Construction 1.13. Let $(R,R^{+})$ be an affinoid $K$ -algebra. The set $\operatorname{Spa}(R,R^{+})$ is the set of equivalence classes of continuous multiplicative valuations $|\cdot |$ (see [Reference HuberHub93, § 3] and [Reference ScholzeSch12, Definitions 2.2, 2.5]). It is endowed with the topology generated by the basis of rational subsets $\{U(f_{1},\ldots ,f_{n}\,\mid \,g)\}$ by letting $f_{1},\ldots ,f_{n},g$ vary among elements in $R$ such that $f_{1},\ldots ,f_{n}$ generate $R$ as an ideal and where the set $U(f_{1},\ldots ,f_{n}\,\mid \,g)$ is the set of those valuations $|\cdot |$ satisfying $|f_{i}|\leqslant |g|$ for all $i$ .

If we define maps of valuation fields over $K$ (see [Reference ScholzeSch12, Definition 2.26]) as maps of affinoid $K$ -algebras $(L,L^{+})\rightarrow (L^{\prime },{L^{\prime }}^{+})$ such that ${L^{\prime }}^{+}\cap L=L^{+}$ then $\operatorname{Spa}(R,R^{+})$ has the following alternative description: it is the set $\varinjlim \operatorname{Hom}((R,R^{+}),(L,L^{+}))$ by letting $(L,L^{+})$ vary in the category of valuation fields over $K$ (see [Reference ScholzeSch12, Proposition 2.27]). Its topology can be defined by declaring the sets $\{\unicode[STIX]{x1D719}:0\neq |\unicode[STIX]{x1D719}(f)|\leqslant |\unicode[STIX]{x1D719}(g)|\}$ to be open, for all pairs of elements $f,g$ in $R$ .

We can associate to a rational subset $U(f_{1},\ldots ,f_{n}\,\mid \,g)$ the affinoid $K$ -algebra the affinoid $K$ -algebra

$$\begin{eqnarray}({\mathcal{O}}(U),{\mathcal{O}}^{+}(U))=(R\langle f_{1}/g,\ldots ,f_{n}/g\rangle ,R\langle f_{1}/g,\ldots ,f_{n}/g\rangle ^{+})\end{eqnarray}$$

defined in [Reference HuberHub94, § 1] and [Reference ScholzeSch12, Defintion 2.13]. This way, we define a presheaf of affinoid $K$ -algebras $({\mathcal{O}}_{X},{\mathcal{O}}_{X}^{+})$ on a basis of $X=\operatorname{Spa}(R,R^{+})$ .

By [Reference HuberHub94, Lemma 1.5, Proposition 1.6] for any $x\in X=\operatorname{Spa}(R,R^{+})$ the valuation at $x$  extends to a valuation on ${\mathcal{O}}_{X,x}$ and the stalk ${\mathcal{O}}_{X,x}^{+}$ is local and corresponds to $\{f\in {\mathcal{O}}_{X,x}:|f(x)|\leqslant 1\}$ .

Assumption 1.18. From now on, we will always assume that $K$ is a perfectoid field. We also make the extra assumption that the invertible element $\unicode[STIX]{x1D70B}$ of $K$ satisfies $|p|\leqslant |\unicode[STIX]{x1D70B}|<1$ and coincides with $(\unicode[STIX]{x1D70B}^{\flat })^{\sharp }$ for a chosen $\unicode[STIX]{x1D70B}^{\flat }$ in $K^{\flat }$ . In particular, $\unicode[STIX]{x1D70B}$ is equipped with a compatible system of $p$ -power roots $\unicode[STIX]{x1D70B}^{1/p^{h}}$ (see [Reference ScholzeSch12, Remark 3.5]).

Proof. Let $(T,T^{+})$ be as in the last claim. We let $W^{\prime }$ be the fiber product of $\operatorname{Spa}(T,T^{+})$ and $\widehat{\mathbb{T}}^{N}\,\times \,\widehat{\mathbb{T}}^{M}$ over $\widehat{\mathbb{T}}^{N}\,\times \,\mathbb{T}^{M}$ . If $\operatorname{char}K=0$ by [Reference ScholzeSch13, Lemma 4.5(i)] it exists and is affinoid perfectoid, represented by a perfectoid pair $(T^{\prime },{T^{\prime }}^{+})$ . The same is true if $\operatorname{char}K=p$ as in this case it coincides with the completed perfection of $X_{0}$ (indeed, the pullback of an étale map over Frobenius is isomorphic to Frobenius, see for example [Sta18, Lemma 0EBS]).

From the strict inclusion $(T,T^{+})\rightarrow (T^{\prime },{T^{\prime }}^{\circ })$ we deduce that $T^{+}$ is bounded since ${T^{\prime }}^{+}$ is, being $T^{\prime }$ perfectoid. By considering rational subspaces of $\operatorname{Spa}(T,T^{+})$ we deduce that this space is stably uniform, and hence sheafy (by [Reference Buzzard and VerberkmoesBV18]).

The proof of the alternative description of $(T,T^{+})$ follows in the same way as [Reference ScholzeSch13, Lemma 4.5(i)].◻

Proof. This follows from Proposition 2.1 and the fact that every étale map is locally (on the source) a composition of rational embeddings and finite étale maps. ◻

Proof. Suppose that $X_{h}=\operatorname{Spa}(R_{h},R_{h}^{\circ })$ and $X=\operatorname{Spa}(R,R^{+})$ . By Proposition 2.1  $R^{+}$ is a ring of definition and is the completion of $\varinjlim R_{h}^{+}$ . A map from $(R,R^{+})$ to $(S,S^{+})$ is uniquely determined by a $K^{\circ }$ -linear map from $\varinjlim R_{h}^{\circ }$ to $S^{+}$ . Similarly, a map from $(R_{h},R_{h}^{\circ })$ to $(S,S^{+})$ is uniquely determined by a $K^{\circ }$ -linear map from $R_{h}^{\circ }$ to $S^{+}$ . From the isomorphism $\operatorname{Hom}_{K^{+}}(\varinjlim R_{h}^{\circ },S^{+})\cong \mathop{\varprojlim }\nolimits_{h}\operatorname{Hom}_{K^{+}}(R_{h}^{\circ },S^{+})$ we then deduce the claim.◻

Proof. This follows from $\widehat{\mathbb{T}}^{N}\sim \mathop{\varprojlim }\nolimits_{h}\operatorname{Spa}K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p^{h}}\rangle$ and from [Reference ScholzeSch12, Proposition 7.16].◻

Proof. The first statement follows from [Reference ScholzeSch12, Lemma 7.5]. For the second, we remark that according to [Reference HuberHub93, Lemma 3.10] any rational subspace $U=(f_{1},\ldots ,f_{n}|g)$ of $X$ can be defined by means of elements $f_{i},g$ lying in $\mathop{\varinjlim }\nolimits_{h}{\mathcal{O}}(X_{h})$ since it is dense in ${\mathcal{O}}(X)$ , and hence the claim.◻

