On the de Rham-Witt complex over perfectoid rings

Christopher Davis* (Corresponding Author), Irakli Patchkoria

*Corresponding author for this work

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Fix an odd prime p. The results in this paper are modeled after work of Hesselholt and Hesselholt-Madsen on the p-typical absolute de Rham-Witt complex in mixed characteristic. We have two primary results. The first is an exact sequence which describes the kernel of the restriction map on the de Rham-Witt complex over A, where A is the ring of integers in an algebraic extension of Qp, or where A is a p-torsion-free perfectoid ring. The second result is a description of the p-power torsion (and related objects) in the deRham-Witt complex over A, where A is a p-torsion-free perfectoid ring containing a compatible system of p-power roots of unity. Both of these results are analogous to results of Hesselholt and Madsen. Our main contribution is the extension of their results to certain perfectoid rings. We also provide algebraic proofs of these results, whereas the proofs of Hesselholt and Madsen used techniques from topology.
Original languageEnglish
Article numberrnab092
Pages (from-to)13897-13983
Number of pages87
JournalInternational Mathematics Research Notices
Issue number18
Early online date27 May 2021
Publication statusPublished - 1 Sept 2022

Bibliographical note

The 1st author is very grateful to Lars Hesselholt, who introduced and explained many aspects of this project to him. (The project began around 2014 when the 1st author was a postdoc of Lars Hesselholt at the University of Copenhagen.) The 1st author would also like to especially thank Bhargav Bhatt for assistance at many different points, especially during a visit to the University of Michigan. Furthermore, both authors thank Johannes Anschütz, Bryden Cais, Dustin Clausen, Elden Elmanto, Kiran Kedlaya, Arthur-César Le Bras, Thomas Nikolaus, Peter Scholze, and David Zureick-Brown for useful conversations regarding this paper. The authors also thank the anonymous referee of an earlier version of this paper; the referee provided careful feedback and many suggestions for improvement, especially in Section 7. Both authors thank the Department of Mathematical Sciences of the University of Copenhagen for its hospitality and
pleasant working environment.


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