A motivic version of the theorem of Fontaine and Wintenberger

Abstract

We establish a tilting equivalence for rational, homotopy-invariant cohomology theories defined over non-archimedean analytic varieties. More precisely, we prove an equivalence between the categories of motives of rigid analytic varieties over a perfectoid field $K$ of mixed characteristic and over the associated (tilted) perfectoid field $K^{\flat }$ of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger, claiming that the Galois groups of $K$ and $K^{\flat }$ are isomorphic.

Introduction

A theorem of Fontaine and Wintenberger [FW79], later expanded by Scholze [Sch12], states that there is an isomorphism between the Galois groups of a perfectoid field $K$ and the associated (tilted) perfect field $K^{\flat }$ of positive characteristic. The standard example of such a pair is formed by the completions of the fields $\mathbb{Q}_{p}(p^{1/p^{\infty }})$ and $\mathbb{F}_{p}((t))(t^{1/p^{\infty }})$ . This theorem is a cornerstone of $p$ -adic Hodge theory, providing a striking equivalence between objects in mixed characteristic (finite étale algebras over $K$ ) and objects in equal characteristic $p$ (finite étale algebras over $K^{\flat }$ ).

In the same spirit, the ‘tilting equivalence’ of Scholze [Sch12] promotes the equivalence above to a certain kind of space of higher dimension, namely the perfectoid spaces over the two fields. These objects are non-noetherian analytic spaces, which can typically be thought as certain infinite covers of classical rigid analytic varieties obtained by adding all $p$ th roots of local coordinates, and their introduction has had many different applications in arithmetic geometry (see [Sch14]).

The aim of this paper is to ‘descend’ Scholze’s tilting equivalence to the level of classical rigid analytic varieties defined over $K$ and $K^{\flat }$ , at least from a (co-)homological point of view. What we actually prove, is an equivalence between the two associated categories of (mixed, effective, with rational coefficients) rigid analytic motives $\mathbf{RigDM}$ defined over the two fields, introduced by Ayoub [Ayo15] adapting the classic construction of $\mathbf{DM}$ by Voevodsky [Voe96].

Theorem 7.10 . Let $K$ be a perfectoid field with tilt $K^{\flat }$ and let $\unicode[STIX]{x1D6EC}$ be a $\mathbb{Q}$ -algebra. There is a monoidal triangulated equivalence of categories

$$\begin{eqnarray}\mathfrak{F}:\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat },\unicode[STIX]{x1D6EC})\xrightarrow[{}]{{\sim}}\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC}).\end{eqnarray}$$

We remark that, from a motivic point of view, the theorem of Fontaine and Wintenberger can be rephrased by saying that the categories of Artin motives over the two fields are equivalent. Conversely, our theorem shows that not only the absolute Galois groups of a perfectoid field and its tilt are isomorphic, but also their absolute local motivic Galois groups (in the sense of the introduction of [Ayo15]). By [Vez17], it is also possible to state the result above with respect to motives without transfers $\mathbf{RigDA}$ defined over the two fields (the rigid analytic analog of the category $\mathbf{DA}$ see [Ayo15]).

The statement above involves only rigid analytic varieties and its proof uses Scholze’s theory of perfectoid spaces only in an auxiliary way. Nonetheless, we can restate our main result highlighting the role of perfectoid spaces as follows.

Theorem 7.11 . Let $K$ be a perfectoid field and let $\unicode[STIX]{x1D6EC}$ be a $\mathbb{Q}$ -algebra. There is a monoidal triangulated equivalence of categories

$$\begin{eqnarray}\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})\xrightarrow[{}]{{\sim}}\mathbf{PerfDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC}).\end{eqnarray}$$

The category $\mathbf{PerfDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ is built in the same way as $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ using as a starting point the big étale site of smooth, small perfectoid spaces, i.e., those that are locally étale over some perfectoid ball $\widehat{\mathbb{B}}^{n}$ .

This last theorem provides a way to ‘perfectoidify’ (and ‘de-perfectoidify’) canonically a rigid analytic variety into a perfectoid motive (and vice versa). The first theorem gives a way to ‘tilt’ (or ‘untilt’) canonically also a rigid analytic varieties over $K$ into a rigid analytic motive defined on the field $K^{\flat }$ (and vice versa).

We now make a rough sketch on how the motivic ‘untilting’ and ‘de-perfectoidification’ procedures work. Start from a smooth rigid variety $X$ over $K^{\flat }$ and associate to it a perfectoid space $\widehat{X}$ obtained by taking the perfection of $X$ . This operation can be performed canonically since $K^{\flat }$ has positive characteristic. We then use Scholze’s theorem to tilt $\widehat{X}$ obtaining a perfectoid space $\widehat{Y}$ in mixed characteristic. Suppose now that $\widehat{Y}$ is the limit of a tower of rigid analytic varieties

$$\begin{eqnarray}\cdots \rightarrow Y_{h+1}\rightarrow Y_{h}\rightarrow \cdots \rightarrow Y_{1}\rightarrow Y_{0}\end{eqnarray}$$

such that $Y_{0}$ is étale over the Tate ball $\mathbb{B}^{n}=\operatorname{Spa}K\langle \unicode[STIX]{x1D710}_{1},\ldots ,\unicode[STIX]{x1D710}_{n}\rangle$ and each $Y_{h}$ is obtained as the pullback of $Y_{0}$ by the map $\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}$ defined by taking the $p^{h}$ -powers of the coordinates $\unicode[STIX]{x1D710}_{i}\mapsto \unicode[STIX]{x1D710}_{i}^{p^{h}}$ . Under such hypotheses (we will actually need slightly stronger conditions on the tower above) we then ‘de-perfectoidify’ $\widehat{Y}$ by associating to it $Y_{\bar{h}}$ for a sufficiently big index $\bar{h}$ .

The main technical problem of this construction is to show that it is independent of the choice of the tower, and on the index $\bar{h}$ . It is also by definition a local procedure, which is not canonically extendable to arbitrary varieties by gluing. In order to overcome these obstacles, we use in a crucial way some techniques of approximating maps between spaces up to homotopy, which are obtained by a generalization of the implicit function theorem in the non-archimedean setting.

We point out that many deep motivic theorems are widely used throughout the article, and they are crucial to prove that the recipe sketched above gives rise to an equivalence (see the proof of 7.11) and not only for the construction of explicit homotopies. Among them, the Cancellation Theorem [Voe10] and the dualizability of compact motives [Rio05]. These results admit a rigid analytic version, proved by Ayoub in [Ayo15]. We also use the parallel between $\mathbf{DA}$ and $\mathbf{DM}$ and in order to do so, we introduce a finer topology in characteristic $p$ (the $\text{Frob}\acute{\text{e}}\text{t}$ topology, also called $\operatorname{quiet}$ in [FJ13]) giving rise to transfers by means of a localization procedure (see [Vez17]).

The following diagram of categories of motives summarizes the situation. The equivalence in the bottom line follows easily from the ‘tilting equivalence’ of Scholze. All notations introduced in the theorems and in the diagram will be described in later sections.

The main result also extends to the categories of non-effective motives, as follows.

Theorem 7.26 . Let $K$ be a perfectoid field with tilt $K^{\flat }$ and let $\unicode[STIX]{x1D6EC}$ be a $\mathbb{Q}$ -algebra. There is a monoidal triangulated equivalence of categories

$$\begin{eqnarray}\mathfrak{F}^{\operatorname{st}}:\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K^{\flat },\unicode[STIX]{x1D6EC})\xrightarrow[{}]{{\sim}}\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC}).\end{eqnarray}$$

Due to the intricate construction of the functor $\mathfrak{F}$ , the theorem above is not immediately obtained from the effective case and requires a little extra inspection of the categories of spectra.

As for concrete applications, our theorem allows the ‘(un-)tilting’ and ‘(de-)perfectoidifying’ of any motivic cohomology theory on rigid or perfectoid spaces satisfying étale descent and homotopy invariance, having coefficients over $\mathbb{Q}$ (it is expected to extend this over $\mathbb{Z}[1/p]$ as well, see Remark 7.28). Such an example is the overconvergent de Rham cohomology for rigid analytic varieties over $K$ . It gives rise to new cohomology theories ‘à la de Rham’ for (small) perfectoid spaces as well as for rigid varieties over local fields of equi-characteristic $p$ , satisfying descent, homotopy invariance, finite-dimensionality and further formal properties (see [Vez18]), which are compatible with rigid cohomology [Vez19]. The relation with $p$ -adic periods is an object of future research. Also, we underline that such realization functors might finally give rise (thanks to the derived tannakian formalism developed in [Ayo14b]) to the absolute local motivic Galois group of $K$ for any valued field $K$ , the case of equi-characteristic zero being described in the introduction of [Ayo15].

It is possible to use our main theorem to answer positively to a conjecture of Ayoub [Ayo15, Conjecture 2.5.72] giving an explicit description of rigid analytic motives of good reduction over a local field $K$ as well as morphisms between compact rigid analytic motives over $K$ in terms of motives and morphisms defined over the residue field, extending these results from the equi-characteristic case to the mixed characteristic case (see [Vez19]).

We also point out that the motivic tilting equivalence can be used to ‘replace’ the base field $K$ with another one $K^{\prime }$ having the same tilt. A classic example is formed by the completions $K$ and $K^{\prime }$ of the fields $\mathbb{Q}_{p}(p^{1/p^{\infty }})$ and $\mathbb{Q}_{p}(\unicode[STIX]{x1D707}_{p^{\infty }})$ , respectively. Similar techniques are exploited by Kedlaya and Liu in [KL16].

The article is organized as follows. In § 1 we recall the basic definitions and the language of adic spaces while in § 2 we define the environment in which we will perform our construction, namely the category of semi-perfectoid spaces $\widehat{\text{Rig}}\operatorname{Sm}$ and we define the étale topology on it. In § 3 we define the categories of motives for $\operatorname{RigSm}$ , $\widehat{\text{Rig}}\operatorname{Sm}$ and $\operatorname{PerfSm}$ adapting the constructions of Voevodsky and Ayoub. Thanks to the general model-categorical tools introduced in this section, we give in § 4 a motivic interpretation of some approximation results of maps valid for non-archimedean Banach algebras. In §§ 5 and 6 we prove the existence of the de-perfectoidification functor $\mathbb{L}\unicode[STIX]{x1D704}_{!}$ from perfectoid motives to rigid motives in zero and positive characteristics, respectively. Finally, we give in § 7 the proof of our main result, both in its effective and stable form.

In the appendix, we collect some technical theorems that are used in our proof. Specifically, we first present a generalization of the implicit function theorem in the rigid setting, and conclude a result about the approximation of maps modulo homotopy as well as its geometric counterpart. We also prove the existence of compatible approximations of a collection of maps $\{f_{1},\ldots ,f_{N}\}$ from a variety in $\widehat{\text{Rig}}\operatorname{Sm}$ of the form $X\,\times \,\mathbb{B}^{n}$ to a rigid variety $Y$ such that the compatibility conditions among the maps $f_{i}$ on the faces $X\,\times \,\mathbb{B}^{n-1}$ are preserved. This fact has important consequences for computing maps to $\mathbb{B}^{1}$ -local complexes of presheaves in the motivic setting.

1 Generalities on adic spaces

We start by recalling the language of adic spaces, as introduced by Huber [Hub93, Hub94] and generalized by Scholze and Weinstein [SW13]. We will always work with adic spaces over a non-archimedean valued field $K$ in the following sense.

Definition 1.1. A non-archimedean field is a topological field $K$ whose topology is induced by a non-trivial valuation of rank one. The associated norm is a multiplicative map that we denote by $|\cdot |:K\rightarrow \mathbb{R}_{{\geqslant}0}$ and its valuation ring is denoted by $K^{\circ }$ .

From now on, we fix a non-archimedean field $K$ and we pick a non-zero element $\unicode[STIX]{x1D70B}\in K$ with $|\unicode[STIX]{x1D70B}|<1$ .

Definition 1.2. A Tate $K$ -algebra is a topological $K$ -algebra $R$ for which there exists a subring $R_{0}$ such that the set $\{\unicode[STIX]{x1D70B}^{k}R_{0}\}$ forms a basis of neighborhoods of $0$ . A subring $R_{0}$ with the above property is called a ring of definition.

Definition 1.3. Let $R$ be a Tate $K$ -algebra.

1. – A subset $S$ of $R$ is bounded if it is contained in the set $\unicode[STIX]{x1D70B}^{-N}R_{0}$ for some integer $N$ . An element $x$ of $R$ is power-bounded if the set $\{x^{n}\}_{n\in \mathbb{N}}$ is bounded. The set of power-bounded elements is a subring of $R$ that we denote by $R^{\circ }$ .

2. – An element $x$ of $R$ is topologically nilpotent if $\lim _{n\rightarrow +\infty }x^{n}=0$ . The set of topologically nilpotent elements is an ideal of $R^{\circ }$ that we denote by $R^{\circ \circ }$ .

Definition 1.4. An affinoid $K$ -algebra is a pair $(R,R^{+})$ where $R$ is a Tate $K$ -algebra and $R^{+}$ is an open and integrally closed $K^{\circ }$ -subalgebra of $R^{\circ }$ . A morphism $(R,R^{+})\rightarrow (S,S^{+})$ of affinoid $K$ -algebras is a pair of compatible $K^{\circ }$ -linear continuous maps of rings $(f,f^{+})$ . An affinoid $K$ -algebra $(R,R^{+})$ is called complete if $R$ (and hence also $R^{+}$ ) is complete.

Remark 1.5. By [Wed12, Proposition 5.30] if $R$ is a Tate $K$ -algebra, then $(R,R^{\circ })$ is an affinoid $K$ -algebra.

Definition 1.6. For any complete Tate $K$ -algebra $R$ we denote by $R\langle \unicode[STIX]{x1D710}_{1},\ldots ,\unicode[STIX]{x1D710}_{n}\rangle$ the Banach algebra of strictly convergent power series in $R[[\unicode[STIX]{x1D710}_{1},\ldots ,\unicode[STIX]{x1D710}_{n}]]$ endowed with the sup-norm (see [BGR84, § 1.4.1]). A topologically of finite type Tate algebra (or simply tft Tate algebra) is a Banach $K$ -algebra $R$ isomorphic to a quotient of the normed $K$ -algebra $K\langle \unicode[STIX]{x1D710}_{1},\ldots ,\unicode[STIX]{x1D710}_{n}\rangle$ for some $n$ .

If $R$ is a tft Tate algebra, the pair $(R,R^{\circ })$ is an affinoid $K$ -algebra, and $R^{\circ }$ is a ring of definition whenever $R$ is reduced (see [BGR84, Theorem 6.2.4/1]).

We now recall the definition of perfectoid pairs, introduced in [Sch12].

Definition 1.7. A perfectoid field $K$ is a complete non-archimedean field whose rank one valuation is non-discrete, whose residue characteristic is $p$ and such that the Frobenius is surjective on $K^{\circ }/p$ . In case $\operatorname{char}K=p$ this last condition amounts to saying that $K$ is perfect.

Definition 1.8. Let $K$ be a perfectoid field.

1. – A perfectoid algebra is a Banach $K$ -algebra $R$ such that $R^{\circ }$ is bounded and the Frobenius map is surjective on $R^{\circ }/p$ .

2. – A perfectoid affinoid $K$ -algebra is an affinoid $K$ -algebra $(R,R^{+})$ over a perfectoid field $K$ such that $R$ is perfectoid.

Remark 1.9. If $R$ is a perfectoid algebra, then $(R,R^{\circ })$ is a perfectoid affinoid $K$ -algebra.

Example 1.10. Suppose that $K$ is a perfectoid field. A basic example of a perfectoid algebra is the following: let $\text{}\underline{\unicode[STIX]{x1D710}}=(\unicode[STIX]{x1D710}_{1},\ldots ,\unicode[STIX]{x1D710}_{N})$ be a $N$ -tuple of coordinates and $K^{\circ }[\text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}]$ be the ring $\mathop{\varinjlim }\nolimits_{h}K^{\circ }[\text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{h}}]$ endowed with the sup-norm induced by the norm on $K$ . We also denote by $K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle$ its $\unicode[STIX]{x1D70B}$ -adic completion. By [Sch12, Proposition 5.20], the ring $K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle [\unicode[STIX]{x1D70B}^{-1}]$ is a perfectoid $K$ -algebra, which we will denote by $K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle$ . The pair $(K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle ,K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle )$ is a perfectoid affinoid $K$ -algebra. We also define in the same way the perfectoid affinoid $K$ -algebra $(K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p^{\infty }}\rangle ,K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p^{\infty }}\rangle )$ (see [Sch13, Example 4.4]).

Remark 1.11. $K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle$ is isomorphic as a $K\langle \text{}\underline{\unicode[STIX]{x1D710}}\rangle$ -topological module to the completion $\widehat{\bigoplus }K\langle \text{}\underline{\unicode[STIX]{x1D710}}\rangle$ of the free module $\bigoplus K\langle \text{}\underline{\unicode[STIX]{x1D710}}\rangle$ with a basis indexed by the set $I=(\mathbb{Z}[1/p]\cap [\!0,1\!))^{N}$ . By [BGR84, Proposition 2.1.5/7] there is an explicit description of this ring as the subring of $\prod _{i\in I}K\langle \text{}\underline{\unicode[STIX]{x1D710}}\rangle$ whose elements are those $(x_{i})_{i\in I}$ such that for any $\unicode[STIX]{x1D700}>0$ the inequality $\Vert x_{i}\Vert <\unicode[STIX]{x1D700}$ holds for almost all $i$ (that is, for all $i$ except for a finite number of them).

The following theorem summarizes some results of Scholze, including the tilting equivalence of perfectoid algebras, which will play a crucial role in our construction.

Theorem 1.12 [Sch12].

Let $K$ be a perfectoid field.

1. (i) [Sch12, Lemma 3.4] The multiplicative monoid $\mathop{\varprojlim }\nolimits_{x\mapsto x^{p}}K$ can be given a structure $K^{\flat }$ of perfectoid field with the norm induced by the multiplicative map $\sharp :K^{\flat }\rightarrow K$ . The field $K^{\flat }$ has characteristic $p$ and coincides with $K$ in case $\operatorname{char}K=p$ .

2. (ii) [Sch12, Theorem 3.7] The functor $L\mapsto L^{\flat }$ for $L$ finite étale over $K$ induces an isomorphism $\operatorname{Gal}(K)\cong \operatorname{Gal}(K^{\flat })$ .

3. (iii) [Sch12, Lemma 6.2] There is an equivalence of categories, the tilting equivalence, from perfectoid affinoid $K$ -algebras to perfectoid affinoid $K^{\flat }$ -algebras denoted by $(R,R^{+})\mapsto (R^{\flat },R^{\flat +})$ such that $R^{\flat }$ is multiplicatively isomorphic to $\mathop{\varprojlim }\nolimits_{x\mapsto x^{p}}R$ and $R^{\flat +}$ is multiplicatively isomorphic to $\mathop{\varprojlim }\nolimits_{x\mapsto x^{p}}R^{+}$ .

4. (iv) [Sch12, Proposition 5.20 and Corollary 6.8] The tilting equivalence associates the perfectoid $K$ -algebra $(K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle ,K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle )$ to $(K^{\flat }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle ,K^{\flat \circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle )$ and the perfectoid $K$ -algebra $(K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p^{\infty }}\rangle ,K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p^{\infty }}\rangle )$ to $(K^{\flat }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p^{\infty }}\rangle ,K^{\flat \circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p^{\infty }}\rangle )$ .

We now recall Huber’s construction of the spectrum of a valuation ring (see [Hub94]).

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 [Hub93, § 3] and [Sch12, 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 [Sch12, 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 [Sch12, 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 [Hub94, § 1] and [Sch12, 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 [Hub94, 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 $\{f\in {\mathcal{O}}_{X,x}:|f(x)|\leqslant 1\}$ .

By [Hub94, Proposition 1.6] there holds ${\mathcal{O}}^{+}(U)=\{f\in {\mathcal{O}}(U):|f(x)|\leqslant 1\text{ for all }x\in U\}$ for any rational subset $U$ of $\operatorname{Spa}(R,R^{+})$ so that ${\mathcal{O}}^{+}$ is a sheaf if and only if ${\mathcal{O}}$ is a sheaf.

