The six-functor formalism for rigid analytic motives

Abstract

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...

On the p-adic weight-monodromy conjecture for complete intersetions in toric varieties

“…We prove this as a consequence of the "spreading out" property of rigid analytic motives shown in [AGV20,Theorem 2.8.15]. In particular, we can take a tubular neighborhood that does not change the ℓ-adic cohomology groups independently on ℓ, which reproves/generalizes, with a completely different method, a smooth intersection case of the main result of [Ito20], in which this ℓ-independence property is proved in an algebraizable situation but including nonsmooth intersections, using the theory of nearby cycles over general bases.…”

The de Rham-Fargues-Fontaine cohomology

“…it can be defined as a contravariant realization functor dR S : RigDA(S) → QCoh(S) op on the (unbounded, derived, stable, étale) category RigDA(S) of rigid analytic motives over S with values in the infinity-category of solid quasi-coherent O S -modules. As a matter of fact, in order to prove the properties above we make extensive use of the theory of motives, and more specifically of their six-functor formalism [AGV20] and of a homotopy-based relative version of Artin's approximation lemma (Theorem 3.8) inspired by the absolute motivic proofs given in [Vez18]. Moreover, if X is a proper smooth rigid variety over S, dR S (X) is a perfect complex, whose cohomology groups are vector bundles.…”

“…Moreover, if X is a proper smooth rigid variety over S, dR S (X) is a perfect complex, whose cohomology groups are vector bundles. To prove this finiteness result, we combine the characterization of dualizable objects in QCoh(S) due to Andreychev, [And21] (see also [Sch20a]), the motivic proper base change and the "continuity" property for rigid analytic motives (see [AGV20]). The latter result, which is based on the use of explicit rigid homotopies, states that whenever one has a weak limit of adic spaces (in the sense of Huber) X ∼ lim ← − X i then any compact motive over X has a model over some X i .…”

“…As a matter of fact, in all what follows one can replace the category Adic with any subcategory of adic spaces over Z p which are locally of finite Krull dimension that is stable under open immersions, finite étale extensions as well as relative discs, and that contains reduced rigid analytic varieties and relative Fargues-Fontaine curves. Alternatively, one may consider the (larger) category of rigid spaces as defined by [FK18] and considered in [AGV20]. In this article, we stick to an adic perspective and we leave it to the reader to extend the statements and definitions of the present article to any more general setting.…”

“…Remark 2.5. Contrarily to [AGV20], we use the notation RigDA (eff) (S) to refer both to the presentable category in Pr L as well as to the structure RigDA (eff) (S) ⊗ of symmetric monoidal category in CAlg(Pr L ) it is equipped with.…”