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Faltings’ main p-adic comparison theorems for non-smooth schemes

Part of: (Co)homology theory

Published online by Cambridge University Press:  12 January 2024

Tongmu He Open the ORCID record for Tongmu He [Opens in a new window]
Tongmu He*
Affiliation:
Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France
*
e-mail: hetm15@ihes.fr
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Abstract

To understand the p-adic étale cohomology of a proper smooth variety over a p-adic field, Faltings compared it to the cohomology of his ringed topos, by the so-called Faltings’ main p-adic comparison theorem, and then deduced various comparisons with p-adic cohomologies originating from differential forms. In this article, we generalize the former to any proper and finitely presented morphism of coherent schemes over an absolute integral closure of $\mathbb {Z}_p$ (without any smoothness assumption) for torsion abelian étale sheaves (not necessarily finite locally constant). Our proof relies on our cohomological descent for Faltings’ ringed topos, using a variant of de Jong’s alteration theorem for morphisms of schemes due to Gabber–Illusie–Temkin to reduce to the relative case of proper log-smooth morphisms of log-smooth schemes over a complete discrete valuation ring proved by Abbes–Gros. A by-product of our cohomological descent is a new construction of Faltings’ comparison morphism, which does not use Achinger’s results on $K(\pi ,1)$-schemes.

Keywords

Cohomological descentFaltings toposv-topologyp-adic Hodge theorycomparison

MSC classification

Primary: 14F30: $p$-adic cohomology, crystalline cohomology
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 33
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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