Professor Saida Filali

Nassima Faraj

Marketing of SMEs in the digital age: Challenges and issues

Keywords: Digital marketing, SME, challenges, issues, Morocco. JEL Classification : M31 Paper type : Empirical research

Digital marketing is revolutionizing organizational structures including businesses, especially Moroccan SMEs, offering a set of opportunities to improve the traditional way of operating while building a more authentic relationship with customers. Indeed, digital marketing brings forward important characteristics for SMEs in search of efficiency and performance, and it is considered as an interesting alternative to traditional approaches. To this end, the interest of our study is to show the main issues and challenges that the digitalization of the marketing function can have on Moroccan SMEs. In this context, the problematic we will try to answer is an original one aiming to explore the impact of the introduction of digital within the marketing function by taking the case of SMEs. To this end, we opted for a sample of 4 consultants to respond to the exploratory phase and 41 respondents for the confirmatory phase. The results have shown that digital marketing is a double-edged sword in that it can have different impacts on SME structures. With respect to the contributions of the study, we first point out the results will be beneficial for SMEs which constitute a large part of the Moroccan economic fabric, and a concise methodology based on the triangulation between the qualitative and quantitative approach. As for the limitations, we mention that the number of respondents is somewhat limited.

Introduction

The digital transition has now become a categorical imperative for any company in order to succeed and achieve better performance in the market (Dudézert, 2018). Indeed, the digitalization of companies is on the way, this is no longer a choice as it disrupts the methods of operation of organizations. Indeed, if the advent of the Internet and its integration into everyday life changes the way people live, its role in the management of small and medium-sized enterprises has become essential. Indeed, SMEs cannot aspire to an efficient management without an adequate and strategic use of ICT (Vacher, 2002). More specifically, the central and growing position of social networking tools evokes its importance, which is reflected in 1.32 billion members on Facebook, 271 million on Twitter and an average of more than 6 hours of use of these networks per day in 2014 (Nakara et al., 2020). These transformations have, at the same time, affected organizations in their managerial approach as well as in the setting of objectives to be achieved with efficiency (Dutot and [START_REF] Castellano | une analyse de l'e-reputation par analogie ou constraste avec la reputation : une approche par les medias sociaux[END_REF]. In this same context, companies have been able to seize the opportunity offered by digitalization, especially via social networks, as a way to expand its network, to perform competitive intelligence and lead to the construction of an even more privileged relationship with consumers (Adam and [START_REF] Deans | Inter-Study Comparisons of Small Business Internet Use in Australia and New Zealand[END_REF]. To this end, the figures mentioned above shows the interest of research to explore the place of digital tools in the proper functioning of companies. As for the empirical study of the subject, the influence of the Internet on the management of companies is a fertile ground for a significant number of research works [START_REF] Miller | the ultimate web marketing guide[END_REF] related to the study of the importance of social networks (Safko, 2010). However, integrating social networks into the implementation of marketing strategies can be a bit risky. Indeed, a distinction must be made regarding the reliability given to social networks as marketing tools. On the other hand, there are companies that integrate and facilitate the integration of social networks in the design of their marketing strategy. For them, social networks are assimilated to lowcost tools allocating communication as well as the diffusion of their identity in a competitive market [START_REF] Dutta | what's your personal social media strategy?[END_REF]. On the other hand, there are companies that are more risk averse and resist the introduction of social networks in the implementation of their strategy. That said, it is interesting to learn about the role of small and medium-sized enterprises. These constitute about 95% of the national productive fabric, contributing to 40% of private investment and 30% of exports. Indeed, and taking into consideration the limit of financial resources, it is wise to question the importance of social networks as marketing tools relatively inexpensive compared to traditional tools [START_REF] Dutta | what's your personal social media strategy?[END_REF]. In this context, the crucial objective of this research is to explore the use of social networks by Moroccan SMEs in the design and implementation of their marketing strategy. In addition, to determine the issues and/or challenges they present. In this context, the key issue we will try to address is the following: To what extent can digital marketing be a source of issues and challenges within Moroccan SMEs?

In order to answer this question, we will adopt a precise methodology based on triangulation, starting with a qualitative study based on interviews with consultants and experts in digital marketing to explore the main issues and challenges of digital www.ijafame.org marketing. Then, a confirmatory study (quantitative) using a questionnaire for 41 respondents in order to confirm the results of the first study.

In order to answer to our essential problem, we will start by citing the main theories to be mobilized, then the research methodology, then the presentation of the result and ending with a conclusion at the end.

Literature review:

It is generally accepted that the basic principles of marketing are universally applicable to both large and small businesses. The study of marketing in SMEs has been recognized as a problem area for researchers for over 20 years (O'Leary, 2011). More recently, the use of Internet marketing or e-commerce has become popular in all types of businesses, including SMEs. However, insufficient knowledge about marketing in small businesses remains and a theory of small business marketing specifically related to understanding and knowledge of strategic marketing is needed. Research on small businesses and their marketing activities has largely been limited to explanations of certain types of behaviors observed in small businesses (Hannon and Atherton, 1998), or to finding factors of barriers missing or present in small businesses, which explains their apparent inability to apply or use marketing ideas and concepts that were often developed for large businesses [START_REF] Freel | barriers to Product Innovation in Small Manufacturing Firms[END_REF]. Indeed, theoretical development in SME research seems to be somewhat limited in general. The work that has been done is more applied in nature, taking the form of prescriptive or descriptive frameworks and "models" on how to apply certain business and management theories to small firms (e.g., Taiminen and Karjaluoto, 2015).

Characteristics of the marketing function in SMEs:

Art The recurrence of the marketing function in the management of companies has aroused the interest of several studies (e.g. [START_REF] Pacitto | les moyennes entreprises pratiquent-elles le marketing ? Une exploration francoquébécoise[END_REF]. The latter is perceived as a secondary function whose presence is not necessary for the current operation of the activity. In order to prove the position of the marketing function within SMEs, two approaches can be distinguished. The first approach considers the theory of marketing in a classical vision and shows the way it is practiced in SMEs. In this sense, Romano and Ratnatunga (1995) have identified three ways of viewing the marketing function within SMEs. First, the marketing function as a strategy that focuses on segmentation, positioning and targeting techniques. Second, the marketing function as a culture that offers the customer a fundamental position within the firm. Finally, marketing as a tactical strategy within the four policies of the marketing mix (Sestino et al. 2020). However, it is important to note that SMEs that do not use marketing tools are more vulnerable in a competitive market [START_REF] Leeflang | challenges and solutions for marketing in a digital era[END_REF]. In this first approach, two categories of SMEs can be identified. First, those that do not practice marketing techniques due to insufficient resources and lack of a clear vision of a return on investment (ROI). Second, those that use marketing in a casual and ad hoc manner (Nakara et al., 2014). In the second approach, the marketing function seems to be a necessity. Yet, its practice within SMEs can be differentiated according to several criteria, including the size of the firm. In this wake, there are other limitations hindering its adoption namely limited resources in terms of funding, human capital or time. In addition, a lack of expertise of the manager in marketing, this can only lead to a botching of the marketing function (Yu et al. 2019). These limitations make the use of marketing techniques in SMEs unsystematic, which largely depends on the manager's decision to take the risk of investing in it despite the lack of means and skills.

The role of social networks in SMEs

By definition, social networks can be equated to media supports and platforms that allow users to maintain a presence and interact online [START_REF] Proulx | media sociaux enjeux[END_REF]). Thus, consumers use these tools to share their experiences, their opinions with other users who are part of the virtual community. To this end, the interest given by firms to social networks as fundamental tools of marketing is important under the pretext that they constitute a lever for the management of projects experienced with the consumer. The latter, thanks to the possibility of online interaction, becomes a co-producer of consumption (O'Leary, 2011). In this context, digital marketing is to be considered as an opportunity since their average cost is much lower than traditional strategies. In addition, they are also a continuation of the networking practice already initiated by SME managers (Ritter and Pedersen, 2020).

Research hypotheses

Digitalization has become an essential part of business life. It assists in upgrading traditional patterns of conducting business activities to interact with stakeholders. The emerging trend of utilization of digital media in business activities has transformed the basis for competition and influence consumer behavior (Sestino et al. 2020). Digitalization has an impact on firms, and influences products and brands through the usage of online services that includes storing, browsing and playing entertainment, email, Facebook and other applications that transform how we interact (Kiili, Leu, Marttunen, Hautala, & Leppänen, 2019). Based on existing theory, we can make the following two hypotheses: H0: The digitalization of the marketing function presents opportunities and threats and has a significant impact on the performance of Moroccan SMEs. H1: The digitalization of the marketing function does not present opportunities and threats and has no significant impact on the performance of Moroccan SMEs.

Research method or methodology

The health crisis of the Corona virus has strongly impacted the activity of SMEs in Morocco given their vulnerability to environmental and systemic risks. In order to remedy the adverse effects of this crisis, firms have invested heavily in their marketing strategy using social networks. The choice of treatment of SMEs is not arbitrary under the pretext that they represent an engine of the Moroccan economy. In this sense, they employ more than 50% of the workforce, distribute 43% of salaries and represent 44% of the turnover. In order to explore the issues and challenges of the marketing strategy of SMEs in the digital era, we have opted for an approach based on the triangulation between the qualitative and quantitative approach. The main advantage of triangulation designs www.ijafame.org in research studies lies in the ability to find agreement and validation of results through a variety of research methods. In addition, filling the gaps of one method by another and approaching the problem through different angles, thus questioning different units. To this end, the methodological design adopted is as follows: First of all, we started our approach with an exploratory study based on semidirective interviews with 4 digital marketing consultants who have worked with SMEs. The interviews were conducted on the basis of an interview guide including 3 essential themes: personal information, knowledge on the digitalization of the marketing function and finally the challenges and issues of the digitalization of the marketing function in relation to Moroccan SMEs. The objective is to discover the main issues and challenges of using digital marketing as part of a global business strategy. The table below reports information about our sample: In order to analyze the data collected through interviews, we opted for a horizontal and vertical analysis.

To complete the first study, a confirmatory study was established by submitting questionnaires to the managers and business leaders of SMEs (41 people) in order to collect data confirming the issues and challenges from the first approach and to propose others if necessary. The questionnaire of this study revolved around 4 fundamental axes: information on the firm and on the marketing function, on the respondents, the challenges and issues of the use of the digital strategy, the obstacles of its adoption.

Results and discussion

To our knowledge, the study we conducted is the first to explore the role of digital marketing in Moroccan SMEs. In this sense, the questionnaire administered to 41 executives and directors of the sales and marketing function, allowed us to perform the following analyses:

Descriptive statistics

The study we conducted involved a sample of 41 respondents. Among them, 22 are general managers of SMEs and 19 are responsible for the sales or marketing function. In addition, these firms are mostly newly created because more than 50% of these companies are created between 2010 and 2019. This can reinforce the adoption of digital marketing within these companies. Among the SMEs that responded to our survey, 34 have a marketing function within the firm and 30 (73%) use digital marketing in their activities (Figure 1).

Figure 1: Use of digital marketing by SMEs

Source: Authors

To this end, the Internet plays a key role in the proper functioning of SMEs. Thus, its use is no longer considered by managers as a choice but a necessity for the survival of the company and the realization of profits.

As for expenses, the results of the study showed that Moroccan SMEs spend an average of 3% of turnover to strengthen the presence of the company on social networks. This remarkable progression was explained by the consultants as the positive relationship between spending on digital marketing and the profit made. This great interest in the transition of the marketing function remains limited because the majority of respondents limit the use to a few tools, namely Facebook at a percentage of 41.5%, e-commerce site at 22.0% and Instagram at 12.2% (Figure 2).

