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A note on Hodge–Tate spectral sequences

Part of: (Co)homology theory Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  22 March 2024

ZHIYOU WU
ZHIYOU WU*
Affiliation:
Morningside center of Mathematics, Chinese Academy of Sciences, No. 55, Zhongguancun East Road, Haidian District, Beijing 100190. e-mail: [email protected]
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Abstract

We prove that the Hodge–Tate spectral sequence of a proper smooth rigid analytic variety can be reconstructed from its infinitesimal $\mathbb{B}_{\text{dR}}^+$-cohomology through the Bialynicki–Birula map. We also give a new proof of the torsion-freeness of the infinitesimal $\mathbb{B}_{\text{dR}}^+$-cohomology independent of Conrad–Gabber spreading theorem, and a conceptual explanation that the degeneration of Hodge–Tate spectral sequences is equivalent to that of Hodge–de Rham spectral sequences.

Keywords

14G45

MSC classification

Secondary: 14F40: de Rham cohomology 14G22: Rigid analytic geometry
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 176 , Issue 3 , May 2024 , pp. 625 - 642
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Allen, P. B., Calegari, F., Caraiani, A., Gee, T., Helm, D., Le Hung, B. V., Newton, J., Scholze, P., Taylor, R. and Thorne, J. A.. Potential automorphy over CM fields. ArXiv:1812.09999 (2018).Google Scholar
Bhatt, B., Morrow, M. and Scholze, P.. Integral p-adic hodge theory. Publ. Math. Inst Hautes Etudes Sci. with an acute accent on the E, 128(1) (2018), 219–397.CrossRefGoogle Scholar
Berthelot, P. and Ogus, A.. Notes on crystalline cohomology . Mathematical Notes 21 (Princeton University Press, Princeton, NJ, 1978).Google Scholar
Caraiani, A. and Scholze, P.. On the generic part of the cohomology of compact unitary shimura varieties. Ann. of Math., 186(3) (2017), 649–766.CrossRefGoogle Scholar
Caraiani, A. and Scholze, P.. On the generic part of the cohomology of non-compact unitary Shimura varieties. ArXiv:1909.01898 (2019).Google Scholar
Guo, H.. Crystalline cohomology of rigid analytic spaces. ArXiv:2112.14304 (2021).Google Scholar
Guo, H.. Prismatic cohomology of rigid analytic spaces over de Rham period ring. ArXiv:2112.14746 (2021).Google Scholar
Guo, H.. Hodge–tate decomposition for non-smooth spaces. J. Eur. Math. Soc. 25(4) (2022), 1553–1625.CrossRefGoogle Scholar
Hansen, D.. Period morphisms and variations of p-adic hodge structure. Preprint, (2016).Google Scholar
Scholze, P.. Perfectoid spaces: a survey. Current Developments in Mathematics 2012(1) (2012), 193–227.CrossRefGoogle Scholar
Scholze, P.. p-adic Hodge theory for rigid-analytic varieties. Forum of Mathematics, Pi 1 (2013).CrossRefGoogle Scholar
Scholze, P.. On torsion in the cohomology of locally symmetric varieties. Ann. of Math. 182 (2015), 945–1066.CrossRefGoogle Scholar
Scholze, P. and Weinstein, J.. Berkeley Lectures on p-adic Geometry . Ann. of Math. Stud. (Princeton University Press, 2020).Google Scholar