Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties

Shiho, Atsushi; Nakkajima, Yukiyoshi

doi:10.1007/978-3-540-70565-9

Cited by 22 publications

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“…By [NS08] Proposition 1.6.4, we have the equality of functors Q * X/S Rθ X/S,crys * Rθ X/S,Rcrys * Q * X • /S . There is an isomorphism Ra crys * Rθ D/S,crys * = Rθ X/S,crys * Ra • crys * by [NS08] (1.6.0.13). Since a and a • are closed immersions, we see…”

Section: Frobenius Compatibilitymentioning

confidence: 99%

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On relative and overconvergent de Rham–Witt cohomology for log schemes

Matsuue

2016

Math. Z.

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Abstract. We construct the relative log de Rham-Witt complex. This is a generalization of the relative de Rham-Witt complex of Langer-Zink to log schemes. We prove the comparison theorem between the hypercohomology of the log de Rham-Witt complex and the relative log crystalline cohomology in certain cases. We construct the p-adic weight spectral sequence for relative proper strict semistable log schemes. When the base log scheme is a log point, We show it degenerates at E 2 after tensoring with the fraction field of the Witt ring. We also extend the definition of the overconvergent de Rham-Witt complex of Davis-Langer-Zink to log schemes (X, D) associated with smooth schemes with simple normal crossing divisor over a perfect field. Finally, we compare its hypercohomology with the rigid cohomology of X \ D.

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Section: Frobenius Compatibilitymentioning

confidence: 99%

“…We also remark that there are other studies to construct a (p-adic) weight filtration. Nakkajima and Shiho [NS08] construct a theory of weights on the log crystalline cohomology of a family of open smooth variety. They used the log de Rham complex of a lift to define a weight filtration.…”

mentioning

confidence: 99%

On relative and overconvergent de Rham–Witt cohomology for log schemes

Matsuue

2016

Math. Z.

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“…an open immersion) of schemes. Then, when f is a smooth morphism, the claim follows from Lemma 2.3.14 of [NS08]. When f is an open immersion, the existence and the uniqueness of f is clear because Spec(A/J) is homeomorphic to Spec(A), and the affinity of U can be proven in the same way as the proof of Lemma 2.3.14 of [NS08].…”

Section: Remark 18mentioning

confidence: 78%

Prismatic and $q$-crystalline sites of higher level

Li¹

2021

Preprint

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In this article, we define the m-prismatic site and the m-q-crystalline site, which is a higher level analogue of the prismatic site and the qcrystalline site respectively. We prove a certain equivalence between the category of crystals on the m-prismatic site (resp. the m-q-crystalline site) and that on the prismatic site (resp. the q-crystalline site), which can be regarded as the prismatic (resp. the q-crystalline) analogue of the Frobenius descent due to Berthelot and the Cartier transform due to Ogus-Vologodsky, Oyama and Xu. We also prove the equivalence between the category of crystals on the m-prismatic site and that on the (m − 1)q-crystalline site. Contents 5 Relation with the results of Xu, Gros-Le Stum-Quirós and Morrow-Tsuji 41 References 47

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“…side, this follows from 1.0.8 or 12.9.1 of Nakkajima (2012). For the right-hand side, we apply Theorem 3.5.4, p. 243, of Nakkajima and Shiho (2008): H i rig (V ) = 0 for i > 2n and the spectral sequence 3.5.4.1 degenerates at E 1 , implying that only finitely many X j contribute to the rigid cohomology of V . The bound on d comes from the arguments following the isomorphism 3.5.0.4 on p. 242 ibid.…”

Section: Smooth Varietiesmentioning

confidence: 99%

THE p-COHOMOLOGY OF ALGEBRAIC VARIETIES AND SPECIAL VALUES OF ZETA FUNCTIONS

Milne¹,

Ramachandran²

2014

J. Inst. Math. Jussieu

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The p-cohomology of an algebraic variety in characteristic p lies naturally in the category D b c (R) of coherent complexes of graded modules over the Raynaud ring (Ekedahl-Illusie-Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc. and the special values of their zeta functions. These results provide compelling evidence that D b c (R) is the correct target for p-cohomology in characteristic p.

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Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties

Cited by 22 publications

References 0 publications

On relative and overconvergent de Rham–Witt cohomology for log schemes

On relative and overconvergent de Rham–Witt cohomology for log schemes

Prismatic and $q$-crystalline sites of higher level

THE p-COHOMOLOGY OF ALGEBRAIC VARIETIES AND SPECIAL VALUES OF ZETA FUNCTIONS

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