Quantitative study results

Before addressing the elements blocking Moroccan SMEs from integrating digital marketing, it would be wise to present the results related to the challenges (opportunities and risks) perceived by the respondents. For the companies interviewed, digitalization offers several opportunities (Figure 3 presents these main opportunities) In this context, 22 out of 41 respondents (or 52%) cited the main opportunity offered by digital marketing as an increase in turnover and consequently profit for SMEs. This finding is consistent with the comments of marketing consultants who have a direct relationship with SMEs. To this effect, they mention that SMEs have duly benefited from the networks in order to improve earnings, something that is metamorphosed in the increase of the budget allocated to digital marketing from one year to another. In addition, 14 respondents stated that the digitalization of marketing allows to increase the company's notoriety insofar as it allocates them the strengthening of the continuous presence with the community as well as making the products and services offered closer to the consumers. The rest of the answers are minimally divided on the remaining opportunity, which is the improvement of the communication with the customers.

In parallel with these opportunities, Moroccan SME managers state that this digitalization also presents threats (Figure 4). First, the loss of financial resources, especially for SMEs that do not opt for digitization or for those who have invested but have not succeeded. Secondly, the information security risk is high (14 out of 41 respondents), especially for companies with customer databases. This risk exposing customer information and consequently not respecting the law on the protection of personal data. Finally, the digitization of the marketing function can be a threat to the competition because companies expose information about their activity on platforms, which gives easy access to competitors wanting to copy them and carry out the same activity (according to 30% of respondents). Digital transformation is a complex issue for companies, and even more so for SMEs, as it requires financial and human resources as well as internal skills and time that SMEs often do not have enough. This can severely limit their integration in their digitalization process. In this vein, our study of SMEs has revealed some challenges in the Moroccan context that restrict these companies from opting for a digital transition of the marketing function (Figure5). First of all, the financial and cultural challenges (68% of respondents). Indeed, an investment in digital marketing can not take place without a preparation of mindset. This means that if the employees' change management is not well prepared, nurtured, anticipated and matured, the risk of rejection is high. In addition to the financial and cultural challenges, the technological challenge is also present (22% of respondents). In this regard, respondents mention that the lack of serious preparation of the tools to be used to disrupt the traditional system may constitute a constraint to this revolution. In this context, the majority of Moroccan SMEs give less importance to investments in technology. Finally, the presence of the organizational challenge at a minimum rate of 9% which means that there is an overlap between the functions within this category of firms and sometimes the commercial and marketing function is neglected.

Conclusions

This research questions the issues and challenges of digital transformation of the marketing function designed by the managers of Moroccan SMEs. Theoretically, the subject of my digitization of the marketing function presents a new subject for which theoretical works are still rare.

To this end, we have opted for an approach based on the triangulation between the qualitative and quantitative approach. The first qualitative study via interviews with 4 marketing consultants in order to explore the Moroccan context. Secondly, the empirical study focused on a sample of 41 Moroccan SMEs shows the importance of digital marketing within these companies.

As mentioned in the literature review and found in the results of our research, the integration of social networks presents huge opportunities such as increased turnover, awareness ... etc. In addition to these benefits, SMEs must also manage threats such as competition, security of customer information. Finally, the success of the digital transition requires support on several aspects such as the culture, the organization of the company and the financial resources to mobilize. The research we have conducted has several interests. First, on the theoretical level, we observed that no study has previously addressed this topic in the Moroccan context. In addition, it will enrich the theory related to the digitalization of the marketing function and SMEs. On the empirical level, it will allow managers and leaders of SMEs to take into consideration the importance of digitalization in improving their performance and seizing investment opportunities.

Basically, the research we have conducted presents several elements: Strengths of the study: the strengthened theoretical and empirical framing, the methodology adopted, the innovative and original results found after the study was conducted.

Limitations: the number of respondents was minimal in both the first qualitative and quantitative studies. Development perspectives: the subject of digital marketing within SMEs is innovative. In this respect, we intend to continue our research in this field in order to assimilate it and make the following research easier. Recommendations: Moroccan SMEs must rethink their strategy while thinking about the inclusion of digitalization considered not as a choice but as a necessity.

Figure 2 :

 2 Figure 2: Digital tools used by SMEs

Figure 3 :

 3 Figure 3: Opportunities of the digitalization of the marketing function

Figure 4 :

 4 Figure 4: Threats of digital marketing

3 ,

 3 Issue 6-2 (2022), pp. 231-241. © Authors: CC BY-NC-ND 239 www.ijafame.org

Figure 5 :

 5 Figure 5: Challenges of digital marketing

  pour la communication, édition Presses de l'université du Québec. (11) RITTER, T., AND C.L. PEDERSEN. (2020). Digitization capability and the digitalization of business models in business-to-business frms: Past, present, and future. Industrial Marketing Management 86: 180-190 (12) ROMANO C. et RATNATUNGA J. (1995), the role of marketing. Its impact on small enterprise research, European Journal of Marketing, 29, 7, pp.42-70. (13) SAFKO L. (2010), the social media bible: tactics, tools and strategies for business success, John Wiley & Sons Ltd. (14) SESTINO, A., M.I. PRETE, L. PIPER, ET G. GUIDO. (2020). Internet of Things and Big Data as enablers for business digitalization strategies. (15) TAIMINEN H. M. et KARJALUOTO H. (2015), the usage of digital marketing channels in SMEs, Journal of Small Business and Enterprise Development, 22, 4, pp.633-65. (16) YU, L., Y. ZHAO, L. TANG, ET Z. YANG. (2019). Online big data-driven oil consumption forecasting with Google trends. International Journal of Forecasting,35 (1): 213-223.

Table 1 : Information about our sample

 1

	Participant	Gender	years of experience Interview Length
	Consultant 1	Male	10	40 minutes
	Consultant 2	Male	5	33 minutes
	Consultant 3	Male	11	20 minutes
	Consultant 4	Female	3	35 minutes
				Source: Authors


Federico Bambozzi

Alberto Vezzani
email: vezzani@math.univ-paris13.fr

Joseph Ayoub

Federico Binda

RIGIDITY FOR RIGID ANALYTIC MOTIVES

Keywords: 1991 Mathematics Subject Classification. Primary 14C15, 14G22; Secondary 11G25, 19F27 Rigidity Theorem, Rigid Analytic Varieties, Motives, Tilting equivalence

In this paper we prove the Rigidity Theorem for motives of rigid analytic varieties over a non-Archimedean valued field K. We prove this theorem both for motives with transfers and without transfers in a relative setting. Applications include the construction of étale realization functors, an upgrade of the known comparison between motives with and without transfers and an upgrade of the rigid analytic motivic tilting equivalence, extending them to Zr1{ps-coefficients.

CONTENTS

Introduction 1. Non-Archimedean étale motives 2. The Rigidity Theorem 2.1. The Embedding Theorems 2.2. The proof of Rigidity 3. Applications 3.1. The étale realization functor 3.2. Rigid motives with and without transfers 3.3. The motivic tilting equivalence with Zr1{ps-coefficients Appendix A. Rigidity for rigid motives with transfers over K References

INTRODUCTION

The history of the Rigidity Theorem for motives traces back (at least) to the computations of algebraic K-theory groups made by Suslin in the eighties to prove the Quillen-Lichtenbaum Conjecture [START_REF] Suslin | On the K-theory of algebraically closed fields[END_REF][START_REF] Suslin | On the K-theory of local fields[END_REF]. In the subsequent decade, Suslin and Voevodsky realized that those computations were special cases of a general phenomenon common to all homotopy-invariant sheaves with finite coefficients over the big étale site over a scheme [START_REF] Voevodsky | Cycles, transfers, and motivic homology theories[END_REF]. In modern terms these results are referred to as "Rigidity Theorems" and can be rephrased using the language of motives by saying that the category DA ét pS, Λq of derived étale stable motives without transfers over a scheme S is equivalent to the derived category DpS ét , Λq of Λ-sheaves on the small étale site over S, if Λ is an N-torsion ring with N P N invertible in S (under some hypotheses on the cohomological dimension of S, see [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF]).

Other instances of "Rigidity" are known. For example, the one for motives with transfers [START_REF] Cisinski | Étale motives[END_REF], or for K-theory and Chow groups (see [START_REF] Binda | Rigidity for relative 0-cycles[END_REF][START_REF] Gabber | K-theory of Henselian local rings and Henselian pairs[END_REF]). This paper treats the Rigidity Theorem in the context of motives of rigid analytic varieties over non-Archimedean valued fields, introduced in [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF] by Ayoub, both in their version with and without transfers. More precisely, given any normal rigid analytic variety S over K, we denote by RigDA ét pS, Λq (resp. RigDM ét pS, Λq) the category of étale motives without transfers (resp. with transfers) over S with coefficients in the ring Λ. The precise definition of these categories is recalled in the first section of the paper. Our main result is the following theorem.

Theorem (2.1). Let S be a normal rigid analytic variety over a non-Archimedean field K with ℓ-finite cohomological dimension, for all primes ℓ invertible in the residue field k of K, and let Λ be a N-torsion ring, where N is a positive integer invertible in k. The functors:

Lι ˚: DpS ét , Λq Ñ RigDA ét pS, Λq Lι ˚: DpS ét , Λq Ñ RigDM ét pS, Λq are equivalences of monoidal DG-categories.

As in the algebraic situation, DpS ét , Λq denotes the derived category of unbounded complexes of étale sheaves of Λ-modules over the small étale site and the functors Lι ˚arise naturally from the inclusion of the small étale topos into the big one.

We remark that the theorem above is a generalization of the usual Rigidity Theorem, corresponding to the case in which K is trivially valued. Nonetheless, to our knowledge the original algebraic proofs can not be adapted easily to the non-Archimedean context. Our strategy is rather to use algebraic Rigidity to deduce the rigid one, by means of the analytification functors and the relation between rigid varieties and formal schemes. We also remark that, even for proving our statement over a field S " Spa K for motives without transfers, the full relative Rigidity Theorem for schemes is used. Indeed, the six functors formalism plays a crucial role in our proof (see Section 2.2). This is no longer true for motives with transfers, as we show in the appendix, for which a more direct and geometric proof is possible in the absolute case.

Just like its algebraic versions, the theorem above has some interesting immediate consequences, discussed in the last section of the paper. They constitute our main motivation for proving the Rigidity Theorem in the non-Archimedean setting.

(1) We can construct the ℓ-adic realization functor for analytic motives, following the approach of Ayoub [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF]. [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF] We can prove an equivalence between rigid analytic motives with and without transfers with coefficients over Zr1{ps where p is the residue exponential characteristic of K.

(3) Over a perfectoid field, the motivic tilting equivalence (cf. [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF]) can be promoted to Zr1{ps-coefficients, and the ℓ-adic realization functors can be shown to be compatible with it.

We remark that, due to the intricate definition of the rigid motivic tilting equivalence, the proof of the last application is more convoluted that the first two. The paper is structured as follows. In Section 1 we recall the basic definitions of the theory of motives of non-Archimedean analytic spaces and we fix the notation used throughout the rest of the paper. The core of the paper is Section 2 where the Rigidity Theorem is stated and proved. The proof is divided in two main steps: the first one, shown in Section 2.1 consists in proving that the functors Lι ˚are fully faithful and the second one, shown in Section 2.2, consists in checking that they are essentially surjective. The applications of the main theorem listed above are discussed in Section 3. Finally, in the Appendix we present a more "geometric" proof of the Rigidity Theorem for RigDM ét pK, Λq for which we use as ingredients the main ideas of the proof of [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF]Théorème 2.5.34], algebraic rigidity over fields, and Temkin's results on alterations.

