Relative log convergent cohomology and relative rigid cohomology I

TL;DR

This paper develops the theory of relative log convergent cohomology, proving its coherence in certain cases and relating it to relative rigid cohomology to support Berthelot's conjecture on overconvergence.

Contribution

It introduces the theory of relative log convergent cohomology and establishes key comparison theorems with existing cohomologies, advancing understanding of their coherence and overconvergence.

Findings

Proves coherence of relative log convergent cohomology in specific cases.

Establishes comparison between relative log convergent and log crystalline cohomology.

Supports Berthelot's conjecture on overconvergence of relative rigid cohomology.

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.