论文标题
实际Weyl代数的估值和顺序
Valuatuions and orderings on the real Weyl algebra
论文作者
论文摘要
$ \ newCommand {\ r} {\ Mathbb r} \ newCommand {\ rweyl} {\ Mathcal {\ Mathcal {a} _1（\ r）} $第一个Weyl Algebra $ \ natercal $ \ Mathcal {a} _1（k）$ k $ y $ k $ y y是$ k $ y- $ [y，x] = 1 $，是在量子力学开发过程中首次引入的。在本文中，我们将所有估值对真实的Weyl代数$ \ Mathcal {a} _1（\ Mathbb {r}）$进行分类，其残基字段为$ \ Mathbb {r} $。然后，我们使用来自真实代数几何形状的Baer-Krull定理的非交通性版本来对$ \ Mathcal {a} _1（\ Mathbb {r}）$上的所有订单进行分类。作为我们研究的副产品，我们解决了非交通评估理论中的两个开放问题。首先，我们表明，并非所有估值都在$ \ mathcal {a} _1（\ mathbb {r}）$上，带有残基字段$ \ r $扩展到较大的环$ r [y; δ] $，其中$ r $是Puiseux系列，由Marshall和Zhang引入，具有相同的残留场，并表征确实具有如此延伸的估值。其次，我们表明，对于对非交通分裂环的估值，卡普兰斯基的定理是通过伪库奇序列扩展的限制，这通常是直接失败的。
$\newcommand{\R}{\mathbb R} \newcommand{\rweyl}{\mathcal{A}_1(\R)}$ The first Weyl algebra $\mathcal{A}_1(k)$ over a field $k$ is the $k$-algebra with two generators $x, y$ subject to $[y, x] = 1$ and was first introduced during the development of quantum mechanics. In this article, we classify all valuations on the real Weyl algebra $\mathcal{A}_1(\mathbb{R})$ whose residue field is $\mathbb{R}$. We then use a noncommutative version of the Baer-Krull theorem from real algebraic geometry to classify all orderings on $\mathcal{A}_1(\mathbb{R})$. As a byproduct of our studies, we settle two open problems in noncommutative valuation theory. First, we show that not all valuations on $\mathcal{A}_1(\mathbb{R})$ with residue field $\R$ extend to a valuation on a larger ring $R[y ; δ]$, where $R$ is the ring of Puiseux series, introduced by Marshall and Zhang, with the same residue field, and characterize the valuations that do have such an extension. Second, we show that for valuations on noncommutative division rings, Kaplansky's theorem that extensions by limits of pseudo-Cauchy sequences are immediate fails in general.
