The de Rham-Fargues-Fontaine cohomology

Abstract

We show how to attach to any rigid analytic variety V over a perfectoid space P a rigid analytic motive over the Fargues-Fontaine curve X (P ) functorially in V and P . We combine this construction with the overconvergent relative de Rham cohomology to produce a complex of solid quasi-coherent sheaves over X (P ), and we show that its cohomology groups are vector bundles if V is smooth and proper over P or if V is quasi-compact and P is a perfectoid field, thus proving and generalizing a conjecture of Scholze. The main ingredients of the proofs are explicit B 1 -homotopies, the motivic proper base change and the formalism of solid quasi-coherent sheaves. CONTENTSThe authors are partially supported by the Agence Nationale de la Recherche (ANR), projects ANR-14-CE25-0002 and ANR-18-CE40-0017.

The six-functor formalism for rigid analytic motives

“…A similar property is also true for DA ét (−; Q), but the proof in the rigid analytic setting is much more involved and relies on approximation techniques as those used in the proof of [Vez19,Proposition 4.5]. We also like to mention that this continuity property for RigDA ét (−; Q) plays a crucial role (along with many of the results described above) in the recent paper [LBV21] where a new relative cohomology theory for rigid analytic varieties over a positive characteristic perfectoid space P is defined and studied. Interestingly, this relative cohomology theory takes values in solid quasi-coherent sheaves over the relative Fargues-Fontaine curve associated to P.…”

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

“…According to Lemma 2.8, the log schemes O × K and O × K ♭ (see Notation 2.7) induce the same log structure on the residue field. Then the compatibility of the motivic tilting equivalence and the Monsky-Washnitzer functor can be stated as follows, which generalizes [Vez19b, Theorem 3.2] (see also [LBV21,Proposition 5.11]).…”

“…From this, the existence of a tubular neighborhood that does not change the motive immediately follows. A similar trick is used in the proof of [LBV21,Theorem 4.46].…”

“…The Frobenius-enrichment F : DA(k) → DA(k) hϕ ω on the category DA(k) (see Remark 2.32) induces Mod χ 0 1 (DA(k) ct ) → Mod F (χ 0 1) (DA(k) ct,hϕ ) → Mod χ 0 1 (DA(k) ct ) hϕ ∼ = RigDA gr (K 0 ) ct,hϕ , where now the Frobenius operator ϕ on the right is by the lift of Frobenius. In particular, by applying we apply the de Rham realization (see [LBV21,Corollary 4.39]) we obtain RΓ dR,K 0 : RigDA gr (K) ct → RigDA gr (K 0 ) ct,hϕ → QCoh(Spa K 0 ) ct,hϕ ∼ = D ϕ (K 0 ) ct . When the residue field k of K is finite, this satisfies the hypotheses of Proposition A.8.…”

“…The key ingredient for this step is the motivic tilting equivalence established in [Vez19a], which provides a natural way to discuss relations between tilting and p-adic cohomology theories (cf. [LBV21]). It is stated in terms of the theory of motives of rigid analytic varieties RigDA(K) introduced by Ayoub [Ayo15], which can be applied to our situation since Hyodo-Kato cohomology can be extended to rigid analytic varieties thanks to the work of Colmez-Nizioł [CN19], and it can be shown to be motivic.…”