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arXiv 2004.06161math.AC
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赋值的分次代数的结构

On the structure of the graded algebra associated to a valuation

M. S. Barnabé, J. Novacoski, M. Spivakovsky

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摘要：

本文的主要目的是研究与赋值相关的分次代数的结构。更具体地说，我们证明了相关的分次代数${\rm gr}_v(R)$的子环$(R,\mathfrak{m})$的赋值环$\mathcal{O}_v$，其中$Kv:=\mathcal{O}_v / \mathfrak{m}_v=R / \mathfrak{m}$，是同构的$Kv[t^{v(R)}]$，其中的乘法是由扭曲。我们证明了在值群是自由的或剩余域被根式封闭的情况下，这种扭曲乘法可以被选择为通常的扭曲乘法。我们还提出了一个例子，表明同构（与平凡扭曲）并不存在。

英文摘要：

The main goal of this paper is to study the structure of the graded algebra associated to a valuation. More specifically, we prove that the associated graded algebra ${\rm gr}_v(R)$ of a subring $(R,\mathfrak{m})$ of a valuation ring $\mathcal{O}_v$, for which $Kv:=\mathcal{O}_v / \mathfrak{m}_v=R / \mathfrak{m}$, is isomorphic to $Kv[t^{v(R)}]$, where the multiplication is given by a twisting. We show that this twisted multiplication can be chosen to be the usual one in the cases where the value group is free or the residue field is closed by radicals. We also present an example that shows that the isomorphism (with the trivial twisting) does not have to exist.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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