)\mapsto (a,x_1,\ldots,x_q) \end{align*}$$

in simplicial degree q constructs the isomorphism (4.7), where $a:=0$ (resp. $a:=1$ ) if $x_0+\cdots +x_q=j$ (resp. $x_0+\cdots +x_q=-j$ ).

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 nondegenerate simplices 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) $$ \begin{align} (S^{\sigma})^{{\mathbb{Z}/2}} \to (\mathrm{B}^{\sigma} \mathbb{Z})^{{\mathbb{Z}/2}} \end{align} $$

is a homotopy equivalence as well.

By [Reference Dotto and Ogle17, 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

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

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

This implies that $\pi _n((\mathrm {B}^{\sigma } \mathbb {Z})^{\mathbb {Z}/2})$ is trivial for every 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) $$ \begin{align} \pi_0((S^{\sigma})^{{\mathbb{Z}/2}}) \to \pi_0((\mathrm{B}^{\sigma} \mathbb{Z})^{{\mathbb{Z}/2}}) \end{align} $$

induced by equation (4.9) is injective. Let $\mathrm {sd}_{\sigma }$ denote Segal’s edgewise subdivision functor in [Reference Segal54]. In simplicial degree $1$ , the two degeneracy maps in $\mathrm {sd}_{\sigma }(\mathrm {N}^{\sigma } \mathbb {Z})$ are given by

$$\begin{align*}(x_1,x_2,x_3)\mapsto x_2,x_1+x_2+x_3. \end{align*}$$

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, equation (4.11) is injective as claimed.

Proof. See [Reference Dotto, Moi, Patchkoria and Reeh16, Example 5.13].

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

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

$$\begin{align*}C_{q+1}\times \operatorname{\mathrm{Hom}}([q],[j-1]) \to \{(x_0,\ldots,x_q)\in \mathbb{N}^{q+1} : x_0+\cdots+x_q=j\}. \end{align*}$$

sending $(t,f)$ to

$$\begin{align*}(f(t(0))-f(t(q)),f(t(1))-f(t(0)),\ldots,f(t(q))-f(t(q-1))). \end{align*}$$

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

$$\begin{align*}(t,f)\mapsto (q+1-t,\rho\circ f), \end{align*}$$

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

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

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

$$\begin{align*}S^1\times \Delta^{j-1} \to (S^1\times \Delta^{j-1})/C_j; \end{align*}$$

see the proof of [Reference Hesselholt28, Lemma 2.2.3] or [Reference Rognes49, 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 equation (4.17) is $O(2)$ -homeomorphic to the quotient map

$$\begin{align*}S^{\sigma} \times \Delta^{j-1}_{\sigma} \to (S^{\sigma} \times \Delta^{j-1}_{\sigma})/C_j \end{align*}$$

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

Proof. We need to show that equation (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 equations (4.20) and (4.22). Hence, the remaining case is $H=D_r$ for every integer $r\geq 1$ . Since equation (4.18) commutes with w, equation (4.21) is ${\mathbb {Z}/2}$ -equivariant. Similarly, equation (4.19) is ${\mathbb {Z}/2}$ -equivariant. Hence, we have the induced homeomorphisms

$$\begin{align*}\mathrm{B}^{\mathrm{di}}(\mathbb{N};j)^{{\mathbb{Z}/2}} \xrightarrow{\cong} \mathrm{B}^{\mathrm{di}}(\mathbb{N};rj)^{D_r} \text{ and } \mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)^{{\mathbb{Z}/2}} \xrightarrow{\cong} \mathrm{B}^{\mathrm{di}}(\mathbb{Z};rj)^{D_r}. \end{align*}$$

Combine with equation (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) $$ \begin{align} \mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)^{{\mathbb{Z}/2}} \simeq S^0. \end{align} $$

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) $$ \begin{align} \pi_0(\mathrm{B}^{\mathrm{di}}(\mathbb{N};j)^{{\mathbb{Z}/2}}) \to \pi_0(\mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)^{{\mathbb{Z}/2}}) \end{align} $$

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

$$\begin{align*}(x_0,x_1,x_2,x_3)\mapsto (x_3+x_0+x_1,x_2),(x_0,x_1+x_2+x_3). \end{align*}$$

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

$$\begin{align*}\pi_0(\mathrm{B}^{\mathrm{di}}(\mathbb{N};j)) \cong \{(x_0,x_1)\in \mathbb{N}^2 : x_0+x_1=j\}/\sim \end{align*}$$

and

$$\begin{align*}\pi_0(\mathrm{B}^{\mathrm{di}}(\mathbb{Z};j)) \cong \{(x_0,x_1)\in \mathbb{Z}^2 : x_0+x_1=j\}/\sim, \end{align*}$$

where $(x_0,x_1)\sim (x_0',x_1')$ if and only if $x_0-x_0'$ is even. This shows that equation (4.25) is a bijection.

Proof. See [Reference Høgenhaven35], and also [Reference Dotto, Moi, Patchkoria and Reeh16, Proposition 5.9].

