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THE SIX-FUNCTOR FORMALISM FOR RIGID ANALYTIC MOTIVES JOSEPH AYOUB, MARTIN GALLAUER, AND ALBERTO VEZZANI Abstract. Weoffer a systematic study of rigid analytic motives over general rigid analytic spaces, and we develop their six-functor formalism. A key ingredient is an extended proper base change theorem that we are able to justify by reducing to the case of algebraic motives. In fact, more generally, we develop a powerful technique for reducing questions about rigid analytic motives to questions about algebraic motives, which is likely to be useful in other contexts as well. We pay special attention to establishing our results without noetherianity assumptions on rigid analytic spaces. This is indeed possible using Raynaud’s approach to rigid analytic geometry. Contents Introduction 2 Notation and conventions 4 1. Formal and rigid analytic geometry 6 1.1. Recollections 6 1.2. Relation with adic spaces 10 1.3. Étale and smooth morphisms 11 1.4. Topologies 17 2. Rigid analytic motives 23 2.1. The construction 23 2.2. Previously available functoriality 29 2.3. Descent 36 2.4. Compact generation 40 2.5. Continuity, I. A preliminary result 46 2.6. Continuity,II.Approximationuptohomotopy 51 2.7. Quasi-compact base change 60 2.8. Stalks 63 2.9. (Semi-)separatedness 69 arXiv:2010.15004v1 [math.AG] 28 Oct 2020 2.10. Rigidity 74 3. Rigidanalyticmotivesasmodulesinformalmotives 79 3.1. Formal and algebraic motives 80 3.2. Descent, continuity and stalks, I. The case of formal motives 83 3.3. Statement of the main result 85 3.4. Construction of ξ⊗ 86 3.5. Descent, continuity and stalks, II. The case of χΛ-modules 91 e Key words and phrases. Motives (algebraic, formal and rigid analytic), six-functor formalism, proper base change theorem. The first author is partially supported by the Swiss National Science Foundation (SNF), project 200020_178729. The third author is partially supported by the Agence Nationale de la Recherche (ANR), projects ANR-14-CE25-0002 and ANR-18-CE40-0017. 1 3.6. Proof of the main result, I. Fully faithfulness 98 3.7. Proof of the main result, II. Sheafification 107 3.8. Complement 118 4. Thesix-functorformalismforrigidanalyticmotives 139 4.1. Extended proper base change theorem 139 4.2. Weak compactifications 147 4.3. The exceptional functors, I. Construction 150 4.4. The exceptional functors, II. Exchange 156 4.5. Projection formula 173 4.6. Compatibilitywiththeanalytificationfunctor 176 References 180 Introduction In this paper, we study rigid analytic motives over general rigid analytic spaces and we develop a six-functor formalism for them. We have tried to free our treatment from unnecessary hypotheses, and many of our main results hold in great generality, with the notable exception of Theorems 3.3.3(2) and 3.8.1 where we impose étale descent. (This is necessary for the former but might be superfluous for the latter.) In this introduction, we restrict to étale rigid analytic motives with rational coefficients, for which our results are the most complete.1 Rigid analytic motives were introduced in [Ayo15] as a natural extension of the notion of a motive associated to a scheme. Given a rigid analytic space S , we denote by RigDA´et(S ; Q) the -category of étale rigid analytic motives over S with rational coefficients. Given a morphism of rigid∞ analytic spaces f : T S , the functoriality of the construction yields an adjunction → f ∗ : RigDA´et(S ; Q) ⇄ RigDA´et(T; Q): f . (0.1) ∗ When f is locally of finite type, we construct in this paper another adjunction ! f! : RigDA´et(T; Q) ⇄ RigDA´et(S ; Q): f , (0.2) ! and show that the functors f ∗, f , f!, f , and Hom satisfy the usual properties of the six-functor formalism. In particular, we have∗ a base⊗ change theorem for direct images with compact support, i.e., an equivalence g∗ f f ′ g′∗ for every Cartesian square of rigid analytic spaces ◦ ! ≃ ! ◦ g′ T ′ / T f ′ f g S ′ / S with f locally of finite type. Also, we have an equivalence f! f , when f is proper, and an ! ≃ ∗ equivalence f f ∗(d)[2d], when f is smooth of pure relative dimension d. Of course, our six- functor formalism≃ matches the one developed by Huber [Hub96] for the étale cohomology of adic spaces. (Similar formalisms for étale cohomology were also developed by Berkovich [Ber93] and de Jong–van der Put [dJvdP96].) 1In fact, everything we say here holds more generally with coefficients in an arbitrary ring when the class of rigid analytic spaces is accordingly restricted. For instance, if one is only considering rigid analytic spaces over Qp of finite 1 étale cohomological dimension, the results discussed in the introduction are valid with Z[p− ]-coefficients. 