Sadly enough, ${\mathcal{O}}$ , ${\mathcal{O}}^{+}$ are not sheaves in general as shown at the end of [Hub94, § 1]. By Tate’s acyclicity theorem [BGR84, Theorem 8.2.1/1] and Scholze’s acyclicity theorem [Sch12, Theorem 6.3], if $(R,R^{+})$ is a tft Tate algebra or a perfectoid affinoid $K$ -algebra, then ${\mathcal{O}}$ , ${\mathcal{O}}^{+}$ are sheaves (see [Hub94, § 2]). Also, if ${\mathcal{O}}^{+}(U)$ is bounded for all rational subspaces $U$ of $X$ (in which case we say that $X$ is stably uniform) then ${\mathcal{O}}$ , ${\mathcal{O}}^{+}$ are sheaves (see [BV18]).

Definition 1.14. Objects of ${\mathcal{V}}$ are triples $(X,{\mathcal{O}}_{X},\{|\cdot |_{x}\}_{x\in X})$ , where $X$ is a topological space, ${\mathcal{O}}_{X}$ is a sheaf of complete topological $K$ -algebras, and $|\cdot |_{x}$ is a continuous valuation on ${\mathcal{O}}_{X,x}$ . Maps are morphisms of ringed spaces, which induce continuous $K$ -algebra morphisms of sheaves and are compatible with the valuations on stalks.

Remark 1.15. By abuse of notation, whenever $R$ is a tft Tate algebra we sometimes denote by $\operatorname{Spa}R$ the object $\operatorname{Spa}(R,R^{\circ })$ of ${\mathcal{V}}$ .

Definition 1.16. Let $X$ be an object of ${\mathcal{V}}$ .

1. – We say that $X$ is an affinoid adic space if it is isomorphic to $\operatorname{Spa}(R,R^{+})$ for some affinoid $K$ -algebra $(R,R^{+})$ .

2. – We say that $X$ is an affinoid rigid variety if it is isomorphic to $\operatorname{Spa}(R,R^{\circ })$ for some tft Tate algebra $R$ .

3. – We say that $X$ is a perfectoid affinoid space if it is isomorphic to $\operatorname{Spa}(R,R^{+})$ for some perfectoid affinoid $K$ -algebra $(R,R^{+})$ .

4. – We say that $X$ is an adic space if it is locally isomorphic to an affinoid adic space.

5. – We say that $X$ is a rigid variety if it is locally isomorphic to an affinoid rigid variety.

6. – We say that $X$ is a perfectoid space if it is locally isomorphic to a perfectoid affinoid space.

There is an apparent clash of definitions between rigid varieties as presented above, and as defined by Tate. In fact, the two categories are canonically isomorphic. We refer to [Hub94, § 4] and [Sch12, § 2] for a more detailed collection of results on the comparison between these theories.

Remark 1.17. By [Hub94, Proposition 2.1(ii)] if $X$ is an adic space and $Y=\operatorname{Spa}(R,R^{+})$ is an affinoid adic space then $\operatorname{Hom}(X,Y)\cong \operatorname{Hom}((R,R^{+}),({\mathcal{O}}_{X}(X),{\mathcal{O}}_{X}^{+}(X)))$ . Moreover, as shown in [Hub93, § 3] and [Hub94, § 2], if $(\widehat{R},\widehat{R}^{+})$ is the completion of $(R,R^{+})$ then it is an affinoid $K$ -algebra and $\operatorname{Spa}(\widehat{R},\widehat{R}^{+})\cong \operatorname{Spa}(R,R^{+})$ .

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 [Sch12, Remark 3.5]).

We now consider some basic examples and fix some notation.

Example 1.19. Let $\text{}\underline{\unicode[STIX]{x1D710}}=(\unicode[STIX]{x1D710}_{1},\ldots ,\unicode[STIX]{x1D710}_{N})$ be a $N$ -tuple of coordinates. The Tate $N$ -ball $\operatorname{Spa}(K\langle \text{}\underline{\unicode[STIX]{x1D710}}\rangle ,K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}\rangle )$ will be denoted by $\mathbb{B}^{N}$ and the $N$ -torus $\operatorname{Spa}(K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1}\rangle ,K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1}\rangle )$ by $\mathbb{T}^{N}$ . It is the rational subspace $U(1\,\mid \,\unicode[STIX]{x1D710}_{1}\ldots \unicode[STIX]{x1D710}_{N})$ of $\mathbb{B}^{N}$ . The map of spaces induced by the inclusion $(K\langle \text{}\underline{\unicode[STIX]{x1D710}}\rangle ,K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}\rangle )\rightarrow (K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{h}}\rangle ,K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{h}}\rangle )$ will be denoted by $\mathbb{B}^{N}\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{h}}\rangle \rightarrow \mathbb{B}^{N}$ . We use the analogous notation $\mathbb{T}^{N}\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{h}}\rangle \rightarrow \mathbb{T}^{N}$ for the torus. These maps are clearly isomorphic to the endomorphism of $\mathbb{B}^{N}$ respectively $\mathbb{T}^{N}$ induced by $\unicode[STIX]{x1D710}_{i}\mapsto \unicode[STIX]{x1D710}_{i}^{p^{h}}$ .

The space defined by the perfectoid affinoid $K$ -algebra $(K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle ,K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle )$ will be denoted by $\widehat{\mathbb{B}}^{N}$ and referred to as the perfectoid $N$ -ball. The space defined by the perfectoid affinoid $K$ -algebra $(K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p^{\infty }}\rangle ,K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p^{\infty }}\rangle )$ coincides with the rational subspace $U(1\,\mid \,\unicode[STIX]{x1D710}_{1}\ldots \unicode[STIX]{x1D710}_{N})$ of $\widehat{\mathbb{B}}^{N}$ and will be denoted by $\widehat{\mathbb{T}}^{N}$ and will be referred to as the perfectoid $N$ -torus.

We now recall the definition of étale maps on the category of adic spaces, taken from [Sch12, § 7].

Definition 1.20. A map of affinoid adic spaces $f:\operatorname{Spa}(S,S^{+})\rightarrow \operatorname{Spa}(R,R^{+})$ is finite étale if the associated map $R\rightarrow S$ is a finite étale map of rings, and if $S^{+}$ is the integral closure of $R^{+}$ in $S$ . A map of adic spaces $f:X\rightarrow Y$ is étale if for any point $x\in X$ there exists an open neighborhood $U$ of $x$ and an affinoid open subset $V$ of $Y$ containing $f(U)$ such that $f|_{U}:U\rightarrow V$ factors as an open embedding $U\rightarrow W$ and a finite étale map $W\rightarrow V$ for some affinoid adic space $W$ .

The previous definitions, when restricted to the case of tft Tate varieties, coincide with the usual ones, as proved in [FvdP04, Proposition 8.1.2].

Remark 1.21. Suppose we are given a diagram of affinoid $K$ -algebras.

In general, it is not possible to define a push-out in the category of affinoid $K$ -algebras. Nonetheless, this can be performed under some hypothesis. For example, if the affinoid $K$ -algebras are tft Tate algebras then the push-out exists and it is the tft Tate algebra associated to the completion $S\,\widehat{\otimes }_{R}\,T$ of $S\,\otimes _{R}\,T$ endowed with the norm of the tensor product (see [BGR84, § 3.1.1]). In case the affinoid $K$ -algebras are perfectoid affinoid, then the push-out exists and is also perfectoid affinoid. It coincides with the completion of $(L,L^{+})$ , where $L$ is the ring $S\,\otimes _{R}\,T$ endowed with the norm of the tensor product and $L^{+}$ is the integral closure of $S^{+}\,\otimes _{R^{+}}\,T^{+}$ in $L$ (see [Sch12, Proposition 6.18]). The same construction holds in case the map $(R,R^{+})\rightarrow (S,S^{+})$ is finite étale and $(T,T^{+})$ is a perfectoid affinoid (see [Sch12, Lemma 7.3]). By Remark 1.17, the constructions above give rise to fiber products in the category of adic spaces.

2 Semi-perfectoid spaces

We can now introduce a convenient generalization of both smooth rigid varieties and smooth perfectoid spaces. We recall that our base field $K$ is a perfectoid field (see Assumption 1.18).

Proposition 2.1. Let $\text{}\underline{\unicode[STIX]{x1D710}}=\unicode[STIX]{x1D710}_{1},\ldots ,\unicode[STIX]{x1D710}_{N}$ and $\text{}\underline{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D708}_{1},\ldots ,\unicode[STIX]{x1D708}_{M}$ be two systems of coordinates. Let $(R_{0},R_{0}^{\circ })$ be a tft Tate algebra and let $f:\operatorname{Spa}(R_{0},R_{0}^{\circ })\rightarrow \mathbb{T}^{N}\,\times \,\mathbb{T}^{M}=\operatorname{Spa}K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1},\text{}\underline{\unicode[STIX]{x1D708}}^{\pm 1}\rangle$ be a map, which is a composition of finite étale maps and rational embeddings. Also let $\operatorname{Spa}(R_{h},R_{h}^{\circ })$ be the affinoid rigid variety $\operatorname{Spa}(R_{0},R_{0}^{\circ })\,\times _{\mathbb{T}^{N}}\,\mathbb{T}^{N}\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{h}}\rangle$ . The $\unicode[STIX]{x1D70B}$ -adic completion $(T,T^{+})$ of $(\mathop{\varinjlim }\nolimits_{h}R_{h},\mathop{\varinjlim }\nolimits_{h}R_{h}^{\circ })$ represents the fiber product $\operatorname{Spa}(R_{0},R_{0}^{\circ })\,\times _{\mathbb{T}^{N}}\,\widehat{\mathbb{T}}^{N}$ and defines a bounded affinoid adic space. Moreover, $(T,T^{+})$ is isomorphic to the completion of $(L,L^{+})$ , where $L$ is the ring $R_{0}\,\otimes _{K\langle \text{}\underline{\unicode[STIX]{x1D710}}\rangle }\,K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{\infty }}\rangle$ endowed with the norm of the tensor product and $L^{+}$ is the integral closure of $R_{0}^{\circ }$ in $L$ .

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 [Sch13, 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 [BV18]).

The proof of the alternative description of $(T,T^{+})$ follows in the same way as [Sch13, Lemma 4.5(i)].◻

Remark 2.2. The statement of the previous proposition is an instance when the second term $T^{+}$ of some affinoid $K$ -algebra $(T,T^{+})$ may not be equal to the ring $T^{\circ }$ .

Corollary 2.3. Let $X$ be a reduced rigid variety with an étale map

$$\begin{eqnarray}f:X\rightarrow \mathbb{T}^{N}\,\times \,\mathbb{T}^{M}=\operatorname{Spa}K\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1},\text{}\underline{\unicode[STIX]{x1D708}}^{\pm 1}\rangle .\end{eqnarray}$$

Then the fiber product $X\,\times _{\mathbb{T}^{N}}\,\widehat{\mathbb{T}}^{N}$ exists in the category of adic spaces.

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. ◻

Definition 2.4. We denote by $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}\!/K$ the full subcategory of adic spaces whose objects are isomorphic to spaces $X=X_{0}\,\times _{\mathbb{T}^{N}}\,\widehat{\mathbb{T}}^{N}$ with respect to a map of affinoid rigid varieties $f:X_{0}\rightarrow \mathbb{T}^{N}\,\times \,\mathbb{T}^{M}$ that is a composition of rational embeddings and finite étale maps. Such spaces will be called smooth semi-perfectoid spaces with good coordinates. Because of Proposition 2.1, such fiber products $X=X_{0}\,\times _{\mathbb{T}^{N}}\,\widehat{\mathbb{T}}^{N}$ exist and are affinoid. Whenever $N=0$ these varieties are rigid analytic varieties and the full subcategory they form will be denoted by $\operatorname{RigSm}^{\operatorname{gc}}\!/K$ and referred to as smooth affinoid rigid varieties with good coordinates. Whenever $M=0$ these varieties are perfectoid affinoid spaces and the full subcategory they form will be denoted by $\operatorname{PerfSm}^{\operatorname{gc}}\!/K$ and referred to as smooth affinoid perfectoids with good coordinates. A perfectoid space $X$ in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}\!/K$ will be sometimes denoted with $\widehat{X}$ .

When $X=X_{0}\,\times _{\mathbb{T}^{N}}\,\widehat{\mathbb{T}}^{N}$ is in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}\!/K$ we denote by $X_{h}$ the fiber product $X_{0}\,\times _{\mathbb{T}^{N}}\,\mathbb{T}^{N}\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{h}}\rangle$ and we will write $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ . We say that a presentation $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ of an object $X$ in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}\!/K$ has good reduction if the map $X_{0}\rightarrow \mathbb{T}^{n}\,\times \,\mathbb{T}^{m}$ has an étale formal model $\mathfrak{X}\rightarrow \operatorname{Spf}(K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1},\text{}\underline{\unicode[STIX]{x1D708}}^{\pm 1}\rangle )$ . We say that a presentation $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ of an object $X$ in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}\!/K$ has potentially good reduction if there exists a finite separable field extension $L/K$ such that $X_{L}=\mathop{\varprojlim }\nolimits_{h}(X_{h})_{L}$ has good reduction in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}\!/L$ . We warn the reader that the association $X\mapsto X_{0}$ is not functorial and the varieties $X_{h}$ are not uniquely determined by $X$ in general.

We denote by $\widehat{\text{Rig}}\operatorname{Sm}/K$ the full subcategory of adic spaces which are locally isomorphic to objects in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}\!/K$ and its objects will be called smooth semi-perfectoid spaces. We denote by $\operatorname{RigSm}/K$ the full subcategory of adic spaces, which are locally isomorphic to objects in $\operatorname{RigSm}^{\operatorname{gc}}\!/K$ and by $\operatorname{PerfSm}/K$ the one of adic spaces, which are locally isomorphic to objects in  $\operatorname{PerfSm}^{\operatorname{gc}}\!/K$ . Its objects will be called smooth perfectoid spaces. Whenever the context allows it, we omit $K$ from the notation.

Remark 2.5. Any smooth rigid variety (see for example [Ayo15, Definition 1.1.41]) has locally good coordinates over $\mathbb{T}^{N}$ by [Ayo15, Corollary 1.1.51]. Hence, $\operatorname{RigSm}$ coincides with the category of smooth rigid varieties.

We remark that the presentations defined above having good reduction are a special case of the objects considered in [And06].

The notation $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ is justified by the following corollary, which is inspired by [SW13, Proposition 2.4.5].

Corollary 2.6. Let $Y=\operatorname{Spa}(S,S^{+})$ be an affinoid such that $S^{+}$ is a ring of definition and let $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ be in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ . Then $\operatorname{Hom}(Y,X)\cong \mathop{\varprojlim }\nolimits_{h}\operatorname{Hom}(Y,X_{h})$ .

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.◻

Let $\{X_{h},f_{h}\}_{h\in I}$ be a cofiltered diagram of rigid varieties and let $\{X\rightarrow X_{h}\}_{h\in I}$ be a collection of compatible maps of adic spaces. We recall that, according to [Hub96, Remark 2.4.5], one writes $X\sim \mathop{\varprojlim }\nolimits_{h}X_{h}$ when the following two conditions are satisfied.

1. (i) The induced map on topological spaces $|X|\rightarrow \mathop{\varprojlim }\nolimits_{h}|X_{h}|$ is a homeomorphism.

2. (ii) For any $x\in X$ with images $x_{h}\in X_{h}$ the map of residue fields $\mathop{\varinjlim }\nolimits_{h}k(x_{h})\rightarrow k(x)$ has a dense image.

The apparent clash of notations is solved by the following fact.

Proposition 2.7. Let $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ be in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ . Then $X\sim \mathop{\varprojlim }\nolimits_{h}X_{h}$ .

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 [Sch12, Proposition 7.16].◻

Étale maps define a topology on $\widehat{\text{Rig}}\operatorname{Sm}$ in the following way.

Definition 2.8. A collection of étale maps of adic spaces $\{U_{i}\rightarrow X\}_{i\in I}$ is an étale cover if the induced map $\bigsqcup _{i\in I}U_{i}\rightarrow X$ is surjective. These covers define a Grothendieck topology on $\widehat{\text{Rig}}\operatorname{Sm}$ called the étale topology.

We pin down the following facts on the topology of the objects $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ .

Proposition 2.9. Let $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ be an object of $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ .

1. (i) Any finite étale map $U\rightarrow X$ is isomorphic to $U_{\bar{h}}\,\times _{X_{\bar{h}}}\,X$ for some integer $\bar{h}$ and some finite étale map $U_{\bar{h}}\rightarrow X_{\bar{h}}$ .

2. (ii) Any rational subspace $U\subset X$ is isomorphic to $U_{\bar{h}}\,\times _{X_{\bar{h}}}\,X$ for some integer $H$ and some rational subspace $U_{\bar{h}}\subset X_{\bar{h}}$ .

Proof. The first statement follows from [Sch12, Lemma 7.5]. For the second, we remark that according to [Hub93, 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.◻

The proposition above can also be used to prove that a finite étale extension of an object $\operatorname{Spa}(T,T^{+})$ in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}\!/K$ also lies in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}\!/K$ , and hence arbitrary fiber products of étale maps in $\widehat{\text{Rig}}\operatorname{Sm}/K$ exist, and are étale.

Corollary 2.10. Let $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ be an object of $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ and let ${\mathcal{U}}:=\{f_{i}:U_{i}\rightarrow X\}$ be an étale covering of adic spaces. There exists an integer $\bar{h}$ and a finite affinoid refinement $\{V_{j}\rightarrow X\}$ of ${\mathcal{U}}$ , which is obtained by pullback of an étale covering $\{V_{\bar{h}j}\rightarrow X_{\bar{h}}\}$ of $X_{\bar{h}}$ and such that $V=\mathop{\varprojlim }\nolimits_{h}V_{hj}$ lies in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ by letting $V_{hj}$ be $V_{\bar{h}j}\,\times _{X_{\bar{h}}}\,X_{h}$ for all $h\geqslant \bar{h}$ .

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.◻

Corollary 2.11. A perfectoid space $X$ lies in $\operatorname{PerfSm}$ if and only if it is locally étale over $\widehat{\mathbb{T}}^{N}$ .

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.◻

Remark 2.12. If $X$ is a smooth affinoid perfectoid space, then it has a finite number of connected components. Indeed, it is quasi-compact and locally isomorphic to a rational subspace of a perfectoid space, which is finite étale over a rational subspace of $\widehat{\mathbb{T}}^{N}$ .

For later use, we record the following simple example of a space $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ for which the varieties $X_{h}$ are easy to understand.

Proposition 2.13. Consider the smooth affinoid variety with good coordinates

$$\begin{eqnarray}X_{0}=U(\unicode[STIX]{x1D710}-1\,\mid \,\unicode[STIX]{x1D70B}){\hookrightarrow}\mathbb{T}^{1}=\operatorname{Spa}(K\langle \unicode[STIX]{x1D710}^{\pm 1}\rangle ).\end{eqnarray}$$

One has $X_{h}\cong \mathbb{B}^{1}$ for all $h$ and $\widehat{X}=\mathop{\varprojlim }\nolimits_{h}X_{h}\cong \widehat{\mathbb{B}}^{1}$ .

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 [Sch12, 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 [Sch12, 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.◻

From the previous proposition we conclude, in particular, that the perfectoid space $\widehat{\mathbb{B}}^{1}$ lies in $\operatorname{PerfSm}^{\operatorname{gc}}$ .

3 Categories of adic motives

From now on, we fix a commutative ring $\unicode[STIX]{x1D6EC}$ and work with $\unicode[STIX]{x1D6EC}$ -enriched categories. In particular, the term ‘presheaf’ should be understood as ‘presheaf of $\unicode[STIX]{x1D6EC}$ -modules’ and similarly for the term ‘sheaf’. The presheaf $\unicode[STIX]{x1D6EC}(X)$ represented by an object $X$ of a category $\mathbf{C}$ sends an object $Y$ of $\mathbf{C}$ to the free $\unicode[STIX]{x1D6EC}$ -module $\unicode[STIX]{x1D6EC}\operatorname{Hom}(Y,X)$ .

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.

