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 1st result is an exact sequence that 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 $\textbf{Q}_p$ or where $A$ is a $p$-torsion-free perfectoid ring. The 2nd 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 the 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.

中文翻译：

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关于完美型环上的 de Rham-Witt 复合体

修正一个奇数素数 $p$。本文中的结果是在 Hesselholt 和 Hesselholt-Madsen 对混合特征中的 $p$-典型绝对 de Rham-Witt 复形的工作之后建模的。我们有两个主要结果。第一个结果是一个精确序列，它描述了 $A$ 上 de Rham-Witt 复数上的限制映射的核，其中 $A$ 是 $\textbf{Q}_p$ 的代数扩展中的整数环或其中 $A$ 是一个 $p$-无扭完美圆环。第二个结果是对 de Rham-Witt 复合体中的 $p$-幂扭转（和相关对象）在 $A$ 上的描述，其中 $A$ 是一个包含兼容系统的无 $p$-torsion-free 完美型环$p$-幂的单位根。这两个结果都类似于 Hesselholt 和 Madsen 的结果。我们的主要贡献是将他们的结果扩展到某些完美的环。