2 A partial six-functor formalism for rigid analytic motives was obtained in [Ayo15, §1.4] at a minimal cost as an application of the theory developed in [Ayo07a, Chapitre 1]. Given a non- archimedean field K and a classical affinoid K-algebra A, the assignment sending a finite type an A-scheme X to the -category RigDA´et(X ; Q) is a stable homotopical functor in the sense of [Ayo07a, Définition∞ 1.4.1]. (Here Xan is the analytification of X.) Applying [Ayo07a, Scholie 1.4.2], we have in particular an adjunction as in (0.2) under the assumption that f is algebraizable, i.e., that f is the analytification of a morphism between finite type A-schemes, for some unspecified classical affinoid K-algebra A. Clearly, it is unnatural and unsatisfactory to restrict to algebraizable morphisms, and it is our objective in this paper to remove this restriction. The key ingredient for doing so is Theorem 4.1.4 which we may consider as an extended proper base change theorem for commuting direct images along proper maps and extension by zero along open immersions. It is worth noting that in the algebraic setting, the extended proper base change theorem is essentially a reformulation of the usual one, but this is far from true in the rigid analytic setting. In fact, the usual proper base change theorem in rigid analytic geometry is a particular case of the so-called quasi-compact base change theorem (see Theorem 2.7.1) which is an easier property. The extended proper base change theorem is already known if one restricts to projective mor- phisms and can be deduced from the partial six-functor formalism developed in [Ayo15, §1.4]. However, in the rigid analytic setting, it is not possible to deduce the general case of proper mor- phisms from the special case of projective morphisms. Indeed, the classical argument used in [SGAIV3, Exposé XII] for reducing the proper case to the projective case relies on Chow’s Lemma for which there is no analogue in rigid analytic geometry. (For instance, there are proper rigid an- alytic tori which are not algebraizable [FvdP04, §6.6].) Therefore, a new approach was necessary for proving Theorem 4.1.4 in general. Our approach is based on another contact point with algebraic geometry: instead of using the analytification functor from schemes to rigid analytic spaces, we go backward and associate to a rigid analytic space X the pro-scheme consisting of the special fibers of the different formal models of X. This was already exploited in [Ayo15, Chapitre 1] when proving that, for K = k((π)) with k a field of characteristic zero, there is an equivalence RigSH(K) quSH(k), with quSH(k) SH(G ) the sub- -category of quasi-unipotent motives; see [Ayo15,≃ Scholie 1.3.26] ⊂ mk ∞ for the precise statement. With K/K the extension of K obtained by adjoining an n-th root of π for every n N×, the previous equivalence gives rise to another equivalence ∈ e RigSH(K) SH(k; χ1), (0.3) ≃ where χ1 is a commutative algebra in SH(ke) and SH(k; χ1) is the -category of χ1-modules. G ∞ Moreover, χ1 is the cohomological motive of the pro-scheme ( mk)n N× where the transition map 1 ∈ corresponding to m n is elevation to the power nm− . In this paper, we will prove a vast gen- eralisation of the equivalence| (0.3). Roughly speaking, for a rigid analytic space S , we obtain a description of the -category RigDA (S ; Q) in terms of modules over commutative algebras in - ∞ ´et ∞ categories of algebraic motives. More precisely, we will show that the functor S RigDA´et(S ; Q) is equivalent to the étale sheaf associated to the functor 7→ S colim DA´et(Sσ; χSQ) S Mdl(S ) 7→ ∈ considered as a presheaf valued in presentable -categories. Here Mdl(S ) is the category of formal ∞ models of S , Sσ is the special fiber of the formal scheme S, χSQ is a commutative algebra in the -category DA´et(Sσ; Q) of motives over Sσ and DA´et(Sσ; χSQ) is the -category of modules ∞ 3 ∞ over χSQ. Moreover, the commutative algebra χSQ admits a concrete description. For precise statements, see Theorems 3.3.3 and 3.8.1. The above description of the -categories RigDA´et(S ; Q) can be used to deduce the extended proper base change theorem in∞ rigid analytic geometry (i.e., Theorem 4.1.4) from its algebraic analogue. In fact, it turns out that we only need a formal consequence of this description which happens to be also a key ingredient in its proof, namely Theorem 3.6.1 (see also Theorem 4.1.3). Once Theorem 4.1.4 is proven, it is easy to construct the adjunction (0.2). Besides the six-functor formalism, the paper contains several foundational results.
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