Proof. Any étale map of adic spaces is locally a composition of rational embeddings and finite étale maps and they descend because of Proposition 2.9. We, therefore, obtain an affinoid refinement $\{V_{j}\rightarrow X\}$ of ${\mathcal{U}}$ such that each $V_{i}$ descends to some $X_{h}$ . Since $X$ is quasi-compact, we can also refine this covering by a finite one, and choose a common index $\bar{h}$ where each $V_{j}$ descends to.◻

Proof. Let $X$ be locally étale over $\widehat{\mathbb{T}}^{N}$ . Then it is locally open in a finite étale space over a rational subspace of $\widehat{\mathbb{T}}^{N}=\mathop{\varprojlim }\nolimits_{h}\mathbb{T}^{N}\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p^{h}}\rangle$ . By Proposition 2.9, we conclude it is locally of the form $X_{0}\,\times _{\mathbb{T}^{N}}\,\widehat{\mathbb{T}}^{N}$ for some étale map $X_{0}\rightarrow \mathbb{T}^{N}=\operatorname{Spa}(K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1}\rangle ,K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1}\rangle )$ , which is the composition of rational embeddings and finite étale maps.◻

Proof. By direct computation, the variety $X_{h}$ is isomorphic to $\operatorname{Spa}(K\langle \unicode[STIX]{x1D710},\unicode[STIX]{x1D714}\rangle /(\unicode[STIX]{x1D714}^{p^{h}}-(\unicode[STIX]{x1D70B}\unicode[STIX]{x1D710}+1)))$ . Since $|p|\leqslant |\unicode[STIX]{x1D70B}|$ we deduce that $|\binom{p^{h}}{i}|\leqslant |\unicode[STIX]{x1D70B}|$ for all $0<i<p^{h}$ . In particular, in the ring $K\langle \unicode[STIX]{x1D710},\unicode[STIX]{x1D714}\rangle /(\unicode[STIX]{x1D714}^{p^{h}}-(\unicode[STIX]{x1D70B}\unicode[STIX]{x1D710}+1))$ one has

$$\begin{eqnarray}|(\unicode[STIX]{x1D714}-1)^{p^{h}}|=\biggl|\unicode[STIX]{x1D70B}\unicode[STIX]{x1D710}+\mathop{\sum }_{i=1}^{p^{h}-1}\binom{p^{h}}{i}\unicode[STIX]{x1D714}^{i}\biggr|=|\unicode[STIX]{x1D70B}|.\end{eqnarray}$$

Analogously, in the ring $K\langle \unicode[STIX]{x1D712}\rangle$ one has

$$\begin{eqnarray}|(\unicode[STIX]{x1D712}+\unicode[STIX]{x1D70B}^{-1/p^{h}})^{p^{h}}-\unicode[STIX]{x1D70B}^{-1}|=\biggl|\unicode[STIX]{x1D712}^{p^{h}}+\mathop{\sum }_{i=1}^{p^{h}-1}\binom{p^{h}}{i}\unicode[STIX]{x1D712}^{p^{h}-1}\unicode[STIX]{x1D70B}^{-i/p^{h}}\biggr|=1.\end{eqnarray}$$

The following maps are, therefore, well defined and clearly mutually inverse:

$$\begin{eqnarray}\displaystyle X_{h}=\operatorname{Spa}(K\langle \unicode[STIX]{x1D710},\unicode[STIX]{x1D714}\rangle /(\unicode[STIX]{x1D714}^{p^{h}}-(\unicode[STIX]{x1D70B}\unicode[STIX]{x1D710}+1))) & \leftrightarrows & \displaystyle \operatorname{Spa}(K\langle \unicode[STIX]{x1D712}\rangle )=\mathbb{B}^{1}\nonumber\\ \displaystyle (\unicode[STIX]{x1D710},\unicode[STIX]{x1D714}) & \mapsto & \displaystyle ((\unicode[STIX]{x1D712}+\unicode[STIX]{x1D70B}^{-1/p^{h}})^{p^{h}}-\unicode[STIX]{x1D70B}^{-1},\unicode[STIX]{x1D70B}^{1/p^{h}}\unicode[STIX]{x1D712}+1)\nonumber\\ \displaystyle \unicode[STIX]{x1D70B}^{-1/p^{h}}(\unicode[STIX]{x1D714}-1) & \leftarrow & \displaystyle \unicode[STIX]{x1D712}.\nonumber\end{eqnarray}$$

Consider the multiplicative map $\sharp :K^{\flat }\langle \unicode[STIX]{x1D710}^{1/p^{\infty }}\rangle =(K\langle \unicode[STIX]{x1D710}^{1/p^{\infty }}\rangle )^{\flat }\rightarrow K\langle \unicode[STIX]{x1D710}^{1/p^{\infty }}\rangle$ defined in [Reference ScholzeSch12, Proposition 5.17]. By our assumptions on $\unicode[STIX]{x1D70B}$ the element $(\unicode[STIX]{x1D710}-1)^{\sharp }-(\unicode[STIX]{x1D710}-1)$ is divisible by $\unicode[STIX]{x1D70B}$ in $K^{\circ }\langle \unicode[STIX]{x1D710}^{1/p^{\infty }}\rangle$ and, therefore, the rational subspace $\widehat{X}\cong U(\unicode[STIX]{x1D710}-1\,\mid \,\unicode[STIX]{x1D70B})$ of  $\widehat{\mathbb{T}}^{1}$ coincides with $U((\unicode[STIX]{x1D710}-1)^{\sharp }\,\mid \,\unicode[STIX]{x1D70B}^{\flat \sharp })$ . From [Reference ScholzeSch12, Theorem 6.3] we conclude $\widehat{X}^{\flat }\cong U(\unicode[STIX]{x1D710}-1\,\mid \,\unicode[STIX]{x1D70B}^{\flat }){\hookrightarrow}\widehat{\mathbb{T}}^{\flat 1}$ , which is isomorphic to $\widehat{\mathbb{B}}^{\flat 1}$ , and hence the claim.◻

Assumption 3.1. Unless otherwise stated, we assume from now on that $\unicode[STIX]{x1D6EC}$ is a $\mathbb{Q}$ -algebra and we omit it from the notations.

Proof. According to [Reference HirschhornHir03, Theorem 4.1.1], any model category that is left proper and cellular (some technical properties that are defined in [Reference HirschhornHir03, Definitions 12.1.1 and 13.1.1]) admits a left Boudsfield localization with respect to a set of maps. The model structure on complexes is left proper and cellular (see [Reference Schwede and ShipleySS00, p. 7]). It follows that the projective model structures in the statement are also left proper and cellular (see [Reference HirschhornHir03, Propositions 12.1.5 and 13.1.14]), and hence the first claim.