NON-ARCHIMEDEAN ÉTALE MOTIVES

The following notation is fixed throughout the paper. We will use the language of rigid analytic varieties following Tate, see [START_REF] Bosch | of Grundlehren der Mathematischen Wissenschaften[END_REF]. On the other hand, we recall that the (topos theoretic) underlying topological space of a rigid analytic variety is described by the adic spectrum (see [START_REF] Huber | Étale cohomology of rigid analytic varieties and adic spaces[END_REF]) and we will typically use notations and results written in the language of adic spaces, following Huber. In particular, we will use the notation Spa to denote the analytic spectrum.

' With K we always denote a field which is complete with respect to a (fixed) non-trivial non-Archimedean valuation || ¨|| : K Ñ R ě0 with a finite ℓ-cohomological dimension, for any prime ℓ which is invertible in its residue field. ' A Tate algebra over K is a topological ring A obtained as a quotient of the completion Kxx 1 , . . . , x n y of the polynomial ring Krx 1 , . . . , x n s with respect to the Gauß norm. Its analytic spectrum is denoted by Spa A. ' An affinoid rigid analytic variety [resp. a rigid analytic variety] is a locally ringed space which is [resp. locally] isomorphic to the analytic spectrum Spa A of a Tate algebra A. ' We denote with B 1 K the closed unit ball. This is the affinoid space Spa KxT y. For any analytic space S we denote by B 1 S " B 1 K ˆK S the (closed) unit disc over S. ' Let A be a Tate algebra. The analytification functor from schemes of finite type over Spec A to analytic varieties over Spa A is denoted p.q an . It is defined by means of the following universal property:

Hom Spa A pSpa B, X an q -Hom Spec A pSpec B, Xq
for any Tate algebra B over A. ' If Λ is a (commutative, unital) ring, and C is a category [resp. a site] we let PshpC, Λq [resp. ShvpC, Λq] be the category of presheaves [resp. sheaves] on C with values in Λ-modules. ' Let S be a rigid analytic variety over K. We denote by S ét the small étale site over S and by Sm S the site of smooth analytic varieties over S equipped with the étale topology. ' For any scheme or rigid analytic variety S, DpS ét , Λq denotes the (unbounded) derived category of the category of sheaves of Λ-modules over the small étale site over S. We also recall the definitions of the (unbounded, derived, étale) motivic categories of algebraic and rigid analytic varieties as defined by Ayoub [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF]. We will make use of the language of model categories (even though everything can be restated in terms of DG-categories or 8categories) for which we refer to [START_REF] Hovey | Model categories[END_REF]. Categories of complexes of presheaves ChpPshpC, Λqq will be endowed with the projective model structure (see [START_REF] Dugger | Universal homotopy theories[END_REF]). Definition 1.1. Let S be a normal rigid analytic variety over K. The category RigDA eff ét pS, Λq (the category of effective étale rigid motives over S) is the homotopy category of the Bousfield localization of ChpPshpSm S , Λqq over the étale weak equivalences (that is, morphisms inducing quasi-isomorphisms of the induced complexes of étale sheaves) and over B 1 -homotopies (shifts of the maps of representable presheaves induced by the projection morphisms B 1 Y Ñ Y with Y P Sm S ). If Y is in Sm S we denote by Λ S pY q the object in RigDA eff ét pS, Λq associated to the presheaf represented by Y .

We refer to Section 2 of [START_REF] Vezzani | The Berkovich realization for rigid analytic motives[END_REF] for more details about RigDA eff ét pS, Λq (in its Nisnevich form). We only point out that this DG-category can be defined as a Verdier quotient of DpPshpSm S , Λqq with respect to étale descent and B 1 -invariance. It therefore enjoys the following universal property: any functor F from Sm S to a Λ-enriched model category M satisfying étale descent and B 1 -invariance admits an extension to a Quillen adjunction LF ˚: RigDA eff ét pS, Λq Õ HopMq : RF ˚. See [START_REF] Choudhury | Homotopy theory of dg sheaves[END_REF][START_REF] Dugger | Universal homotopy theories[END_REF] for details. Remark 1.2. The category RigDA eff ét pS, Λq is endowed with a monoidal structure, for which Λ S pY q b Λ S pY 1 q -Λ S pY ˆS Y 1 q for any Y, Y 1 in Sm S (see [START_REF] Ayoub | Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique[END_REF]Propositions 4.2.76 and 4.4.63]). The same is true for the category DpS ét , Λq. Definition 1.3. We denote with T " T S P DpPshpSm S , Λqq the complex of sheaves Cofib pΛrHom p´, G an m,S qs Ñ ΛrHom p´, A an S qsq. The object Λ S p1q :" T S r´2s P RigDA eff ét pS, Λq is called the Tate (twisting) motive. The category RigDA ét pS, Λq (the category of étale rigid motives without transfers over S) is defined to be the monoidal T S -stabilization of RigDA eff ét pS, Λq. Also in this category, we denote by Λ S pY q the motive associated to the presheaf represented by Y . The endofunctor M Þ Ñ M bT bn will be written as M Þ Ñ Mpnq and its quasi-inverse by M Þ Ñ Mp´nq for any n P N. The motives Mpnq are called the twists of M.

For the general theory of stabilization of categories the reader is referred to [START_REF] Hovey | Spectra and symmetric spectra in general model categories[END_REF] and [START_REF] Ayoub | Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique[END_REF]. We only remark that RigDA ét pS, Λq is once again the homotopy category of the model category of symmetric T S -spectra of (the pB 1 , étq-localization of) ChpPshpSm S , Λqq and that there is a natural (left Quillen) functor RigDA eff ét pS, Λq Ñ RigDA ét pS, Λq which is a monoidal functor also enjoying a universal property, with respect to making the endo-functor p´q b T S invertible. We will typically use spectra (rather than symmetric spectra) as a model of RigDA ét pS, Λq in proofs, which is allowed by means of [START_REF] Ayoub | Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique[END_REF]Théorème 4.3.79].

Remark 1.4. In the case when K has is equipped with the trivial valuation, the theory of rigid analytic varieties collapses to the usual theory of algebraic varieties. The motivic categories defined above coincide then with the classical categories of étale motives, denoted by DA eff ét pS, Λq and DA ét pS, Λq (see [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF]). In this work we also deal with motives with transfers, whose definition we now recall (see also [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF] and [START_REF] Vezzani | Effective motives with and without transfers in characteristic p[END_REF]). Definition 1.5. Let S be a normal variety over K. We let Nor S be the category of quasicompact normal varieties over S and we let the fh-topology be the one generated by those covering families tf i : X i Ñ Xu iPI such that I is finite, and the induced map

Ů f i : Ů iPI X i Ñ X is fi-
nite and surjective. We define the category Cor S as the category whose objects are those of Sm S and whose morphisms Hom pX, Y q are computed in Shv fh pNor S q. We let RigDM eff ét pS, Λq (the category of effective étale rigid motives with transfers over S) be the homotopy category of the Bousfield localization of Ch PshpCor S , Λq over those morphisms f which are étale weak equivalences as morphisms in Ch PshpSm S , Λq, and over B 1 -homotopies (shifts of the maps of representable presheaves induced by the projection morphisms B 1 Y Ñ Y with Y P Sm S ). If Y is in Sm S we denote by Λ tr S pY q the object in RigDM eff ét pS, Λq associated to the presheaf represented by Y . Definition 1.6. We consider the object T tr P DpShpCor S , Λqq given by the complex Cofib pHom Cor S p´, G an m,S q b Λ Ñ Hom Cor S p´, A an S q b Λq. The category RigDM ét pS, Λq (briefly, étale rigid motives with transfers over S) is defined to be the T tr S -stabilization of RigDM eff ét pS, Λq. If Y is in Sm S we denote by Λ tr S pY q the motive in RigDM ét pS, Λq associated to the presheaf represented by Y . The endofunctor M Þ Ñ M b pT tr q bn will be written as M Þ Ñ Mpnq and its quasi-inverse as M Þ Ñ Mp´nq for any n P N.

The motives Mpnq are called the twists of M.

Remark 1.7. Once again, whenever K is endowed with the trivial valuation, the definitions above recover the usual categories of (derived, étale) motives with transfers over S, denoted by DM eff ét pS, Λq and DM ét pS, Λq (see [START_REF] Cisinski | Étale motives[END_REF]). Remark 1.8. There is a more down-to-earth description of the category Cor S in terms of multivalued functions (or rather, Zariski closed subvarieties of the product which are finite over the first component) see [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF]Remarque 2.2.21].

We recall that in a triangulated category, an object S is compact if the functor Hom pS, ´q commutes with direct colimits. A triangulated category is compactly generated if there is a set of compact objects S for which the smallest triangulated subcategory with small sums containing them is the whole category. This property is technically very convenient, and the following fact will be used several times throughout the paper. We now summarize the basic functors between the various motivic categories introduced so far.

Proposition 1.10. Let S be a normal rigid analytic variety and Λ be a ring.

( Take Λ to be N-torsion with N invertible in K. The classic Rigidity Theorem can be restated by saying that algebraic Artin motives (i.e. Artin motives with respect to a trivial valuation of K) are equivalent to DpS ét , Λq as well as to the categories of stable motives DA ét pS, Λq and DM ét pS, Λq, under mild hypotheses on S (see [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF]).

Also in this paper, we will focus on the case of N-torsion coefficients, with N coprime to the residual (exponential) characteristic. Also in this setting, algebraic Artin motives are easily seen to embed in the effective categories of motives. Proof. We remark that DpS ét , Λq can be seen as a full subcategory of DpShvpSm S q, Λq and RigDA eff ét pS, Λq can be described as the full subcategory of DpShvpSm S q, Λq of B 1 S -homotopy invariant objects. In order to prove the first claim, it then suffices to check that the objects of DpS ét , Λq are B 1 S -homotopy invariant. In order to do so, one can adapt the proof of [2, Souslemme 4.7] using the analytic version of the acyclicity theorem [20, Example 0.1.1(ii)]. For the case with transfers, one can similarly adapt the proof of [START_REF] Cisinski | Étale motives[END_REF]Theorem 3.1.7].

THE RIGIDITY THEOREM

From now on, we fix a normal rigid analytic variety S over K. Once again, we recall that K is assumed to have finite ℓ-cohomological dimension, for any prime ℓ which is invertible in its residue field. The main aim of this section is the proof of the Rigidity Theorem, for motives with and without transfers. We state it here.

Theorem 2.1. Let S be a normal rigid analytic variety over K and let Λ be a N-torsion ring, where N is invertible in the residue field of K. The functors:

Lι ˚: DpS ét , Λq Ñ RigDA ét pS, Λq Lι ˚: DpS ét , Λq Ñ RigDM ét pS, Λq
are equivalences of monoidal DG-categories. Moreover the canonical functor between RigDM eff ét pS, Λq and RigDM ét pS, Λq is an equivalence. We will divide the proof in two steps: we first show that the two functors are fully faithful, and we then prove that they are essentially surjective.

2.1. The Embedding Theorems. We now show that the functors above are fully faithful. As in the algebraic proofs (see [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF] and [START_REF] Cisinski | Étale motives[END_REF]) one of the key points is the equivalence between the Tate twisting motive and the sheaf of N-th roots of unity.

Proposition 2.2. Let Λ be a N-torsion ring, where N is invertible in the residue field of K.

There is a natural morphism

Λ S p1q -T S r´2sѵ S,N
in RigDA eff ét pS, Λq where the motive on the right is the one induced by the locally constant sheaf of N-th roots of unity. It becomes invertible in RigDA ét pS, Λq after applying L Sus.

