We study Tate motives with integral coefficients through the lens of tensor triangular geometry. For some base fields, including Q and F p , we arrive at a complete description of the tensor triangular spectrum and a classification of the thick tensor ideals.

In this note we study the local projective model structure on presheaves of complexes on a site, i. e. we describe its classes of cofibrations, fibrations and weak equivalences. In particular, we prove that the fibrant objects are those satisfying descent with respect to all hypercovers. We also describe cofibrant and fibrant replacement functors with pleasant properties.

Abstract. In characteristic 0 there are essentially two approaches to the conjectural theory of mixed motives, one due to Nori and the other one due to, independently, Hanamura, Levine, and Voevodsky. Although these approaches are apriori quite different it is expected that ultimately they can be reduced to one another. In this article we provide some evidence for this belief by proving that their associated motivic Galois groups are canonically isomorphic.

We offer 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 1.2. Relation with adic spaces 1.3. Étale and smooth morphisms 1.4. Topologies 2. Rigid analytic motives 2.1. The construction 2.2. Previously available functoriality 2.3. Descent 2.4. Compact generation 2.5. Continuity, I. A preliminary result 2.6. Continuity, II. Approximation up to homotopy 2.7. Quasi-compact base change 2.8. Stalks 2.9.(Semi-)separatedness 2.10. Rigidity 3. Rigid analytic motives as modules in formal motives 3.1. Formal and algebraic motives 3.2. Descent, continuity and stalks, I. The case of formal motives 3.3. Statement of the main result 3.4. Construction of ξ ⊗ 3.5. Descent, continuity and stalks, II. The case of χΛ-modules 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. The six-functor formalism for rigid analytic motives 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. Compatibility with the analytification functor 176 References 180 Proof. This follows immediately from [Gro67, Chapitre IV, Théorème 18.1.2].Notation 1.4.3. Given a rigid analytic space S , we denote by Ét/S the category of étale rigid analytic S -spaces. We denote by Ét gr /S the full subcategory of Ét/S spanned by those étale rigid analytic S -spaces with good reduction (in the sense of Definition 1.1.13).Definition 1.4.4. Let (Y i → X) i be a family of étale morphisms of rigid analytic varieties. We say that this family is a Nisnevich cover if, locally on X and after refinement, it admits a formal model (Y i → X) i which is a Nisnevich cover. Nisnevich covers generate a topology on rigid analytic spaces which we call the Nisnevich topology.Definition 1.4.5. Let ( f : Y i → X) i be a family of étale morphisms of rigid analytic varieties. We say that this family is an étale cover if it is jointly surjective, i.e., |X| = i f (|Y i |). Étale covers generate the...

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