Relative log convergent cohomology and relative rigid cohomology II

TL;DR

This paper extends the theory of relative log convergent cohomology, compares it with other cohomologies, and proves a version of Berthelot's conjecture on overconvergence for proper smooth families.

Contribution

It develops the theory of relative log convergent cohomology of radius λ and proves a conjecture on overconvergence in the context of rigid cohomology.

Findings

Established a generalized theory of relative log convergent cohomology.

Compared this cohomology with other types like crystalline and rigid cohomology.

Proved a version of Berthelot's conjecture on overconvergence for proper smooth families.

Abstract

In this paper, we develop the theory of relative log convergent cohomology of radius $\lambda$ ($0 < \lambda \leq 1$), which is a generalization of the notion of relative log convergent cohomology in the previous paper. By comparing this cohomology with relative log crystalline cohomology, relative rigid cohomology and its variants and by using some technique of hypercovering, we prove a version of Berthelot's conjecture on the overconvergence of relative rigid cohomology for proper smooth families.