A motivic version of the theorem of Fontaine and Wintenberger

Vezzani, Alberto

doi:10.1112/s0010437x18007595

Cited by 15 publications

(68 citation statements)

References 46 publications

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“…In the case when char K " p, the perfection functor induces an adjunction (3.12.1) L Perf˚: RigDAé t pK, Λq Ô PerfDAé t pK, Λq : R Perfẘ hich is shown to be invertible if Λ is a Q-algebra (see [27,Theorem 6.9] and the proof of Theorem 3.8 to deduce the stable statement without Frob-localization).…”

Section: 2mentioning

confidence: 99%

“…which is shown to be invertible if Λ is a Q-algebra (see [27]). The contribution of this paper to this topic is the following theorem.…”

Section: 2mentioning

confidence: 99%

“…(2) We can prove an equivalence between rigid analytic motives with and without transfers with coefficients over Zr1{ps where p is the residue exponential characteristic of K. (3) Over a perfectoid field, the motivic tilting equivalence (cf. [27]) can be promoted to Zr1{ps-coefficients, and the ℓ-adic realization functors can be shown to be compatible with it.…”

Section: Introductionmentioning

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: 2mentioning

confidence: 99%

“…which is shown to be invertible if Λ is a Q-algebra (see [27]). The contribution of this paper to this topic is the following theorem.…”

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rigidity for Rigid Analytic Motives

Bambozzi¹,

Vezzani²

2019

J. Inst. Math. Jussieu

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“…We will now decompose this diagram in some sub-squares following the picture of [41,Page 40]. We recall (see [41,Theorem 7.11…”

Section: Compatibility With the Tilting Equivalencementioning

confidence: 99%

The Berkovich realization for rigid analytic motives

Vezzani¹

2019

Journal of Algebra

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“…Indeed, for rigid analytic varieties as well as for perfectoid spaces, the de Rham complex is still problematic (its cohomology groups can be oddly infinite-dimensional for smooth, affinoid rigid analyitic varieties). Nonetheless, the results of [15] and [23] show that some natural de Rham cohomology groups for both smooth rigid analytic varieties as well as smooth perfectoid spaces can truly be defined.…”

Section: Introductionmentioning

confidence: 99%

Rigid cohomology via the tilting equivalence

Vezzani

2019

Journal of Pure and Applied Algebra

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We define a de Rham cohomology theory for analytic varieties over a valued field K ♭ of equal characteristic p with coefficients in a chosen untilt of the perfection of K ♭ by means of the motivic version of Scholze's tilting equivalence. We show that this definition generalizes the usual rigid cohomology in case the variety has good reduction. We also prove a conjecture of Ayoub yielding an equivalence between rigid analytic motives with good reduction and unipotent algebraic motives over the residue field, also in mixed characteristic.

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A motivic version of the theorem of Fontaine and Wintenberger

Cited by 15 publications

References 46 publications

Rigidity for Rigid Analytic Motives

Rigidity for Rigid Analytic Motives

The Berkovich realization for rigid analytic motives

Rigid cohomology via the tilting equivalence

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