We make extensive use of the theory of model categories and localization, following the approach of Ayoub in [Ayo15, Ayo07]. Fix a site $(\mathbf{C},\unicode[STIX]{x1D70F})$ . In our situation, this will be the étale site of $\operatorname{RigSm}$ or $\widehat{\text{Rig}}\operatorname{Sm}$ . The category of complexes of presheaves $\mathbf{Ch}(\mathbf{Psh}(\mathbf{C}))$ can be endowed with the projective model structure for which weak equivalences are quasi-isomorphisms and fibrations are maps ${\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime }$ such that ${\mathcal{F}}(X)\rightarrow {\mathcal{F}}^{\prime }(X)$ is a surjection for all $X$ in $\mathbf{C}$ (cfr [Hov99, § 2.3] and [Ayo07, Proposition 4.4.16]).

Also the category of complexes of sheaves $\mathbf{Ch}(\mathbf{Sh}_{\unicode[STIX]{x1D70F}}(\mathbf{C}))$ can be endowed with the projective model structure defined in [Ayo07, Definition 4.4.40]. In this structure, weak equivalences are quasi-isomorphisms of complexes of sheaves.

Remark 3.2. Let $\mathbf{C}$ be a category. As shown in [Fau06] any projectively cofibrant complex ${\mathcal{F}}$ in $\mathbf{Ch}\,\mathbf{Psh}(\mathbf{C})$ is a retract of a complex that is the filtered colimit of bounded above complexes, each constituted by presheaves that are direct sums of representable ones.

Just like in [Jar87, MVW06, MV99] or [Rio07], we consider the left Bousfield localization of the model category $\mathbf{Ch}(\mathbf{Psh}(\mathbf{C}))$ with respect to the topology we select, and a chosen ‘contractible object’. We recall that left Bousfield localizations with respect to a class of maps $S$ (see [Hir03, ch. 3]) is the universal model categories in which the maps in $S$ become weak equivalences. The existence of such structures is granted only under some technical hypotheses, as shown in [Hir03, Theorem 4.1.1] and [Ayo07, Theorem 4.2.71].

Proposition 3.3. Let $(\mathbf{C},\unicode[STIX]{x1D70F})$ be a site with finite direct products and let $\mathbf{C}^{\prime }$ be a full subcategory of $\mathbf{C}$ such that every object of $\mathbf{C}$ has a covering by objects of $\mathbf{C}^{\prime }$ . Also let $I$ be an object of $\mathbf{C}^{\prime }$ .

1. (i) The projective model category $\mathbf{Ch}\,\mathbf{Psh}(\mathbf{C})$ admits a left Bousfield localization $\mathbf{Ch}_{I}\mathbf{Psh}(\mathbf{C})$ with respect to the set $S_{I}$ of all maps $\unicode[STIX]{x1D6EC}(I\,\times \,X)[i]\rightarrow \unicode[STIX]{x1D6EC}(X)[i]$ as $X$ varies in $\mathbf{C}$ and $i$ varies in $\mathbb{Z}$ .

2. (ii) The projective model categories $\mathbf{Ch}\,\mathbf{Psh}(\mathbf{C})$ and $\mathbf{Ch}\,\mathbf{Psh}(\mathbf{C}^{\prime })$ admit left Bousfield localizations $\mathbf{Ch}_{\unicode[STIX]{x1D70F}}\mathbf{Psh}(\mathbf{C})$ and $\mathbf{Ch}_{\unicode[STIX]{x1D70F}}\mathbf{Psh}(\mathbf{C}^{\prime })$ with respect to the class $S_{\unicode[STIX]{x1D70F}}$ of maps ${\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime }$ inducing isomorphisms on the $\acute{\text{e}}\text{t}$ -sheaves associated to $H_{i}({\mathcal{F}})$ and $H_{i}({\mathcal{F}}^{\prime })$ for all $i\in \mathbb{Z}$ . Moreover, the two localized model categories are Quillen equivalent and the sheafification functor induces a Quillen equivalence to the projective model category $\mathbf{Ch}\,\mathbf{Sh}_{\unicode[STIX]{x1D70F}}(\mathbf{C})$ .

3. (iii) The model categories $\mathbf{Ch}_{\unicode[STIX]{x1D70F}}\mathbf{Psh}(\mathbf{C})$ and $\mathbf{Ch}_{\unicode[STIX]{x1D70F}}\mathbf{Psh}(\mathbf{C}^{\prime })$ admit left Bousfield localizations $\mathbf{Ch}_{\unicode[STIX]{x1D70F},I}\mathbf{Psh}(\mathbf{C})$ and $\mathbf{Ch}_{\unicode[STIX]{x1D70F},I}\mathbf{Psh}(\mathbf{C}^{\prime })$ with respect to the set $S_{I}$ defined above. Moreover, the two localized model categories are Quillen equivalent.

Proof. According to [Hir03, Theorem 4.1.1], any model category that is left proper and cellular (some technical properties that are defined in [Hir03, 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 [SS00, p. 7]). It follows that the projective model structures in the statement are also left proper and cellular (see [Hir03, 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 [Ayo07, 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 [Ayo07, Corollary 4.4.42, Proposition 4.4.55].

Since by [Ayo07, Proposition 4.4.31] the $\unicode[STIX]{x1D70F}$ -localization coincides with the Bousfield localization with respect to a set, we conclude by [Ayo07, 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 [Hir03, Theorem 4.1.1] and the second claim.◻

In the situation above, we will denote by $S_{(\unicode[STIX]{x1D70F},I)}$ the union of the class $S_{\unicode[STIX]{x1D70F}}$ and the set $S_{I}$ .

Remark 3.4. A geometrically relevant situation is induced when $I$ is endowed with a multiplication map $\unicode[STIX]{x1D707}:I\,\times \,I\rightarrow I$ and maps $i_{0}$ and $i_{1}$ from the terminal object to $I$ satisfying the relations of a monoidal object with $0$ as in the definition of an interval object (see [MV99, § 2.3]). Under these hypotheses, we say that the triple $(\mathbf{C},\unicode[STIX]{x1D70F},I)$ is a site with an interval.

Example 3.5. The affinoid rigid variety with good coordinates $\mathbb{B}^{1}=\operatorname{Spa}K\langle \unicode[STIX]{x1D712}\rangle$ is an interval object with respect to the natural multiplication $\unicode[STIX]{x1D707}$ and maps $i_{0}$ and $i_{1}$ induced by the substitution $\unicode[STIX]{x1D712}\mapsto 0$ and $\unicode[STIX]{x1D712}\mapsto 1$ , respectively.

We now apply the constructions above to the sites introduced in the previous sections. We recall that we consider adic spaces defined over a perfectoid field $K$ .

Corollary 3.6. The following pairs of model categories are Quillen equivalent.

1. – $\mathbf{Ch}_{\acute{\text{e}}\text{t}}\mathbf{Psh}(\operatorname{RigSm})$ and $\mathbf{Ch}_{\acute{\text{e}}\text{t}}\mathbf{Psh}(\operatorname{RigSm}^{\operatorname{gc}})$ .

2. – $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\operatorname{RigSm})$ and $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\operatorname{RigSm}^{\operatorname{gc}})$ .

3. – $\mathbf{Ch}_{\acute{\text{e}}\text{t}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ and $\mathbf{Ch}_{\acute{\text{e}}\text{t}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}})$ .

4. – $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ and $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}})$ .

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.◻

Definition 3.7. For $\unicode[STIX]{x1D702}\in \{\acute{\text{e}}\text{t},\mathbb{B}^{1},(\acute{\text{e}}\text{t},\mathbb{B}^{1})\}$ we say that a map in $\mathbf{Ch}\,\mathbf{Psh}(\operatorname{RigSm})$ [respectively $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ ] is a $\unicode[STIX]{x1D702}$ -weak equivalence if it is a weak equivalence in the model structure $\mathbf{Ch}_{\unicode[STIX]{x1D702}}\mathbf{Psh}(\operatorname{RigSm})$ [respectively $\mathbf{Ch}_{\unicode[STIX]{x1D702}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ ]. The triangulated homotopy category associated to the localization $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\operatorname{RigSm})$ [respectively to the localization $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ ] is denoted by $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ [respectively $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ ]. We omit $\unicode[STIX]{x1D6EC}$ from the notation whenever the context allows it. The image of a variety $X$ in one of these categories is denoted by $\unicode[STIX]{x1D6EC}(X)$ . We say that an object ${\mathcal{F}}$ of the derived category $\mathbf{D}=\mathbf{D}(\mathbf{Psh}(\operatorname{RigSm}))$ [respectively $\mathbf{D}=\mathbf{D}(\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}))$ ] is $\unicode[STIX]{x1D702}$ -local if the functor $\operatorname{Hom}_{\mathbf{D}}(\cdot ,{\mathcal{F}})$ sends maps in $S_{\unicode[STIX]{x1D702}}$ (see Proposition 3.3) to isomorphisms. This amounts to saying that ${\mathcal{F}}$ is quasi-isomorphic to a $\unicode[STIX]{x1D702}$ -fibrant object.

We need to keep track of $\mathbb{B}^{1}$ in the notation of $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ since later we will perform a localization on $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ with respect to a different interval object.

Remark 3.8. Using the language of [BV08], the localizations defined above induce endofunctors $C^{\unicode[STIX]{x1D702}}$ of the derived categories $\mathbf{D}(\mathbf{Psh}(\operatorname{RigSm}))$ , $\mathbf{D}(\mathbf{Psh}(\operatorname{RigSm}^{\operatorname{gc}}))$ , $\mathbf{D}(\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}))$ and $\mathbf{D}(\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}))$ such that $C^{\unicode[STIX]{x1D702}}{\mathcal{F}}$ is $\unicode[STIX]{x1D702}$ -local for all ${\mathcal{F}}$ and there is a natural transformation $C^{\unicode[STIX]{x1D702}}\rightarrow \operatorname{id}$ , which is a pointwise $\unicode[STIX]{x1D702}$ -weak equivalence. The functor $C^{\unicode[STIX]{x1D702}}$ restricts to a triangulated equivalence on the objects ${\mathcal{F}}$ that are $\unicode[STIX]{x1D702}$ -local and one can compute the Hom set $\operatorname{Hom}({\mathcal{F}},{\mathcal{F}}^{\prime })$ in the homotopy category of the $\unicode[STIX]{x1D702}$ -localization as $\mathbf{D}({\mathcal{F}},C^{\unicode[STIX]{x1D702}}{\mathcal{F}}^{\prime })$ .

Remark 3.9. According to Proposition 3.3, for any $X$ in $\widehat{\text{Rig}}\operatorname{Sm}$ and any integer $i$ , one has $\operatorname{Hom}_{\mathbf{D}}(\unicode[STIX]{x1D6EC}(X)[-i],C^{\acute{\text{e}}\text{t}}{\mathcal{F}})\cong \operatorname{Hom}_{\mathbf{D}(\mathbf{Sh}_{\acute{\text{e}}\text{t}}(\widehat{\text{Rig}}\operatorname{Sm}))}(\unicode[STIX]{x1D6EC}(X)[-i],{\mathcal{F}})$ . The latter group is the étale hypercohomology group $\mathbb{H}_{\acute{\text{e}}\text{t}}^{i}(X,{\mathcal{F}})$ , which can be computed with respect to the small étale site over $X$ . The property $\operatorname{Hom}_{\mathbf{D}}(\unicode[STIX]{x1D6EC}(X)[-i],C^{\acute{\text{e}}\text{t}}{\mathcal{F}})\cong \mathbb{H}^{i}(X_{\acute{\text{e}}\text{t}},{\mathcal{F}})$ characterizes $C^{\acute{\text{e}}\text{t}}{\mathcal{F}}$ up to quasi-isomorphisms (and holds true for more general topologies, see [Ayo07, Proposition 4.4.58]).

We now show that the étale localization can alternatively be described in terms of étale hypercoverings ${\mathcal{U}}_{\bullet }\rightarrow X$ (see for example [DHI04]). Any such datum defines a simplicial presheaf $n\mapsto \bigoplus _{i}\unicode[STIX]{x1D6EC}(U_{ni})$ whenever ${\mathcal{U}}_{n}=\bigsqcup _{i}h_{U_{ni}}$ is the sum of the presheaves of sets $h_{U_{ni}}$ represented by $U_{ni}$ . This simplicial presheaf can be associated to a normalized chain complex that we denote by $\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet })$ . It is endowed with a map to $\unicode[STIX]{x1D6EC}(X)$ .

Proposition 3.10. The localization over $S_{\acute{\text{e}}\text{t}}$ on $\mathbf{Ch}\,\mathbf{Psh}(\operatorname{RigSm}^{\operatorname{gc}})$ [respectively $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}})$ ] coincides with the localization over the set $\unicode[STIX]{x1D6EC}({\mathcal{U}}_{\bullet })[i]\rightarrow \unicode[STIX]{x1D6EC}(X)[i]$ as ${\mathcal{U}}_{\bullet }\rightarrow X$ varies among bounded étale hypercoverings of the objects $X$ of $\operatorname{RigSm}^{\operatorname{gc}}$ [respectively $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ ] and $i$ varies in $\mathbb{Z}$ .

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 [MVW06, Proposition 14.23] holds also here). By means of [Ayo15, Corollary 1.2.21] we conclude that any rigid variety $X$ has a finite cohomological dimension. By [SGA4, Theorem V.7.4.1] and [SV00, 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 [SGA4, 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.◻

As the following proposition shows, there are also alternative presentations of the homotopy categories introduced so far, which we will later use.

Proposition 3.11. Let $\unicode[STIX]{x1D6EC}$ be a $\mathbb{Q}$ -algebra. The natural inclusion induces a Quillen equivalence

$$\begin{eqnarray}L_{S}\mathbf{Ch}(\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}))\leftrightarrows \mathbf{Ch}_{\acute{\text{e}}\text{t}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}),\end{eqnarray}$$

where $L_{S}$ denotes the Bousfield localization with respect to the set $S$ of shifts of the maps of complexes induced by étale Cech hypercoverings ${\mathcal{U}}_{\bullet }\rightarrow X$ of objects $X$ in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ such that for some presentation $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ the covering ${\mathcal{U}}_{0}\rightarrow X$ descends to a covering of $X_{0}$ .

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 [SS03, § 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 [DHI04, 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 [DHI04, Corollary A.8]. We remark that [DHI04, 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 [DHI04, 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.◻

Remark 3.12. It is shown in the proof that the statements of Propositions 3.10 and 3.11 hold true without any assumptions on $\unicode[STIX]{x1D6EC}$ under the condition that all varieties $X$ have finite cohomological dimension with respect to the étale topology.

As we pointed out in Remark 3.9, there is a characterization of $C^{\acute{\text{e}}\text{t}}{\mathcal{F}}$ for any complex ${\mathcal{F}}$ . This is also true for the $\mathbb{B}^{1}$ -localization, described in the following part.

Definition 3.13. We denote by $\Box$ the $\unicode[STIX]{x1D6F4}$ -enriched cocubical object (see [Ayo14b, Appendix A]) defined by putting $\Box ^{n}=\mathbb{B}^{n}=\operatorname{Spa}K\langle \unicode[STIX]{x1D70F}_{1},\ldots ,\unicode[STIX]{x1D70F}_{n}\rangle$ and considering the morphisms $d_{r,\unicode[STIX]{x1D716}}$ induced by the maps $\mathbb{B}^{n}\rightarrow \mathbb{B}^{n+1}$ corresponding to the substitution $\unicode[STIX]{x1D70F}_{r}=\unicode[STIX]{x1D716}$ for $\unicode[STIX]{x1D716}\in \{0,1\}$ and the morphisms $p_{r}$ induced by the projections $\mathbb{B}^{n}\rightarrow \mathbb{B}^{n-1}$ . For any variety $X$ and any presheaf ${\mathcal{F}}$ with values in an abelian category, we can therefore consider the $\unicode[STIX]{x1D6F4}$ -enriched cubical object ${\mathcal{F}}(X\,\times \,\Box )$ (see [Ayo14b, Appendix A]). Associated to any $\unicode[STIX]{x1D6F4}$ -enriched cubical object ${\mathcal{F}}$ there are the following complexes: the complex $C_{\bullet }^{\sharp }{\mathcal{F}}$ defined as $C_{n}^{\sharp }{\mathcal{F}}={\mathcal{F}}_{n}$ and with differential $\sum (-1)^{r}(d_{r,1}^{\ast }-d_{r,0}^{\ast })$ ; the simple complex $C_{\bullet }{\mathcal{F}}$ defined as $C_{n}{\mathcal{F}}=\bigcap _{r=1}^{n}\ker d_{r,0}^{\ast }$ and with differential $\sum (-1)^{r}d_{r,1}^{\ast }$ ; the normalized complex $N_{\bullet }{\mathcal{F}}$ defined as $N_{n}{\mathcal{F}}=C_{n}\cap {\mathcal{F}}\bigcap _{r=2}^{n}\ker d_{r,1}^{\ast }$ and with differential $-d_{1,1}^{\ast }$ . By [Ayo14c, Lemma A.3, Propositions A.8, A.11], the inclusion $N_{\bullet }{\mathcal{F}}{\hookrightarrow}C_{\bullet }{\mathcal{F}}$ is a quasi-isomorphism and both inclusions $C_{\bullet }{\mathcal{F}}{\hookrightarrow}C_{\bullet }^{\sharp }{\mathcal{F}}$ and $N_{\bullet }{\mathcal{F}}{\hookrightarrow}C_{\bullet }{\mathcal{F}}$ split.

For any complex of presheaves ${\mathcal{F}}$ we define the singular complex of ${\mathcal{F}}$ , denoted by $\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}}$ , to be the total complex of the simple complex associated to the $\text{}\underline{\text{Hom}}(\unicode[STIX]{x1D6EC}(\Box ),{\mathcal{F}})$ . It sends the object $X$ to the total complex of the simple complex associated to ${\mathcal{F}}(X\,\times \,\Box )$ .

The following lemma is the cocubical version of [MVW06, Lemma 2.18].

Lemma 3.14. For any presheaf ${\mathcal{F}}$ the two maps of cubical sets $i_{0}^{\ast },i_{1}^{\ast }:{\mathcal{F}}(\Box \,\times \,\mathbb{B}^{1})\rightarrow {\mathcal{F}}(\Box )$ induce chain homotopic maps on the associated simple and normalized complexes.

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.◻

The following proposition is the rigid analytic analog of [Ayo14b, Theorem 2.23], or the cocubical analog of [Ayo15, Lemma 2.5.31].

Proposition 3.15. Let ${\mathcal{F}}$ be a complex in $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ . Then $\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}}$ is $\mathbb{B}^{1}$ -local and $\mathbb{B}^{1}$ -weak equivalent to ${\mathcal{F}}$ in $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ .

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 [Ayo15, 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 [Ayo15, Corollary 2.5.36]), which is in turn quasi-isomorphic to ${\mathcal{F}}$ .◻

Corollary 3.16. Let $\unicode[STIX]{x1D6EC}$ be a $\mathbb{Q}$ -algebra. For any ${\mathcal{F}}$ in $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ the localization $C^{\mathbb{B}^{1}}{\mathcal{F}}$ is quasi-isomorphic to $\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}}$ and the localization $C^{\acute{\text{e}}\text{t},\mathbb{B}^{1}}{\mathcal{F}}$ is quasi-isomorphic to $\operatorname{Sing}^{\mathbb{B}^{1}}(C^{\acute{\text{e}}\text{t}}{\mathcal{F}}^{\bullet })$ .

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 [SV00, 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.◻

We now investigate some of the natural Quillen functors that arise between the model categories introduced so far. We start by considering the natural inclusion of categories $\operatorname{RigSm}\rightarrow \widehat{\text{Rig}}\operatorname{Sm}$

Proposition 3.17. The inclusion $\operatorname{RigSm}{\hookrightarrow}\widehat{\text{Rig}}\operatorname{Sm}$ induces a Quillen adjunction

Moreover, the functor $\mathbb{L}\unicode[STIX]{x1D704}^{\ast }:\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)\rightarrow \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K)$ is fully faithful.

Proof. The first claim is a special instance of [Ayo07, 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.◻

We are now interested in finding a convenient set of compact objects that generate the categories above, as triangulated categories with small sums. This will simplify many definitions and proofs in what follows.

Proposition 3.18. The category $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ [respectively $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K)$ ] is compactly generated (as a triangulated category with small sums) by motives $\unicode[STIX]{x1D6EC}(X)$ associated to rigid varieties $X$ , which are in $\operatorname{RigSm}^{\operatorname{gc}}$ [respectively $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ ].

