A motivic proof of the finiteness of the relative de Rham cohomology (Q6622524)

It is a classical result that goes back to \textit{P. Deligne} [Publ. Math., Inst. Hautes Étud. Sci. 35, 107--126 (1968; Zbl 0159.22501)] that for \(X\to S\) a smooth and proper morphism of schemes over \(\mathbb{Q}\), the relative algebraic de Rham cohomology groups \(H^i_{dR}(X/S)\) are vector bundles on the scheme \(S\). See [\textit{N. M. Katz}, Publ. Math., Inst. Hautes Étud. Sci. 39, 175--232 (1970; Zbl 0221.14007); \textit{Y. André}, Ann. Sci. Éc. Norm. Supér. (4) 47, No. 2, 449--467 (2014; Zbl 1344.12001)] for a proof in the situation where \(S\) is smooth. Deligne's proof uses quite a bit of machinery (the relative Hodge-de Rham spectral sequence and various reductions to the case of a field). In this paper, Vezzani gives a short alternative proof, that instead uses motivic homotopy theory.\N\NFor a smooth morphism \(f:X\to S\) of schemes, Vezzani considers the de Rham complex \(R\Gamma_{dR}(X/S)\), and then shows that the functor \(X\mapsto R\Gamma_{dR}(X/S)\) extends to a functor from the category of dualizable étale rational motives to the opposite category of perfect complexes on \(S\). Then for a dualizable motive \(M\), Vezzani proves that \(R\Gamma_{dR}(M/S)\) is a perfect complex of which the cohomology groups are vector bundles over \(S\). The proof uses a reduction to the case where \(S\) is affine and the complex is computed on the completion of the stalk over a closed point \(s\in S\), together with a comparison to the corresponding complex over \(\bar{\mathbb{Q}}\). Using dualizability of smooth and proper maps as established by \textit{J. Riou} [C. R., Math., Acad. Sci. Paris 340, No. 6, 431--436 (2005; Zbl 1068.14021)], the desired result can be deduced from this.\N\NEven though the paper is very short (only five pages if one excludes the reference list at the end), the concepts used are introduced nicely, with helpful descriptions of their properties, and references to the literature. It is fascinating that motivic homotopy theory can be applied to give new proofs of classical results in algebraic geometry, and this paper provides a wonderful example of such a proof.