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Completed prismatic F-crystals and crystalline Zp-local systems

Part of: Discontinuous groups and automorphic forms (Co)homology theory

Published online by Cambridge University Press:  18 April 2024

Heng Du Open the ORCID record for Heng Du [Opens in a new window] ,
Tong Liu ,
Yong Suk Moon Open the ORCID record for Yong Suk Moon [Opens in a new window]  and
Koji Shimizu Open the ORCID record for Koji Shimizu [Opens in a new window]
Heng Du
Affiliation:
Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, PR China hengdu@mail.tsinghua.edu.cn
Tong Liu
Affiliation:
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, USA tongliu@purdue.edu
Yong Suk Moon
Affiliation:
Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, PR China ysmoon@bimsa.cn
Koji Shimizu
Affiliation:
Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, PR China shimizu@tsinghua.edu.cn and Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, PR China
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Abstract

We introduce the notion of completed $F$-crystals on the absolute prismatic site of a smooth $p$-adic formal scheme. We define a functor from the category of completed prismatic $F$-crystals to that of crystalline étale $\mathbf {Z}_p$-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a mixed characteristic complete discrete valuation ring with perfect residue field.

Keywords

p-adic Hodge theorycrystalline local systemsprismatic F-crystals

MSC classification

Primary: 14F30: $p$-adic cohomology, crystalline cohomology
Secondary: 11F80: Galois representations
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 5 , May 2024 , pp. 1101 - 1166
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Copyright
© 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

The third and fourth authors are partially supported by an AMS–Simons Travel Grant.

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