arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2003.03911math.NTmath.ATmath.KT
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关于拟完全环上的de Rham-Witt复形

On the de Rham-Witt complex over perfectoid rings

Christopher Davis, Irakli Patchkoria

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摘要：

修复奇素数$p$。本文的结果是在Hesselholt和Hesselholt-Madsen关于$p$ --典型的混合特征绝对de Rham-Witt复形的工作的基础上建立的。我们有两个主要结果。第一个是描述$A$上的de Rham-Witt复形上的限制映射的核的正合序列，其中$A$是$\mathbb{Q}_p$的代数扩张中的整数环，或者其中$A$是$p$ -无挠的完全环。第二个结果是描述了$A$上的de Rham-Witt复形中的$p$ -幂挠（及相关对象），其中$A$是一个包含相容的$p$ -幂单位根系统的$p$ -无挠全拟环.这两个结果都类似于Hesselholt和Madsen的结果。我们的主要贡献是推广他们的结果，某些perfectoid环。我们还提供了这些结果的代数证明，而Hesselholt和马德森的证明使用拓扑技术。

英文摘要：

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.

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