For the first part of the second claim, it suffices to apply [Reference AyoubAyo07, Proposition 4.4.31, Lemma 4.4.34] showing that the localization over $S_{\unicode[STIX]{x1D70F}}$ is equivalent to a localization over a set of maps. The second part is a restatement of [Reference AyoubAyo07, Corollary 4.4.42, Proposition 4.4.55].

Since by [Reference AyoubAyo07, Proposition 4.4.31] the $\unicode[STIX]{x1D70F}$ -localization coincides with the Bousfield localization with respect to a set, we conclude by [Reference AyoubAyo07, Theorem 4.2.71] that the model category $\mathbf{Ch}_{\unicode[STIX]{x1D70F}}\mathbf{Psh}(\mathbf{C})$ is still left proper and cellular. The last statement then follows from [Reference HirschhornHir03, Theorem 4.1.1] and the second claim.◻

Proof. It suffices to apply Proposition 3.3 to the sites with interval $(\operatorname{RigSm},\acute{\text{e}}\text{t},\mathbb{B}^{1})$ and $(\widehat{\text{Rig}}\operatorname{Sm},\acute{\text{e}}\text{t},\mathbb{B}^{1})$ , where $\mathbf{C}^{\prime }$ is in both cases the subcategory of varieties with good coordinates.◻

Proof. Any $\acute{\text{e}}\text{t}$ -local object ${\mathcal{F}}$ is also local with respect to the maps of the statement. We are left to prove that a complex ${\mathcal{F}}$ , which is local with respect to the maps of the statement, is also $\acute{\text{e}}\text{t}$ -local.

Since $\unicode[STIX]{x1D6EC}$ contains $\mathbb{Q}$ the étale cohomology of an étale sheaf ${\mathcal{F}}$ coincides with the Nisnevich cohomology (the same proof of [Reference Mazza, Voevodsky and WeibelMVW06, Proposition 14.23] holds also here). By means of [Reference AyoubAyo15, Corollary 1.2.21] we conclude that any rigid variety $X$ has a finite cohomological dimension. By [Reference Artin, Grothendieck and VerdierSGA4, Theorem V.7.4.1] and [Reference Suslin and VoevodskySV00, Theorem 0.3], we obtain for any rigid variety $X$ and any complex of presheaves ${\mathcal{F}}$ an isomorphism

$$\begin{eqnarray}\mathbb{H}_{\acute{\text{e}}\text{t}}^{n}(X,{\mathcal{F}})\cong \underset{{\mathcal{U}}_{\bullet }\in \mathit{HR}_{\infty }(X)}{\varinjlim }H_{-n}\text{Hom}_{\bullet }(\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet }),{\mathcal{F}}),\end{eqnarray}$$

where $\mathit{HR}_{\infty }(X)$ is the category of bounded étale hypercoverings of $X$ (see [Reference Artin, Grothendieck and VerdierSGA4, V.7.3]) and $\text{Hom}_{\bullet }$ is the $\operatorname{Hom}$ -complex computed in the unbounded derived category of presheaves. Suppose now ${\mathcal{F}}$ is local with respect to the maps of the statement. Then $\text{Hom}_{\bullet }(\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet }),{\mathcal{F}})$ is quasi-isomorphic to $\text{Hom}_{\bullet }(X,{\mathcal{F}})$ for every bounded hypercovering ${\mathcal{U}}_{\bullet }$ , and hence $H_{-n}{\mathcal{F}}(X)\cong \mathbb{H}_{\acute{\text{e}}\text{t}}^{n}(X,{\mathcal{F}})$ by the formula above. We then conclude that the map ${\mathcal{F}}\rightarrow C^{\acute{\text{e}}\text{t}}{\mathcal{F}}$ is a quasi-isomorphism, proving the proposition.◻

Proof. Using Proposition 3.10, it suffices to prove that the map $\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet })\rightarrow \unicode[STIX]{x1D6EC}(X)$ is an isomorphism in the homotopy category $L_{S}\mathbf{Ch}(\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}))$ for a fixed bounded étale hypercovering ${\mathcal{U}}_{\bullet }$ of an object $X$ in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ .

Since the inclusion functor $\mathbf{Ch}_{{\geqslant}0}\rightarrow \mathbf{Ch}$ is a Quillen functor, it suffices to prove that $\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet })\rightarrow \unicode[STIX]{x1D6EC}(X)$ is a weak equivalence in $L_{T}\mathbf{Ch}_{{\geqslant}0}(\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}))$ , where $T$ is the set of shifts of the maps of complexes induced by étale Cech hypercoverings descending at finite level. Let $L_{\tilde{T}}\mathbf{sPsh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}})$ be the Bousfield localization of the projective model structure on simplicial presheaves of sets with respect to the set $\tilde{T}$ formed by maps induced by étale Cech hypercoverings ${\mathcal{U}}_{\bullet }\rightarrow X$ descending at finite level. We remark that the Dold–Kan correspondence (see [Reference Schwede and ShipleySS03, § 4.1]) and the $\unicode[STIX]{x1D6EC}$ -enrichment also define a left Quillen functor from $L_{\tilde{T}}\mathbf{sPsh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}})$ to the category $L_{T}\mathbf{Ch}_{{\geqslant}0}(\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}))$ . It, therefore, suffices to prove that ${\mathcal{U}}_{\bullet }\rightarrow X$ is a weak equivalence in $L_{\tilde{T}}\mathbf{sPsh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}})$ and this follows from the fact that bounded hypercoverings define the same localization as Cech hypercoverings (see [Reference Dugger, Hollander and IsaksenDHI04, Theorem A.6]) together with the fact that coverings descending to finite level define the same topology (Corollary 2.10), and hence the same localization [Reference Dugger, Hollander and IsaksenDHI04, Corollary A.8]. We remark that [Reference Dugger, Hollander and IsaksenDHI04, Corollary A.8] applies in our case even if the coverings ${\mathcal{U}}\rightarrow X$ descending to the finite level do not form a basis of the topology, as their pullback via an arbitrary map $Y\rightarrow X$ may not have the same property. However, the proof of the statement relies on [Reference Dugger, Hollander and IsaksenDHI04, Proposition A.2], where it is only used that the chosen family of coverings ${\mathcal{U}}\rightarrow X$ generates the topology and that the fiber product ${\mathcal{U}}\,\times _{X}\,{\mathcal{U}}$ is defined.◻