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 [Ayo07, Proposition 4.5.62].◻

Remark 3.19. The above proof shows that the statement of Proposition 3.18 holds true without any assumptions on $\unicode[STIX]{x1D6EC}$ under the condition that all varieties $X$ have finite cohomological dimension with respect to the étale topology.

We now introduce the category of motives associated to smooth perfectoid spaces, using the same formalism as before. In this category, the canonical choice of the ‘interval object’ for defining homotopies is the perfectoid ball $\widehat{\mathbb{B}}^{1}$ .

Example 3.20. The perfectoid ball $\widehat{\mathbb{B}}^{1}=\operatorname{Spa}(K\langle \unicode[STIX]{x1D712}^{1/p^{\infty }}\rangle ,K^{\circ }\langle \unicode[STIX]{x1D712}^{1/p^{\infty }}\rangle )$ is an interval object with respect to the natural multiplication $\unicode[STIX]{x1D707}$ and maps $i_{0}$ and $i_{1}$ induced by the substitution $\unicode[STIX]{x1D712}^{1/p^{h}}\mapsto 0$ and $\unicode[STIX]{x1D712}^{1/p^{h}}\mapsto 1$ , respectively.

The perfectoid variety $\widehat{\mathbb{B}}^{1}$ naturally lives in $\widehat{\text{Rig}}\operatorname{Sm}$ and has good coordinates by Proposition 2.13. It can therefore be used to define another homotopy category out of $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ and $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}})$ .

Corollary 3.21. The following pairs of model categories are Quillen equivalent.

1. – $\mathbf{Ch}_{\acute{\text{e}}\text{t}}\mathbf{Psh}(\operatorname{PerfSm})$ and $\mathbf{Ch}_{\acute{\text{e}}\text{t}}\mathbf{Psh}(\operatorname{PerfSm}^{\operatorname{gc}})$ .

2. – $\mathbf{Ch}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}\mathbf{Psh}(\operatorname{PerfSm})$ and $\mathbf{Ch}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}\mathbf{Psh}(\operatorname{PerfSm}^{\operatorname{gc}})$ .

3. – $\mathbf{Ch}_{\acute{\text{e}}\text{t}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ and $\mathbf{Ch}_{\acute{\text{e}}\text{t}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}})$ .

4. – $\mathbf{Ch}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ and $\mathbf{Ch}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}})$ .

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.◻

Definition 3.22. For $\unicode[STIX]{x1D702}\in \{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1},(\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1})\}$ we say that a map in $\mathbf{Ch}\,\mathbf{Psh}(\operatorname{PerfSm})$ [respectively $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ ] is a $\unicode[STIX]{x1D702}$ -weak equivalence if it is a weak equivalence in the model structure $\mathbf{Ch}_{\unicode[STIX]{x1D702}}\mathbf{Psh}(\operatorname{PerfSm})$ [respectively $\mathbf{Ch}_{\unicode[STIX]{x1D702}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ ]. We say that an object ${\mathcal{F}}$ of the derived category $\mathbf{D}=\mathbf{D}(\mathbf{Psh}(\operatorname{PerfSm}))$ [respectively $\mathbf{D}=\mathbf{D}(\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}))$ ] is $\unicode[STIX]{x1D702}$ -local if the functor $\operatorname{Hom}_{\mathbf{D}}(\cdot ,{\mathcal{F}})$ sends maps in $S_{\unicode[STIX]{x1D702}}$ (see Proposition 3.3) to isomorphisms. This amounts to say that ${\mathcal{F}}$ is quasi-isomorphic to a $\unicode[STIX]{x1D702}$ -fibrant object. The triangulated homotopy category associated to the localization $\mathbf{Ch}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}\mathbf{Psh}(\operatorname{PerfSm})$ [respectively $\mathbf{Ch}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ ] will be denoted by $\mathbf{PerfDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ [respectively $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ ]. We will omit $\unicode[STIX]{x1D6EC}$ whenever the context allows it. The image of a variety $X$ in one of these categories will be denoted by $\unicode[STIX]{x1D6EC}(X)$ .

We recall one of the main results of Scholze [Sch12], reshaped in our derived homotopical setting. It will constitute the bridge to pass from characteristic $p$ to characteristic $0$ . As summarized in Theorem 1.12 there is an equivalence of categories between perfectoid affinoid $K$ -algebras and perfectoid affinoid $K^{\flat }$ -algebras, extending to an equivalence between the categories of perfectoid spaces over $K$ and over $K^{\flat }$ (see [Sch12, Proposition 6.17]). We refer to this equivalence as the tilting equivalence.

Proposition 3.23. There exists an equivalence of triangulated categories

induced by the tilting equivalence.

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 [Sch12, 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.◻

We now investigate the triangulated functor between the categories of motives induced by the natural embedding $\operatorname{PerfSm}\rightarrow \widehat{\text{Rig}}\operatorname{Sm}$ in the same spirit of what we did previously in Proposition 3.17.

Proposition 3.24. The inclusion $\operatorname{PerfSm}{\hookrightarrow}\widehat{\text{Rig}}\operatorname{Sm}$ induces a Quillen adjunction

Moreover, the functor $\mathbb{L}j^{\ast }:\mathbf{PerfDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)\rightarrow \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}(K)$ is fully faithful.

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

Also in this framework, the $\widehat{\mathbb{B}}^{1}$ -localization has a very explicit construction. Most proofs are straightforward analogs of those relative to the $\mathbb{B}^{1}$ -localizations, and will therefore be omitted.

Definition 3.25. We denote by $\widehat{\Box }$ the $\unicode[STIX]{x1D6F4}$ -enriched cocubical object (see [Ayo14c, Appendix A]) defined by putting $\widehat{\Box }^{n}=\widehat{\mathbb{B}}^{n}=\operatorname{Spa}K\langle \unicode[STIX]{x1D70F}_{1}^{1/p^{\infty }},\ldots ,\unicode[STIX]{x1D70F}_{n}^{1/p^{1/\infty }}\rangle$ and considering the morphisms $d_{r,\unicode[STIX]{x1D716}}$ induced by the maps $\widehat{\mathbb{B}}^{n}\rightarrow \widehat{\mathbb{B}}^{n+1}$ corresponding to the substitution $\unicode[STIX]{x1D70F}_{r}^{1/p^{h}}=\unicode[STIX]{x1D716}$ for $\unicode[STIX]{x1D716}\in \{0,1\}$ and the morphisms $p_{r}$ induced by the projections $\widehat{\mathbb{B}}^{n}\rightarrow \widehat{\mathbb{B}}^{n-1}$ . For any complex of presheaves ${\mathcal{F}}$ we let $\operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}{\mathcal{F}}$ be the total complex of the simple complex associated to $\text{}\underline{\text{Hom}}(\widehat{\Box },{\mathcal{F}})$ . It sends the object $X$ to the total complex of the simple complex associated to ${\mathcal{F}}(X\,\times \,\widehat{\Box })$ .

Proposition 3.26. Let ${\mathcal{F}}$ be a complex in $\mathbf{Ch}\,\mathbf{Psh}(\operatorname{PerfSm})$ [respectively in $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ ]. Then $\operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}{\mathcal{F}}$ is $\widehat{\mathbb{B}}^{1}$ -local and $\widehat{\mathbb{B}}^{1}$ -weak equivalent to ${\mathcal{F}}$ .

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.◻

The following lemmas are used in the previous proof.

Lemma 3.27. A presheaf ${\mathcal{F}}$ in $\mathbf{Psh}(\operatorname{Sm}\operatorname{Perf})$ [respectively in $\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ ] is $\widehat{\mathbb{B}}^{1}$ -invariant if and only if $i_{0}^{\ast }=i_{1}^{\ast }:{\mathcal{F}}(X\,\times \,\widehat{\mathbb{B}}^{1})\rightarrow {\mathcal{F}}(X)$ for all $X$ in $\operatorname{Sm}\operatorname{Perf}$ [respectively in $\widehat{\text{Rig}}\operatorname{Sm}$ ].

Proof. This follows in the same way as [MVW06, Lemma 2.16]. ◻

Lemma 3.28. For any presheaf ${\mathcal{F}}$ the two maps of cubical sets $i_{0}^{\ast },i_{1}^{\ast }:{\mathcal{F}}(\widehat{\Box }\,\times \,\widehat{\mathbb{B}}^{1})\rightarrow {\mathcal{F}}(\widehat{\Box })$ induce chain homotopic maps on the associated simple and normalized complexes.

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

Corollary 3.29. Let ${\mathcal{F}}$ be in $\mathbf{Ch}\,\mathbf{Psh}(\operatorname{PerfSm})$ [respectively in $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ ] the $(\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1})$ -localization $C^{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}{\mathcal{F}}$ is quasi-isomorphic to $\operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}(C^{\acute{\text{e}}\text{t}}{\mathcal{F}})$ .

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

Proposition 3.30. The category $\mathbf{PerfDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ [respectively $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}(K)$ ] is compactly generated (as a triangulated category with small sums) by motives $\unicode[STIX]{x1D6EC}(X)$ associated to rigid varieties $X$ , which are in $\operatorname{PerfSm}^{\operatorname{gc}}$ [respectively $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ ].

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

Remark 3.31. The above proof shows that the statement of Proposition 3.30 holds true without any assumptions on $\unicode[STIX]{x1D6EC}$ under the condition that all varieties $X$ have finite cohomological dimension with respect to the étale topology.

So far, we have defined two different Bousfield localizations on complexes of presheaves on $\widehat{\text{Rig}}\operatorname{Sm}$ according to two different choices of intervals: $\mathbb{B}^{1}$ and $\widehat{\mathbb{B}}^{1}$ . We remark that the second constitutes a further localization of the first, in the following sense.

Proposition 3.32. $\mathbb{B}^{1}$ -weak equivalences in $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ are $\widehat{\mathbb{B}}^{1}$ -weak equivalences.

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 $\mathbb{B}^{1}$ .◻

Corollary 3.33. The triangulated category $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}(K)$ is equivalent to the full triangulated subcategory of $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K)$ formed by $\widehat{\mathbb{B}}^{1}$ -local objects.

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 $\mathbb{Z}$ .◻

We end this section by recalling the definition of rigid motives with transfers. The notion of finite correspondence plays an important role in Voevodsky’s theory of motives. In the case of rigid varieties over a field $K$ correspondences give rise to the category $\operatorname{RigCor}(K)$ as defined in [Ayo15, Definition 2.2.29].

Definition 3.34. Additive presheaves over $\operatorname{RigCor}(K)$ are called presheaves with transfers, and the category they form is denoted by $\mathbf{PST}(\operatorname{RigSm}/K,\unicode[STIX]{x1D6EC})$ or simply by $\mathbf{PST}(\operatorname{RigSm})$ when the context allows it.

By [Ayo15, Definition 2.5.15], the projective model category $\mathbf{Ch}\,\mathbf{PST}(\operatorname{RigSm})$ admits a Bousfield localization $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{PST}(\operatorname{RigSm})$ with respect to the union of the class of maps ${\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime }$ inducing isomorphisms on the $\acute{\text{e}}\text{t}$ -sheaves associated to $H_{i}({\mathcal{F}})$ and $H_{i}({\mathcal{F}}^{\prime })$ for all $i\in \mathbb{Z}$ and the set of all maps $\unicode[STIX]{x1D6EC}(\mathbb{B}_{X}^{1})[i]\rightarrow \unicode[STIX]{x1D6EC}(X)[i]$ as $X$ varies in $\operatorname{RigSm}$ and $i$ varies in $\mathbb{Z}$ .

Definition 3.35. The triangulated homotopy category associated to $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{PST}(\operatorname{RigSm})$ will be denoted by $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ . We will omit $\unicode[STIX]{x1D6EC}$ from the notation whenever the context allows it. The image of a variety $X$ in will be denoted by $\unicode[STIX]{x1D6EC}_{\operatorname{tr}}(X)$ .

Remark 3.36. Since $\unicode[STIX]{x1D6EC}$ is a $\mathbb{Q}$ -algebra, one can equivalently consider the Nisnevich topology in the definition above and obtain a homotopy category $\mathbf{RigDM}_{\operatorname{Nis}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ , which is equivalent to $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ .

Remark 3.37. The faithful embedding of categories $\operatorname{RigSm}\rightarrow \operatorname{RigCor}$ induces a Quillen adjunction (see [Ayo15, Proposition 2.5.19]):

such that $a_{\text{tr}}\unicode[STIX]{x1D6EC}(X)=\unicode[STIX]{x1D6EC}_{\text{tr}}(X)$ for any $X\in \operatorname{RigSm}$ and $o_{\text{tr}}$ is the functor of forgetting transfers. These functors induce an adjoint pair:

which is investigated in [Vez17].

4 Motivic interpretation of approximation results

Throughout this section, $K$ is a perfectoid field of arbitrary characteristic. We begin by presenting an approximation result whose proof is postponed to Appendix A.

Proposition 4.1. Let $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ be in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ . Also let $Y$ be an affinoid rigid variety endowed with an étale map $Y\rightarrow \mathbb{B}^{m}$ . For a given finite set of maps $\{f_{1},\ldots ,f_{N}\}$ in $\operatorname{Hom}(X\,\times \,\mathbb{B}^{n},Y)$ we can find corresponding maps $\{H_{1},\ldots ,H_{N}\}$ in $\operatorname{Hom}(X\,\times \,\mathbb{B}^{n}\,\times \,\mathbb{B}^{1},Y)$ and an integer $\bar{h}$ such that the following are true.

1. (i) For all $1\leqslant k\leqslant N$ it holds that $i_{0}^{\ast }H_{k}=f_{k}$ and $i_{1}^{\ast }H_{k}$ factors over the canonical map $X\rightarrow X_{\bar{h}}$ .

2. (ii) If $f_{k}\,\circ \,d_{r,\unicode[STIX]{x1D716}}=f_{k^{\prime }}\,\circ \,d_{r,\unicode[STIX]{x1D716}}$ for some $1\leqslant k,k^{\prime }\leqslant N$ and some $(r,\unicode[STIX]{x1D716})\in \{1,\ldots ,n\}\,\times \,\{0,1\}$ then $H_{k}\,\circ \,d_{r,\unicode[STIX]{x1D716}}=H_{k^{\prime }}\,\circ \,d_{r,\unicode[STIX]{x1D716}}$ .

3. (iii) If for some $1\leqslant k\leqslant N$ and some $h\in \mathbb{N}$ the map $f_{k}\,\circ \,d_{1,1}\in \operatorname{Hom}(X\,\times \,\mathbb{B}^{n-1},Y)$ lies in $\operatorname{Hom}(X_{h}\,\times \,\mathbb{B}^{n-1},Y)$ then the element $H_{k}\,\circ \,d_{1,1}$ of $\operatorname{Hom}(X\,\times \,\mathbb{B}^{n-1}\,\times \,\mathbb{B}^{1},Y)$ is constant on $\mathbb{B}^{1}$ equal to $f_{k}\,\circ \,d_{1,1}$ .

The statement above has the following interpretation in terms of complexes.

Proposition 4.2. Let $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ be in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ and let $Y$ be in $\operatorname{RigSm}^{\operatorname{gc}}$ . The natural map

$$\begin{eqnarray}\unicode[STIX]{x1D719}:\underset{h}{\varinjlim }(\operatorname{Sing}^{\mathbb{B}^{1}}\unicode[STIX]{x1D6EC}(Y))(X_{h})\rightarrow (\operatorname{Sing}^{\mathbb{B}^{1}}\unicode[STIX]{x1D6EC}(Y))(X)\end{eqnarray}$$

is a quasi-isomorphism.

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.◻

Corollary 4.3. Let ${\mathcal{F}}$ be a projectively cofibrant complex in $\mathbf{Ch}\,\mathbf{Psh}(\operatorname{RigSm}^{\operatorname{gc}})$ . For any $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ the natural map

$$\begin{eqnarray}\unicode[STIX]{x1D719}:\underset{h}{\varinjlim }(\operatorname{Sing}^{\mathbb{B}^{1}}{\mathcal{F}})(X_{h})\rightarrow (\operatorname{Sing}^{\mathbb{B}^{1}}\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})(X)\end{eqnarray}$$

is a quasi-isomorphism.

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.◻

The following technical proposition is actually a crucial point of our proof, as it allows some explicit computations of morphisms in the category $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ .

Proposition 4.4. Let ${\mathcal{F}}$ be a cofibrant and $(\mathbb{B}^{1},\acute{\text{e}}\text{t})$ -fibrant complex in $\mathbf{Ch}\,\mathbf{Psh}(\operatorname{RigSm}^{\operatorname{gc}})$ . Then $\operatorname{Sing}^{\mathbb{B}^{1}}(\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})$ is $(\mathbb{B}^{1},\acute{\text{e}}\text{t})$ -local in $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}})$ .

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.◻

We are finally ready to state the main result of this section.

Proposition 4.5. Let $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ be in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ . For any complex of presheaves ${\mathcal{F}}$ on $\operatorname{RigSm}^{\operatorname{gc}}$ the natural map

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

is an isomorphism.

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. ◻

5 The de-perfectoidification functor in characteristic $0$

The results proved in § 4 are valid both for $\operatorname{char}K=0$ and $\operatorname{char}K=p$ . In contrast, the results of this section require that $\operatorname{char}K=0$ . We will present later their variant for the case $\operatorname{char}K=p$ .

We start by considering the adjunction between motives with and without transfers (see Remark 3.37). Thanks to the following theorem, we are allowed to add or ignore transfers according to the situation.

Theorem 5.1 [Vez17].

Suppose that $\operatorname{char}K=0$ . The functors $(a_{\text{tr}},o_{\text{tr}})$ induce an equivalence:

Remark 5.2. The proof of the statement above uses in a crucial way the fact that the ring of coefficients $\unicode[STIX]{x1D6EC}$ is a $\mathbb{Q}$ -algebra.

Proposition 5.3. Suppose $\operatorname{char}K=0$ . Let $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ be in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ . If $h$ is big enough, then the map $\unicode[STIX]{x1D6EC}(X_{h+1})\rightarrow \unicode[STIX]{x1D6EC}(X_{h})$ is an isomorphism in $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ .

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 [Ayo15, 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.

We remark that the norm of $Q(\unicode[STIX]{x1D70E}-\bar{\unicode[STIX]{x1D70E}})$ in the circle of radius $\unicode[STIX]{x1D70C}\leqslant 1$ around $\bar{\unicode[STIX]{x1D70E}}$ is bounded by $\max \{\unicode[STIX]{x1D70C}^{p},|p|\}\leqslant \max \{\unicode[STIX]{x1D70C},|p|\}$ . Suppose that $F_{h}$ converges in a circle of radius $\unicode[STIX]{x1D70C}$ with $0<\unicode[STIX]{x1D70C}\leqslant 1$ around $\bar{\unicode[STIX]{x1D70E}}$ and in there it takes values in power-bounded elements. By the expression above, the same holds true for $F_{h+1}$ in the circle of radius $\min \{\unicode[STIX]{x1D70C}|p|^{-1},1\}$ around $\bar{\unicode[STIX]{x1D70E}}$ . By induction we conclude that for a sufficiently big $h$ the power series $F_{h}$ converges in a circle of radius $\unicode[STIX]{x1D6FF}>|p|^{1/(p-1)}$ around $\bar{\unicode[STIX]{x1D70E}}$ and its values in it are power bounded. Up to rescaling indices, we suppose that this holds for $h=1$ .

