Cohomology of rigid curves with semi-stable coverings

TL;DR

This paper constructs a semi-stable formal model for wide open rigid curves with semi-stable coverings, analyzing their l-adic cohomology via components and graph homology, and establishing functoriality under compatible morphisms.

Contribution

It provides a new method to describe the l-adic cohomology of rigid curves using semi-stable models and graph homology, including functorial properties.

Findings

Explicit description of l-adic cohomology in terms of components and graphs

Construction of semi-stable formal models for wide open rigid curves

Proof of functoriality under compatible finite flat morphisms

Abstract

We construct a semi-stable formal model of a wide open rigid curve with a semi-stable covering, and study the l-adic cohomology of the rigid curve. We describe the l-adic cohomology of the rigid curve using the l-adic cohomology of the irreducible components of a semi-stable reduction, and homology and cohomology of some graphs. We also prove the functoriality of the description for a finite flat morphism that is compatible with semi-stable coverings of wide open rigid curves.