Proof. Consider the isomorphism $s_{n}:\mathbb{B}^{n+1}\rightarrow \mathbb{B}^{n}\,\times \,\mathbb{B}^{1}$ defined on points by separating the last coordinate and let $s_{n}^{\ast }$ be the induced map ${\mathcal{F}}(\Box ^{n}\,\times \,\mathbb{B}^{1})\rightarrow {\mathcal{F}}(\Box ^{n+1})$ . We have $s_{n-1}^{\ast }\,\circ \,d_{r,\unicode[STIX]{x1D716}}^{\ast }=d_{r,\unicode[STIX]{x1D716}}^{\ast }\,\circ \,s_{n}^{\ast }$ for all $1\leqslant r\leqslant n$ and $\unicode[STIX]{x1D716}\in \{0,1\}$ . We conclude that

$$\begin{eqnarray}\displaystyle & & \displaystyle s_{n-1}^{\ast }\,\circ \,\mathop{\sum }_{r=1}^{n}(-1)^{r}(d_{r,1}^{\ast }-d_{r,0}^{\ast })+\mathop{\sum }_{r=1}^{n+1}(-1)^{r}(d_{r,1}^{\ast }-d_{r,0}^{\ast })\,\circ \,(-s_{n}^{\ast })\nonumber\\ \displaystyle & & \displaystyle \quad =(-1)^{n}(d_{n+1,1}^{\ast }\,\circ \,s_{n}^{\ast }-d_{n+1,0}^{\ast }\,\circ \,s_{n}^{\ast })=(-1)^{n}(i_{1}^{\ast }-i_{0}^{\ast }).\nonumber\end{eqnarray}$$

Therefore, the maps $\{(-1)^{n}s_{n}^{\ast }\}$ define a chain homotopy from $i_{0}^{\ast }$ to $i_{1}^{\ast }$ as maps of complexes $C_{\bullet }^{\sharp }{\mathcal{F}}(\Box \,\times \,\mathbb{B}^{1})\rightarrow C_{\bullet }^{\sharp }{\mathcal{F}}(\Box )$ .

We automatically deduce that if an inclusion $C_{\bullet }^{\prime }{\mathcal{F}}\rightarrow C_{\bullet }^{\sharp }{\mathcal{F}}$ has a functorial retraction, then the maps $i_{0}^{\ast },i_{1}^{\ast }:C_{\bullet }^{\prime }{\mathcal{F}}(\Box \,\times \,\mathbb{B}^{1})\rightarrow C_{\bullet }^{\prime }{\mathcal{F}}(\Box )$ are also chain homotopic.◻

Proof. In order to prove that $\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}}$ is $\mathbb{B}^{1}$ -local in $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ we need to check that each homology presheaf $H_{n}(\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}})$ is homotopy-invariant. By means of [Reference AyoubAyo15, Proposition 2.2.46] it suffices to show that the maps $i_{0}^{\ast },i_{1}^{\ast }:N_{\bullet }{\mathcal{F}}(\Box \,\times \,\mathbb{B}^{1})\rightarrow N_{\bullet }{\mathcal{F}}(\Box )$ are chain homotopic, and this follows from Lemma 3.14.

We now prove that $\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}}$ is $\mathbb{B}^{1}$ -weak equivalent to ${\mathcal{F}}$ . We first prove that the canonical map $a:{\mathcal{F}}\rightarrow \text{}\underline{\text{Hom}}(\unicode[STIX]{x1D6EC}(\Box ^{n}),{\mathcal{F}})$ has an inverse up to homotopy for a fixed $n$ . Consider the map $b:\text{}\underline{\text{Hom}}(\unicode[STIX]{x1D6EC}(\Box ^{n}),{\mathcal{F}})\rightarrow {\mathcal{F}}$ induced by the zero section of $\Box ^{n}$ . It holds that $b\,\circ \,a=\operatorname{id}$ and $a\,\circ \,b$ is homotopic to $\operatorname{id}$ via the map

$$\begin{eqnarray}H:\unicode[STIX]{x1D6EC}(\mathbb{B}^{1})\,\otimes \,\text{}\underline{\text{Hom}}(\unicode[STIX]{x1D6EC}(\Box ^{n}),{\mathcal{F}})\rightarrow \text{}\underline{\text{Hom}}(\unicode[STIX]{x1D6EC}(\Box ^{n}),{\mathcal{F}}),\end{eqnarray}$$

which is deduced from the adjunction $(\unicode[STIX]{x1D6EC}(\mathbb{B}^{1})\,\otimes \,\cdot ,\text{}\underline{\text{Hom}}(\unicode[STIX]{x1D6EC}(\mathbb{B}^{1}),\cdot ))$ and the map

$$\begin{eqnarray}\text{}\underline{\text{Hom}}(\unicode[STIX]{x1D6EC}(\Box ^{n}),{\mathcal{F}})\rightarrow \text{}\underline{\text{Hom}}(\unicode[STIX]{x1D6EC}(\mathbb{B}^{1}\,\times \,\Box ^{n}),{\mathcal{F}})\end{eqnarray}$$

defined via the homothety of $\mathbb{B}^{1}$ on $\Box ^{n}$ . As $\mathbb{B}^{1}$ -weak equivalences are stable under filtered colimits and cones, we also conclude that the total complex associated to the simple complex of $\text{}\underline{\text{Hom}}(\unicode[STIX]{x1D6EC}(\Box ),{\mathcal{F}})$ is $\mathbb{B}^{1}$ -equivalent to the one associated to the constant cubical object ${\mathcal{F}}$ (see, for example, the argument of [Reference AyoubAyo15, Corollary 2.5.36]), which is in turn quasi-isomorphic to ${\mathcal{F}}$ .◻

Proof. The first claim follows from Proposition 3.15. We are left to prove that the complex $\operatorname{Sing}^{\mathbb{B}^{1}}(C^{\acute{\text{e}}\text{t}}{\mathcal{F}}^{\bullet })$ is $\acute{\text{e}}\text{t}$ -local. To this aim, we use the description given in Proposition 3.10 and we show that $\operatorname{Sing}^{\mathbb{B}^{1}}(C^{\acute{\text{e}}\text{t}}{\mathcal{F}}^{\bullet })$ is local with respect to shifts of maps $\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet })\rightarrow \unicode[STIX]{x1D6EC}(X)$ induced by bounded hypercoverings ${\mathcal{U}}_{\bullet }\rightarrow X$ .

Fix such a hypercovering ${\mathcal{U}}_{\bullet }\rightarrow X$ . From the isomorphisms

$$\begin{eqnarray}H_{p}\text{Hom}_{\bullet }(\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet }\,\times \,\Box ^{q}),C^{\acute{\text{e}}\text{t}}{\mathcal{F}})\cong H_{p}\text{Hom}_{\bullet }(\unicode[STIX]{x1D6EC}(X\,\times \,\Box ^{q}),C^{\acute{\text{e}}\text{t}}{\mathcal{F}})\end{eqnarray}$$

valid for all $p,q$ and a spectral sequence argument (see [Reference Suslin and VoevodskySV00, Theorem 0.3]) we deduce that

$$\begin{eqnarray}\mathbf{D}(\unicode[STIX]{x1D6EC}(X)[n],\operatorname{Sing}^{\mathbb{B}^{1}}C^{\acute{\text{e}}\text{t}}{\mathcal{F}})\cong \mathbf{D}(\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet })[n],\operatorname{Sing}^{\mathbb{B}^{1}}C^{\acute{\text{e}}\text{t}}{\mathcal{F}})\end{eqnarray}$$

for all $n$ as wanted.◻

Proof. The first claim is a special instance of [Reference AyoubAyo07, Proposition 4.4.45].

