Almost Coherence of Higher Direct Images
Tongmu He
TL;DR
This work generalizes the preservation of coherence under higher direct images to the almost algebra setting. By combining Noetherian approximation with truncated Čech hypercoverings and a notion of pseudo-coherence up to bounded torsion, it proves that for a flat, proper, finite-presented morphism $f$ with $\,\mathcal{O}_X$ and $\mathcal{O}_S$ almost coherent, the cohomology $R^q f_*\mathcal{M}$ remains quasi-coherent and almost coherent for any quasi-coherent and almost coherent $\mathcal{O}_X$-module $\mathcal{M}$. This result extends Abbes–Gros' relative $p$-adic comparison to the proper setting and underpins the relative Hodge–Tate spectral sequence in broader contexts. The paper also develops a robust toolkit for almost coherence, including descent on truncated Čech hypercovers and controlled torsion techniques, enabling coherent descent and cohomology estimates in the almost setting.
Abstract
For a flat proper morphism of finite presentation between schemes with almost coherent structural sheaves (in the sense of Faltings), we prove that the higher direct images of quasi-coherent and almost coherent modules are quasi-coherent and almost coherent. Our proof uses Noetherian approximation, inspired by Kiehl's proof of the pseudo-coherence of higher direct images. Our result allows us to extend Abbes-Gros' proof of Faltings' main $p$-adic comparison theorem in the relative case for projective log-smooth morphisms of schemes to proper ones, and thus also their construction of the relative Hodge-Tate spectral sequence.