Proof. We first produce the natural morphism between the two motives. The presheaf of abelian groups O ˆis represented by G an m,S . There is an obvious map of presheaves Λ S pG an m,S q " Hom p´, G an m,S q b Λ Ñ O ˆbZ Λ where the first tensor is the free Λ-module over the set, while the second is a base change over Z Ñ Λ. The induced map on the associated sheaves factors over T S r´1s. On the other hand, we remark that by the Kummer exact sequence, we can write µ S,N r1s -O ˆbZ Λ in RigDA eff ét pS, Λq. By shifting on both sides, we therefore obtain a morphism T S r´2sѵ S,N in RigDA eff ét pS, Λq as wanted. We now pass to the stable categories and we prove that this morphism becomes invertible. Using (the obvious rigid analogue of) [2, Lemma 3.4] it suffices to prove the statement in the case when S is affinoid, equal to Spa A. We remark that Tate algebras are excellent (see [START_REF] Fresnel | Rigid analytic geometry and its applications[END_REF]Remarks 3.5.2]) and therefore (see the paragraph after [2, Theorem 4.1]) one can apply [2, Proposition 4.10] and claim that there is an isomorphism

(2.2.1) µ S,N -Λ Spec A p1q
in DA ét pSpec A, Λq induced by the analogous natural morphism between the two motives.

Applying the analytification functor LAn ˚: DA ét pSpec A, Λq Ñ RigDA ét pS, Λq to the isomorphism (2.2.1) we obtain the statement (using Proposition 1.10, ( 5)).

There is a stronger statement for motives with transfers.

Proposition 2.3. Let Λ be a N-torsion ring, where N is invertible in the residue field of K.

There is a natural isomorphism

Λ tr S p1q -T tr S r´2sѵ S,N in RigDM eff
ét pS, Λq where the motive on the right is the one induced by the locally constant sheaf of N-th roots of unity.

Proof. The natural morphism between the two motives is deduced by the morphism of Proposition 2.2. In order to prove that it is invertible, using (the obvious rigid analogue of) [ Proof. The sheaf µ S,N is invertible in DpS ét , Λq (étale locally, it is isomorphic to Λ) and hence (as the functor Lι ˚is monoidal) the functor p´qbµ S,N is already invertible in RigDM eff ét pS, Λq. We conclude the first claim using Proposition 2.3. The second claim follows at once from Proposition 1.12.

The case of motives without transfers is more complicated. On the other hand, the proof of the next theorem is the straightforward rigid analytic analogue of [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF]Corollaire 4.11]. We reproduce here its proof for the convenience of the reader. Proof. We divide the proof in several steps.

Step Hom pM b T bn , Lι ˚pK q b T bn q colim nPN Hom pM, RHom pT bn , Lι ˚pK q b T bn qq.

As M is compact, it is enough to show that the map

Lι ˚K Ñ hocolim nPN
RHom pT bn , Lι ˚pK q b T bn q is invertible. We make a variable change and we swap RHom with homotopy colimits (see Lemma 2.6) and we rewrite the second term above as:

(2.5.2)

hocolim nPN RHom pT bn , Lι ˚pK q b T bn q -hocolim n,mPN
RHom pT bn`m , Lι ˚pK q b T bn`m q hocolim nPN RHom pT bn , hocolim mPN RHom pT bm , Lι ˚pK q b T bn`m q.

Step 2: We now prove that

hocolim mPN RHom pT bm , Lι ˚pK q b T bn`m q appearing in (2.5.2) is isomorphic to hocolim mPN RHom pT bm , Lι ˚pK pnqr2nsqq b T bm q.
By [4, Théoreme 4.3.61] we know that we can compute morphisms between two spectra pS n q and pY n q in the (derived) category of N-sequences by taking level-wise the Hom -groups

Hom pS n , hocolim kPN RHom pT bk , Y k`n qq.
In particular, an equivalence of spectra pY n q and pY 1 n q induces equivalences in the effective category (2.5.3) hocolim kPN RHom pT bk , Y k`n qhocolim kPN RHom pT bk , Y 1 k`n q. We deduce that the isomorphism of the claim follows from the equivalence of spectra

Lι ˚pK q b T bn -Lι ˚pK pnqr2nsqq
which is a consequence of Proposition 2.2.

Step 3: We now prove that for any m the object hocolim nPN RHom pT bn , Lι ˚pK pnqqr2ns b T bm q is canonically equivalent (in the effective category) to:

hocolim nPN RHom pT bn , Lι ˚pK pn `mqqr2n `2msq.
Using again the line of reasoning of the previous step, and more specifically the equation (2.5.3) with k " 0, we see that it suffices to give to L :" pLι ˚pK pnqqr2nsq nPN a structure of a spectrum, and prove the existence of a compatible equivalence

L b T bm -L b ι ˚pK pmqr2msq
for any m. This second fact follows from Proposition 2.2 while for the first fact one can use the transition maps induced by the following equivalence in RigDA ét pS, Λq (Poincaré duality in étale cohomology, see [20, Section 7.5]):

Lι ˚pK pnqqr2ns -RHom pT, Lι ˚pK pn `1qqr2n `2sq and eventually replacing Lι ˚pK pnqqr2ns with a cofibrant-fibrant replacement in order to have genuine maps between complexes of presheaves.

Step 4: We now finish the proof. We take again (2.5. Proof. As RigDA eff ét pS, Λq is compactly generated (see Proposition 1.9) it suffices to test that for any compact object M attached to an affinoid variety X " Spa A smooth over S, the natural map Hom pM b T bn , hocolim U i q / / Hom pM, hocolim RHom pT bn , U i qq " colim Hom pM b T bn , U i q is invertible. It suffices then to prove that X b T is also compact. In order to do this, we can prove (see [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF]Page 54]) that the motive attached to Spa AxT ˘y is compact, which follows again from Proposition 1.9.

We pin down the following consequence of Theorems 2.4 and 2.5.

Corollary By abuse of notation, under the hypotheses of the previous Corollary, we will refer to rigid analytic Artin motives as well as to the objects of DpS ét , Λq simply as Artin motives.

2.2.

The proof of Rigidity. We can finally achieve the proof of Theorem 2.1. To this aim, we will use a (limited) version of the six functor formalism for rigid analytic motives, which is already present in [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF]. We recall here the main properties that we will use.

Remark 2.8. For the sake of readability, we will drop L and R for derived functors, whenever the context allows it. In particular, we will write ι ˚for the functors introduced in Definition 1.11. Theorem 2.9. Fix an affinoid algebra A. Let S be Spa A. These functors coincide with those induced by the map of analytic varieties f an according to Theorem 2.9 [START_REF] Adiprasito | Log smoothness and polystability over valuation rings[END_REF]. We can therefore use the notation pf an ˚, f an ˚q unambiguously. On the other hand we point out that the full six functor formalism for rigid analytic varieties is not yet available in the literature (progress in this direction is being done by Ayoub, Gallauer and the second named author). For this paper the limited version stated above will be enough. More precisely, we will need the following corollary: Corollary 2.11. Let f : W Ñ S be a smooth map of affinoid rigid analytic varieties and let p : P Ñ S be a composition of a Zariski closed embedding and the canonical projection P N,an S Ñ S for some N P N. Consider the following cartesian diagram

P 1 p 1 / / f 1 W f P p / / S The natural transformation f ˚p˚ñ p 1 ˚f 1i
s invertible.

Proof. It suffices to consider separately the case when p is a Zariski closed immersion and when it is the projection P N S Ñ S over an affinoid variety. In the latter case, we have (see Theorem 2.9 p ˚-pan ! p 7 p´b Λp´Nqr´2Nsq where p is the projection of schemes P N S Ñ S and where S is Spec OpSq. The commutation between this functor and f ˚follows from the commutation of p 7 and f ˚as shown below:

p 1 ˚f 1˚M -p 1 7 pf 1˚M p´Nqr´2Nsq -f ˚p7 pMp´Nqr´2Nsq -f ˚p˚M .
We now suppose that p is a Zariski closed embedding with complement j : U Ñ S. By means of (3) of Theorem 2.9 it suffices to show that the two natural transformations

p 1˚f ˚p˚ñ p 1˚p1 ˚f 1˚j1˚f ˚p˚ñ j 1˚p1 ˚f 1å
re invertible, where j 1 : U 1 Ñ W is the open complement of P 1 . Using the commutation of f ˚with p 1˚a nd j 1˚a nd the equivalences p ˚p˚-Id , p 1˚p1 ˚-Id , j ˚p˚-0 and j 1˚p1 ˚-0 the result is immediate.

The following result is an adaptation of [START_REF] Cisinski | Étale motives[END_REF]Proposition 4.4.3] to the rigid setting and follows formally from the previous versions of the proper base change. We recall that we write simply ι ˚for the derived functor that was previously denoted by Lι ˚.

Proposition 2.12. Let S be an affinoid rigid analytic variety and let p : P Ñ S be a map which is a composition of a Zariski closed immersion and the canonical projection P N,an S Ñ S. The natural transformation ι ˚˝p ˚ñ p ˚˝ι is invertible.

Proof. We first prove an auxiliary result, namely that for any complex L in DpP ét , Λq, we have Hom pΛ, p ˚ι˚L q -Hom pΛ, ι ˚p˚L q.

By the fully faithfulness of ι ˚the second term can be re-written as Hom pΛ, ι ˚p˚L q -Hom pΛ, p ˚Lq.

On the other hand, the first term can be re-written as Hom pΛ, p ˚ι˚L q -Hom pp ˚Λ, ι ˚Lq -Hom pp ˚ι˚Λ , ι ˚Lq -Hom pι ˚p˚Λ , ι ˚Lq -Hom pp ˚Λ, Lq -Hom pΛ, p ˚Lq where we used the adjunction pp ˚, p ˚q, the fact that ι ˚is fully faithful and the fact that p ˚ι˚Λι ˚p˚Λ because both sheaves are isomorphic to the constant sheaf on P . Moreover, the natural transformation induces the identity on these groups, as wanted. We now prove the general statement. By Proposition 1.9 the category RigDA ét pS, Λq is generated by the motives of smooth affinoid varieties over S. It suffices then to prove that for each f : W Ñ S smooth and each L P DpP ét , Λq the natural transformation of the statement induces an isomorphism between the group Hom pf 7 Λ, p ˚ι˚L q and Hom pf 7 Λ, ι ˚p˚L q. We let L 1 be f 1˚L where the map f 1 (resp. p 1 ) can be introduced as the base change of f over p (resp. of p over f ). By means of the projective base change for rigid motives (see Corollary 2.11) we know that the former is canonically isomorphic to Hom pf 7 Λ, p ˚ι˚L q -Hom pΛ, f ˚p˚ι ˚Lq -Hom pΛ, p 1 ˚f 1˚ι˚L q -Hom pΛ, p 1 ˚ι˚L1 q. On the other hand, by means of the proper base change for Artin motives (see [20, Proposition 4.4.1]) we see that the latter is isomorphic to Hom pf 7 Λ, ι ˚p˚L q -Hom pΛ, f ˚ι˚p ˚Lq -Hom pΛ, ι ˚f ˚p˚L q -Hom pΛ, ι ˚p1 ˚L1 q. The two are therefore canonically isomorphic by what we proved in the first part.

We end our list of preliminary statements with the following algebraic approximation result. Proposition 2.13. Let X " Spa A be a (rig-)smooth affinoid space over Spa R, then there exist finite coverings tX i u of X and tSpa R i u of Spa R by admissible open affinoid subdomains such that each X i admits an embedding j i : X i Ñ Z i where Z i is the analytification of a smooth algebraic variety over R i .

Proof. By [5, Corollaire 1.1.51] X admits a finite covering by admissible open affinoid subdomains X i each of which admits an étale map to B n i R i with R i as in the statement. Then, applying [5, Lemme 1.1.52] to the étale maps X i Ñ B n i R i we obtain that X i " Spa pA i q with A i a K-affinoid algebra that admits a presentation of the form

A i " R i xT 1 , .
. . , T n , U 1 , . . . , U m y pP 1 , . . . , P m q with P j P R i rT 1 , . . . , T n , U 1 , . . . , U m s polynomials and d :" detp BP i BU k q invertible in A i . Therefore, we can embed X i as an admissible open affinoid of the analytification of the smooth affine R i -variety

Spec

ˆRi rT 1 , . . . , T n , U 1 , . . . , U m s pP 1 , . . . , P m q " 1 d

˙.