We prove the second claim by showing that $\mathbb{R}\unicode[STIX]{x1D704}_{\ast }\mathbb{L}\unicode[STIX]{x1D704}^{\ast }$ is isomorphic to the identity. Let ${\mathcal{F}}$ be a cofibrant object in $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\operatorname{RigSm})$ . We need to prove that the map ${\mathcal{F}}\rightarrow \unicode[STIX]{x1D704}_{\ast }(\operatorname{Sing}^{\mathbb{B}^{1}}C^{\acute{\text{e}}\text{t}}(\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}}))$ is an $(\acute{\text{e}}\text{t},\mathbb{B}^{1})$ -weak equivalence. Since $\unicode[STIX]{x1D704}_{\ast }$ commutes with $\operatorname{Sing}^{\mathbb{B}^{1}}$ we are left to prove that the map $\unicode[STIX]{x1D704}_{\ast }\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}}={\mathcal{F}}\rightarrow \unicode[STIX]{x1D704}_{\ast }C^{\acute{\text{e}}\text{t}}(\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})$ is an $\acute{\text{e}}\text{t}$ -weak equivalence. This follows since $\unicode[STIX]{x1D704}_{\ast }$ preserves $\acute{\text{e}}\text{t}$ -weak equivalences, as it commutes with $\acute{\text{e}}\text{t}$ -sheafification.◻

Proof. The statements are analogous, and we only consider the case of $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K)$ . It is clear that the set of functors $H_{i}\text{Hom}_{\bullet }(\unicode[STIX]{x1D6EC}(X),\cdot )$ detect quasi-isomorphisms between étale local objects, by letting $X$ vary in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ and $i$ vary in $\mathbb{Z}$ . We are left to prove that the motives $\unicode[STIX]{x1D6EC}(X)$ with $X$ in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ are compact. Since $\unicode[STIX]{x1D6EC}(X)$ is compact in $\mathbf{D}(\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}))$ and $\operatorname{Sing}^{\mathbb{B}^{1}}$ commutes with direct sums, it suffices to prove that if $\{{\mathcal{F}}_{i}\}_{i\in I}$ is a family of $\acute{\text{e}}\text{t}$ -local complexes, then also $\bigoplus _{i}{\mathcal{F}}_{i}$ is $\acute{\text{e}}\text{t}$ -local. If $I$ is finite, the claim follows from the isomorphisms $H_{-n}\text{Hom}_{\bullet }(X,\bigoplus _{i}{\mathcal{F}}_{i})\cong \bigoplus _{i}\mathbb{H}^{n}(X,{\mathcal{F}}_{i})\cong \mathbb{H}^{n}(X,\bigoplus _{i}{\mathcal{F}}_{i})$ . A coproduct over an arbitrary family is a filtered colimit of finite coproducts, and hence the claim follows from [Reference AyoubAyo07, Proposition 4.5.62].◻

Proof. It suffices to apply Proposition 3.3 to the sites with interval $(\operatorname{PerfSm},\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1})$ and $(\widehat{\text{Rig}}\operatorname{Sm},\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1})$ , where $\mathbf{C}^{\prime }$ is in both cases the subcategory of affinoid rigid varieties with good coordinates.◻

Proof. The tilting equivalence (see Theorem 1.12) induces an equivalence of the étale sites on perfectoid spaces over $K$ and over $K^{\flat }$ (see [Reference ScholzeSch12, Theorem 7.12]). Moreover $(\widehat{\mathbb{T}}^{n})^{\flat }=\widehat{\mathbb{T}}^{n}$ and $(\widehat{\mathbb{B}}^{n})^{\flat }=\widehat{\mathbb{B}}^{n}$ . It therefore induces an equivalence of sites with interval $(\operatorname{PerfSm}/K,\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1})\cong (\operatorname{PerfSm}/K^{\flat },\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1})$ , and hence the claim.◻

Proof. The result follows in the same way as Proposition 3.17. ◻

Proof. The fact that $\operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}{\mathcal{F}}$ is $\widehat{\mathbb{B}}^{1}$ -local in $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ can be deduced by Lemmas 3.27 and 3.28. We are left to prove that $\operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}{\mathcal{F}}$ is $\widehat{\mathbb{B}}^{1}$ -weak equivalent to ${\mathcal{F}}$ and this follows in the same way as in the proof of Proposition 3.15.◻

Proof. This follows in the same way as [Reference Mazza, Voevodsky and WeibelMVW06, Lemma 2.16]. ◻

Proof. This follows in the same way as Lemma 3.14. ◻

Proof. This follows in the same way as Corollary 3.16. ◻

Proof. This follows in the same way as Proposition 3.18. ◻

Proof. It suffices to prove that $X\,\times \,\mathbb{B}^{1}\rightarrow X$ induces a $\widehat{\mathbb{B}}^{1}$ -weak equivalence, for any variety $X$ in $\widehat{\text{Rig}}\operatorname{Sm}$ . This follows as the multiplicative homothety $\widehat{\mathbb{B}}^{1}\,\times \,\mathbb{B}^{1}\rightarrow \mathbb{B}^{1}$ induces a homotopy between the zero map and the identity on $\mathbb{B}^{1}$ .◻

Proof. Because of Proposition 3.32, the triangulated category $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}(K)$ coincides with the localization of $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K)$ with respect to the set generated by the maps $\unicode[STIX]{x1D6EC}(\widehat{\mathbb{B}}_{X}^{1})[n]\rightarrow \unicode[STIX]{x1D6EC}(X)[n]$ as $X$ varies in $\widehat{\text{Rig}}\operatorname{Sm}$ and $n$ in $\mathbb{Z}$ .◻

Proof. We need to prove that the natural map

$$\begin{eqnarray}\unicode[STIX]{x1D719}:\underset{h}{\varinjlim }C_{\bullet }\unicode[STIX]{x1D6EC}\operatorname{Hom}(X_{h}\,\times \,\Box ,Y)\rightarrow C_{\bullet }\unicode[STIX]{x1D6EC}\operatorname{Hom}(X\,\times \,\Box ,Y)\end{eqnarray}$$

defines bijections on homology groups.

