The Monsky–Washnitzer and the overconvergent realizations

Vezzani, Alberto

doi:10.1093/imrn/rnw335

Cited by 15 publications

(23 citation statements)

References 43 publications

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“…As for concrete applications, our theorem allows the '(un-)tilting' and '(de-)perfectoidifying' of any motivic cohomology theory on rigid or perfectoid spaces satisfyingétale descent and homotopy invariance, having coefficients over Q (it is expected to extend this over Z[1/p] as well, see Remark 7.28). Such an example is the overconvergent de Rham cohomology for rigid analytic varieties over K. It gives rise to new cohomology theories 'à la de Rham' for (small) perfectoid spaces as well as for rigid varieties over local fields of equi-characteristic p, satisfying descent, homotopy invariance, finite-dimensionality and further formal properties (see [Vez18]), which are compatible with rigid cohomology [Vez19]. The relation with p-adic periods is an object of future research.…”

Section: A Motivic Version Of the Theorem Of Fontaine And Wintenbergermentioning

confidence: 99%

“…Such an example is the overconvergent de Rham cohomology for rigid analytic varieties over . It gives rise to new cohomology theories ‘à la de Rham’ for (small) perfectoid spaces as well as for rigid varieties over local fields of equi-characteristic , satisfying descent, homotopy invariance, finite-dimensionality and further formal properties (see [Vez18]), which are compatible with rigid cohomology [Vez19]. The relation with -adic periods is an object of future research.…”

Section: Introductionmentioning

confidence: 99%

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A motivic version of the theorem of Fontaine and Wintenberger

Vezzani¹

2018

Compositio Math.

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We establish a tilting equivalence for rational, homotopy-invariant cohomology theories defined over non-archimedean analytic varieties. More precisely, we prove an equivalence between the categories of motives of rigid analytic varieties over a perfectoid field K of mixed characteristic and over the associated (tilted) perfectoid field K ♭ of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger, claiming that the Galois groups of K and K ♭ are isomorphic.

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Section: A Motivic Version Of the Theorem Of Fontaine And Wintenbergermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A motivic version of the theorem of Fontaine and Wintenberger

Vezzani¹

2018

Compositio Math.

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“…Overconvergent Hyodo-Kato cohomology via presentations of dagger structures. In this section we introduce a definition of overconvergent Hyodo-Kato cohomology using presentations of dagger structures (see [42,Appendix], [16,Sec. 6.3]).…”

Section: 21mentioning

confidence: 99%

On the cohomology of $p$-adic analytic spaces, I: The basic comparison theorem

Colmez¹,

Nizioł²

2021

Preprint

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The purpose of this paper is to prove a basic p-adic comparison theorem for smooth rigid analytic and dagger varieties over the algebraic closure C of a p-adic field: p-adic pro-étale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over B + dR ). The key computation is the passage from absolute crystalline cohomology to Hyodo-Kato cohomology and the construction of the related Hyodo-Kato isomorphism. We also "geometrize" our comparison theorem by turning p-adic pro-étale and syntomic cohomologies into sheaves on the category Perf C of perfectoid spaces over C (this geometrization will be crucial in our proof of the Cst-conjecture in the sequel to this paper). ContentsPIERRE COLMEZ AND WIESŁAWA NIZIOŁ 7.2. Rigid analytic varieties, period morphism 55 7.3. Dagger varieties 63 Index 66 References 66This proposition is proved by representing, using distinguished triangles (1.5) and (1.9), both sides of the morphism by means of the rigid analytic and the overconvergent Hyodo-Kato cohomology, respectively, then passing through the rigid analytic and the overconvergent Hyodo-Kato quasi-isomorphisms (that are compatible by construction) to the de Rham cohomology, where the result is known.Remark 1.11. The approach we have taken here to deal with dagger varieties is very different from the one in [14] or [16] (these two approaches also differing between themselves). That is, we do not use Grosse-Klönne's overconvergent Hyodo-Kato cohomology nor the related Hyodo-Kato morphism (which is difficult to work with and is also very different from the rigid analytic version making checking the overconvergent-rigid analytic compatibility a bit of a nightmare). Instead, we induce all the overconvergent cohomologies from their rigid analytic analogs; hence, by definition, the two constructions are compatible. This was only possible because we have constructed a functorial, ∞-category version of the Hyodo-Kato morphism.Structure of the paper. Sections 2 and 4 are devoted to a definition of a functorial, ∞-categorical Hyodo-Kato quasi-isomorphism. In Section 3 we present our definition of B + dR -cohomology. Section 5 puts the above things together and introduces overconvergent geometric syntomic cohomology. In Section 6 we define comparison morphisms and in Section 7 we put a geometric structure on them.

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“…Proof. By construction, it suffices to prove the statement for the K-de Rham cohomology of analytic K-varieties, that is the overconvergent de Rham cohomology with coefficients in K. All the properties above are proved in [24,Section 5].…”

Section: De Rham Cohomology Over Local Fields Of Positive Characteristicmentioning

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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The Monsky–Washnitzer and the overconvergent realizations

Cited by 15 publications

References 43 publications

A motivic version of the theorem of Fontaine and Wintenberger

A motivic version of the theorem of Fontaine and Wintenberger

On the cohomology of $p$-adic analytic spaces, I: The basic comparison theorem

Rigid cohomology via the tilting equivalence

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