We are finally able to prove our main result.

Proof of Theorem 2.1. In light of the Embedding Theorems 2.5 and 2.4 we are left to prove that a set of generators (as triangulated category with infinite sums) for RigDA ét pS, Λq [resp. RigDM ét pS, Λq] lies in the image of the functors Lι ˚. Since the functor La tr : RigDA ét pS, Λq Ñ RigDM ét pS, Λq sends a set of compact generators to a set of compact generators, we deduce that it suffices to prove the claim for RigDA ét pS, Λq.

The category RigDA ét pS, Λq is generated by motives of the form Λ S pY q with Y affinoid and smooth over S. We can consider an arbitrary covering tS i u iPI of S (resp. an arbitrary covering tY j u jPJ of Y ) and write Λ S pY q as the homotopy colimit of the Cech hypercover induced by the covering tS i ˆS Y u iPI (resp. tY j u jPJ ). Since Λ S pS i ˆS Y q " pj i q 7 Λ S i pS i ˆS Y q, with j i : S i Ñ S being the open inclusion, and pj i q 7 sends Artin motives to Artin motives, it suffices to prove that Λ S pY q is Artin, locally on S and Y .

According to Proposition 2.13, we can then suppose that ' S -Spa R is affinoid; ' that f : Y " Spa pAq Ñ S is affinoid; ' Y can be embedded as an admissible open inside the analytification of a smooth affine scheme f 1 : Y 1 Ñ Spec R over R of finite type, of pure dimension d.

From now on, all analytification functors are over Spec R. We can embed Y 1 into a closed subvariety Z of P N R by means of an open immersion j 1 : Y 1 Ñ Z. We let p be the structural morphism p : Z Ñ Spec R. The induced open immersion j : Y Ñ Z an has a formal model j : Y Ñ Z so that Λ Z an pY q " ξp j7 Λ Y pY qq, where we let FormDA ét pZ , Λq be the category of étale motives over smooth formal schemes topologically of finite type over Z (see [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF]Section 1.4.2]) and ξ : FormDA ét pZ , Λq Ñ RigDA ét pZ an , Λq is induced by the generic fiber functor X Þ Ñ X η . We recall that by the algebraic Rigidity Theorem [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF] and [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF]Corollary 1.4.23] the special fiber functor X Þ Ñ X σ determines an equivalence FormDA ét pZ , Λq -DApZ σ , Λq -DppZ σ q ét , Λq

By the canonical equivalence (see [START_REF] Grothendieck | Éléments de géométrie algébrique IV. Étude locale des schémas et des morphismes de schémas[END_REF]Théorème 18.1.2]) DpZ ét , Λq -DppZ σ q ét , Λq we deduce that j7 Λ Y pY q is an Artin motive. Since ξ preserves Artin motives (the generic fiber of an étale scheme over S is étale over S η ), we deduce that Λ Z an pY q is Artin, equal, say, to ι ˚N with N P DpZ an , Λq.

Since the analytification of an étale extension is still étale, the functor An ˚preserves Artin motives. Also the functor j 1 7 does by its explicit description (Theorem 2.9(2)). We deduce then that An ˚pj 1 7 pΛ Y 1 pY 1 qpdqqr2dsq is ι ˚M for some Artin motive M over Y 1an . We let α be the immersion Y Ñ Y 1an . We now give an alternative description of the motive Λ S pY q. By Theorem 2.9 (1) and ( 2) we have that f 1 ! anp an ! j 1 ! anp an ˚j1 an 7 and up to considering coverings of S and Y , we can suppose that f 1an 7 pα 7 Λ Y pY qqf 1an ! pα 7 Λ Y pY q b Λpdqr2dsq. We then deduce

Λ S pY q -f 7 Λ Y pY q -f 1an 7 α 7 Λ Y pY q -f 1an ! pα 7 Λ Y pY q b Λ Y 1an pY 1an qpdqr2dsq -p an ˚j1an 7 pα 7 Λ Y pY q b Λ Y 1an pY 1an qpdqr2dsq -p an ˚pj 7 Λ Y pY q b j 1an 7 pΛ Y 1an pY 1an qpdqqr2dsq -p an ˚pι ˚N b An ˚pj 1 7 pΛ Y 1 pY 1 qpdqqr2dsqq -p an
˚ι˚p N b Mq where we used the fact that the functor j 1an 7 commutes with tensor products of effective motives (see its definition -Theorem 2.9( 2)) together with the equality j " j 1an ˝α and, for the last equality, the monoidality of the functor ι ˚(the product in S ét is the same as the one in Sm S ). In order to prove that Λ S pY q is Artin it suffices then to show that p an ˚commutes with ι ˚which is proven in Proposition 2.12.

APPLICATIONS

In this concluding section we describe three applications of our main theorems. We keep the hypothesis used so far, with the difference that now Λ is supposed to be any ring. We keep the notation that K is a complete non-Archimedean non-trivially valued field with finite ℓ-cohomological dimension, and S a normal rigid analytic variety over it.

3.1. The étale realization functor. We follow [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF] and we show how to use the Rigidity Theorem to produce étale realization functors. In particular, we can produce an ℓ-adic realization functor for the category RigDA ét pK, Qq completing the picture of "classical" realizations for rigid analytic motives (for a p-adic realization for this category see [START_REF] Vezzani | The Monsky-Washnitzer and the overconvergent realizations[END_REF] and for a Betti-like realization see [START_REF] Vezzani | The Berkovich realization for rigid analytic motives[END_REF]).

We first introduce the target category, which is the derived category of ℓ-adic étale sheaves, following Ekedhal. We refer to [2, Section 5] for details on this construction, which we simply adapt to the rigid analytic situation. Definition 3.1. Let J Ă Λ be an ideal. The objects of the category Λ{J ˚-Mod are diagrams of Λ-modules We define the category p DpS ét , Λ J q to be the full triangulated subcategory of DpS ét , Λ{J ˚q whose objects are complexes K such that the canonical map ps `1q ˚K b Λ{J s`1 Λ{J s Ñ s ˚K is invertible for each s P N.

M ' " ¨¨¨Ñ M s`1 Ñ M s Ñ ¨¨¨Ñ M 1 Ñ M 0 such that
We define the category p D ct pS ét , Λ J q (where ct stands for constructible) to be the full triangulated subcategory of p DpS ét , Λ J q whose objects are complexes K such that for each s P N, the étale sheaves associated to H n ps ˚K q are constructible in the sense of [START_REF] Huber | Étale cohomology of rigid analytic varieties and adic spaces[END_REF]Definition 2.7.2] for all n P Z, and zero for |n| " 0.

We define the category p D ct pS ét , Λ J b Qq to be the triangulated category obtained from p D ct pS ét , Λ J q by tensoring all Hom -groups with Q (it is still a pseudo-abelian triangulated category, see [START_REF] Cisinski | Étale motives[END_REF]Corollary B.2.3]).

We can make a direct-limit argument on coefficients and use Rigidity to produce ℓ-adic realizations. For a triangulated category T we denote by T comp the full subcategory of compact objects.

Theorem 3.2. Let J be an ideal of Λ for which Λ{J is N-torsion, with N invertible in the residue field of K. For any normal rigid analytic variety S over K there are monoidal triangulated functors R S,J :

RigDA ét pS, Λq Ñ p DpS ét , Λ J q R Q S,J : RigDA comp ét pS, Λ b Qq Ñ p D ct pS ét , Λ J b
Qq which have the following properties (for the six functors formalism, see Theorem 2.9 for the motivic side, and [START_REF] Huber | Étale cohomology of rigid analytic varieties and adic spaces[END_REF] for the étale side).

(1) The composition of the first with p DpS ét , Λ J q Ñ DpS ét , Λ{J s q gives the functor

RigDA ét pS, Λq p´qb L Λ{J s ÝÑ RigDA ét pS, Λ{J s q Rι- DpS ét , Λ{J s q
where the second equivalence is the one from the Rigidity Theorem. (2) They commute with the functors f ˚and f ˚, for any morphism f of normal rigid-analytic varieties.

(3) They commute with the functors f 7 (left adjoint to f ˚in case f is smooth). ( 4) They commute with the functors f an ! and f ! an for any morphism f of quasi-projective schemes over a Tate algebra.

(5) They commute with the bi-functor RHom when restricted to compact objects in the first variable.

Proof. For the functor R S,J , it suffices to copy verbatim the arguments of [2, Théorème 6.9]. Indeed, Rigidity gives rise to a canonical equivalence of tensor DG-categories RigDA ét pS, Λ{J ˚q -DpS ét , Λ{J ˚q. The realization functor is then defined by means of

RigDA ét pS, Λq Ñ RigDA ét pS, Λ{J ˚q -DpS ét , Λ{J ˚q
where the first functor is induced by the obvious functor M Þ Ñ pM b Λ{J ˚q from Λ -Mod to Λ{J ˚-Mod. It is easily seen to take values in p DpS ét , Λ J q and to commute with f 7 , f ˚. As for f ˚one can use the proof of [2, Théorème 6.3] while for RHom one can use the proof of [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF]Théorème 6.4]. The point p4q deals with algebraic morphisms, so the statement for f an ! follows at once from [5, Théorème 1.4.33] and [6, Théorème 3.4(b)] while for f ! an one can consider separately the case where f is smooth (in which [6, Théorème 3.4(c)] holds) and the case where f is a closed immersion, which follows from the localization triangle [3, Proposition 1.4.9] and point [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF]. We leave the details to the reader. Following [2, Théorème 9.7], in order to extend the functor R S,J (and its properties) to the functor R Q S,J one needs to prove that R S,J sends compact objects to the category p D ct pS ét , Λ J q. As already remarked in Proposition 2.13, a class of compact generators for RigDA ét pS, Λq is obtained by f 7 pΛqpnq with f : X Ñ Spa A Ă S is a smooth map of an affinoid variety X over an affinoid open subvariety Spa A of S. We can also suppose that X is open in a smooth variety j : X Ă Y an which is the analytification of a smooth A-scheme of finite type g : Y Ñ Spec A. By the commutation of R S,J with the six operations and Theorem 2.9 we obtain R Y an ,J j 7j ! R X,J and R Spa A,J g an 7 g an ! pR Y an ,J pnqr2nsq (the last isomorphism holding locally on X, see Theorem 2.9(4i)). It therefore suffices to show that for a smooth morphism f : X Ñ S of rigid analytic varieties, the functor f ! : p DpX ét , Λ{J s q Ñ p DpS ét , Λ{J s q preserves constructible complexes, which follows from [20, Theorem 6.2.2].

Remark 3.3. Suppose Λ " Z and J " pℓq with ℓ a prime invertible in the residue field of K. We point out that the realization functor R computes ℓ-adic (co-)homology. In order to do so we first put Λ 1 " Z{ℓ N and compute, for any p : X Ñ S in Sm S using Rigidity:

R n p ˚Λ1 -H n RHom pΛ 1 S pXq, Lι ˚Λ1 q -H n RHom pRι ˚Λ1 S pXq, Λ 1 q
We deduce, by Theorem 3.2(1), that RHom pR S,ℓ Z S pXq, Z ℓ q -Rp ˚Zℓ .