From the relation $F_{h+1}(\unicode[STIX]{x1D70E})=\unicode[STIX]{x1D719}_{h}(F_{h}(\unicode[STIX]{x1D70E}^{p}))$ we also conclude $F_{1}(\unicode[STIX]{x1D701}\bar{\unicode[STIX]{x1D70E}})=F_{1}(\bar{\unicode[STIX]{x1D70E}})=\bar{\unicode[STIX]{x1D70F}}$ . Therefore, since $|\unicode[STIX]{x1D701}_{i}-1|=|p|^{1/(p-1)}$ for all $i$ , the map

$$\begin{eqnarray}\displaystyle X_{1}=\operatorname{Spa}(S\langle \unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\rangle /P(\unicode[STIX]{x1D70E}^{p},\unicode[STIX]{x1D70F})) & \leftarrow & \displaystyle X_{1}\,\times \,\mathbb{B}^{1}=\operatorname{Spa}(S\langle \bar{\unicode[STIX]{x1D70E}},\bar{\unicode[STIX]{x1D70F}},\unicode[STIX]{x1D712}\rangle /(P(\bar{\unicode[STIX]{x1D70E}}^{p},\bar{\unicode[STIX]{x1D70F}})))\nonumber\\ \displaystyle (\unicode[STIX]{x1D70E}_{i},\unicode[STIX]{x1D70F}_{j}) & \mapsto & \displaystyle (\bar{\unicode[STIX]{x1D70E}}_{i}+(\unicode[STIX]{x1D701}_{i}-1)\bar{\unicode[STIX]{x1D70E}}_{i}\unicode[STIX]{x1D712},F_{1}(\bar{\unicode[STIX]{x1D70E}}+(\unicode[STIX]{x1D701}-1)\bar{\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D712}))\nonumber\end{eqnarray}$$

is a well-defined map, inducing a homotopy between $\operatorname{id}_{X_{1}}$ and $f_{\unicode[STIX]{x1D701}}$ , as claimed.◻

It cannot be expected that all maps $X_{h+1}\rightarrow X_{h}$ are isomorphisms in $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ : consider, for example, $X_{0}=\mathbb{T}^{1}\langle \unicode[STIX]{x1D708}^{1/p}\rangle \rightarrow \mathbb{T}^{1}$ . Then $X_{0}$ is a connected variety, while $X_{1}$ is not. That said, there is a particular class of objects $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ for which this happens: this is the content of the following proposition, which nevertheless will not be used in the following.

We recall that a presentation $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ of an object in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ is of good reduction if the map $X_{0}\rightarrow \mathbb{T}^{N}\,\times \,\mathbb{T}^{M}$ has a formal model, which is an étale map over $\operatorname{Spf}K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1},\text{}\underline{\unicode[STIX]{x1D708}}^{\pm 1}\rangle$ and is of potentially good reduction if this happens after base change by a separable finite field extension $L/K$ .

Proposition 5.4. Let $\operatorname{char}K=0$ and let $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ be a presentation of a variety in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ of potentially good reduction. The maps $\unicode[STIX]{x1D6EC}(X_{h+1})\rightarrow \unicode[STIX]{x1D6EC}(X_{h})$ are isomorphisms in $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ for all $h$ .

Proof. If the map $X_{0}\rightarrow \mathbb{T}^{N}\,\times \,\mathbb{T}^{M}$ has an étale formal model, then also the map $X_{h}\rightarrow \mathbb{T}^{N}\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p^{h}}\rangle \,\times \,\mathbb{T}^{M}$ does. It is then sufficient to consider only the case $h=0$ . Since $L/K$ is finite and $\unicode[STIX]{x1D6EC}$ is a $\mathbb{Q}$ -algebra, by the same argument of the proof of Proposition 5.3 we can assume that $\mathop{\varprojlim }\nolimits_{h}X_{h}$ has good reduction. Also, by means of Theorem 5.1 and the Cancellation Theorem [Ayo15, Corollary 2.5.49], we can equally prove the statement in the stable category $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}(K)$ defined in [Ayo15, Definition 1.3.19].

Let $\mathfrak{X}_{0}\rightarrow \operatorname{Spf}K^{\circ }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1},\text{}\underline{\unicode[STIX]{x1D708}}^{\pm 1}\rangle$ be a formal model of the map $X_{0}\rightarrow \mathbb{T}^{n}\,\times \,\mathbb{T}^{m}$ . We let $\bar{X}_{0}$ be the special fiber over the residue field $k$ of $K$ . The variety $X_{1}$ has also a smooth formal model $\mathfrak{X}_{1}$ whose special fiber is $\bar{X}_{1}$ . By definition, the natural map $\bar{X}_{1}\rightarrow \bar{X}_{0}$ is the push-out of the (relative) Frobenius map $\mathbb{A}_{k}^{\dim X}\rightarrow \mathbb{A}_{k}^{\dim X}$ , which is isomorphic to the relative Frobenius map, and hence an isomorphism of correspondences as $p$ is invertible in $\unicode[STIX]{x1D6EC}$ . We conclude that $\unicode[STIX]{x1D6EC}_{\operatorname{tr}}(\bar{X}_{1})\rightarrow \unicode[STIX]{x1D6EC}_{\operatorname{tr}}(\bar{X}_{0})$ is an isomorphism in $\mathbf{DM}_{\acute{\text{e}}\text{t}}(k)$ .

Let $\mathbf{FormDA}_{\acute{\text{e}}\text{t}}(K^{\circ })$ be the stable category of motives of formal varieties $\mathbf{FSH}_{\mathfrak{M}}(K^{\circ })$ defined in [Ayo15, Definition 1.4.15] associated to the model category $\mathfrak{M}=\mathbf{Ch}(\unicode[STIX]{x1D6EC}\boldsymbol{-Mod})$ . Using [Ayo14a, Theorem B.1] we deduce that the map $\unicode[STIX]{x1D6EC}(\bar{X}_{1})\rightarrow \unicode[STIX]{x1D6EC}(\bar{X}_{0})$ is an isomorphism in $\mathbf{DA}_{\acute{\text{e}}\text{t}}(k)$ as is its image via the following functor (see [Ayo15, Remark 1.4.30]) induced by the special fiber functor and the generic fiber functor:

This morphism is precisely the map $\unicode[STIX]{x1D6EC}(X_{1})\rightarrow \unicode[STIX]{x1D6EC}(X_{0})$ proving the claim.◻

We are now ready to present the main result of this section.

Theorem 5.5. Let $\operatorname{char}K=0$ . The functor $\mathbb{L}\unicode[STIX]{x1D704}^{\ast }:\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)\rightarrow \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ has a left adjoint $\mathbb{L}\unicode[STIX]{x1D704}_{!}$ and the counit map $\operatorname{id}\rightarrow \mathbb{L}\unicode[STIX]{x1D704}_{!}\mathbb{L}\unicode[STIX]{x1D704}^{\ast }$ is invertible. Whenever $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ is an object of $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ then $\mathbb{L}\unicode[STIX]{x1D704}_{!}\unicode[STIX]{x1D6EC}(X)\cong \unicode[STIX]{x1D6EC}(X_{h})$ for a sufficiently large index $h$ . If moreover $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ is of potentially good reduction, then $\mathbb{L}\unicode[STIX]{x1D704}_{!}\unicode[STIX]{x1D6EC}(X)\cong \unicode[STIX]{x1D6EC}(X_{0})$ .

Proof. We start by proving that the canonical map

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

is an isomorphism, for every $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ and for $\bar{h}$ big enough. By Proposition 4.5, it suffices to prove that the natural map

$$\begin{eqnarray}\mathbf{RigDA}^{\operatorname{eff}}(K)(\unicode[STIX]{x1D6EC}(X_{\bar{h}}),\mathbb{L}a_{\text{tr}}{\mathcal{F}})\rightarrow \underset{h}{\varinjlim }\mathbf{RigDA}^{\operatorname{eff}}(K)(\unicode[STIX]{x1D6EC}(X_{h}),\mathbb{L}a_{\text{tr}}{\mathcal{F}})\end{eqnarray}$$

is an isomorphism for some $\bar{h}$ . This follows from Proposition 5.3 since the maps $\unicode[STIX]{x1D6EC}(X_{h+1})\rightarrow \unicode[STIX]{x1D6EC}(X_{h})$ are isomorphisms if $h\geqslant \bar{h}$ for some big enough $\bar{h}$ . In case $\mathop{\varprojlim }\nolimits_{h}X_{h}$ is of potentially good reduction, then Proposition 5.4 ensures that we can choose $\bar{h}=0$ .

We conclude that the subcategory $\mathbf{T}$ of $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}(K)$ formed by the objects $M$ such that the functor $N\mapsto \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}(K)(M,\mathbb{L}\unicode[STIX]{x1D704}^{\ast }N)$ is corepresentable contains all motives $\unicode[STIX]{x1D6EC}(X)$ with $X$ any object of $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ . Since these objects form a set of compact generators of $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}(K)$ by Proposition 3.18, we deduce the existence of the functor $\mathbb{L}\unicode[STIX]{x1D704}_{!}$ by Lemma 5.6.

The formula $\mathbb{L}\unicode[STIX]{x1D704}_{!}\mathbb{L}\unicode[STIX]{x1D704}^{\ast }\cong \operatorname{id}$ is a formal consequence of the fact that $\mathbb{L}\unicode[STIX]{x1D704}_{!}$ is the left adjoint of a fully faithful functor $\mathbb{L}\unicode[STIX]{x1D704}^{\ast }$ (see Proposition 3.17).◻

Lemma 5.6. Let $\mathfrak{G}:\mathbf{T}\rightarrow \mathbf{T}^{\prime }$ be a triangulated functor of triangulated categories. The full subcategory $\mathbf{C}$ of $\mathbf{T}^{\prime }$ of objects $M$ such that the functor $a_{M}:N\mapsto \operatorname{Hom}(M,\mathfrak{G}N)$ is corepresentable is closed under cones and small direct sums.

Proof. For any object $M$ in $\mathbf{C}$ we denote by $\mathfrak{F}M$ the object corepresenting the functor $a_{M}$ . Now let $\{M_{i}\}_{i\in I}$ be a set of objects in $\mathbf{C}$ . It is immediate to check that $\bigoplus _{i}\mathfrak{F}M_{i}$ corepresents the functor $a_{\bigoplus _{i}M_{i}}$ .

Now let $M_{1}$ , $M_{2}$ be two objects of $\mathbf{C}$ and $f:M_{1}\rightarrow M_{2}$ be a map between them. There are canonical maps $\unicode[STIX]{x1D702}_{i}:M_{i}\rightarrow \mathfrak{G}\mathfrak{F}M_{i}$ induced by the identity $\mathfrak{F}M_{i}\rightarrow \mathfrak{F}M_{i}$ and the universal property of $\mathfrak{F}M_{i}$ . By composing with $\unicode[STIX]{x1D702}_{2}$ we obtain a morphism $\operatorname{Hom}(M_{1},M_{2})\rightarrow \operatorname{Hom}(M_{1},\mathfrak{G}\mathfrak{F}M_{2})\cong \operatorname{Hom}(\mathfrak{F}M_{1},\mathfrak{F}M_{2})$ sending $f$ to a map $\mathfrak{F}f$ . Let $C$ be the cone of $f$ and $D$ be the cone of $\mathfrak{F}f$ . We claim that $D$ represents $a_{C}$ . From the triangulated structure we obtain a map of distinguished triangles

inducing for any object $N$ of $\mathbf{T}$ the following maps of long exact sequences.

Since the vertical compositions are isomorphisms for $M_{1}$ and $M_{2}$ we deduce that they all are, proving that $D$ corepresents $a_{C}$ , as desired.◻

We remark that we used the fact that $\unicode[STIX]{x1D6EC}$ is a $\mathbb{Q}$ -algebra at least twice in the proof of Theorem 5.5: to allow for field extensions and correspondences using Theorem 5.1 as well as to invert the map defined by multiplication by $p$ .

The following fact is a straightforward corollary of Theorem 5.5.

Proposition 5.7. Let $\operatorname{char}K=0$ . The motive $\mathbb{L}\unicode[STIX]{x1D704}_{!}\unicode[STIX]{x1D6EC}(\widehat{\mathbb{B}}^{1})$ is isomorphic to $\unicode[STIX]{x1D6EC}$ .

Proof. In order to prove the claim, it suffices to prove that $\mathbb{L}\unicode[STIX]{x1D704}_{!}\unicode[STIX]{x1D6EC}(\widehat{\mathbb{B}}^{1})\cong \unicode[STIX]{x1D6EC}(\mathbb{B}^{1})$ . This follows from Proposition 2.13 and the description of $\mathbb{L}\unicode[STIX]{x1D704}_{!}$ given in Theorem 5.5.◻

We recall that all the homotopy categories we consider are monoidal (see [Ayo07, Propositions 4.2.76 and 4.4.62]), and the tensor product $\unicode[STIX]{x1D6EC}(X)\,\otimes \,\unicode[STIX]{x1D6EC}(X^{\prime })$ of two motives associated to varieties $X$ and $Y$ coincides with $\unicode[STIX]{x1D6EC}(X\,\times \,X^{\prime })$ . The unit object is obviously the motive $\unicode[STIX]{x1D6EC}$ . Due to the explicit description of the functor $\mathbb{L}\unicode[STIX]{x1D704}_{!}$ we constructed above, it is easy to prove that it respects the monoidal structures.

Proposition 5.8. Let $\operatorname{char}K=0$ . The functor $\mathbb{L}\unicode[STIX]{x1D704}_{!}:\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K)\rightarrow \mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ is a monoidal functor.

Proof. Since $\mathbb{L}\unicode[STIX]{x1D704}_{!}$ is the left adjoint of a monoidal functor $\mathbb{L}\unicode[STIX]{x1D704}^{\ast }$ there is a canonical natural transformation of bifunctors $\mathbb{L}\unicode[STIX]{x1D704}_{!}(M\,\otimes \,M^{\prime })\rightarrow \mathbb{L}\unicode[STIX]{x1D704}_{!}M\,\otimes \,\mathbb{L}\unicode[STIX]{x1D704}_{!}M^{\prime }$ . In order to prove it is an isomorphism, it suffices to check it on a set of generators of $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}$ such as motives of semi-perfectoid varieties $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ , $X^{\prime }=\mathop{\varprojlim }\nolimits_{h}X_{h}^{\prime }$ . Up to rescaling, we can suppose that $\mathbb{L}\unicode[STIX]{x1D704}_{!}\unicode[STIX]{x1D6EC}(X)=\unicode[STIX]{x1D6EC}(X_{0})$ and $\mathbb{L}\unicode[STIX]{x1D704}_{!}\unicode[STIX]{x1D6EC}(X^{\prime })=\unicode[STIX]{x1D6EC}(X_{0}^{\prime })$ by Theorem 5.5. In this case, by definition of the tensor product, we obtain the following isomorphisms:

$$\begin{eqnarray}\mathbb{L}\unicode[STIX]{x1D704}_{!}(\unicode[STIX]{x1D6EC}(X)\,\otimes \,\unicode[STIX]{x1D6EC}(X^{\prime }))\cong \mathbb{L}\unicode[STIX]{x1D704}_{!}\unicode[STIX]{x1D6EC}(X\,\times \,X^{\prime })\cong \unicode[STIX]{x1D6EC}(X_{0}\,\times \,X_{0}^{\prime })\cong \unicode[STIX]{x1D6EC}(X_{0})\,\otimes \,\unicode[STIX]{x1D6EC}(X_{0}^{\prime })\cong \mathbb{L}\unicode[STIX]{x1D704}_{!}\unicode[STIX]{x1D6EC}(X)\,\otimes \,\mathbb{L}\unicode[STIX]{x1D704}_{!}\unicode[STIX]{x1D6EC}(X^{\prime }),\end{eqnarray}$$

proving our claim. ◻

The following proposition can be considered to be a refinement of Theorem 5.5.

Proposition 5.9. Let $\operatorname{char}K=0$ . The functor $\mathbb{L}\unicode[STIX]{x1D704}_{!}$ factors through $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}\rightarrow \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}$ and the image of the functor $\mathbb{L}\unicode[STIX]{x1D704}^{\ast }:\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)\rightarrow \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K)$ lies in the subcategory of $\widehat{\mathbb{B}}^{1}$ -local objects. In particular, the triangulated adjunction

restricts to a triangulated adjunction

Proof. By Propositions 5.7 and 5.8, $\mathbb{L}\unicode[STIX]{x1D704}_{!}$ is a monoidal functor sending $\unicode[STIX]{x1D6EC}(\widehat{\mathbb{B}}^{1})$ to $\unicode[STIX]{x1D6EC}$ . This proves the first claim.

From the adjunction $(\mathbb{L}\unicode[STIX]{x1D704}_{!},\mathbb{L}\unicode[STIX]{x1D704}^{\ast })$ we then obtain the following isomorphisms, for any $X$ in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ and any $M$ in $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ :

$$\begin{eqnarray}\displaystyle \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K)(\unicode[STIX]{x1D6EC}(X\,\times \,\widehat{\mathbb{B}}^{1}),\mathbb{L}\unicode[STIX]{x1D704}^{\ast }M) & \cong & \displaystyle \mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)(\mathbb{L}\unicode[STIX]{x1D704}_{!}\unicode[STIX]{x1D6EC}(X)\,\otimes \,\unicode[STIX]{x1D6EC},M)\nonumber\\ \displaystyle & \cong & \displaystyle \mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)(\mathbb{L}\unicode[STIX]{x1D704}_{!}\unicode[STIX]{x1D6EC}(X),M)\nonumber\\ \displaystyle & \cong & \displaystyle \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K)(\unicode[STIX]{x1D6EC}(X),\mathbb{L}\unicode[STIX]{x1D704}^{\ast }M),\nonumber\end{eqnarray}$$

proving the second claim. ◻

Remark 5.10. In the statement of the proposition above, we make a slight abuse of notation when denoting with $(\mathbb{L}\unicode[STIX]{x1D704}_{!},\mathbb{L}\unicode[STIX]{x1D704}^{\ast })$ both adjoint pairs. It will be clear from the context which one we consider at each instance.

6 The de-perfectoidification functor in characteristic $p$

We now consider the case of a perfectoid field $K^{\flat }$ of characteristic $p$ and try to generalize the results of § 5. We will need to perform an extra localization on the model structure, and in return we will prove a stronger result. In this section, we always assume that the base perfectoid field has characteristic $p$ . In order to emphasize this hypothesis, we denote it with $K^{\flat }$ .

In positive characteristic, we are not able to prove Theorem 5.1 as it is stated, and it is therefore not clear that the maps $X_{h+1}\rightarrow X_{h}$ associated to an object $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ of $\widehat{\text{Rig}}\operatorname{Sm}$ are isomorphisms in $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })$ for a sufficiently big $h$ . In order to overcome this obstacle, we localize our model category further.

For any variety $X$ over $K^{\flat }$ we denote by $X^{(1)}$ the pullback of $X$ over the Frobenius map $\unicode[STIX]{x1D6F7}:K^{\flat }\rightarrow K^{\flat }$ , $x\mapsto x^{p}$ . The absolute Frobenius morphism induces a $K^{\flat }$ -linear map $X\rightarrow X^{(1)}$ . Since $K^{\flat }$ is perfect, we can also denote by $X^{(-1)}$ the pullback of $X$ over the inverse of the Frobenius map $\unicode[STIX]{x1D6F7}^{-1}:K^{\flat }\rightarrow K^{\flat }$ and $X\cong (X^{(-1)})^{(1)}$ . There is, in particular, a canonical map $X^{(-1)}\rightarrow X$ , which is isomorphic to the map $X^{\prime }\rightarrow X$ induced by the absolute Frobenius, where we denote by $X^{\prime }$ the same variety $X$ endowed with the structure map $X\rightarrow \operatorname{Spa}K\xrightarrow[{}]{\unicode[STIX]{x1D6F7}}\operatorname{Spa}K$ .

Proposition 6.1. The model category $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\operatorname{RigSm}/K^{\flat })$ admits a left Bousfield localization denoted by $\mathbf{Ch}_{\text{Frob}\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\operatorname{RigSm}/K^{\flat })$ with respect to the set $S_{\operatorname{Frob}}$ of relative Frobenius maps $\unicode[STIX]{x1D6F7}:\unicode[STIX]{x1D6EC}(X^{(-1)})[i]\rightarrow \unicode[STIX]{x1D6EC}(X)[i]$ as $X$ varies in $\operatorname{RigSm}$ and $i$ varies in $\mathbb{Z}$ .

Proof. Since by [Ayo07, Proposition 4.4.31] the $\unicode[STIX]{x1D70F}$ -localization coincides with the Bousfield localization with respect to a set, we conclude by [Ayo07, Theorem 4.2.71] that the model category $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\operatorname{RigSm}/K^{\flat })$ is still left proper and cellular. We can then apply [Hir03, Theorem 4.1.1].◻

Definition 6.2. We denote by $\mathbf{RigDA}_{\text{Frob}\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat },\unicode[STIX]{x1D6EC})$ the homotopy category associated to $\mathbf{Ch}_{\text{Frob}\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\operatorname{RigSm}/K^{\flat })$ . We omit $\unicode[STIX]{x1D6EC}$ whenever the context allows it. The image of a rigid variety $X$ in this category is denoted by $\unicode[STIX]{x1D6EC}(X)$ .

