Rigidity for Rigid Analytic Motives

Abstract

fields, introduced in [5] by Ayoub, both in their version with and without transfers. More precisely, given any normal rigid analytic variety S over K, we denote by RigDAé t pS, Λq (resp. RigDMé t pS, Λq) the category ofétale 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:As in the algebraic situation, DpSé t , Λq denotes the derived category of unbounded complexes ofétale sheaves of Λ-modules over the smallétale site and the functors Lι˚arise naturally from the inclusion of the smallétale 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.

The Berkovich realization for rigid analytic motives

“…The main statement and the first property follow from [12,Theorem 3.2]. The second property is proved in [12,Remark 3.3].…”

“…The main statement and the first property follow from [12,Theorem 3.2]. The second property is proved in [12,Remark 3.3]. The third property can be proved at an integral level, be inspecting the functor Dé t (K, Z) cp → D cṕ et (K, Z ℓ ) induced by the integral version of Ré t,ℓ (see [12,Theorem 3.2]).…”

“…In [40] we constructed a de Rham-like realization, giving rise to the rigid realization on DAé t (k, Λ). In [12] we construct the ℓ-adic realizations compatible with the ones of DAé t (k, Λ). In this article, we deal with a Betti-like realization functor.…”

“…The second property is proved in [12,Remark 3.3]. The third property can be proved at an integral level, be inspecting the functor Dé t (K, Z) cp → D cṕ et (K, Z ℓ ) induced by the integral version of Ré t,ℓ (see [12,Theorem 3.2]). By its construction, based on the Rigidity Theorem [12, Theorem 2.1], we see that it is canonically equivalent to the functor induced by extending coefficients, as wanted.…”

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

“…By taking the motivic ℓ-adic realization instead of the p-adic one, the proof above coincides with (a motivic version of) Scholze's proof of the ℓ-adic weight monodromy conjecture for scheme-theoretical complete intersections in toric varieties. For a motivic rigid analytic ℓ-adic realization functor, one can use [BV21].…”