The Scheme of Monogenic Generators II: Local Monogenicity and Twists

TL;DR

This paper explores various notions of local monogenicity of number rings using a moduli-theoretic approach, relating local and global generators, and introduces the concept of twisted monogenerators linked to class number one.

Contribution

It extends the study of monogenicity by analyzing local conditions, constructing universal spaces for étale cases, and defining twisted monogenerators, connecting these to class number properties.

Findings

Characterization of local monogenicity in Zariski and finer topologies.

Construction of a universal monogenicity space for étale extensions.

Equivalence of class number one with the global nature of twisted monogenerators.

Abstract

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 $\theta$ 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 \'etale, where the local structure of \'etale 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 $\theta$.