The triangulated category $\mathbf{RigDA}_{\text{Frob}\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ is canonically isomorphic to the full triangulated subcategory of $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ formed by $\operatorname{Frob}$ -local objects, i.e., objects that are local with respect to the maps in $S_{\operatorname{Frob}}$ . Modulo this identification, there is an obvious functor $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })\rightarrow \mathbf{RigDA}_{\text{Frob}\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })$ associating to ${\mathcal{F}}$ a $\operatorname{Frob}$ -local object $C^{\operatorname{Frob}}{\mathcal{F}}$ .

Inverting Frobenius morphisms is enough to obtain an analog of Theorem 5.1 in characteristic  $p$ .

Theorem 6.3 [Vez17].

Let $\operatorname{char}K^{\flat }=p$ . The functors $(a_{\text{tr}},o_{\text{tr}})$ induce an equivalence of triangulated categories:

$$\begin{eqnarray}\mathbb{L}a_{\text{tr}}:\mathbf{RigDA}_{\text{Frob}\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })\cong \mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat }).\end{eqnarray}$$

Remark 6.4. The proof of the statement above uses in a crucial way the fact that the ring of coefficients $\unicode[STIX]{x1D6EC}$ is a $\mathbb{Q}$ -algebra.

We now investigate the relations between the category $\mathbf{RigDA}_{\text{Frob}\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })$ we have just defined, and the other categories of motives introduced so far.

Proposition 6.5. Let $X_{0}$ be in $\operatorname{RigSm}/K^{\flat }$ endowed with an étale map $X_{0}\rightarrow \mathbb{T}^{N}\,\times \,\mathbb{T}^{M}=\operatorname{Spa}(K^{\flat }\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1},\text{}\underline{\unicode[STIX]{x1D708}}^{\pm 1}\rangle )$ . The map $X_{1}=X_{0}\,\times _{\mathbb{T}^{N}}\,\mathbb{T}^{N}\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p}\rangle \rightarrow X_{0}$ is invertible in $\mathbf{RigDA}_{\text{Frob}\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })$ .

Proof. The map of the claim is a factor of $X_{0}\,\times _{(\mathbb{B}^{N}\times \mathbb{B}^{M})}\,(\mathbb{B}^{N}\langle \text{}\underline{\unicode[STIX]{x1D710}}^{1/p}\rangle \,\times \,\mathbb{B}^{M}\langle \text{}\underline{\unicode[STIX]{x1D708}}^{1/p}\rangle )\rightarrow X_{0}$ , which is isomorphic to the relative Frobenius map $X_{0}^{(-1)}\rightarrow X_{0}$ (see for example [GR03, Theorem 3.5.13]). If we consider the diagram

$$\begin{eqnarray}X_{1}^{(-1)}\xrightarrow[{}]{a}X_{0}^{(-1)}\xrightarrow[{}]{b}X_{1}\xrightarrow[{}]{c}X_{0}\end{eqnarray}$$

we conclude that the two compositions $ba$ and $cb$ are isomorphisms, and hence $c$ is also an isomorphism, as claimed.◻

Proposition 6.6. The image via $\mathbb{L}\unicode[STIX]{x1D704}^{\ast }$ of a $\operatorname{Frob}$ -local object of $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })$ is $\widehat{\mathbb{B}}^{1}$ -local. In particular, the functor $\mathbb{L}\unicode[STIX]{x1D704}^{\ast }$ restricts to a functor $\mathbb{L}\unicode[STIX]{x1D704}^{\ast }:\mathbf{RigDA}_{\text{Frob}\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })\rightarrow \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}(K^{\flat })$ .

Proof. Let $X^{\prime }=\mathop{\varprojlim }\nolimits_{h}X_{h}^{\prime }$ be in $\widehat{\text{Rig}}\operatorname{Sm}^{\operatorname{gc}}$ . We consider the object $X^{\prime }\,\times \,\widehat{\mathbb{B}}^{1}=\mathop{\varprojlim }\nolimits_{h}(X_{h}^{\prime }\,\times \,X_{h})$ , where we use the description $\widehat{\mathbb{B}}^{1}=\mathop{\varprojlim }\nolimits_{h}X_{h}$ of Proposition 2.13. Let $M$ be a $\operatorname{Frob}$ -local object of $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })$ . From Propositions 4.5 and 6.5, we then deduce the following isomorphisms:

$$\begin{eqnarray}\displaystyle \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K^{\flat })(X^{\prime }\,\times \,\widehat{\mathbb{B}}^{1},\mathbb{L}\unicode[STIX]{x1D704}^{\ast }M) & \cong & \displaystyle \underset{h}{\varinjlim }\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })(X_{h}^{\prime }\,\times \,X_{h},M)\nonumber\\ \displaystyle & \cong & \displaystyle \mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })(X_{0}^{\prime }\,\times \,\mathbb{B}^{1},M)\nonumber\\ \displaystyle & \cong & \displaystyle \mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })(X_{0}^{\prime },M)\nonumber\\ \displaystyle & \cong & \displaystyle \underset{h}{\varinjlim }\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })(X_{h}^{\prime },M)\nonumber\\ \displaystyle & \cong & \displaystyle \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K^{\flat })(X^{\prime },\mathbb{L}\unicode[STIX]{x1D704}^{\ast }M),\nonumber\end{eqnarray}$$

proving the claim. ◻

We remark that in positive characteristic the perfection $\operatorname{Perf}:X\mapsto \varprojlim X^{(-i)}$ is functorial. This makes the description of various functors a lot easier. We recall that we denote by

the adjoint pair induced by the inclusion of categories $j:\operatorname{PerfSm}\rightarrow \widehat{\text{Rig}}\operatorname{Sm}$ .

Proposition 6.7. The perfection functor $\operatorname{Perf}:\widehat{\text{Rig}}\operatorname{Sm}\rightarrow \operatorname{PerfSm}$ induces an adjunction

and $\mathbb{L}\operatorname{Perf}^{\ast }$ factors through $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K^{\flat })\rightarrow \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}(K^{\flat })$ . Moreover, the functor $\mathbb{L}\operatorname{Perf}^{\ast }$ coincides with $\mathbb{R}j_{\ast }$ on $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}(K^{\flat })$ .

Proof. The perfection functor is continuous with respect to the étale topology and maps $\mathbb{B}^{1}$ and $\widehat{\mathbb{B}}^{1}$ to $\widehat{\mathbb{B}}^{1}$ , and hence the first claim.

We now consider the functors $j:\operatorname{PerfSm}\rightarrow \widehat{\text{Rig}}\operatorname{Sm}$ and $\operatorname{Perf}:\widehat{\text{Rig}}\operatorname{Sm}\rightarrow \operatorname{PerfSm}$ . They induce two Quillen pairs $(j^{\ast },j_{\ast })$ and $(\operatorname{Perf}^{\ast },\operatorname{Perf}_{\ast })$ on the associated $(\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1})$ -localized model categories of complexes. Since $\operatorname{Perf}$ is a right adjoint of $j$ we deduce that $\operatorname{Perf}^{\ast }$ is a right adjoint of $j^{\ast }$ , and hence we obtain an isomorphism $j_{\ast }\cong \operatorname{Perf}^{\ast }$ , which shows the second claim.◻

Proposition 6.8. Let $\unicode[STIX]{x1D6EC}$ be a $\mathbb{Q}$ -algebra. The functor

$$\begin{eqnarray}\mathbb{L}\operatorname{Perf}^{\ast }\mathbb{L}\unicode[STIX]{x1D704}^{\ast }:\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })\rightarrow \mathbf{PerfDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })\end{eqnarray}$$

factors over $\mathbf{RigDA}_{\text{Frob}\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })$ and is isomorphic to $\mathbb{R}j_{\ast }\mathbb{L}\unicode[STIX]{x1D704}^{\ast }C^{\operatorname{Frob}}$ .

Proof. The first claim follows as the perfection of $X^{(-1)}$ is canonically isomorphic to the perfection of $X$ for any object $X$ in $\operatorname{RigSm}$ .

The second part of the statement follows from the first claim and the commutativity of the following diagram, which is ensured by Propositions 6.6 and 6.7.

Theorem 6.9. Let $\unicode[STIX]{x1D6EC}$ be a $\mathbb{Q}$ -algebra. The functor

$$\begin{eqnarray}\mathbb{L}\operatorname{Perf}^{\ast }:\mathbf{RigDA}_{\text{Frob}\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })\rightarrow \mathbf{PerfDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })\end{eqnarray}$$

defines a monoidal, triangulated equivalence of categories.

Proof. Let $X_{0}$ and $Y$ be objects of $\operatorname{RigSm}^{\operatorname{gc}}$ . Suppose $X_{0}$ is endowed with an étale map over $\mathbb{T}^{N}$ , which is a composition of finite étale maps and inclusions, and let $\widehat{X}$ be $\mathop{\varprojlim }\nolimits_{h}X_{h}$ . We can identify $\widehat{X}$ with $\operatorname{Perf}X_{0}$ . Since $C^{\operatorname{Frob}}\unicode[STIX]{x1D6EC}(Y)$ is $\operatorname{Frob}$ -local, by Proposition 6.5 the maps

$$\begin{eqnarray}\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })(\unicode[STIX]{x1D6EC}(X_{h}),C^{\operatorname{Frob}}\unicode[STIX]{x1D6EC}(Y))\rightarrow \mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })(\unicode[STIX]{x1D6EC}(X_{h+1}),C^{\operatorname{Frob}}\unicode[STIX]{x1D6EC}(Y))\end{eqnarray}$$

are isomorphisms for all $h$ . Using Propositions 4.5 and 6.6, we obtain the following sequence of isomorphisms for any $n\in \mathbb{Z}$ :

$$\begin{eqnarray}\displaystyle \mathbf{RigDA}_{\text{Frob}\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })(\unicode[STIX]{x1D6EC}(X_{0}),\unicode[STIX]{x1D6EC}(Y)[n]) & \cong & \displaystyle \mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })(\unicode[STIX]{x1D6EC}(X_{0}),C^{\operatorname{Frob}}\unicode[STIX]{x1D6EC}(Y)[n])\nonumber\\ \displaystyle & \cong & \displaystyle \underset{h}{\varinjlim }\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })(\unicode[STIX]{x1D6EC}(X_{h}),C^{\operatorname{Frob}}\unicode[STIX]{x1D6EC}(Y)[n])\nonumber\\ \displaystyle & \cong & \displaystyle \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}^{\operatorname{eff}}(K^{\flat })(\unicode[STIX]{x1D6EC}(\widehat{X}),\mathbb{L}\unicode[STIX]{x1D704}^{\ast }C^{\operatorname{Frob}}\unicode[STIX]{x1D6EC}(Y)[n])\nonumber\\ \displaystyle & \cong & \displaystyle \widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}^{\operatorname{eff}}(K^{\flat })(\unicode[STIX]{x1D6EC}(\widehat{X}),\mathbb{L}\unicode[STIX]{x1D704}^{\ast }C^{\operatorname{Frob}}\unicode[STIX]{x1D6EC}(Y)[n])\nonumber\\ \displaystyle & \cong & \displaystyle \mathbf{PerfDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })(\unicode[STIX]{x1D6EC}(\widehat{X}),\mathbb{R}j_{\ast }\mathbb{L}\unicode[STIX]{x1D704}^{\ast }C^{\operatorname{Frob}}\unicode[STIX]{x1D6EC}(Y)[n])\nonumber\\ \displaystyle & \cong & \displaystyle \mathbf{PerfDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })(\mathbb{L}\operatorname{Perf}^{\ast }(X_{0}),\mathbb{L}\operatorname{Perf}^{\ast }(Y)[n]),\nonumber\end{eqnarray}$$

where the last isomorphism follows from the identification $\widehat{X}\cong \operatorname{Perf}X_{0}$ and Proposition 6.8. In particular, we deduce that the triangulated functor $\mathbb{L}\operatorname{Perf}^{\ast }$ maps a set of compact generators to a set of compact generators (see Propositions 3.18 and 3.30) and on these objects it is fully faithful. By means of [Ayo15, Lemma 1.3.32], we then conclude it is a triangulated equivalence of categories, as claimed. ◻

Remark 6.10. From the proof of the previous claim, we also deduce that the inverse $\mathbb{R}\operatorname{Perf}_{\ast }$ of $\mathbb{L}\operatorname{Perf}^{\ast }$ sends the motive associated to an object $X=\mathop{\varprojlim }\nolimits_{h}X_{h}$ to the motive of $X_{0}$ . This functor is then analogous to the de-perfectoidification functor $\mathbb{L}j^{\ast }\,\circ \,\mathbb{L}\unicode[STIX]{x1D704}_{!}$ of Theorem 5.5.

7 The main theorem

Thanks to the results of the previous sections, we can reformulate Theorem 5.5 in terms of motives of rigid varieties. We will always assume that $\operatorname{char}K=0$ since the results of this section are tautological when $\operatorname{char}K=p$ .

Corollary 7.1. There exists a triangulated adjunction of categories

such that $\mathfrak{F}$ is a monoidal functor.

Proof. From Theorem 5.5 and Proposition 5.8, we can define an adjunction

by putting $\mathfrak{F}^{\prime }:=\mathbb{L}\unicode[STIX]{x1D704}_{!}\,\circ \,\mathbb{L}j^{\ast }\,\circ \,(-)^{\sharp }\,\circ \,\mathbb{L}\operatorname{Perf}^{\ast }$ . We remark that by Proposition 5.8, $\mathfrak{F}^{\prime }$ is also monoidal. The claim then follows from the equivalence of motives with and without transfers (Theorems 5.1 and 6.3).◻

Our goal is to prove that the adjunction of Corollary 7.1 is an equivalence of categories. To this aim, we recall the construction of the stable versions of the rigid motivic categories given in [Ayo15, Definition 2.5.27].

Definition 7.2. Let $T$ be the cokernel in $\mathbf{PST}(\operatorname{RigSm}/K)$ of the unit map $\unicode[STIX]{x1D6EC}_{\operatorname{tr}}(K)\rightarrow \unicode[STIX]{x1D6EC}_{\operatorname{tr}}(\mathbb{T}^{1})$ . We denote by $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})$ or simply by $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K)$ the homotopy category of the stable $(\acute{\text{e}}\text{t},\mathbb{B}^{1})$ -local model structure on symmetric spectra $\mathbf{Spt}_{T}^{\unicode[STIX]{x1D6F4}}(\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{PST}(\operatorname{RigSm}/K))$ .

As explained in [Ayo15, § 2.5], $T$ is cofibrant and the cyclic permutation induces the identity on $T^{\otimes 3}$ in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}$ . Moreover, by [Hov01, Theorem 9.3], $T\,\otimes \,-$ is a Quillen equivalence in this category, which is actually the universal model category where this holds (in some weak sense made precise by [Hov01, Theorem 5.1, Proposition 5.3 and Corollary 9.4]). We recall that the canonical functor $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)\rightarrow \mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K)$ is fully faithful, as proved in [Ayo15, Corollary 2.5.49] as a corollary of the cancellation theorem [Ayo15, Theorem 2.5.38].

Definition 7.3. We denote by $\unicode[STIX]{x1D6EC}(1)$ the motive $T[-1]$ in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ . For any positive integer $d$ we let $\unicode[STIX]{x1D6EC}(d)$ be $\unicode[STIX]{x1D6EC}(1)^{\otimes d}$ . The functor $(\cdot )(d):=(\cdot )\,\otimes \,\unicode[STIX]{x1D6EC}(d)$ is an auto-equivalence of $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K)$ and its inverse will be denoted with $(\cdot )(-d)$ .

Definition 7.4. We denote by $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K,\unicode[STIX]{x1D6EC})$ or simply by $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K)$ the full triangulated subcategory of $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})$ whose objects are the compact ones. They are of the form $M(d)$ for some compact object $M$ in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ and some $d$ in $\mathbb{Z}$ . This category is called the category of constructible motives.

We now present an important result that is a crucial step toward the proof of our main theorem. The motivic property it induces will be given right afterwards.

Proposition 7.5. Let $\widehat{X}$ be a smooth affinoid perfectoid. The natural map of complexes

$$\begin{eqnarray}\operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}(\unicode[STIX]{x1D6EC}(\widehat{\mathbb{T}}^{d}))(\widehat{X})\rightarrow \operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}(\unicode[STIX]{x1D6EC}(\mathbb{T}^{d}))(\widehat{X})\end{eqnarray}$$

is a quasi-isomorphism.

Proof. We let $\widehat{X}$ be $\operatorname{Spa}(R,R^{+})$ . A map $f$ in $\operatorname{Hom}(\widehat{X}\,\times \,\widehat{\mathbb{B}}^{n},\mathbb{T}^{d})$ [respectively in $\operatorname{Hom}(\widehat{X}\,\times \,\widehat{\mathbb{B}}^{n},\widehat{\mathbb{T}}^{d})$ ] corresponds to $d$ elements $f_{1},\ldots ,f_{d}$ in the group $(R^{+}\langle \unicode[STIX]{x1D70F}_{1}^{1/p^{\infty }},\ldots ,\unicode[STIX]{x1D70F}_{n}^{1/p^{\infty }}\rangle )^{\times }$ [respectively in the group $(R^{\flat +}\langle \unicode[STIX]{x1D70F}_{1}^{1/p^{\infty }},\ldots ,\unicode[STIX]{x1D70F}_{n}^{1/p^{\infty }}\rangle )^{\times }$ ] and the map between the two objects is induced by the multiplicative tilt map $R^{\flat +}\langle \unicode[STIX]{x1D70F}_{1}^{1/p^{\infty }},\ldots ,\unicode[STIX]{x1D70F}_{n}^{1/p^{\infty }}\rangle \rightarrow R^{+}\langle \unicode[STIX]{x1D70F}_{1}^{1/p^{\infty }},\ldots ,\unicode[STIX]{x1D70F}_{n}^{1/p^{\infty }}\rangle$ .

We now present some facts about homotopy theory for cubical objects, which mirror classical results for simplicial objects (see for example [May67, ch. IV]). We remark that the map of the statement is induced by a map of enriched cubical $\unicode[STIX]{x1D6EC}$ -vector spaces (see [Ayo14b, Definition A.6]), which is obtained by adding $\unicode[STIX]{x1D6EC}$ -coefficients to a map of enriched cubical sets

$$\begin{eqnarray}\operatorname{Hom}(\widehat{X}\,\times \,\widehat{\Box },\widehat{\mathbb{T}}^{d})\rightarrow \operatorname{Hom}(\widehat{X}\,\times \,\widehat{\Box },\mathbb{T}^{d}).\end{eqnarray}$$

Any enriched cubical object has connections in the sense of [BH81, § 1.2], induced by the maps $m_{i}$ in [Ayo14b, Definition A.6]. We recall that the category of cubical sets with connections can be endowed with a model structure by which all objects are cofibrant and weak equivalences are defined through the geometric realization (see [Jar02]). Moreover, its homotopy category is canonically equivalent to the one of simplicial sets, as cubical sets with connections form a strict test category by [Mal09].

The two cubical sets appearing above are abelian groups on each level and the maps defining their cubical structure are group homomorphisms. They therefore are cubical groups. By [Ton92], they are fibrant objects and their homotopy groups $\unicode[STIX]{x1D70B}_{i}$ coincide with the homology $H_{i}N$ of the associated normalized complexes of abelian groups (see Definition 3.13). The $\unicode[STIX]{x1D6EC}$ -enrichment functor is tensorial with respect to the monoidal structure of cubical sets introduced in [BHS11, § 11.2]. Also, the cubical Dold–Kan functor that associates to a cubical $\unicode[STIX]{x1D6EC}$ -module with connections its normalized complex (see [BHS11, § 14.8]) is a left Quillen functor. We deduce that in order to prove the statement of the proposition it suffices to show that the two normalized complexes of abelian groups are quasi-isomorphic. We also remark that it suffices to consider the case $d=1$ .

