Relative log convergent cohomology and relative rigid cohomology I

Abstract

In this paper, we develop the theory of relative log convergent cohomology. We prove the coherence of relative log convergent cohomology in certain case by using the comparison theorem between relative log convergent cohomlogy and relative log crystalline cohomology, and we relates relative log convergent cohomology to relative rigid cohomology to show the validity of Berthelot's conjecture on the coherence and the overconvergence of relative rigid cohomology for proper smooth families when they admit nice proper log smooth compactification to which the coefficient extends logarithmically.

arXiv comment69 pages. Errors and typos fixed

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Cited by in corpus (15)

• Incarnations of Berthelot's conjecture
• Maximal tori of monodromy groups of $F$-isocrystals and an application to abelian varieties
• Comparison between rigid syntomic and crystalline syntomic cohomology for strictly semistable log schemes with boundary
• Rigid analytic reconstruction of Hyodo--Kato theory
• Clemens-Schmid exact sequence in characteristic p
• Perfect points of abelian varieties
• Full faithfulness without Frobenius structure and partially overcoherent isocrystals
• Locally analytic representations of $\mathrm{GL}(2,L)$ via semistable models of ${\mathbb{P}}^1$
• Hyodo-Kato theory with syntomic coefficients
• A crystalline incarnation of Berthelot's conjecture and Künneth formula for isocrystals
• Monodromy Criterion for the Good Reduction of K3 Surfaces
• On logarithmic extension of overconvergent isocrystals
• Poincaré duality for rigid analytic Hyodo--Kato theory
• On extension of overconvergent log isocrystals on log smooth varieties
• Around the nearby cycle functor for arithmetic $\mathcal{D}$-modules