We start by proving surjectivity. As $\Box$ is a $\unicode[STIX]{x1D6F4}$ -enriched cocubical object, the complexes above are quasi-isomorphic to the associated normalized complexes $N_{\bullet }$ , which we consider instead. Suppose that $\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6EC}\operatorname{Hom}(X\,\times \,\Box ^{n},Y)$ defines a cycle in $N_{n}$ , i.e.,  $\unicode[STIX]{x1D6FD}\,\circ \,d_{r,\unicode[STIX]{x1D716}}=0$ for $1\leqslant r\leqslant n$ and $\unicode[STIX]{x1D716}\in \{0,1\}$ . This means that $\unicode[STIX]{x1D6FD}=\sum \unicode[STIX]{x1D706}_{k}f_{k}$ with $\unicode[STIX]{x1D706}_{k}\in \unicode[STIX]{x1D6EC}$ , $f_{k}\in \operatorname{Hom}(X\,\times \,\Box ^{n},Y)$ and $\sum \unicode[STIX]{x1D706}_{k}f_{k}\,\circ \,d_{r,\unicode[STIX]{x1D716}}=0$ . This amounts to saying that for every $k,r,\unicode[STIX]{x1D716}$ the sum $\sum \unicode[STIX]{x1D706}_{k^{\prime }}$ over the indices $k^{\prime }$ such that $f_{k^{\prime }}\,\circ \,d_{r,\unicode[STIX]{x1D716}}=f_{k}\,\circ \,d_{r,\unicode[STIX]{x1D716}}$  is zero. By Proposition 4.1, we can find an integer $h$ and maps $H_{k}\in \operatorname{Hom}(X\,\times \,\Box ^{n}\,\times \,\mathbb{B}^{1},Y)$ such that $i_{0}^{\ast }H=f_{k}$ , $i_{1}^{\ast }H=\unicode[STIX]{x1D719}(\tilde{f}_{k})$ with $\tilde{f}_{k}\in \operatorname{Hom}(X_{h}\,\times \,\Box ^{n},Y)$ and $H_{k}\,\circ \,d_{r,\unicode[STIX]{x1D716}}=H_{k^{\prime }}\,\circ \,d_{r,\unicode[STIX]{x1D716}}$ whenever $f_{k}\,\circ \,d_{r,\unicode[STIX]{x1D716}}=f_{k^{\prime }}\,\circ \,d_{r,\unicode[STIX]{x1D716}}$ . If we denote by $H$ the cycle $\sum \unicode[STIX]{x1D706}_{k}H_{k}\in \unicode[STIX]{x1D6EC}\operatorname{Hom}(X\,\times \,\Box ^{n}\,\times \,\mathbb{B}^{1},Y)$ we therefore have $d_{r,\unicode[STIX]{x1D716}}^{\ast }H=0$ for all $r,\unicode[STIX]{x1D716}$ .

By Lemma 3.14, we conclude that $i_{1}^{\ast }H$ and $i_{0}^{\ast }H$ define the same homology class, and therefore $\unicode[STIX]{x1D6FD}$ defines the same class as $i_{1}^{\ast }H$ , which is the image of a class in $\unicode[STIX]{x1D6EC}\operatorname{Hom}(X_{h}\,\times \,\Box ^{n},Y)$ , as desired.

We now turn to injectivity. Consider an element $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6EC}\operatorname{Hom}(X_{0}\,\times \,\Box ^{n},Y)$ such that $\unicode[STIX]{x1D6FC}\,\circ \,d_{r,\unicode[STIX]{x1D716}}=0$ for all $r,\unicode[STIX]{x1D716}$ and suppose there exists an element $\unicode[STIX]{x1D6FD}=\sum \unicode[STIX]{x1D706}_{i}f_{i}\in \unicode[STIX]{x1D6EC}\operatorname{Hom}(X\,\times \,\Box ^{n+1},Y)$ such that $\unicode[STIX]{x1D6FD}\,\circ \,d_{r,0}=0$ for $1\leqslant r\leqslant n+1$ , $\unicode[STIX]{x1D6FD}\,\circ \,d_{r,1}=0$ for $2\leqslant r\leqslant n+1$ and $\unicode[STIX]{x1D6FD}\,\circ \,d_{1,1}=\unicode[STIX]{x1D719}(\unicode[STIX]{x1D6FC})$ . Again, by Proposition 4.1, we can find an integer $\bar{h}$ and maps $H_{k}\in \operatorname{Hom}(X\,\times \,\Box ^{n+1}\,\times \,\mathbb{B}^{1},Y)$ such that $H:=\sum \unicode[STIX]{x1D706}_{k}H_{k}$ satisfies $i_{1}^{\ast }H=\unicode[STIX]{x1D719}(\unicode[STIX]{x1D6FE})$ for some $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6EC}\operatorname{Hom}(X_{\bar{h}}\,\times \,\Box ^{n+1},Y)$ , $H\,\circ \,d_{r,0}=0$ for $1\leqslant r\leqslant n+1$ , $H\,\circ \,d_{r,1}=0$ for $2\leqslant r\leqslant n+1$ and $H\,\circ \,d_{1,1}$ is constant on $\mathbb{B}^{1}$ and coincides with $\unicode[STIX]{x1D719}(\unicode[STIX]{x1D6FC})$ . We conclude that $\unicode[STIX]{x1D6FE}\in N_{n}$ and $d\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}$ . In particular, $\unicode[STIX]{x1D6FC}=0$ in the homology group, as desired.◻

Proof. As homology commutes with filtered colimits, by means of Remark 3.2 we can assume that ${\mathcal{F}}$ is a bounded above complex formed by sums of representable presheaves. For any $X$ in $\widehat{\text{Rig}}\operatorname{Sm}$ the homology of $\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}}(X)$ coincides with the homology of the total complex associated to $C_{\bullet }({\mathcal{F}}(X\,\times \,\Box ))$ . The result then follows from Proposition 4.2 and the convergence of the spectral sequence associated to the double complex above, which is concentrated in one quadrant.◻

Proof. The difficulty lies in showing that the object $\operatorname{Sing}^{\mathbb{B}^{1}}(\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})$ is $\acute{\text{e}}\text{t}$ -local. By Propositions 3.11 and 3.15, it suffices to prove that $\operatorname{Sing}^{\mathbb{B}^{1}}(\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})$ is local with respect to the étale-Cech hypercoverings ${\mathcal{U}}_{\bullet }\rightarrow X$ in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ of $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ descending at finite level. Let ${\mathcal{U}}_{\bullet }\rightarrow X$ be one of them. Without loss of generality, we assume that it descends to an étale covering of $X_{0}$ . In particular, we conclude that ${\mathcal{U}}_{n}=\mathop{\varprojlim }\nolimits_{h}{\mathcal{U}}_{nh}$ is a disjoint union of objects in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ .

We need to show that $\text{Hom}_{\bullet }(\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet }),\operatorname{Sing}^{\mathbb{B}^{1}}(\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}}))$ is quasi-isomorphic to $\operatorname{Sing}^{\mathbb{B}^{1}}(\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})(X)$ . Using Corollary 4.3, we conclude that for each $n\in \mathbb{N}$ the complex $(\operatorname{Sing}^{\mathbb{B}^{1}}\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})({\mathcal{U}}_{n})$ is quasi-isomorphic to $\mathop{\varinjlim }\nolimits_{h}(\operatorname{Sing}^{\mathbb{B}^{1}}\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})({\mathcal{U}}_{nh})$ . Passing to the homotopy limit on $n$ on both sides, we deduce that $\text{Hom}_{\bullet }(\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet }),\operatorname{Sing}^{\mathbb{B}^{1}}\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})$ is quasi-isomorphic to $\mathop{\varinjlim }\nolimits_{h}\text{Hom}_{\bullet }(\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet h}),\operatorname{Sing}^{\mathbb{B}^{1}}\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})$ . Using again Corollary 4.3, we also obtain that $(\operatorname{Sing}^{\mathbb{B}^{1}}\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})(X)$ is quasi-isomorphic to $\mathop{\varinjlim }\nolimits_{h}(\operatorname{Sing}^{\mathbb{B}^{1}}\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})(X_{h})$ .

From the exactness of $\varinjlim$ it suffices then to prove that the maps

$$\begin{eqnarray}\text{Hom}_{\bullet }(\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet h}),\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}})\rightarrow \text{Hom}_{\bullet }(\unicode[STIX]{x1D6EC}(X_{h}),\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}})\end{eqnarray}$$

are quasi-isomorphisms. This follows once we show that the complex $\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}}$ is $\acute{\text{e}}\text{t}$ -local.

We point out that since ${\mathcal{F}}$ is $\mathbb{B}^{1}$ -local, then the canonical map ${\mathcal{F}}\rightarrow \operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}}$ is a quasi-isomorphism. As ${\mathcal{F}}$ is $\acute{\text{e}}\text{t}$ -local we conclude that $\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}}$ also is, and hence the claim.◻

Proof. Since any complex ${\mathcal{F}}$ has a fibrant-cofibrant replacement in $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\operatorname{RigSm}^{\operatorname{gc}})$ we can assume that ${\mathcal{F}}$ is cofibrant and $(\acute{\text{e}}\text{t},\mathbb{B}^{1})$ -fibrant. Since it is $\mathbb{B}^{1}$ -local, it is quasi-isomorphic to $\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}}$ . By Corollary 4.3, for any integer $i$ one has

$$\begin{eqnarray}\underset{h}{\varinjlim }\operatorname{Hom}(\unicode[STIX]{x1D6EC}(X_{h})[i],\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}})\cong \operatorname{Hom}(\unicode[STIX]{x1D6EC}(X)[i],\operatorname{Sing}^{\mathbb{B}^{1}}\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}}).\end{eqnarray}$$

As $\unicode[STIX]{x1D6EC}(X)$ is a cofibrant object in $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}})$ and $\operatorname{Sing}^{\mathbb{B}^{1}}\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}}$ is a $(\mathbb{B}^{1},\acute{\text{e}}\text{t})$ -local replacement of ${\mathcal{F}}$ in $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}})$ by Proposition 4.4, we conclude that the previous isomorphism can be rephrased in the following way:

$$\begin{eqnarray}\underset{h}{\varinjlim }\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)(\unicode[STIX]{x1D6EC}(X_{h})[i],{\mathcal{F}})\cong \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K)(\unicode[STIX]{x1D6EC}(X)[i],\mathbb{L}\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}}),\end{eqnarray}$$

proving the claim. ◻

Proof. By means of Theorem 5.1, we can equally prove the statement in the category $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ . We claim that we can also make an arbitrary finite field extension $L/K$ . Indeed the transpose of the natural map $Y_{L}\rightarrow Y$ is a correspondence from $Y$ to $Y_{L}$ . Since $\unicode[STIX]{x1D6EC}$ is a $\mathbb{Q}$ -algebra, we conclude that $\unicode[STIX]{x1D6EC}_{\operatorname{tr}}(Y)$ is a direct factor of $\unicode[STIX]{x1D6EC}_{\operatorname{tr}}(Y_{L})=\mathbb{L}e_{\sharp }\unicode[STIX]{x1D6EC}_{\operatorname{tr}}(Y_{L})$ for any variety $Y$ , where $\mathbb{L}e_{\sharp }$ is the functor $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(L)\rightarrow \mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ induced by restriction of scalars. In particular, if $\unicode[STIX]{x1D6EC}_{\operatorname{tr}}((X_{h+1})_{L})\rightarrow \unicode[STIX]{x1D6EC}_{\operatorname{tr}}((X_{h})_{L})$ is an isomorphism in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(L)$ then $\unicode[STIX]{x1D6EC}_{\operatorname{tr}}((X_{h+1})_{L})\rightarrow \unicode[STIX]{x1D6EC}_{\operatorname{tr}}((X_{h})_{L})$ is an isomorphism in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ and therefore also $\unicode[STIX]{x1D6EC}_{\operatorname{tr}}(X_{h+1})\rightarrow \unicode[STIX]{x1D6EC}_{\operatorname{tr}}(X_{h})$ is.

