  Fix an odd prime $p$. The results in this paper are modeled after work of
Hesselholt and Hesselholt-Madsen on the $p$-typical absolute de Rham-Witt
complex in mixed characteristic. We have two primary results. The first is an
exact sequence which describes the kernel of the restriction map on the de
Rham-Witt complex over $A$, where $A$ is the ring of integers in an algebraic
extension of $\mathbb{Q}_p$, or where $A$ is a $p$-torsion-free perfectoid
ring. The second result is a description of the $p$-power torsion (and related
objects) in the de Rham-Witt complex over $A$, where $A$ is a $p$-torsion-free
perfectoid ring containing a compatible system of $p$-power roots of unity.
Both of these results are analogous to results of Hesselholt and Madsen. Our
main contribution is the extension of their results to certain perfectoid
rings. We also provide algebraic proofs of these results, whereas the proofs of
Hesselholt and Madsen used techniques from topology.

Number Theory;Algebraic Topology;K-Theory and Homology