We prove the following claim: the $n$ th homology of the complex $N((R\,\widehat{\otimes }\,{\mathcal{O}}(\widehat{\Box }))^{+\times })$ is $0$ for $n>0$ . Let $f$ be invertible in $R^{+}\langle \unicode[STIX]{x1D70F}_{1}^{1/p^{\infty }},\ldots ,\unicode[STIX]{x1D70F}_{n}^{1/p^{\infty }}\rangle$ with $d_{r,\unicode[STIX]{x1D716}}f=1$ for all $(r,\unicode[STIX]{x1D716})$ . We claim that $f-1$ is topologically nilpotent. Up to adding a topological nilpotent element, we can assume that $f\in R^{+}[\text{}\underline{\unicode[STIX]{x1D70F}}]$ . Since $f$ is invertible, its image in $(R^{+}/R^{\circ \circ })[\text{}\underline{\unicode[STIX]{x1D70F}}^{1/p^{\infty }}]$ is invertible as well. Invertible elements in this ring are just the invertible constants. We deduce that all coefficients of $f\,-\,f(0)=f-1$ are topologically nilpotent, and hence $f-1$ is topologically nilpotent. In particular, the element $H=f+\unicode[STIX]{x1D70F}_{n+1}(1-f)$ in $R^{+}\langle \text{}\underline{\unicode[STIX]{x1D70F}}^{1/p^{\infty }},\unicode[STIX]{x1D70F}_{n+1}^{1/p^{\infty }}\rangle$ is invertible, satisfies $d_{r,\unicode[STIX]{x1D716}}H=1$ for all $\unicode[STIX]{x1D716}$ and all $1\leqslant r\leqslant n$ and determines a homotopy between $f$ and $1$ . This proves the claim.

We can also prove that the $0$ th homology of the complex $N((R\,\widehat{\otimes }\,{\mathcal{O}}(\widehat{\Box }))^{+\times })$ coincides with $R^{+\times }/(1+R^{\circ \circ })$ . This amounts to showing that the image of the ring map

$$\begin{eqnarray}\displaystyle \{f\in R^{+}\langle \unicode[STIX]{x1D70F}^{1/p^{\infty }}\rangle ^{\times }:f(0)=1\} & \rightarrow & \displaystyle R^{+\times }\nonumber\\ \displaystyle f & \mapsto & \displaystyle f(1)\nonumber\end{eqnarray}$$

coincides with $1+R^{\circ \circ }$ . Let $f$ be invertible in $R^{+}\langle \unicode[STIX]{x1D70F}^{1/p^{\infty }}\rangle$ with $f(0)=1$ . As proved above, $f-1$ is topologically nilpotent so that also $f(1)-1$ is. Vice versa if $a\in R$ is topologically nilpotent then the element $1+a\unicode[STIX]{x1D70F}\in R^{+}\langle \unicode[STIX]{x1D70F}^{1/p^{\infty }}\rangle$ is invertible, satisfies $f(0)=1$ and $f(1)=1+a$ , proving the claim.

We are left to prove that the multiplicative map $\sharp$ induces an isomorphism $(R^{\flat +})^{\times }/(1+R^{\flat \circ \circ })\rightarrow (R^{+})^{\times }/(1+R^{\circ \circ })$ . We start by proving it is injective. Let $a\in R^{\flat +}$ such that $(a^{\sharp }-1)$ is topologically nilpotent. Since $(a^{\sharp }-1)=(a-1)^{\sharp }$ in $R^{+}/\unicode[STIX]{x1D70B}$ we deduce that the element $(a-1)^{\sharp }-(a^{\sharp }-1)$ is also topologically nilpotent. We conclude that $(a-1)^{\sharp }$ , as well as $(a-1)$ , are topologically nilpotent, as desired.

We now prove surjectivity. Let $a$ be invertible in $R^{+}$ . In particular, both $a$ and $a^{-1}$ are power-bounded. From the isomorphism $R^{\flat +}/\unicode[STIX]{x1D70B}^{\flat }\cong R^{+}/\unicode[STIX]{x1D70B}$ we deduce that there exists an element $b\in R^{\flat +}$ such that $b^{\sharp }=a+\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FC}=a(1+\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FC}a^{-1})$ for some (power-bounded) element $\unicode[STIX]{x1D6FC}\in R^{+}$ . We deduce that $(1\,+\,\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FC}a^{-1})$ lies in $1\,+\,R^{\circ \circ }$ and that $b^{\sharp }$ is invertible. Since the multiplicative structure of $R^{\flat }$ is isomorphic to $\mathop{\varprojlim }\nolimits_{x\mapsto x^{p}}R$ and $\sharp$ is given by the projection to the last component, we deduce that as $b^{\sharp }$ is invertible, then also $b$ is. In particular, the image of $b\in (R^{\flat +})^{\times }$ in $(R^{+})^{\times }/(1+R^{\circ \circ })$ is equal to $a$ , as desired.◻

We recall that by Corollary 7.1 there is an adjunction

and our goal is to prove it is an equivalence.

Proposition 7.6. The motive $\mathfrak{G}\unicode[STIX]{x1D6EC}(d)$ is isomorphic to $\unicode[STIX]{x1D6EC}(d)$ for any positive integer $d$ .

Proof. The natural map $\unicode[STIX]{x1D6EC}(d)\rightarrow \mathfrak{G}\unicode[STIX]{x1D6EC}(d)$ is induced by the isomorphism $\mathfrak{F}\unicode[STIX]{x1D6EC}(d)\cong \unicode[STIX]{x1D6EC}(d)$ . We need to prove it is an isomorphism. The motive $\unicode[STIX]{x1D6EC}(d)$ is a direct factor of the motive $\unicode[STIX]{x1D6EC}(\mathbb{T}^{d})[-d]$ and the map above is induced by $\unicode[STIX]{x1D6EC}(\mathbb{T}^{d})\rightarrow \mathfrak{G}\unicode[STIX]{x1D6EC}(\mathbb{T}^{d})$ . It suffices then to prove that the map $\unicode[STIX]{x1D6EC}(\mathbb{T}^{d})\rightarrow \mathfrak{G}\unicode[STIX]{x1D6EC}(\mathbb{T}^{d})$ is an isomorphism.

By the definition of the adjoint pair $(\mathfrak{F},\mathfrak{G})$ given in Corollary 7.1, we can equivalently consider the adjunction

and prove that $\unicode[STIX]{x1D6EC}(\widehat{\mathbb{T}}^{d})\rightarrow (\mathbb{R}j_{\ast }\,\circ \,\mathbb{L}\unicode[STIX]{x1D704}^{\ast })\unicode[STIX]{x1D6EC}(\mathbb{T}^{d})$ is an isomorphism in $\mathbf{PerfDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ .

From Proposition 7.5 we deduce that the complexes $\operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}\unicode[STIX]{x1D6EC}(\widehat{\mathbb{T}}^{d})$ and $j_{\ast }\operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}\unicode[STIX]{x1D6EC}(\mathbb{T}^{d})$ are quasi-isomorphic in $\mathbf{Ch}\,\mathbf{Psh}(\operatorname{PerfSm})$ . Since $j_{\ast }$ commutes with $\operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}$ and with $\acute{\text{e}}\text{t}$ -sheafification, the quasi-isomorphism above can be restated as

$$\begin{eqnarray}\operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}\unicode[STIX]{x1D6EC}(\widehat{\mathbb{T}}^{d})\cong \mathbb{R}j_{\ast }\operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}\unicode[STIX]{x1D6EC}(\mathbb{T}^{d}).\end{eqnarray}$$

Due to Proposition 3.26 and the isomorphism $\mathbb{L}\unicode[STIX]{x1D704}^{\ast }\unicode[STIX]{x1D6EC}(\mathbb{T}^{d})\cong \unicode[STIX]{x1D6EC}(\mathbb{T}^{d})$ this implies $\unicode[STIX]{x1D6EC}(\widehat{\mathbb{T}}^{d})\cong \mathbb{R}j_{\ast }\mathbb{L}\unicode[STIX]{x1D704}^{\ast }\unicode[STIX]{x1D6EC}(\mathbb{T}^{d})$ , as desired.◻

Remark 7.7. Along the proof of the previous proposition, we showed, in particular, that $\mathbb{R}j_{\ast }\mathbb{L}\unicode[STIX]{x1D704}^{\ast }\unicode[STIX]{x1D6EC}(\mathbb{T}^{d})\cong \unicode[STIX]{x1D6EC}(\widehat{\mathbb{T}}^{d})$ .

Remark 7.8. Since $j_{\ast }$ commutes with $\acute{\text{e}}\text{t}$ -sheafification, it preserves $\acute{\text{e}}\text{t}$ -weak equivalences. It also commutes with $\operatorname{Sing}^{\widehat{\mathbb{B}}^{1}}$ and therefore preserves $\mathbb{B}^{1}$ -weak equivalences. We conclude that $\mathbb{R}j_{\ast }=j_{\ast }$ and, in particular, $\mathbb{R}j_{\ast }$ commutes with small direct sums.

Definition 7.9. A rigid analytic variety with good coordinates $X_{0}\rightarrow \mathbb{T}^{N}$ , such that the induced maps $\unicode[STIX]{x1D6EC}(X_{h+1})\rightarrow \unicode[STIX]{x1D6EC}(X_{h})$ are invertible in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\mathbb{Q})$ , is called a variety with very good coordinates.

We are finally ready to present the proof of our main result.

Theorem 7.10. Let $K$ be a perfectoid field and $\unicode[STIX]{x1D6EC}$ be a $\mathbb{Q}$ -algebra. The adjunction

is a monoidal, triangulated equivalence of categories.

Before the proof, we remark that by putting Theorems 3.23 and 6.9 together, the theorem above has the following restatement.

Theorem 7.11. Let $K$ be a perfectoid field and $\unicode[STIX]{x1D6EC}$ be a $\mathbb{Q}$ -algebra. The adjunction

is a monoidal, triangulated equivalence of categories.

Proof. By Theorem 5.5 the functor $\mathbb{L}\unicode[STIX]{x1D704}_{!}\mathbb{L}j^{\ast }:\mathbf{PerfDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)\rightarrow \mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ sends the motive $\unicode[STIX]{x1D6EC}(\widehat{X})$ associated to a perfectoid $\widehat{X}=\mathop{\varprojlim }\nolimits_{h}X_{h}$ to the motive $\unicode[STIX]{x1D6EC}(X_{0})$ associated to $X_{0}$ up to rescaling indices. It is triangulated, commutes with sums, and its essential image contains motives $\unicode[STIX]{x1D6EC}(X_{0})$ of varieties $X_{0}$ having very good coordinates $X_{0}\rightarrow \mathbb{T}^{N}$ (see Definition 7.9). By Proposition 5.3, for every rigid variety with good coordinates $X_{0}\rightarrow \mathbb{T}^{N}$ there exists an index $h$ such that $X_{h}=X_{0}\,\times _{\mathbb{T}^{N}}\,\mathbb{T}^{N}\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p^{h}}\rangle$ has very good coordinates. Since $\operatorname{char}K=0$ the map $\mathbb{T}^{N}\langle \text{}\underline{\unicode[STIX]{x1D710}}^{\pm 1/p^{h}}\rangle \rightarrow \mathbb{T}^{N}$ is finite étale, and therefore also the map $X_{h}\rightarrow X_{0}$ is. We conclude that any rigid variety with good coordinates  has a finite étale covering with very good coordinates, and hence the motives associated to varieties with very good coordinates generate the étale topos. In particular, the motives associated to them generate $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ , and hence the functor $\mathbb{L}\unicode[STIX]{x1D704}_{!}\,\circ \,\mathbb{L}j^{\ast }$ maps a set of compact generators to a set of compact generators.

Since $\mathfrak{F}$ is monoidal and $\mathfrak{F}(\unicode[STIX]{x1D6EC}(1))=\unicode[STIX]{x1D6EC}(1)$ it extends formally to a monoidal functor from the category $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K^{\flat })$ to $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}(K)$ by putting $\mathfrak{F}(M(-d))=\mathfrak{F}(M)(-d)$ . Now let $M$ , $N$ in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K^{\flat })$ be twists of the motives associated to the analytification of smooth projective varieties $X$ , respectively, $X^{\prime }$ . They are strongly dualizable objects of $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K^{\flat })$ since $\unicode[STIX]{x1D6EC}_{\operatorname{tr}}(X)$ and $\unicode[STIX]{x1D6EC}_{\operatorname{tr}}(X^{\prime })$ are strongly dualizable in $\mathbf{DM}_{\acute{\text{e}}\text{t}}(K^{\flat })$ . Fix an integer $d$ such that $N^{\vee }(d)$ lies in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })$ . The objects $M$ , $N$ , $M^{\vee }$ and $N^{\vee }$ lie in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K^{\flat })$ and moreover $\mathfrak{F}(N^{\vee })=\mathfrak{F}(N)^{\vee }$ . From Lemma 7.12 we also deduce that the functor $\mathfrak{F}$ induces a bijection

$$\begin{eqnarray}\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K^{\flat })(M\,\otimes \,N^{\vee },\unicode[STIX]{x1D6EC})\cong \mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K)(\mathfrak{F}(M)\,\otimes \,\mathfrak{F}(N)^{\vee },\unicode[STIX]{x1D6EC}).\end{eqnarray}$$

By means of the Cancellation Theorem [Ayo15, Corollary 2.5.49], the first set is isomorphic to $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })(M,N)$ and the second is isomorphic to $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)(\mathfrak{F}(M),\mathfrak{F}(N))$ . We then deduce that all motives $M$ associated to the analytification of smooth projective varieties lie in the left orthogonal of the cone of the map $N\rightarrow \mathfrak{G}\mathfrak{F}N$ , which is closed under direct sums and cones. Since $\unicode[STIX]{x1D6EC}$ is a $\mathbb{Q}$ -algebra, such motives generate $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })$ by means of [Ayo15, Theorem 2.5.35]. We conclude that $N\cong \mathfrak{G}\mathfrak{F}N$ . Therefore, the category $\mathbf{T}$ of objects $N$ such that $N\cong \mathfrak{G}\mathfrak{F}N$ contains all motives associated to the analytification of smooth projective varieties. It is clear that $\mathbf{T}$ is closed under cones. The functors $\mathfrak{F}$ and $\mathbb{L}\unicode[STIX]{x1D704}^{\ast }$ commute with direct sums as they are left adjoint functors. As pointed out in Remark 7.8 also the functor $\mathbb{R}j_{\ast }$ does. Since $\mathfrak{G}$ is a composite of $\mathbb{R}j_{\ast }\mathbb{L}\unicode[STIX]{x1D704}^{\ast }$ with equivalences of categories, it commutes with small sums as well. We conclude that $\mathbf{T}$ is closed under direct sums. Using again [Ayo15, Theorem 2.5.35] we deduce $\mathbf{T}=\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })$ , proving that $\mathfrak{F}$ is fully faithful. This is enough to prove that it is an equivalence of categories, by applying [Ayo15, Lemma 1.3.32].◻

Lemma 7.12. Let $M$ be an object of $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K^{\flat })$ . The functor $\mathfrak{F}$ induces an isomorphism

$$\begin{eqnarray}\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K^{\flat })(M,\unicode[STIX]{x1D6EC})\cong \mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K)(\mathfrak{F}(M),\unicode[STIX]{x1D6EC}).\end{eqnarray}$$

Proof. Suppose that $d$ is an integer such that $M(d)$ lies in $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })$ . One has $\mathfrak{F}\unicode[STIX]{x1D6EC}(d)\cong \unicode[STIX]{x1D6EC}(d)$ and by Proposition 7.6 the unit map $\unicode[STIX]{x1D702}:\unicode[STIX]{x1D6EC}(d)\rightarrow \mathfrak{G}\mathfrak{F}\unicode[STIX]{x1D6EC}(d)$ is an isomorphism. In particular, from the adjunction $(\mathfrak{F},\mathfrak{G})$ we obtain a commutative square

in which the top arrow is then an isomorphism. By the Cancellation Theorem [Ayo15, Corollary 2.5.49] we also obtain the following commutative square

and hence also the top arrow is an isomorphism. We conclude the claim from the following commutative square, whose vertical arrows are isomorphisms since the functor $(\cdot )(d)$ is invertible in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{}(K)$ .

Remark 7.13. In the proof of Theorem 7.10 we again used the hypothesis that $\unicode[STIX]{x1D6EC}$ is a $\mathbb{Q}$ -algebra in order to apply [Ayo15, Theorem 2.5.35], which states that the motives associated to the analytification of smooth projective varieties generate $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat })$ .

Remark 7.14. In the proof, we also showed that the motives $\unicode[STIX]{x1D6EC}(X)[i]$ where $X$ has very good coordinates and $i\in \mathbb{Z}$ generate $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K)$ as a triangulated category with small sums.

Corollary 7.15. Let $X_{0}\rightarrow \mathbb{T}^{N}$ be a variety with very good coordinates. Then $\mathbb{R}j_{\ast }\mathbb{L}\unicode[STIX]{x1D704}^{\ast }\unicode[STIX]{x1D6EC}(X_{0})\cong \unicode[STIX]{x1D6EC}(\mathop{\varprojlim }\nolimits_{h}X_{h})$ .

Proof. By the description given in Theorem 5.5, we conclude that $\mathbb{R}\unicode[STIX]{x1D704}_{!}\mathbb{L}j^{\ast }(\unicode[STIX]{x1D6EC}(\mathop{\varprojlim }\nolimits_{h}X_{h}))\cong \unicode[STIX]{x1D6EC}(X_{0})$ . The claim then follows from Theorem 7.10, which shows that $\mathbb{R}\unicode[STIX]{x1D704}_{!}\,\circ \,\mathbb{L}j^{\ast }$ is a quasi-inverse of $\mathbb{R}j_{\ast }\,\circ \,\mathbb{L}\unicode[STIX]{x1D704}^{\ast }$ .◻

We remark that the proof of Theorem 7.10 also induces the following result.

Corollary 7.16. The functor

$$\begin{eqnarray}\displaystyle \mathfrak{F}:\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K^{\flat }) & \rightarrow & \displaystyle \mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K)\nonumber\\ \displaystyle M(d) & \mapsto & \displaystyle (\mathfrak{F}M)(d)\nonumber\end{eqnarray}$$

is a monoidal equivalence of categories.

We can refine the previous corollary by stating the stable version of our main result (Theorem 7.26), which is based on the following intermediate results. For the definitions of spectra of model categories, we refer to [Ayo07, § 4.3] and [Hov01].

Definition 7.17. Let $T$ be the cokernel of the unit map $\unicode[STIX]{x1D6EC}\rightarrow \unicode[STIX]{x1D6EC}(\mathbb{T}^{1})$ in $\mathbf{Psh}(\operatorname{RigSm}/K)$ and let $\widehat{T}$ be the cokernel of the unit map $\unicode[STIX]{x1D6EC}\rightarrow \unicode[STIX]{x1D6EC}(\widehat{\mathbb{T}}^{1})$ in $\mathbf{Psh}(\operatorname{PerfSm}/K)$ . They are direct factors of cofibrant objects, and hence cofibrant.

1. (i) We denote by $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})$ the homotopy category of the stable $(\acute{\text{e}}\text{t},\mathbb{B}^{1})$ -local model structure on spectra $\mathbf{Spt}_{T}(\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\operatorname{RigSm}/K))$ . We omit $\unicode[STIX]{x1D6EC}$ whenever the context allows it.

2. (ii) We denote by $\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}^{1}}}(K,\unicode[STIX]{x1D6EC})$ the homotopy category of the stable $(\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1})$ -local model structure on spectra $\mathbf{Spt}_{\unicode[STIX]{x1D704}^{\ast }T}(\mathbf{Ch}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}/K))$ . We omit $\unicode[STIX]{x1D6EC}$ whenever the context allows it.

3. (iii) We denote by $\mathbf{PerfDA}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})$ the homotopy category of the stable $(\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1})$ -local model structure on spectra $\mathbf{Spt}_{\widehat{T}}(\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\operatorname{PerfSm}/K))$ . We omit $\unicode[STIX]{x1D6EC}$ whenever the context allows it.

4. (iv) For $\mathbf{C}$ equal to $\operatorname{RigSm}$ , $\widehat{\text{Rig}}\operatorname{Sm}$ or $\operatorname{PerfSm}$ , we denote by $\operatorname{Sus}_{k}$ the $k$ th suspension functor from the model category of complexes of presheaves on $\mathbf{C}$ to the associated spectra, which is the left adjoint of the functor

   $$\begin{eqnarray}\operatorname{Ev}_{k}:(M_{n})_{n\in \mathbb{N}}\mapsto M_{k}.\end{eqnarray}$$

Remark 7.18. The functor $-\,\otimes \,T$ [respectively $-\,\otimes \,\unicode[STIX]{x1D704}^{\ast }T$ respectively $-\,\otimes \,\widehat{T}$ ] has a prolongation to a left Quillen endofunctor on the associated spectra, which is furthermore a Quillen equivalence. The category of spectra is the universal model category with this property, in a weak sense made precise by [Hov01, Theorem 5.1 and Proposition 5.3].

