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.

中文翻译：

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关于与估值相关的分级代数的结构

本文的主要目标是研究与估值相关的分级代数的结构。更具体地说，我们证明了估值环 $\mathcal{O}_v$ 的子环 $(R,\mathfrak{m})$ 的相关分级代数 ${\rm gr}_v(R)$，其中$Kv:=\mathcal{O}_v / \mathfrak{m}_v=R / \mathfrak{m}$，同构于 $Kv[t^{v(R)}]$，其中乘法由下式给出一个扭曲。我们表明，在值组是自由的或残差域被部首封闭的情况下，可以选择这种扭曲的乘法作为通常的乘法。我们还提供了一个例子，表明同构（带有微不足道的扭曲）不一定存在。