The crystalline comparison of Ainf-cohomology: the case of good reduction

Yao, Zijian

doi:10.48550/arxiv.1908.06366

Cited by 2 publications

(3 citation statements)

References 7 publications

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“…where the left vertical arrow is given by [BS19, Theorem 17.2] and the bottom horizontal arrow is given by [BMS18, Theorem 12.1] or [Yao19].…”

Section: This Induces a Surjection Of Shifted Cotangent Complexes: L ...mentioning

confidence: 99%

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Comparison of prismatic cohomology and derived de Rham cohomology

Shizhang¹,

Liu²

2020

Preprint

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We establish a comparison isomorphism between prismatic cohomology and derived de Rham cohomology respecting various structures, such as their Frobenius actions and filtrations. As an application, when X is a proper smooth formal scheme over O K with K being a p-adic field, we improve Breuil-Caruso's theory on comparison between torsion crystalline cohomology and torsion étale cohomology. Contents 1. Introduction Acknowledgement 2. Preliminaries 2.1. Transversal prisms 2.2. Envelopes and derived de Rham cohomology 2.3. Frobenii 2.4. Naïve comparison 3. Comparing prismatic and derived de Rham cohomology 3.1. The comparison 3.2. Functorial endomorphisms of derived de Rham complex 4. Filtrations 4.1. Hodge filtration on dR ∧ R/A 4.2. Nygaard filtration 4.3. Comparing Hodge and Nygaard filtrations 5. Connection on dR ∧ −/S and structure of torsion crystalline cohomology 5.1. Connection on dR ∧ −/S 5.2. Structures of torsion crystalline cohomology 5.3. Galois action on torsion crystalline cohomology 6. Torsion Kisin module, Breuil module and associated Galois representations 6.1. (Generalized) Kisin modules 6.2. Galois representation attached to étale Kisin modules 6.3. Torsion Breuil modules 7. Torsion cohomology and comparison with étale cohomology 7.1. Prismatic cohomology and (generalized) Kisin modules 7.2. Nygaard filtration and Breuil-Kisin filtration 7.3. Torsion crystalline cohomology 8. Some calculations on T S 8.1. Identification on (6.20) and (6.19) 8.2. T S and T st, References

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“…where the left vertical arrow is given by [BS19, Theorem 17.2] and the bottom horizontal arrow is given by [BMS18, Theorem 12.1] or [Yao19].…”

Section: This Induces a Surjection Of Shifted Cotangent Complexes: L ...mentioning

confidence: 99%

“…Then our comparison here becomes the one established by [BMS18, Theorem 1.8. (iii)] (see also [Yao19]).…”

Section: Introductionmentioning

confidence: 96%

Comparison of prismatic cohomology and derived de Rham cohomology

Shizhang¹,

Liu²

2020

Preprint

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“…A variant of the infinitesimal site for smooth rigid spaces has been considered by Zijian Yao in [Yao19, Section 5], where crystalline cohomology [BMS,Section 13] is reconstructed conceptually, and is compared with pro-étale cohomology of the de Rham period sheaf. Using the Čech-Alexander complex, it can be shown that our RΓ inf (X/B + dR ) for smooth rigid spaces coincides with crystalline cohomology of [Yao19].…”

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confidence: 99%

Crystalline cohomology of rigid analytic spaces

Guo¹

2021

Preprint

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In this article, we introduce infinitesimal cohomology for rigid analytic spaces that are not necessarily smooth, with coefficients in a p-adic field or Fontaine's de Rham period ring B + dR . Contents1. Introduction 1.1. Background 1.2. Main results 1.3. Summary of sections 1.4. Acknowledgements 2. Infinitesimal geometry over B + dR,e 2.1. de Rham period ring and infinitesimal sites 2.2. Envelopes 2.3. Infinitesimal and rigid topology 2.4. Functoriality 3. Crystals 3.1. Crystals and their connections 3.2. A criterion for crystal in vector bundles 3.3. Integrable connections over envelope 4. Cohomology over B + dR,e 4.1. Cohomology of crystals over affinoid spaces 4.2. Global result 5. Derived de Rham cohomology over B + dR,e 5.1. Topological algebras over A inf,e 5.2. Analytic cotangent complex: affinoid case 5.3. Derived de Rham complex: affinoid case 5.4. Global constructions 5.5. Infinitesimal cohomology and derived de Rham complex 6. Éh descent 6.1. Descent for blowup coverings 6.2. Éh-hyperdescent in general 6.3. Finiteness 6.4. Algebraic and analytic infinitesimal cohomology 7. Cohomology over B + dR 7.1. Infinitesimal sites and topoi over B + dR 7.2. Cohomology of crystals over X/Σ inf 7.3. Comparison with pro-étale cohomology References

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The crystalline comparison of Ainf-cohomology: the case of good reduction

Abstract: We provide a simple approach for the crystalline comparison of A inf -cohomology, and reprove the comparison between crystalline and p-adic étale cohomology for formal schemes in the case of good reduction.

Cited by 2 publications

References 7 publications

Comparison of prismatic cohomology and derived de Rham cohomology

Comparison of prismatic cohomology and derived de Rham cohomology

Crystalline cohomology of rigid analytic spaces

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