A motivic proof of the finiteness of relative de Rham cohomology

TL;DR

This paper provides a concise proof that the relative de Rham cohomology groups of smooth, proper schemes over Q are vector bundles, using A1-homotopy theory instead of traditional Hodge-theoretic methods.

Contribution

It introduces a motivic approach to prove the finiteness of relative de Rham cohomology, replacing classical transcendental techniques with A1-homotopy theory.

Findings

Relative de Rham cohomology groups are vector bundles on the base.

The proof avoids Hodge-theoretic and transcendental methods.

A1-homotopy theory effectively establishes the finiteness result.

Abstract

We give a quick proof of the fact that the relative de Rham cohomogy groups of a smooth and proper map X/S between schemes over Q are vector bundles on the base, replacing Hodge-theoretic and transcendental methods with A1-homotopy theory