The Berkovich realization for rigid analytic motives

Vezzani, Alberto

doi:10.1016/j.jalgebra.2019.02.026

Cited by 2 publications

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“…We follow [4] and we show how to use the Rigidity Theorem to produceétale realization functors. In particular, we can produce an -adic realization functor for the category RigDA´e t (K , Q) completing the picture of 'classical' realizations for rigid analytic motives (for a p-adic realization for this category see [27] and for a Betti-like realization see [30]).…”

Section: Theétale Realization Functormentioning

confidence: 99%

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Rigidity for Rigid Analytic Motives

Bambozzi¹,

Vezzani²

2019

J. Inst. Math. Jussieu

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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.

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Section: Theétale Realization Functormentioning

confidence: 99%

“…We refer to Section 2 of [28] for more details about RigDA eff ét pS, Λq (in its Nisnevich form). We only point out that this DG-category can be defined as a Verdier quotient of DpPshpSm S , Λqq with respect to étale descent and B 1 -invariance.…”

Section: Introductionmentioning

confidence: 99%