Abstract
We give in this note an overview of a recent work (Abbes and Gros, Les suites spectrales de Hodge–Tate. Preprint, 2020. https://arxiv.org/abs/2003.04714.) leading to a generalization of the Hodge–Tate spectral sequence to morphisms. The latter takes place in Faltings topos, but its construction requires the introduction of a relative variant of this topos which is the main novelty of our work.
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Notes
- 1.
We treat in [2] schemes with toric singularities using logarithmic geometry, but for simplicity, we consider in this overview only the smooth case.
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Acknowledgements
We would like first to convey our deep gratitude to G. Faltings for the continuing inspiration coming from his work on p-adic Hodge theory. We also thank very warmly O. Gabber and T. Tsuji for the exchanges we had on various aspects discussed in this work. Their invaluable expertise has enabled us to avoid long and unnecessary detours. The first author (A.A) thanks the University of Tokyo and Tsinghua University for their hospitality during several visits where parts of this work have been developed and presented. He expresses his gratitude to T. Saito and L. Fu for their invitations. The authors sincerely thank also B. Bhatt and M. Olsson for their invitation to the second Simons symposium (April 28–May 4, 2019) on p-adic Hodge theory.
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Abbes, A., Gros, M. (2023). The Relative Hodge–Tate Spectral Sequence: An Overview. In: Bhatt, B., Olsson, M. (eds) p-adic Hodge Theory, Singular Varieties, and Non-Abelian Aspects. SISYPHT 2019. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-031-21550-6_1
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