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

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 equation (2.7) and Propositions Appendix A.1.10, Appendix A.2.7(3) and 2.3.3, we obtain equivalences

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

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

Proof. By Propositions 2.1.5 and 4.2.13, we have equivalences

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

Proposition 4.2.14 finishes the proof.

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 co-Cartesian 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:

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 equation (4.4) and Propositions 4.2.6, 4.2.8 and 4.2.11, we obtain the following ${\mathbb {Z}/2}$ -equivariant (homotopy) co-Cartesian square:

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

and the result follows.

Proof. Consider the co-Cartesian square of ${\mathbb {Z}/2}$ -spaces

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 Appendix A.2.4, we obtain a natural equivalence

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

By adjunction, we obtain the desired natural equivalence.

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

(5.2)

as in [Reference Carmody12, 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

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

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 $i^*\mathbb {G}_{m,\mathbb {S}}^{\sigma }\cong i^*\mathbb {G}_{m,\mathbb {S}}$ . The induced map $\mathrm {THR}(\mathbb {G}_{m,\mathbb {S}}^{\sigma })\to \mathrm {THR}(\mathbb {Z}/2\times \mathbb {G}_{m,\mathbb {S}}^{\sigma })$ can be identified with the map

$$\begin{align*}\mathrm{THR}(\mathbb{G}_{m,\mathbb{S}}^{\sigma}) \to i_*i^*\mathrm{THR}(\mathbb{G}_{m,\mathbb{S}}^{\sigma}) \end{align*}$$

obtained by the unit of the adjunction pair $(i^*,i_*)$ . As in the proof of Theorem 5.1.2, use Propositions 4.2.6, 4.2.8, 4.2.11 and 4.2.12 to see that the induced map $\mathrm {THR}(\mathbb {A}_{\mathbb {S}}^1)\to \mathrm {THR}(\mathbb {G}_{m,\mathbb {S}})$ can be identified with the map

$$\begin{align*}\mathbb{S} \oplus \bigoplus_{j>0} \Sigma_+^{\infty} S^{\sigma} \to \mathbb{S} \oplus \bigoplus_{j>0} i_*i^*\Sigma_+^{\infty} S^{\sigma} \end{align*}$$

obtained by the unit of the adjunction pair $(i^*,i_*)$ . By equation (4.13), we obtain an equivalence

$$\begin{align*}\mathrm{THR}(\mathbb{G}_{m,\mathbb{S}}^{\sigma}) \simeq \mathbb{S} \oplus \bigoplus_{j>0} i_*i^*\Sigma_+^{\infty} S^1. \end{align*}$$

Applying $\mathrm {THR}$ to equation (5.2) and combining with the above discussion yield the following homotopy co-Cartesian square:

(5.3)

Consider the commutative square

(5.4)

extracted from equation (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 Appendix 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

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

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)

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

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

Together with Lemma 5.1.3, we obtain the desired equivalence.

Proof. Combine Theorems 5.1.2 and 5.1.4.

Proof. This is again obtained by dualizing the arguments for the corresponding one [Reference Binda, Park and Østvær8, Proposition A.6.7]. 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.

Proof. Use Proposition 5.2.3 repeatedly.

Proof. Let us use the equivalence $\mathrm {N}(\mathbf {P}(S))\simeq (\Delta ^1)^n$ . We include the description of this equivalence if $S=\{1\}$ . We regard the partially ordered set $\mathbf {P}(\{1\})$ as the category associated with the diagram $\emptyset \to \{1\}$ . The nerve of this category is $\Delta ^1$ .

We proceed by induction on n. The claim is clear if $n=1$ . Assume $n>1$ . By induction, we fiber sequences

$$\begin{align*}\mathrm{tfib}(\mathrm{THR}(Q|_{\{0\}\times (\Delta^1)^{n-1}})) \to \mathrm{THR}(X) \to \mathrm{THR}(U_2\cup \cdots \cup U_n) \end{align*}$$

and

$$\begin{align*}\mathrm{tfib}(\mathrm{THR}(Q|_{\{1\}\times (\Delta^1)^{n-1}})) \to \mathrm{THR}(U_1) \to \mathrm{THR}(U_1\cap (U_2\cup \cdots \cup U_n)). \end{align*}$$

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

is Cartesian. This follows from Corollary 3.4.9.