Remark 7.19. In contrast with Definition 7.2, we use above the categories of non-symmetric spectra. On the other hand, we remark that by [Ayo07, Proposition 4.3.47] the model categories of symmetric and non-symmetric spectra are Quillen equivalent. We then conclude by Proposition 3.23, Theorem 6.3, Proposition 6.9 and [Hov01, Theorem 5.5] that the canonical adjunctions

$$\begin{eqnarray}\displaystyle & \displaystyle \mathbf{RigDA}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})\leftrightarrows \mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K^{\flat },\unicode[STIX]{x1D6EC})\leftrightarrows \mathbf{PerfDA}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC}) & \displaystyle \nonumber\end{eqnarray}$$

are equivalences of categories, and the prolongation of $-\,\otimes \,T$ [respectively $-\,\otimes \,\widehat{T}$ ] corresponds to the functor $M\mapsto M(1)$ .

Remark 7.20. We point out that the functor $\operatorname{Sus}_{0}$ has an explicit description. For example, in the case of $\mathbf{Ch}_{\acute{\text{e}}\text{t},\mathbb{B}^{1}}\mathbf{Psh}(\operatorname{RigSm}/K)$ it sends a complex ${\mathcal{F}}$ to the spectrum $({\mathcal{F}}\,\otimes \,T^{\otimes n})_{n\in \mathbb{N}}$ with the obvious transition maps. The functor $\operatorname{Sus}_{k}$ is the composition $t^{k}\operatorname{Sus}_{0}$ , where $t$ is the shift functor such that $\operatorname{Ev}_{0}tM=0$ and $\operatorname{Ev}_{k+1}tM=\operatorname{Ev}_{k}M$ for any spectrum $M$ . The pair $(\operatorname{Sus}_{k},\operatorname{Ev}_{k})$ is a Quillen adjunction (see [Ayo07, Lemma 4.3.24]) as well as the pair $(t,s)$ , where $s$ is such that $\operatorname{Ev}_{k}sM=\operatorname{Ev}_{k+1}M$ for any spectrum $M$ (see [Hov01, Definition 3.7]).

Definition 7.21. We denote by

the Quillen adjunction induced by the pair $(\unicode[STIX]{x1D704}^{\ast },\unicode[STIX]{x1D704}_{\ast })$ by means of [Hov01, Proposition 5.3].

The natural map $\widehat{\mathbb{T}}^{1}\rightarrow \unicode[STIX]{x1D704}^{\ast }\mathbb{T}^{1}$ induces a map $\widehat{T}\rightarrow j_{\ast }\unicode[STIX]{x1D704}^{\ast }T$ , which, in turn, defines a natural transformation $\unicode[STIX]{x1D70F}:j^{\ast }(-\,\otimes \,\widehat{T})\rightarrow (j^{\ast }-)\,\otimes \,\unicode[STIX]{x1D704}^{\ast }T$ . Nevertheless, it is not clear that for a smooth affinoid perfectoid $\widehat{X}$ the map $\unicode[STIX]{x1D70F}:j^{\ast }(\unicode[STIX]{x1D6EC}(\widehat{X})\,\otimes \,\widehat{T})\rightarrow (j^{\ast }\unicode[STIX]{x1D6EC}(\widehat{X}))\,\otimes \,\unicode[STIX]{x1D704}^{\ast }T$ is a weak equivalence, so that we can not apply the criterion of [Hov01, Proposition 5.3] to define a Quillen adjunction between $\mathbf{Spt}_{\widehat{T}}(\mathbf{Ch}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}\mathbf{Psh}(\operatorname{PerfSm}/K))$ and $\mathbf{Spt}_{\unicode[STIX]{x1D704}^{\ast }T}(\mathbf{Ch}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}/K))$ . Therefore, the following proposition, albeit easy to prove, has a non-trivial content.

Proposition 7.22. The functor

$$\begin{eqnarray}\displaystyle \operatorname{Spt}j_{\ast }:\mathbf{Spt}_{\unicode[STIX]{x1D704}^{\ast }T}(\mathbf{Ch}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm}/K)) & \rightarrow & \displaystyle \mathbf{Spt}_{\widehat{T}}(\mathbf{Ch}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}\mathbf{Psh}(\operatorname{PerfSm}/K))\nonumber\\ \displaystyle (M_{n})_{n\in \mathbb{N}} & \mapsto & \displaystyle (j_{\ast }M_{n})_{n\in \mathbb{N}}\nonumber\end{eqnarray}$$

is well defined, preserves stable weak equivalences and induces a triangulated functor

$$\begin{eqnarray}\mathbb{R}\operatorname{Spt}j_{\ast }:\widehat{\mathbf{Rig}}\mathbf{DA}_{\acute{\text{e}}\text{t},\widehat{\mathbb{B}}^{1}}(K,\unicode[STIX]{x1D6EC})\,\rightarrow \,\mathbf{PerfDA}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC}).\end{eqnarray}$$

Proof. If $(M_{n})$ is a spectrum we can define the transition maps $j_{\ast }M_{n}\,\otimes \,\widehat{T}\rightarrow j_{\ast }M_{n+1}$ with the following composition

$$\begin{eqnarray}j_{\ast }M_{n}\,\otimes \,\widehat{T}\xrightarrow[{}]{\unicode[STIX]{x1D70F}}j_{\ast }M_{n}\,\otimes \,j_{\ast }\unicode[STIX]{x1D704}^{\ast }T\cong j_{\ast }(M_{n}\,\otimes \,\unicode[STIX]{x1D704}^{\ast }T)\rightarrow j_{\ast }M_{n+1}\end{eqnarray}$$

deduced by the transition maps $M_{n}\,\otimes \,\unicode[STIX]{x1D704}^{\ast }T\rightarrow M_{n+1}$ . Moreover, as shown in the first part of the proof of [Hov01, Proposition 5.3] $\operatorname{Spt}j_{\ast }$ is a right Quillen functor with respect to the projective model structures on spectra, whose weak equivalences are level-wise weak equivalences. It also preserves such weak equivalences, as proved in Remark 7.8.

The stable model structure on spectra is obtained as a left Bousfield localization with respect to the maps $\operatorname{Sus}_{n+1}({\mathcal{F}}\,\otimes \,\unicode[STIX]{x1D704}^{\ast }T)\rightarrow \operatorname{Sus}_{n}{\mathcal{F}}$ [respectively $\operatorname{Sus}_{n+1}({\mathcal{F}}\,\otimes \,\widehat{T})\rightarrow \operatorname{Sus}_{n}{\mathcal{F}}$ ] as ${\mathcal{F}}$ runs among cofibrant objects (see for example [Ayo07, Definition 4.3.29]). We now prove that $\operatorname{Spt}j_{\ast }$ also preserves stable weak equivalences.

Consider the natural map

$$\begin{eqnarray}\operatorname{Ev}_{k}(\operatorname{Spt}j_{\ast }\operatorname{Sus}_{n+1}({\mathcal{F}}\,\otimes \,\unicode[STIX]{x1D704}^{\ast }T))\rightarrow \operatorname{Ev}_{k}(\operatorname{Spt}j_{\ast }\operatorname{Sus}_{n}{\mathcal{F}}).\end{eqnarray}$$

For $k>n$ it is an equality, and hence the map

$$\begin{eqnarray}\operatorname{Spt}j_{\ast }\operatorname{Sus}_{n+1}({\mathcal{F}}\,\otimes \,\unicode[STIX]{x1D704}^{\ast }T)\rightarrow \operatorname{Spt}j_{\ast }\operatorname{Sus}_{n}{\mathcal{F}}\end{eqnarray}$$

is a stable weak equivalence by [Ayo07, Lemma 4.3.59]. We then deduce that $\operatorname{Spt}j_{\ast }$ preserves stable weak equivalences as claimed.

The fact that $\mathbb{R}\operatorname{Spt}j_{\ast }$ is triangulated follows from the fact that it coincides with the restriction to stably local spectra of the functor induced by $\operatorname{Spt}j_{\ast }$ on the homotopy categories of the projective model structures on spectra, which is triangulated being a right derived Quillen functor.◻

Remark 7.23. The functor $\mathbb{R}\operatorname{Spt}j_{\ast }$ commutes with small direct sums by its explicit description and Remark 7.8.

Proposition 7.24. There is an invertible natural transformation of functors from the category $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ to $\mathbf{PerfDA}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})$ :

$$\begin{eqnarray}\unicode[STIX]{x1D702}:\mathbb{L}\operatorname{Sus}_{0}\,\circ \,\mathbb{R}j_{\ast }\,\circ \,\mathbb{L}\unicode[STIX]{x1D704}^{\ast }\xrightarrow[{}]{{\sim}}\mathbb{R}\operatorname{Spt}j_{\ast }\,\circ \,\mathbb{L}\operatorname{Spt}\unicode[STIX]{x1D704}^{\ast }\,\circ \,\mathbb{L}\operatorname{Sus}_{0}\!.\end{eqnarray}$$

Proof. For any cofibrant object ${\mathcal{F}}$ in $\mathbf{Ch}\,\mathbf{Psh}(\operatorname{RigSm})$ one has

$$\begin{eqnarray}\displaystyle & \displaystyle \operatorname{Ev}_{k}(\mathbb{L}\operatorname{Sus}_{0}\,\circ \,\mathbb{R}j_{\ast }\,\circ \,\mathbb{L}\unicode[STIX]{x1D704}^{\ast })({\mathcal{F}})\cong Q(j_{\ast }\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})\,\otimes \,\widehat{T}^{\otimes k}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \operatorname{Ev}_{k}(\mathbb{R}\operatorname{Spt}j_{\ast }\,\circ \,\mathbb{L}\operatorname{Spt}\unicode[STIX]{x1D704}^{\ast }\,\circ \,\mathbb{L}\operatorname{Sus}_{0})({\mathcal{F}})\cong j_{\ast }\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}}\,\otimes \,j_{\ast }(\unicode[STIX]{x1D704}^{\ast }T)^{\otimes k}, & \displaystyle \nonumber\end{eqnarray}$$

where $Q$ is a cofibrant-replacement functor on $\mathbf{Ch}\,\mathbf{Psh}(\widehat{\text{Rig}}\operatorname{Sm})$ . The natural transformation $\unicode[STIX]{x1D702}$ of the statement is then induced by the canonical maps $Q(j_{\ast }\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}})\rightarrow j_{\ast }\unicode[STIX]{x1D704}^{\ast }{\mathcal{F}}$ and $\widehat{T}\rightarrow j_{\ast }\unicode[STIX]{x1D704}^{\ast }T$ .

The two functors of the statement are triangulated and commute with small direct sums by Remarks 7.8, 7.23 and Proposition 7.22. We deduce that the subcategory of $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ on which $\unicode[STIX]{x1D702}$ is invertible is triangulated and closed under direct sums.

By Proposition 3.18 and Remark 7.14 we conclude that it suffices to prove that $\unicode[STIX]{x1D702}$ is invertible on a fixed motive $\unicode[STIX]{x1D6EC}(X)[i]$ with $X$ with very good coordinates and $i\in \mathbb{Z}$ . For such an object, using Corollary 7.15, we have an explicit description of a cofibrant replacement of $j_{\ast }\unicode[STIX]{x1D704}^{\ast }\unicode[STIX]{x1D6EC}(X)[i]$ namely $\unicode[STIX]{x1D6EC}(\widehat{X})[i]$ , where $\widehat{X}=\mathop{\varprojlim }\nolimits_{h}X_{h}$ is built with respect to some very good coordinates $X\rightarrow \mathbb{T}^{N}$ of $X$ . We are left to prove that the maps $\unicode[STIX]{x1D6EC}(\widehat{X})\,\otimes \,\widehat{T}^{\otimes k}[i]\rightarrow j_{\ast }\unicode[STIX]{x1D704}^{\ast }\unicode[STIX]{x1D6EC}(X)\,\otimes \,j_{\ast }(\unicode[STIX]{x1D704}^{\ast }T)^{\otimes k}[i]$ induce a weak equivalence of spectra. We claim that each one of them is a weak equivalence.

The object $T$ [respectively $\widehat{T}$ ] is a direct summand of $\unicode[STIX]{x1D6EC}(\mathbb{T}^{1})$ [respectively $\unicode[STIX]{x1D6EC}(\widehat{\mathbb{T}}^{1})$ ] and the decomposition is compatible with the map $\widehat{T}\rightarrow j_{\ast }\unicode[STIX]{x1D704}^{\ast }T$ . Therefore, in order to prove the claim we can show that the maps

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(\widehat{X}\,\times \,\widehat{\mathbb{T}}^{k})\cong \unicode[STIX]{x1D6EC}(\widehat{X})\,\otimes \,\unicode[STIX]{x1D6EC}(\widehat{\mathbb{T}}^{k})\rightarrow j_{\ast }\unicode[STIX]{x1D704}^{\ast }\unicode[STIX]{x1D6EC}(X)\,\otimes \,j_{\ast }\unicode[STIX]{x1D6EC}(\mathbb{T}^{k})\cong j_{\ast }\unicode[STIX]{x1D704}^{\ast }\unicode[STIX]{x1D6EC}(X\,\times \,\mathbb{T}^{k})\end{eqnarray}$$

are weak equivalences, and this follows from Corollary 7.15 and Remark 7.7. ◻

We define the following composite functor (see Remark 7.19):

$$\begin{eqnarray}\mathfrak{G}^{\operatorname{st}}:\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K)\cong \mathbf{RigDA}_{\acute{\text{e}}\text{t}}(K)\xrightarrow[{}]{\mathbb{R}\operatorname{Spt}j_{\ast }\,\circ \,\mathbb{L}\operatorname{Spt}\unicode[STIX]{x1D704}^{\ast }}\mathbf{PerfDA}_{\acute{\text{e}}\text{t}}(K)\cong \mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K^{\flat }),\end{eqnarray}$$

which can be inserted in the diagram

whose vertical functors are fully faithful by the Cancellation Theorem [Ayo15, Corollary 2.5.49]. It is commutative, up to a natural transformation, by means of Proposition 7.24.

Proposition 7.25. The category $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})$ [respectively $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})$ ] is generated, as a triangulated category with small sums, by the objects of $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K,\unicode[STIX]{x1D6EC})$ [respectively $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K^{\flat },\unicode[STIX]{x1D6EC})$ ].

Proof. We prove the statement only for $K$ . From the Cancellation Theorem [Ayo15, Corollary 2.5.49] we deduce that any object $M$ of $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})$ is isomorphic to $\operatorname{hocolim}_{n}M_{n}(-n)$ with $M_{n}$ in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ (see the proof of [Kah13, Proposition 7.4.1]). In particular, it sits in a distinguished triangle

Since $\mathbf{RigDM}_{\operatorname{eff}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})$ is generated by compact objects by Proposition 3.18, we conclude that each $M_{n}(-n)$ lies in the triangulated category with small sums generated by the objects of $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K,\unicode[STIX]{x1D6EC})$ so that $M$ also does.◻

We can finally prove the stable version of the main result of this article.

Theorem 7.26. The functor

$$\begin{eqnarray}\mathfrak{G}^{\operatorname{st}}:\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})\rightarrow \mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K^{\flat },\unicode[STIX]{x1D6EC})\end{eqnarray}$$

is a monoidal triangulated equivalence of categories and, up to a natural transformation, it restricts to the equivalence

$$\begin{eqnarray}\mathfrak{G}:\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\unicode[STIX]{x1D6EC})\rightarrow \mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K^{\flat },\unicode[STIX]{x1D6EC})\end{eqnarray}$$

of Theorem 7.10 on the subcategories of effective motives.

Proof. By its definition, the functor $\mathbb{R}\operatorname{Spt}j_{\ast }$ commutes with the derived shift functor $\mathbb{L}t$ (see Remark 7.20). We claim that $\mathbb{L}\operatorname{Spt}\unicode[STIX]{x1D704}^{\ast }$ also does. We can equivalently show that $\mathbb{R}\operatorname{Spt}\unicode[STIX]{x1D704}_{\ast }$ commutes with $\mathbb{R}s$ and this again follows from their definitions (see Remark 7.20 and [Hov01, Theorem 5.3]).

By [Hov01, Theorem 3.8] the functor $\mathbb{L}t$ on $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})$ [respectively on $\mathbf{PerfDA}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})$ ] is a quasi-inverse of the prolongation of $\mathbb{L}(-\,\otimes \,T)$ [respectively of $\mathbb{L}(-\,\otimes \,\widehat{T})$ ]. Therefore, we conclude that for any $M$ in $\mathbf{RigDA}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})$ :

$$\begin{eqnarray}(\mathbb{R}\operatorname{Spt}j_{\ast }\,\circ \,\mathbb{L}\operatorname{Spt}\unicode[STIX]{x1D704}^{\ast })(\mathbb{L}(-\,\otimes \,T))^{-1}M\cong (\mathbb{L}(-\,\otimes \,\widehat{T}))^{-1}(\mathbb{R}\operatorname{Spt}j_{\ast }\,\circ \,\mathbb{L}\operatorname{Spt}\unicode[STIX]{x1D704}^{\ast })M,\end{eqnarray}$$

which implies for any $M$ in $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})$ the following canonical isomorphism:

$$\begin{eqnarray}\mathfrak{G}^{\operatorname{st}}(M(-1))\cong (\mathfrak{G}^{\operatorname{st}}M)(-1).\end{eqnarray}$$

Since we already showed in Proposition 7.24 that $\mathfrak{G}^{\operatorname{st}}$ restricts to $\mathfrak{G}$ on effective motives, we conclude that it restricts on $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K,\unicode[STIX]{x1D6EC})$ to a quasi-inverse of the functor $\mathfrak{F}$ of Corollary 7.16. In particular, this restriction is fully faithful and its essential image contains $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}^{\operatorname{ct}}(K^{\flat },\unicode[STIX]{x1D6EC})$ . By Remark 7.23 we also deduce that $\mathfrak{G}^{\operatorname{st}}$ commutes with small direct sums. The claim of the theorem then follows from Proposition 7.25 and [Ayo15, Lemma 1.3.32].◻

Remark 7.27. It is worth noticing that along the proof of our main theorem, we have not used any result on almost algebra (which nonetheless has a critical role in the theory of perfectoid spaces).

Remark 7.28. The reader may wonder if the equivalence of categories $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})\cong \mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K^{\flat },\unicode[STIX]{x1D6EC})$ still holds true for more general rings of coefficients $\unicode[STIX]{x1D6EC}$ . The hypothesis $\mathbb{Q}\subset \unicode[STIX]{x1D6EC}$ has been used several times along the proof of the main statement, and most crucially in the following two instances. First, it was used to invoke the results of [Vez17] on the equivalence of motives with and without transfers (see Remarks 5.2 and 6.4). Secondly, the hypothesis $\mathbb{Q}\subset \unicode[STIX]{x1D6EC}$ was used in order to apply the result of Ayoub [Ayo15, Theorem 2.5.35] about a generating set for the triangulated category $\mathbf{RigDM}(K^{\flat },\unicode[STIX]{x1D6EC})$ (see Remark 7.13).

On the other hand, we conjecture that the tilting equivalence $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K,\unicode[STIX]{x1D6EC})\cong \mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K^{\flat },\unicode[STIX]{x1D6EC})$ can be generalized to a ring of coefficients $\unicode[STIX]{x1D6EC}$ over $\mathbb{Z}[1/p]$ . As a matter of fact, for any prime $\ell \neq p$ we expect the category $\mathbf{RigDM}_{\acute{\text{e}}\text{t}}(K,\mathbb{Z}/\ell )$ to be equivalent to the derived category of Galois representations over $\mathbb{Z}/\ell$ , similarly to the case of $\mathbf{DM}_{\acute{\text{e}}\text{t}}^{\operatorname{eff}}(K,\mathbb{Z}/\ell )$ (see [MVW06, Theorem 9.35]). The tilting equivalence in the case $\unicode[STIX]{x1D6EC}=\mathbb{Z}/\ell$ would then follow from the classic theorem of Fontaine and Wintenberger.