On the p-typical de Rham–Witt complex over W(k)

Davis, Christopher

doi:10.2140/ant.2019.13.1597

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On the $p$-typical de Rham–Witt complex over $W(k)$

Christopher Davis

Vol. 13 (2019), No. 7, 1597–1631

DOI: 10.2140/ant.2019.13.1597

Abstract

Hesselholt and Madsen (2004) define and study the (absolute, $p$ -typical) de Rham–Witt complex in mixed characteristic, where $p$ is an odd prime. They give as an example an elementary algebraic description of the de Rham–Witt complex over $ℤ_{(p)}$ , $W_{\cdot} Ω_{ℤ_{(p)}}^{∙}$ . The main goal of this paper is to construct, for $k$ a perfect ring of characteristic $p > 2$ , a Witt complex over $A = W (k)$ with an algebraic description which is completely analogous to Hesselholt and Madsen’s description for $ℤ_{(p)}$ . Our Witt complex is not isomorphic to the de Rham–Witt complex; instead we prove that, in each level, the de Rham–Witt complex over $W (k)$ surjects onto our Witt complex, and that the kernel consists of all elements which are divisible by arbitrarily high powers of $p$ . We deduce an explicit description of $W_{n} Ω_{A}^{∙}$ for each $n \geq 1$ . We also deduce results concerning the de Rham–Witt complex over certain $p$ -torsion-free perfectoid rings.

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Keywords

Witt vectors, de Rham–Witt complex, perfectoid rings

Mathematical Subject Classification 2010

Primary: 13F35

Secondary: 13N05, 14F30

Milestones

Received: 22 June 2017

Revised: 21 April 2019

Accepted: 13 June 2019

Published: 21 September 2019

Authors

Christopher Davis
Department of Mathematics University of California, Irvine CA United States

Vol. 13, No. 7, 2019