论文标题
单基因发电机的方案II：局部单基因和曲折
The Scheme of Monogenic Generators II: Local Monogenicity and Twists
论文作者
论文摘要
这是ARXIV的续集：2108.07185，从模量理论的角度继续研究数字环的单基因。根据本系列的第一篇论文的结果，对于$ a $ a $ -Algebra $ b $的生成器$θ$的选择是方案$ \ MATHCAL {M} _ {B/A} $的点。在本文中，我们从这个角度研究并联系了几个局部单基因概念。我们首先考虑扩展$ b/a $在Zariski和更精细拓扑的本地接收单生物的条件，从而恢复了一种愉悦定理作为特殊情况。接下来，我们考虑$ b/a $是étale的情况，étale地图的本地结构使我们能够构建通用的单基因空间并将其与无序的配置空间联系起来。最后，我们考虑$ b/a $何时接纳仅因某些组的作用而不同的本地单生物（通常是$ \ mathbb {g} _m $或$ \ mathrm {aff}^1 $），从而引起了扭曲的单生物的概念。特别是，当且仅当每个扭曲的单基因器实际上都是全球单子发电机$θ$时，我们显示了一个数字环$ A $具有第一类。
This is the sequel paper to arXiv:2108.07185, continuing a study of monogenicity of number rings from a moduli-theoretic perspective. By the results of the first paper in this series, a choice of a generator $θ$ for an $A$-algebra $B$ is a point of the scheme $\mathcal{M}_{B/A}$. In this paper, we study and relate several notions of local monogenicity that emerge from this perspective. We first consider the conditions under which the extension $B/A$ admits monogenerators locally in the Zariski and finer topologies, recovering a theorem of Pleasants as a special case. We next consider the case in which $B/A$ is étale, where the local structure of étale maps allows us to construct a universal monogenicity space and relate it to an unordered configuration space. Finally, we consider when $B/A$ admits local monogenerators that differ only by the action of some group (usually $\mathbb{G}_m$ or $\mathrm{Aff}^1$), giving rise to a notion of twisted monogenerators. In particular, we show a number ring $A$ has class number one if and only if each twisted monogenerator is in fact a global monogenerator $θ$.
