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

## TL;DR

This paper constructs an algebraic Witt complex over W(k) for perfect rings k of characteristic p > 2, providing explicit descriptions and relating it to the de Rham-Witt complex, with implications for perfectoid rings.

## Contribution

It introduces a new algebraic Witt complex over W(k) that parallels existing descriptions, and clarifies its relationship with the de Rham-Witt complex.

## Key findings

- Surjection from de Rham-Witt complex to the constructed Witt complex
- Explicit description of the de Rham-Witt complex over W(k)
- Results applicable to perfectoid rings

## Abstract

Hesselholt and Madsen in [7] 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). 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 the de Rham-Witt complex over W(k). We also deduce results concerning the de Rham-Witt complex over certain perfectoid rings.