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

$$\begin{align*}M_j:=\{(x_1,\ldots,x_n)\in \mathbb{Z}^n : x_j\geq 0\} \end{align*}$$

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

$$\begin{align*}M_{n+1}:=\{(x_1,\ldots,x_n)\in \mathbb{Z}^n : x_1+\cdots+ x_n \leq 0\}. \end{align*}$$

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 subsets $I:=\{i_1,\ldots ,i_q\}$ 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 equation (4.4), we have a canonical decomposition

$$\begin{align*}\mathrm{B}^{\mathrm{di}} M_I \simeq \coprod_{v\in \mathbb{Z}^n} \mathrm{B}^{\mathrm{di}}(M_I;v), \end{align*}$$

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

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

is an equivalence for every 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

$$\begin{align*}\mathrm{B}^{\mathrm{di}}(M_{I\cup \{j\}};v) \to \mathrm{B}^{\mathrm{di}}(M_I;v) \end{align*}$$

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 nonunits of $M_j$ . Use Proposition 5.2.2 repeatedly to show

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

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

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

Together with equation (5.5), we have an equivalence

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

since $\mathrm {B}^{\mathrm {di}}(M_{\{1,\ldots ,n\}};O)\cong \mathrm {B}^{\mathrm {di}}(M_{\{1,\ldots ,n\}})$ . For every subset I of $\{1,\ldots ,n\}$ , we have the canonical decomposition

$$\begin{align*}\mathrm{B}^{\mathrm{di}}(M_I;O) \cong V_I\amalg *, \end{align*}$$

where $V_I$ is obtained by removing the base point $*$ of $\mathrm {B}^{\mathrm {di}}(M_I;O)$ corresponding to the element $0\in M_I$ in simplicial degree $0$ . This yields the canonical decomposition

$$\begin{align*}\mathbb{S}[\mathrm{B}^{\mathrm{di}} (M;O)] \simeq Q\oplus Q', \end{align*}$$

where every entry of $Q'$ is $\Sigma ^{\infty }_+ * \simeq \mathbb {S}$ . Use Proposition 5.2.2 repeatedly to have an equivalence $\mathrm {tfib}(Q') \simeq 0$ . This implies that we have an equivalence

$$\begin{align*}\lim Q'|_{(\Delta^1)^n-\{(0,\ldots,0)\}} \simeq \mathbb{S}. \end{align*}$$

On the other hand, we have $Q(0,\ldots ,0)=0$ since $M_{\{1,\ldots ,n\}}=0$ . This implies that we have an equivalence

$$\begin{align*}\mathrm{tfib}(Q) \simeq \Sigma^{-1} \lim Q|_{(\Delta^1)^n-\{(0,\ldots,0)\}}. \end{align*}$$

Combine what we have discussed above to have an equivalence

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

We claim that

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

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

$$\begin{align*}P_{a_1}(e_1-e_{d+1})\oplus \cdots \oplus P_{a_d}(e_d-e_{d+1})\oplus \mathbb{N} (-e_{d+1}) \oplus \mathbb{N}(e_{d+2}-e_{d+1})\oplus \cdots \oplus \mathbb{N} (e_{n}-e_{d+1}), \end{align*}$$

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

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

Use Proposition 5.2.2 repeatedly to deduce an equivalence

$$\begin{align*}\mathrm{tfib}(Q'|_{(\Delta^1)^{d}\times \{1\} \times \{0\}^{n-d}}) \simeq 0. \end{align*}$$

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

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

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

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

If d is odd, then we obtain a fiber sequence

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

by induction. Since $\pi _{-1}(\Sigma ^0)=0$ , 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 $0$ , that is, equation (5.7) splits. This completes the induction argument for odd d.

If d is even, we obtain a fiber sequence

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

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

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

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

$$ \begin{align*} &\mathbb{Z} (u_1-u_d)\oplus \cdots \oplus \mathbb{Z} (u_{d-1}-u_d) \oplus \mathbb{N}(-e_d) \oplus \mathbb{N}(e_{d+1}-e_d) \oplus \cdots \oplus \mathbb{N}(e_n-e_d) \\ \to & \mathbb{Z} u_1\oplus \cdots \oplus \mathbb{Z} u_d \oplus \mathbb{N} u_{d+1} \oplus \cdots \oplus \mathbb{N} u_{n}, \end{align*} $$

where $u_i:=e_i-e_{d+1}$ 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

$$\begin{align*}(a_1,\ldots,a_{d-1}) \mapsto (a_1,\ldots,a_{n-1},-a_1-\cdots-a_{d-1}). \end{align*}$$

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

$$\begin{align*}(x_1,\ldots,x_{d-1})\mapsto (x_1,\ldots,x_{d-1},x_1^{-1}\cdots x_{d-1}^{-1}). \end{align*}$$

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 pushout 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 equation (A.15). This completes the induction argument for even d.

Together with equation (5.6), we obtain an equivalence

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

Use Propositions 4.2.15 and 5.2.5 for the standard cover of $\mathbb {P}^n$ to obtain an equivalence

$$\begin{align*}\mathrm{THR}(X\times \mathbb{P}^n) \simeq \mathrm{THR}(X)\wedge \mathrm{THR}(\mathbb{P}_{\mathbb{S}}^n) \end{align*}$$

whenever X is an affine scheme with involution. Use Proposition 5.2.5 again to generalize this equivalence to the case when X is a separated scheme with involution. Hence, to obtain the desired equivalence, it suffices to obtain an equivalence

$$\begin{align*}\mathrm{THR}(X)\wedge i_{\sharp}\mathbb{S} \simeq i_{\sharp}{\mathrm{THH}}(X). \end{align*}$$

This follows from Propositions Appendix A.1.10 and 2.1.3.