#### 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
##### 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 ${ℤ}_{\left(p\right)}$, ${W}_{\cdot }{\Omega }_{{ℤ}_{\left(p\right)}}^{\bullet }$. The main goal of this paper is to construct, for $k$ a perfect ring of characteristic $p>2$, a Witt complex over $A=W\left(k\right)$ with an algebraic description which is completely analogous to Hesselholt and Madsen’s description for ${ℤ}_{\left(p\right)}$. 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 $W\left(k\right)$ 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 ${W}_{n}{\Omega }_{A}^{\bullet }$ for each $n\ge 1$. We also deduce results concerning the de Rham–Witt complex over certain $p$-torsion-free perfectoid rings.

##### Keywords
Witt vectors, de Rham–Witt complex, perfectoid rings
##### Mathematical Subject Classification 2010
Primary: 13F35
Secondary: 13N05, 14F30