  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.
