Vol. 13, No. 7, 2019

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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

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 (p), WΩ(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 (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 WnΩA for each n 1. We also deduce results concerning the de Rham–Witt complex over certain p-torsion-free perfectoid rings.

Witt vectors, de Rham–Witt complex, perfectoid rings
Mathematical Subject Classification 2010
Primary: 13F35
Secondary: 13N05, 14F30
Received: 22 June 2017
Revised: 21 April 2019
Accepted: 13 June 2019
Published: 21 September 2019
Christopher Davis
Department of Mathematics
University of California, Irvine
United States