论文标题
关于与估值相关的分级代数的结构
On the structure of the graded algebra associated to a valuation
论文作者
论文摘要
本文的主要目的是研究与估值相关的分级代数的结构。更具体地说，我们证明了一个子$（r，\ mathfrak {Mathfrak {m}）$的关联分级为代数$ {\ rm gr} _v（r）$的评估环$ \ mathcal {o} _v $的$（r，\ mathfrak {m}）$ \ mathfrak {m} $，是同构至$ kv [t^{v（r）}] $，其中乘法通过扭曲给出。我们表明，在值组是免费的，或者用自由基封闭的情况下，可以选择这种扭曲的乘法为通常的乘法。我们还提出了一个示例，该例子表明同构（带有微不足道的扭曲）不必存在。
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.