Remark 3.4. We point out that the ℓ-adic realization can alternatively be constructed using the universal property of rigid motives and the pro-étale cohomology [START_REF] Bhatt | The pro-étale topology for schemes[END_REF][START_REF] Scholze | p-adic Hodge theory for rigid-analytic varieties[END_REF]. This is particularly convenient in order to define an infinity-category DpS proét , Λ b Qq containing p DpS ét , Λ J b Qq as a full subcategory, allowing one to extend the functor R Q S,J to a functor

R Q S,J : RigDA ét pS, Λ b Qq Ñ DpS proét , Λ b Qq
by means of the universal property of RigDA ét pS, Λ b Qq. To this aim, it suffices to define a functor from the category Sm S of smooth affinoid varieties over S to DpS proét , Λ b Qq having étale descent and homotopy invariance. One can then take the functor X Þ Ñ R Q S,J pΛ S pXqq defined above. Remark 3.5. In order to prove that R S,J sends compact objects to constructible complexes, one could also argue by considering the composition with each s ˚which is canonically equivalent to the change of coefficients RigDA ét pS, Λq Ñ RigDA ét pS, Λ{J s q (see Theorem 3.2(1)). This functor has a right adjoint which commutes with sums, hence it preserves compact objects. The result then follows from the comparison between compact and constructible complexes see [START_REF] Scholze | Etale cohomology of diamonds[END_REF]Proposition 20.17].

In case the base variety is the analytification of a quasi-projective scheme S over a Tate algebra A, one can consider the analytification functor between algebraic and analytic motives. It is known that the étale cohomology of S coincides with the one of S an . We have the following motivic strengthening of this fact.

Corollary 3.6. Let S be a normal scheme of finite presentation over a Tate algebra over K and let Λ and J be as in Theorem 3.2. The analytification functor LAn ˚: DA ét pS, Λq Ñ RigDA ét pS an , Λq is compatible with the étale realization functors, i.e. there is a commutative diagram

DA ét pS, Λq LAn ˚/ / R S,J
RigDA ét pS an , Λq R S an ,J p DpS ét , Λ J q LAn ˚/ / p DpS an ét , Λ J q Proof. By construction, it suffices to prove that the analytification functor is compatible with the functors arising with the change of coefficients Λ Ñ Λ{ℓ N which is obvious. Proof. By [START_REF] Vezzani | Effective motives with and without transfers in characteristic p[END_REF]Corollary 4.20] we can deduce the statement for the categories RigDM ét pK, Λq and RigDA Frobét pK, Λq the latter being the localization of RigDA ét pK, Λq over the relative Frobenius maps (see [START_REF] Vezzani | Effective motives with and without transfers in characteristic p[END_REF]Section 2.1] for the definition of the Frob-étale topology and its relation with motives). We claim that the relative Frobenius maps are already equivalences in RigDA ét pK, Λq. Indeed, they induce an endofunctor Φ and a natural transformation Φ ñ Id in RigDA ét pK, Λq. We let T be the full triangulated subcategory of those objects where the natural transformation is invertible. By the separatedness property of DA ét pK, Λq (see [2, Théorème 3.9]) the category T, which is obviously closed under sums and cones, contains the motives of the form LAn ˚M with M P DA ét pK, Λq. But such motives generate the whole category (by [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF]Théorème 2.5.35]) so that T " RigDA ét pK, Λq as wanted.

Using the same strategy as in [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF] we can promote the comparison above to motives with Zr1{ps-coefficients using the Rigidity Theorem. Theorem 3.9. Let Λ be a Zr 1 p s-algebra, where p is the exponential residual characteristic of K. The canonical functor La tr : RigDA ét pK, Λq Ñ RigDM ét pK, Λq is an equivalence of monoidal DG-categories.

Proof. It suffices to adapt the proof of [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF]Corollaire B.3] to rigid motives. We recall this argument for the convenience of the reader.

We remark that the functor La tr : RigDA ét pK, Λq Ñ RigDM ét pK, Λq sends a generating set of compact objects of the first category (motives of the form ΛpSqrnspkq with S affinoid, étale over B N K ) to a generating set of compact objects of the second (motives Λ tr pSqrnspkq with S as before). By means of [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF][ Lemme 1.3.32] it then suffices to show that for any such M, N compact in RigDA ét pK, Λq the map (3.9.1)

Hom RigDA ét pK,Λq pM, Nq Ñ Hom RigDM ét pK,Λq pLa tr M, La tr Nq is bijective. We remark that for any map of rings Λ Ñ Λ 1 and any object N of RigDA ét pK, Λ 1 q (resp. RigDM ét pK, Λq) we have the following adjunctions Hom RigDA ét pK,Λ 1 q pM b L Λ Λ 1 , Nq -Hom RigDA ét pK,Λq pM, Nq Hom RigDM ét pK,Λ 1 q pM b L Λ Λ 1 , Nq -Hom RigDM ét pK,Λq pM, Nq and since we may assume that M " ΛpSqrnspkq -Zr1{pspSqrnspkqb L Λ it suffices to consider the case Λ " Zr1{ps.

We can consider the exact triangle

Zr1{ps Ñ Q Ñ Q{Zr1{ps -colim Z{ℓ k Z Ñ pZr1{psqr1s
and tensor it with N, where the colimit ranges over primes ℓ ‰ p and natural numbers k. Since M as well as La tr M are compact, we are reduced to proving the bijectivity of (3.9.1) for the cases N " N b Q and N " N b Z{ℓ k separately. Using the adjunctions above once more, one can test the fully faithfulness of La tr in the cases Λ " Q and Λ " Z{ℓ k which follow from Theorem 3.8 and Theorem 2.1 respectively.

Remark 3.10. Even though there is a version of Theorem 3.8 which holds also for effective motives (see [START_REF] Vezzani | Effective motives with and without transfers in characteristic p[END_REF]Theorem 4.1]) there is no effective version of the Rigidity Theorem without transfers, so that Theorem 3.9 cannot be stated for effective motives.

3.3.

The motivic tilting equivalence with Zr1{ps-coefficients. In this section we focus on rigid motives over perfectoid fields and we answer positively to the conjecture stated in [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF]Remark 7.28]. We recall that a perfectoid field K is a non-Archimedean field such that |p| ă 1 for a prime p and that the Frobenius morphism

O K {p Ñ O K {p is surjective (classical examples include C p or the completion of Q p pµ p 8 q)
. The interest about these objects comes from the tilting functor K Þ Ñ K 5 which associates to K a perfectoid field of characteristic p (in the examples above, one obtains the complete algebraic closure of F p ppT qq and the completion of F p ppT 1{p 8 qq respectively).

Theorem 3.11 ([22, Theorem 1.3]). The tilting functor induces an equivalence between the small étale site over K and the small étale site over K 5 .

It can be restated motivically as an equivalence between Artin motives over the two fields. The main result of [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF] is the extension of this result to a motivic tilting equivalence, where we write RigDA ét pK, Λq for RigDA ét pSpa K, Λq. We refer to [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF] for a thorough discussion of this result and its meaning. We summarize here the construction of the connecting functor.

One can consider the category of perfectoid motives PerfDA ét pKq which is obtained by considering motives over the étale site of (geometrically) smooth perfectoid spaces over K (homotopies are defined over the perfectoid disc p B 1 " Spa KxT 1{p 8 y and the twist is defined by considering the cokernel of the map of sheaves Λ Ñ ΛpSpa KxT ˘1{p 8 yq).

Scholze's tilting equivalence [22, Theorems 1.9 and 1.11] straightforwardly implies that the adjunction induced by the tilting functor L5 ˚: PerfDA ét pK, Λq Ô PerfDA ét pK 5 , Λq : R5 is an equivalence (for any Λ).

In the case when char K " p, the perfection functor induces an adjunction (3.12.1)

L Perf ˚: RigDA ét pK, Λq Ô PerfDA ét pK, Λq : R Perf ẘhich is shown to be invertible if Λ is a Q-algebra (see [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF]Theorem 6.9] and the proof of Theorem 3.8 to deduce the stable statement without Frob-localization).

Suppose now char K " 0. As a further auxiliary category, one can also introduce sPerfDA ét pK, Λq which is the motivic category over the étale site of "semi-perfectoid spaces", that is: those adic spaces which are locally étale over B n ˆp B m (homotopies are considered over p B 1 and the twist is defined by considering the cokernel of the map of sheaves Λ Ñ ΛpSpa KxT ˘1yq (we follow here the notation of [START_REF] Vezzani | Rigid cohomology via the tilting equivalence[END_REF]. This category is denoted y

RigDA ét, p B 1 pK, Λq in [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF]). The canonical inclusion of étale sites induces a Quillen adjunction:

Lι ˚: that can be described as follows: it maps a spectrum tM i u iPN to the spectrum tj ˚Mi u iPN where pj ˚, j ˚q is the Quillen equivalence on effective motives induced by the inclusion of sites from perfectoid spaces in semi-perfectoid spaces.

In particular, even in characteristic 0, one has a connecting functor (3.12.2) Rj ˚Lι ˚: RigDA ét pK, Λq Ñ PerfDA ét pK, Λq which is shown to be invertible if Λ is a Q-algebra (see [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF]). The contribution of this paper to this topic is the following theorem.

Theorem 3.13. Let Λ be a Zr 1 p s-algebra, where p is the residual exponential characteristic of K. The functor R Perf ˚L5 ˚Rj ˚Lι ˚induces an equivalence RigDA ét pK, Λq -RigDA ét pK 5 , Λq.

Moreover, the equivalence above is compatible with the ℓ-adic étale realization functors of Theorem 3.2.

Proof. The proof is divided in several steps.

Step 1. We first claim that the perfection functor (3.12.1) is an equivalence. Indeed, the functor L Perf ˚sends a set of compact generators to a set of compact generators (see [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF]Proposition 3.30]). We can therefore argue like in the proof of Theorem 3.9 (see also [2, Corollaire B.3]) and consider separately the case Λ " Q and Λ " Z{ℓ N with ℓ prime different from p. The first case is known to be invertible (see [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF]Theorem 6.9]). We can assume now that Λ is a torsion ring. The isomorphism Λp1qµ S,ℓ N holds in RigDA ét pK 5 , Λq -DpK 5 ét , Λq and L Perf ˚induces the analogue isomorphism in PerfDA ét pK 5 , Λq. As in the case of Theorem 2.5, one can show that DpK 5 ét , Λq Ñ PerfDA ét pK 5 , Λq is fully faithful, proving the claim. Step 2. We now assume char K " 0 and we prove that F :" Rj ˚Lι ˚is also an equivalence. We cannot argue as in the previous step since it is not clear that the functor F sends compact objects to compact objects. On the other hand, we remark that it commutes with small sums (see [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF]Remark 7.23]) so that, by Brown's representability theorem, it has a triangulated right adjoint G. As in the proof of Theorem 3.9 we may and do suppose that Λ " Zr1{ps. We use the same letters F and G to indicate these functors defined when Λ " Q and Λ " Z{ℓ in which cases we know they are equivalences of categories (by [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF]Theorem 7.10] and by Rigidity together with Step 1, respectively). We now prove that F and G commute with the functors of change of coefficients ρ ˚(in both cases Q or Z{ℓ). From the conservativity stated in (1) one deduces the following: if G commutes with ρ ˚(which amounts to say that F commutes with ρ ˚) then it commutes with ρ ˚if and only if it commutes with ρ ˚ρ˚w hich is ´b Q or ´b Z{ℓ respectively (property (2)). From the property (3) any triangulated additive functor (like G) commutes with ´b Z{ℓ. Also, in order to show GpS b Qq -GS b Q it suffices to check that for any compact object A one has Hom pA, GpS b Qqq -Hom pA, GS b Qq and this follows from adjunction and the property (4). It then suffices to show that F commutes with ρ ˚. We recall that F is the composition of Lι ˚(which obviously commutes with ρ ˚) and Rj ˚which is termwise defined by means of j ˚which is the effective Quillen right adjoint induced by the inclusion of perfectoid spaces in semi-perfectoid ones. We are left to prove that this functor commutes with ρ ˚. Since its left adjoint obviously does, we deduce the claim by the adjunction arguments above.