By lemma [Reference AyoubAyo15, 1.1.52], we can suppose that  $X_{0}=\operatorname{Spa}(R_{0},R_{0}^{\circ })$  with  $R_{0}=S\langle \unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\rangle /(P(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}))$ , where $S={\mathcal{O}}(\mathbb{T}^{M})$ , $\unicode[STIX]{x1D70E}=(\unicode[STIX]{x1D70E}_{1},\ldots ,\unicode[STIX]{x1D70E}_{N})$ is a $N$ -tuple of coordinates, $\unicode[STIX]{x1D70F}=(\unicode[STIX]{x1D70F}_{1},\ldots ,\unicode[STIX]{x1D70F}_{m})$ is a $m$ -tuple of coordinates and $P$ is a set of $m$ polynomials in $S[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]$ with $\det (\unicode[STIX]{x2202}P/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F})\in R_{0}^{\times }$ . In particular, $X_{1}=\operatorname{Spa}(R_{1},R_{1}^{\circ })$ with $R_{1}=S\langle \unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\rangle /(P(\unicode[STIX]{x1D70E}^{p},\unicode[STIX]{x1D70F}))$ and the map $f:X_{1}\rightarrow X_{0}$ is induced by $\unicode[STIX]{x1D70E}\mapsto \unicode[STIX]{x1D70E}^{p}$ , $\unicode[STIX]{x1D70F}\mapsto \unicode[STIX]{x1D70F}$ . Since the map $f$ is finite and surjective, we can also consider the transpose correspondence $f^{T}\in \operatorname{RigCor}(X_{0},X_{1})$ . The composition $f\,\circ \,f^{T}$ is associated to the correspondence $X_{0}\xleftarrow[{}]{\;f}X_{1}\xrightarrow[{}]{f}X_{0}$ , which is the cycle $\deg (f)X_{0}=p^{N}\,\cdot \,\operatorname{id}_{X_{0}}$ . The composition $f^{T}\,\circ \,f$ is associated to the correspondence $X_{1}\xleftarrow[{}]{\;p_{1}}X_{1}\,\times _{X_{0}}\,X_{1}\xrightarrow[{}]{p_{2}}X_{1}$ . Since $\mathbb{T}^{N}\langle \unicode[STIX]{x1D70E}^{1/p}\rangle \,\times _{\mathbb{T}^{N}}\,\mathbb{T}^{N}\langle \unicode[STIX]{x1D70E}^{1/p}\rangle \cong \mathbb{T}^{N}\langle \unicode[STIX]{x1D70E}^{1/p}\rangle \,\times \,\unicode[STIX]{x1D707}_{p}^{N}$ we conclude that the above correspondence is $X_{1}\xleftarrow[{}]{\;p_{1}}X_{1}\,\times \,(\unicode[STIX]{x1D707}_{p})^{N}\xrightarrow[{}]{\unicode[STIX]{x1D702}}X_{1}$ , where $\unicode[STIX]{x1D702}$ is induced by the multiplication map $\mathbb{T}^{N}\,\times \,\unicode[STIX]{x1D707}_{p}^{N}\rightarrow \mathbb{T}^{N}$ . Up to a finite field extension, we can assume that $K$ has $p$ th roots of unity. The above correspondence is then equal to $\sum f_{\unicode[STIX]{x1D701}}$ , where each $f_{\unicode[STIX]{x1D701}}$ is a map $X_{1}\rightarrow X_{1}$ defined by $\unicode[STIX]{x1D70E}_{i}\mapsto \unicode[STIX]{x1D701}_{i}\unicode[STIX]{x1D70E}_{i}$ , $\unicode[STIX]{x1D70F}\mapsto \unicode[STIX]{x1D70F}$ for each $N$ -tuple $\unicode[STIX]{x1D701}=(\unicode[STIX]{x1D701}_{i})$ of $p$ th roots of unity. If we prove that each $f_{\unicode[STIX]{x1D701}}$ is homotopically equivalent to $\operatorname{id}_{X_{1}}$ then we get $(1/p^{N})f^{T}\,\circ \,f=\operatorname{id}$ , $f\,\circ \,(1/p^{N})f^{T}=\operatorname{id}$ in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}$ , as desired.

We are left to find a homotopy between $\operatorname{id}$ and $f_{\unicode[STIX]{x1D701}}$ for a fixed $\unicode[STIX]{x1D701}=(\unicode[STIX]{x1D701}_{1},\ldots ,\unicode[STIX]{x1D701}_{n})$ up to considering higher indices $h$ . For the sake of clarity, we consider them as maps $\operatorname{Spa}\bar{R}_{1}\rightarrow \operatorname{Spa}R_{1}$ , where we put $\bar{R}_{h}=S\langle \bar{\unicode[STIX]{x1D70E}},\bar{\unicode[STIX]{x1D70F}}\rangle /(P(\bar{\unicode[STIX]{x1D70E}}^{p^{h}},\bar{\unicode[STIX]{x1D70F}}))$ for any integer $h$ . The first map is induced by $\unicode[STIX]{x1D70E}\mapsto \bar{\unicode[STIX]{x1D70E}}$ , $\unicode[STIX]{x1D70F}\mapsto \bar{\unicode[STIX]{x1D70F}}$ and the second induced by $\unicode[STIX]{x1D70E}\mapsto \unicode[STIX]{x1D701}\bar{\unicode[STIX]{x1D70E}}$ , $\unicode[STIX]{x1D70F}\mapsto \bar{\unicode[STIX]{x1D70F}}$ . Let $F_{h}=\sum _{n}a_{n}(\unicode[STIX]{x1D70E}-\bar{\unicode[STIX]{x1D70E}})^{n}$ be the unique array of formal power series in $\bar{R}_{h}[[\unicode[STIX]{x1D70E}-\bar{\unicode[STIX]{x1D70E}}]]$ centered in $\bar{\unicode[STIX]{x1D70E}}$ associated to the polynomials $P(\unicode[STIX]{x1D70E}^{p^{h}},\unicode[STIX]{x1D70F})$ in $\bar{R}_{h}[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}]$ via Corollary A.2. Also let $\unicode[STIX]{x1D719}_{h}$ be the map $\bar{R}_{h}\rightarrow \bar{R}_{h+1}$ . From the formal equalities $P(\unicode[STIX]{x1D70E}^{p^{(h+1)}},F_{h+1}(\unicode[STIX]{x1D70E}))=0$ , $P(\unicode[STIX]{x1D70E}^{p^{h}},\unicode[STIX]{x1D719}(F_{h}(\unicode[STIX]{x1D70E})))=\unicode[STIX]{x1D719}_{h}(P(\unicode[STIX]{x1D70E}^{p^{h}},F_{h}(\unicode[STIX]{x1D70E})))=0$ and the uniqueness of $F_{h+1}$ we deduce $F_{h+1}(\unicode[STIX]{x1D70E})=\unicode[STIX]{x1D719}_{h}(F_{h}(\unicode[STIX]{x1D70E}^{p}))$ .

We therefore have

$$\begin{eqnarray}\displaystyle F_{h+1}(\unicode[STIX]{x1D70E}) & = & \displaystyle \mathop{\sum }_{n}\unicode[STIX]{x1D719}_{h}(a_{n})(\unicode[STIX]{x1D70E}^{p}-\bar{\unicode[STIX]{x1D70E}}^{p})^{n}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{n}\unicode[STIX]{x1D719}_{h}(a_{n})\biggl((\unicode[STIX]{x1D70E}-\bar{\unicode[STIX]{x1D70E}})^{p-1}+\mathop{\sum }_{j=1}^{p-1}\binom{p}{j}(\unicode[STIX]{x1D70E}-\bar{\unicode[STIX]{x1D70E}})^{j-1}\bar{\unicode[STIX]{x1D70E}}^{p-j}\biggr)^{n}(\unicode[STIX]{x1D70E}-\bar{\unicode[STIX]{x1D70E}})^{n}.\nonumber\end{eqnarray}$$

The expression

$$\begin{eqnarray}Q(x)=x^{p}+\mathop{\sum }_{j=1}^{p-1}\binom{p}{j}x^{j}\bar{\unicode[STIX]{x1D70E}}^{p-j}\end{eqnarray}$$

is a polynomial in $x$ and it is easy to show that the mapping $x\mapsto Q(x)$ extends to a map $\bar{R}_{h+1}\langle x\rangle \rightarrow \bar{R}_{h+1}\langle x\rangle$ . We deduce that we can read off the convergence in the circle of radius $1$ around $\bar{\unicode[STIX]{x1D70E}}$ and the values of $F_{h+1}$ on its expression given above.