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
We investigate the maximal finite length submodule of the Breuil-Kisin prismatic cohomology of a smooth proper formal scheme over a p-adic ring of integers. This submodule governs pathology phenomena in integral p-adic cohomology theories. Geometric applications include a control, in low degrees and mild ramifications, of (1) the discrepancy between two naturally associated Albanese varieties in characteristic p, and (2) kernel of the specialization map in p-adic étale cohomology. As an arithmetic application, we study the boundary case of the theory due to Fontaine-Laffaille, Fontaine-Messing, and Kato. Also included is an interesting example, generalized from a construction in Bhatt-Morrow-Scholze's work, which (1) illustrates some of our theoretical results being sharp, and (2) negates a question of Breuil. Contents1. Introduction 2. Various modules and their Galois representations 2.1. Kisin modules 2.2. Breuil modules 2.3. Fontaine-Laffaille modules 2.4. Relations to Galois representations 3. Boundary degree prismatic cohomology 3.1. Structure of u ∞ -torsion 3.2. Comparing Frobenius and Verschiebung 3.3. Induced Nygaard filtration 4. Geometric applications 4.1. The discrepancy of Albanese varieties 4.2. The p-adic specialization maps 4.3. Revisiting the integral Hodge-de Rham spectral sequence 5. Crystalline cohomology in boundary degree 5.1. Understand filtrations 5.2. Compute divided Frobenius 5.3. The connection 5.4. Fontaine-Laffaille and Breuil modules 5.5. Comparison to étale cohomology 6. An example 6.1. Raynaud's theorem on prolongations References
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