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arXiv · 2212.01797

Almost Coherence of Higher Direct Images

Tongmu He
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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.

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Tongmu He. 2022-12-04. Almost Coherence of Higher Direct Images. https://doi.org/10.2969/jmsj%2F90809080

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