A motivic proof of the finiteness of the relative de Rham cohomology

Abstract

We give a quick proof of the fact that the relative de Rham cohomology groups \(H^i_{{{\,\textrm{dR}\,}}}(X/S)\) of a smooth and proper map X/S between schemes over \({\mathbb {Q}}\) are vector bundles on the base, replacing Hodge-theoretic and transcendental methods with \({\mathbb {A}}^1\)-homotopy theory.