Step 4. We can finally prove the statement of the theorem. Fix S in RigDA ét pK, Zr1{psq. We want to prove that F GS -S -GF S. By the property (1) it suffices to show this after applying ρ Q and ρ l . Since F and G commute with them as shown in Step 3, the claim follows from [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF]Theorem 7.10] and Rigidity paired up with Step 1. We remark that along the proof we have shown the compatibility of F with the functor ρ ˚over Z{ℓ N . By construction, this is enough to prove that F is compatible with the ℓ-adic realization functors. Remark 3.14. Let Λ be a Zr 1 p s-algebra, where p is the residual exponential characteristic of K. We remark that there is also an effective version of the motivic tilting equivalence with transfers (see [START_REF] Vezzani | A motivic version of the theorem of Fontaine and Wintenberger[END_REF]). Once it is paired up with the Cancellation Theorem [5, Théorème 2.5.38] and Rigidity, it enables one to prove that the equivalence of Theorem 3.13 (which can also be stated for RigDM using Theorem 3.9) restricts to an equivalence: RigDM eff ét pK, Λq -RigDM eff ét pK 5 , Λq which is the effective version of the previous theorem (with transfers).

We recall that the fields K and K 5 have the same residue field k. In particular, for both of them we have a canonical Quillen adjunction ξ : DA ét pk, Λq Ô RigDA ét pK, Λq : χ ξ 5 : DA ét pk, Λq Ô RigDA ét pK 5 , Λq : χ 5

It is obtained in the following way: first, we can consider the following equivalence, induced by the special fiber functor X Þ Ñ X k (see [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF]Corollaire 1.4.24]):

DA ét pk, Λq -FormDA ét pO K , Λq where we let FormDA ét pK, Λq be the category of étale motives of smooth formal schemes topologically of finite type over O K . Then, we can consider the (Quillen) adjunction This equivalence induces the equivalence RigDM ét pK, Λq Ñ RigDM ét pL, Λq of the claim, which obviously preserves Artin motives.

Lemma A.4. Let S be a smooth rigid variety over K, L{K a finite separable extension and S L the base change of S to L. Then Λ tr L pS L q is an Artin motive if and only if Λ tr K pSq is an Artin motive.

Proof. We can consider the adjoint pair e 7 : RigDM ét pL, Λq Õ RigDM ét pK, Λq : e induced by the smooth map Spec L Ñ Spec K and observe that it restricts to Artin motives. In particular, if Λ tr pSq is Artin then also Λ tr pS L q is. Vice-versa, we can assume that L{K is a normal extension and we suppose that Λ tr pS L q is Artin over L. Then, also e # pΛ tr pS L qq " Λ tr K pS L q is Artin. We can then consider the Cech hypercover U ' of S induced by S L Ñ S and remark that at each stage it is isomorphic to a disjoint union of copies of S L . In particular the simplicial motive Λ tr pU ' q is lewel-wise Artin, and hence also ΛpSqhocolim Λ tr pU ' q is, since Artin motives are closed under sums and cones.

Proposition 1 . 12 .

 112 If Λ is a N-torsion ring with N coprime to the residual (exponential) characteristic of K, then the functors Lι ˚: DpS ét , Λq Õ RigDA eff ét pS, Λq :Rι ånd Lι ˚: DpS ét , Λq Õ RigDM eff ét pS, Λq :Rι åre fully faithful.

Theorem 2 . 5 (

 25 Embedding Theorem for RigDA). Let Λ be a N-torsion ring, where N is invertible in the residue field of K. The functor Lι ˚: DpS ét , Λq Ñ RigDA ét pS, Λq is fully faithful.

( 1 ) 7 :

 17 Any morphism f : Y Ñ Z between rigid analytic varieties over S induces a (Quillen) adjunction f ˚: RigDA ét pZ, Λq Õ RigDA ét pY, Λq :f such that f ˚ΛZ pUq " Λ Y pU ˆZ Y q for any U P Sm Z . (2) If the morphism f : Y Ñ Z is smooth, then f ˚has left adjoint f RigDA ét pY, Λq Õ RigDA ét pZ, Λq : f such that f 7 Λ Y pUq " Λ Z pUq for any U P Sm Y . (3) If i : Z Ñ S isa Zariski closed immersion of varieties over S with open complement j : U Ñ S, then the pair pi ˚, j ˚q is conservative. More precisely, there is an exact triangle j 7 j ˚Ñ Id Ñ i ˚i˚Ñ r1s and canonical equivalences j ˚i˚-0, i ˚i˚-Id . (4) The functor from quasi-projective schemes over Spec A to DG-categories defined as follows Y Þ Ñ RigDA ét pY an , Λq has the structure of a stable homotopic 2-functor in the sense of [3, Définition 1.4.1]. In particular, any morphism f : Y Ñ Z between quasi-projective schemes over Spec A induces an adjoint pair f an ! : RigDA ét pY an , Λq Õ RigDA ét pZ an , Λq : f an ! with the following properties. (i) If f : W Ñ Y is smooth of relative dimension d then, locally on W , we have f an ! p´qf an 7 p´b Λp´dqr´2dsq. (ii) There is a natural transformation f an ! ñ f an ˚which is invertible if f is proper. Proof. See (the étale versions of) [5, Proposition 1.4.9, Corollaire 1.4.28, Théorème 1.4.33] and [3, Scholie 1.4.2]. Remark 2.10. According to the structure of a stable homotopic 2-functor in the sense of [3, Définition 1.4.1], any morphism f : Y Ñ Z between quasi-projective schemes over Spec A also induces an adjoint pair f an ˚: RigDA ét pZ an , Λq Õ RigDA ét pY an , Λq : f an ˚.

Remark 3 . 7 . 3 . 2 .

 3732 By putting S " Spa K the Rigidity Theorem gives equivalences DpK ét , Λq -DApK, Λq -RigDA ét pK, Λq.In particular, ℓ-adic étale cohomology is insensitive to analytification, which is already shown in[START_REF] Huber | Étale cohomology of rigid analytic varieties and adic spaces[END_REF] Corollary 3.8.1]. Rigid motives with and without transfers. The Rigidity Theorem permits to improve the known comparison results about the categories RigDA ét pK, Λq and RigDM ét pK, Λq, similarly to the algebraic case (see[START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF] Annexe B]). The current state of the art in the setting of rigid analytic motives is the following. Theorem 3.8. Let Λ be a Q-algebra. The canonical functor La tr : RigDA ét pK, Λq Ñ RigDM ét pK, Λq is an equivalence of monoidal DG-categories.

Theorem 3 . 12 .

 312 Let Λ be a Q-algebra. There is an equivalence RigDA ét pK, Λq -RigDA ét pK 5 , Λq, Proof. cf. [27, Theorem 7.26] paired up with [29, Corollary 4.20]. See the proof Theorem 3.8 on how to avoid the Frob-localization.

Step 3 .

 3 By (the obvious analogues of) [13, Proposition 5.4.3, Paragraph 5.4.4, Proposition 5.4.12] we know the following facts: (1) Each (Quillen) functor induced by change of coefficients ρ Q : RigDA ét pK, Zr1{psq Ñ RigDA ét pK, Qq and ρ l : RigDA ét pK, Zr1{psq Ñ RigDA ét pK, Z{ℓq with ℓ ‰ p prime has a conservative right adjoint ρ Q˚r esp. ρ ℓ˚a nd the set tρ Q, ρ l u ℓ‰p is a conservative family. (2) ρ ℓ˚ρ l M -M b Z Z{l and ρ Q˚ρ QM -M b Z Q.

( 3 )

 3 M b Z Z{l -ConepM ˆℓ Ñ Mq. (4) If A is compact, then Hom pA, M b Qq -Hom pA, Mq b Q.

  Proposition 1.9. Let S be a normal rigid analytic variety over K. The motivic categories RigDA eff ét pS, Λq [resp. RigDM eff ét pS, Λq] are compactly generated by shifts of motives of the form Λ S pAq [resp. Λ tr S pAq] with A smooth affinoid over S. Similarly, the motivic categories RigDA ét pS, Λq [resp. RigDM ét pS, Λq] are compactly generated by shifts and twists of motives of the form Λ S pAq [resp. Λ tr S pAq] with A smooth affinoid over S. Proof. It suffices to adapt [2, Proposition 3.19] to the rigid setting. We remark that a bound on the cohomological dimension of affinoid rigid analytic varieties can be found in [20, Proposition 0.5.7].

The map of sites Sm S Ñ Cor S induces a monoidal Quillen adjunction La tr : RigDA eff ét pS, Λq Õ RigDM eff ét pS, Λq :Ro tr . (3) The T -stabilization and T tr -stabilization functors induce monoidal Quillen adjunctions

  1) The map of sites S ét Ñ Sm S induces a monoidal Quillen adjunction For the proof of (2) one can easily adapt the argument of[START_REF] Cisinski | Étale motives[END_REF] Paragraph 2.1.7] to the rigid setting. For the other points, the existence of the Quillen pairs follows at once from the universal property of the categories of motives, stabilizations and localizations (to see that the analytification functor preserves A 1 -homotopies we refer to [5, Proposition 1.3.6 and Theoréme 2.5.24]). For the last isomorphisms, see [5, Lemme 2.5.18] and [5, Proposition 1.4.17].We now introduce the canonical functors that we will be mostly interested in. Definition 1.11. By composing the functors of the previous proposition, we obtain the following monoidal Quillen adjunctions (all denoted by Lι ˚and Rι ˚by abuse of notation) Lι ˚: DpS ét , Λq Õ RigDA eff ét pS, Λq :Rι Λq] generated by the essential image of Lι ˚will be called the category of rigid analytic Artin motives over S [with transfers].

	Proof.
	Lι ˚: DpS ét , Λq Õ RigDA eff ét pS, Λq :Rι ˚.
	(2) L Sus: RigDA eff ét pS, Λq Õ RigDA ét pS, Λq :R Ev
	L Sus: RigDM eff ét pS, Λq Õ RigDM ét pS, Λq :R Ev .
	(4) Let S be an affinoid rigid analytic variety S " Spa A. The analytification functor
	induces monoidal Quillen adjunctions
	LAn ˚: DA eff ét pSpec A, Λq Õ RigDA eff ét pS, Λq :RAn LAn
	˚: DM eff ét pSpec A, Λq Õ RigDM eff ét pS, Λq :RAn ˚.
	(5) There is a canonical isomorphism
	La tr T S -T tr S
	in RigDM eff ét pS, Λq and in case S " Spa A is affinoid, there are canonical isomor-
	phisms
	LAn ˚pT Spec A q -T S	LAn ˚pT tr Spec A q -T tr S
	in RigDA eff ét pS, Λq and RigDM eff ét pS, Λq respectively.

Lι ˚: DpS ét , Λq Õ RigDA ét pS, Λq :Rι Lι ˚: DpS ét , Λq Õ RigDM eff ét pS, Λq :Rι ånd Lι ˚: DpS ét , Λq Õ RigDM ét pS, Λq :Rι The full triangulated subcategory with small sums of RigDA ét pS, Λq [resp. of RigDM ét pS,

  13, Proposition 3.2.8], one can assume S is affinoid, equal to Spa A. We remark that Tate algebras are Noetherian (see[START_REF] Fresnel | Rigid analytic geometry and its applications[END_REF] Remarks 3.5.2]). One can then apply[START_REF] Cisinski | Étale motives[END_REF] Proposition 3.2.3] and Λq induced by the analogous natural morphism. Applying the analytification functor LAn ˚: DM eff ét pSpec A, Λq Ñ RigDM eff ét pS, Λq to the isomorphism (2.3.1) we obtain the statement (using Proposition 1.10, (5)).From now on, we will adopt the usual notation for Tate twists in DpS ét , Λq, hence, for any object K in it, we will write Kpnq for the object K b µ bn N . The previous proposition can be rephrased by saying that Lι ˚: DpS ét , Λq Ñ RigDA ét pS, Λq and Lι ˚: DpS ét , Λq Ñ RigDM eff ét pS, Λq preserve the twists. For the case of motives with transfers, the previous proposition easily shows that the functor Lι ˚is fully faithful, and proves the last claim of Theorem 2.1.Theorem 2.4 (Embedding Theorem for RigDM). Let Λ be a N-torsion ring, where N is invertible in the residue field of K. The categories RigDM eff ét pS, Λq and RigDM ét pS, Λq are canonically equivalent. In particular, the functor Lι ˚: DpS ét , Λq Ñ RigDM ét pS, Λq is fully faithful.

	obtain an isomorphism
	(2.3.1)	µ S,N -Λ tr Spec A p1q
	in DM eff ét pSpec A,

1 :

 1 The functor Lι ˚is obtained as the composition of the functor (2.5.1) Lι ˚: DpS ét , Λq Ñ RigDA eff ét pS, Λq and the left adjoint of the adjunction L Sus: RigDA eff ét pS, Λq Õ RigDA ét pS, Λq :R Ev . As the functor (2.5.1) is fully faithful (cf. Proposition 1.10), it is enough to show that R Ev ˝L Sus ˝Lι ˚-Lι ˚. Since RigDA eff ét pS, Λq is compactly generated, it is enough to check that for each compact object M P RigDA eff ét pS, Λq one has that Hom pM, Lι ˚K q -Hom pM, R Ev ˝L Sus ˝Lι ˚pK qq for all K P DpS ét , Λq. By [4, Théoreme 4.3.61] we know that Hom pM, R Ev ˝L Sus ˝Lι ˚pK qq -Hom pL Sus M, L Sus ˝Lι ˚pK qq colim

	nPN

  2) and we replace using Step 2: Lι ˚pK pnqr2nsqq b T bm qq.We now swap RHom with hocolim (using again Lemma 2.6) and Step 3 to obtain hocolim

	hocolim

nPN RHom pT bn , hocolim mPN RHom pT bm , Lι ˚pK q b T bn`m q hocolim nPN RHom pT bn , hocolim mPN RHom pT bm , nPN RHom pT bn , hocolim mPN RHom pT bm , Lι ˚pK pnqr2nsqq b T bm qq hocolim mPN RHom pT bm , hocolim nPN RHom pT bn , Lι ˚pK pnqr2nsqq b T bm qq hocolim mPN RHom pT bm , hocolim nPN RHom pT bn , Lι ˚pK pn `mqqr2n `2msqq hocolim nPN RHom pT bn , Lι ˚pK pnqqr2nsq which is in turn canonically isomorphic to Lι ˚K (by Poincaré duality). The following technical lemma was used in the previous proof. Lemma 2.6. The functor RHom RigDA eff ét pS,Λq pT bn S , ´q commutes with homotopy colimits.

  2.7. Under the hypotheses of Theorem 2.1, the subcategory of rigid analytic Artin motives in RigDA ét pS, Λq [resp. RigDM ét pS, Λq] (see Definition 1.11) is equivalent to the category DpS ét , Λq.

  DpS ét , Λ{J ˚q and RigDA ét pS, Λ{J ˚q defined as in Definition 1.3. The functors M ' Þ Ñ M s induce left Quillen functors s ˚: RigDA ét pS, Λ{J ˚q Ñ RigDA ét pS, Λ{J s q s ˚: DpS ét , Λ{J ˚q Ñ DpS ét , Λ{J s q which are jointly conservative by (the obvious analogues of) [2, Lemme 5.3 and 5.4].

J s ¨Ms " 0 for all s P N. In particular, each M s is canonically a Λ{J s -module. Morphisms are defined level-wise. The category ChpΛ{J ˚-Modq can be endowed with a (projective) model structure, giving rise to the category

  RigDA ét pK, Λq Ô sPerfDA ét pK, Λq : Rι

	ånd
	one can produce explicitly a functor (see [27, Proposition 7.22])
	Rj

˚: sPerfDA ét pK, Λq Ñ PerfDA ét pK, Λq.

  Lp´q η : FormDA ét pO K , Λq Ô RigDA ét pK, Λq : Rp´q η2 the diagram Dpk ét , Λq DA ét pk, Λq DpO K ét , Λq FormDA ét pO K , Λq RigDA art pK, Λq RigDA ét pK, Λq ΛpXq lies in the category generated by the image of Lι ˚˝Lp´q η ˝ξ which is contained in Artin motives. This shows in particular that Λ tr pXq is an Artin motive. Lemma A.3. Let L{K be a finite purely inseparable extension of fields, then there is a commutative diagram of functors DpK ét , Λq RigDM ét pK, Λq DpL ét , Λq RigDM ét pL, Λq where the vertical maps are equivalences and the horizontal are fully faithful functors. Proof. The fact that the horizontal functors are fully faithful can be shown by means of Proposition 2.3 (see the proof of Theorem 2.4). By Proposition 2.2.22 of [5] there is an equivalence of categories CorpK, Λq

	Lι ξ
	ξ
	Lι Lp´qη
	Lp´qη
	Lι

it follows that " Ñ CorpL, Λq.

The first author acknowledges the University of Regensburg with the support of the DFG funded CRC "Higher Invariants. Interactions between Arithmetic Geometry and Global Analysis". The second author was partially supported by the ANR Grant PERCOLATOR: ANR-14-CE25-0002-01 and by the ANR JCJC Grant PERGAMO: ANR-18-CE40-0017.

induced by generic fiber functor X Þ Ñ X η . In particular ξpΛpX k qq -ΛpX η q for any smooth formal scheme topologically of finite type X over O K .

In [START_REF] Vezzani | Rigid cohomology via the tilting equivalence[END_REF] it is shown that these functors commute with the motivic tilting equivalence, whenever Q Ă Λ. With Rigidity, we can now complement this result for any Λ where p is invertible. Corollary 3.15. Let K be a perfectoid field of residual characteristic p and let Λ be a Zr1{psalgebra. Up to a natural transformation, the following diagram of monoidal DG-categories is commutative. Proof. We can assume char K " 0 and we follow the notation and the proof of [START_REF] Vezzani | Rigid cohomology via the tilting equivalence[END_REF]Theorem 3.6]. In particular we use the notion of perfectoid space over k introduced in Section 2 of [START_REF] Vezzani | Rigid cohomology via the tilting equivalence[END_REF] and the relative category of motives. With no hypotheses on Λ, one can define a natural transformation ξ ˝Rj ˚˝Lι ˚ñ Rj ˚˝Lι ˚˝ξ between the two sides of the following square / / PerfDApK, Λq inducing a natural transformation

We therefore obtain a natural transformation

It can be pre-composed with the natural transformation

from Lj ˚˝L Perf ˚ñ Lι ˚(which is induced by the canonical projection to X from the perfection X Perf ). We recall that the tilting equivalence is given from left to right by the functor R Perf ˚˝L5 ˚˝Rj ˚˝Lι ˚. It suffices then to show that α ˝β : ξ 5 ñ R Perf ˚˝L5 ˚˝Rj ˚˝Lι ˚˝ξ is invertible. Using the same strategy as the proof of Theorem 3.13 we can consider separately the case Λ " Q and the case Λ " Z{ℓ. The first case is dealt with in [START_REF] Vezzani | Rigid cohomology via the tilting equivalence[END_REF] while for the second we can invoke the Rigidity Theorem, and the fact that in this case R Perf ˚˝L5 ˚˝Rj ˚˝Lι ˚is just the tilting of Artin motives DpK ét , Λq -DpK 5 ét , Λq.

APPENDIX A. RIGIDITY FOR RIGID MOTIVES WITH TRANSFERS OVER K

In this appendix, we present an alternative, more "geometric" proof of the Rigidity Theorem for RigDM ét pK, Λq with Λ " Z{ℓ N , where we write as usual RigDM ét pK, Λq for RigDM ét pSpa K, Λq. This proof has the advantage of being independent on the full algebraic Rigidity Theorem in its relative form, but relies only on its version over fields for DM ét pK, Λq [21, Corollary 4.8, Theorem 9.35] and for DA ét pK, Λq [2, Lemme 4.6].

We remark that the following proof is just an adaptation of a proof by Ayoub [5, Théorème 2.5.34] mixed with the results on ℓ 1 -alterations of Gabber and Temkin. We will heavily refer to [5, Théorème 2.5.34] and only explain the points where the argument needs to be adapted to our situation.

As usual, we let K be a complete non-Archimedean valued field, we suppose that ℓ is a prime which is invertible in the residue field of K and that K has a finite ℓ-cohomological dimension. From now on, we let Λ be Z{ℓ N for some N P N ą0 .

Theorem A.1. The functor Lι ˚: DpK ét , Λq Ñ RigDM ét pK, Λq is an equivalence of categories.

Proof. Rigidity for DM ét pK, Λq implies Proposition 2.3 in the case S " Spa K using the derived analytification functor LAn ˚. We then know that Lι ˚is fully faithful (cf. Theorem 2.4). In order to conclude the theorem, it suffices to show that a motive of the form Λ tr pXq is Artin, for any given smooth rigid variety X over K.

We remark that motives which are potentially of good reduction (see the definition before [5, Théorème 2.5.34]) are Artin: this follows by combining Lemma A.4 with Lemma A.2. It then suffices to show that Λ tr pXq is in the triangulated category with small sums generated by motives which are potentially of good reduction.

To this aim, it suffices to follow verbatim the proof of [5, Théorème 2.5.34] with the following slight changes.

-By Lemmas A.3 and A.4 a motive is Artin if and only if its base change over a finite extension K 1 {K is. This replaces the first step of [5, Théorème 2.5.34].

-The category RigDM ét pK, Λq is a Verdier localization of RigDM Nis pK, Λq. In particular, Nisnevich weak equivalences are étale weak equivalences and Nisnevich squares give rise to exact triangles also in RigDM ét pK, Λq (see [START_REF] Ayoub | Motifs des variétés analytiques rigides[END_REF]Théorème 2.5.12]).

-Artin motives are closed under tensor products, as Lι ˚is a monoidal functor.

-In the third step of the proof of loc.cit. one can require that the alteration e : Y Ñ X (following the notation in loc.cit. ) has degree d which is coprime to ℓ on the dense open sets where it is finite, using [START_REF] Adiprasito | Log smoothness and polystability over valuation rings[END_REF]Theorem 5.2.18] in place of the alteration proved in [START_REF] Vladimir | Smooth p-adic analytic spaces are locally contractible[END_REF] and using Lemma A.4. We remark that if e 1 : Y Ñ X is a finite morphism of analytic varieties of degree d coprime to ℓ, then the motive Λ tr pXq is a direct summand of Λ tr pY q since e 1 ˝e1 tr " d ¨Id and d P Λ ˚, where e 1 tr is the transpose of e 1 lying in CorpX, Y q. In particular, Λ tr pXq is Artin if Λ tr pY q is. Then, the fact that Λ tr pY q is an Artin motive when Y has poly-stable reduction can be proved in the same way as it is proved in the third step of the proof of [5, Théorème 2.5.34].

The following lemmas were used in the previous proof.

Lemma A.2. Let X be the generic fiber of a smooth formal scheme over O K . The motive Λ tr pXq P RigDM ét pK, Λq is an Artin motive.

Proof. We follow the notation introduced before Corollary 3.15. By hypothesis the motive ΛpXq P RigDA ét pK, Λq lies in the image of the functor Lp´q η : FormDA ét pO K , Λq Ñ RigDA ét pK, Λq. We remark that the category FormDA ét pO K , Λq -DA ét pk, Λq is generated by Artin motives as a consequence of [START_REF] Ayoub | La réalisation étale et les opérations de Grothendieck[END_REF]Lemme 4.6]. Therefore, by the commutativity of