The Scheme of Monogenic Generators II: Local Monogenicity and Twists

• Published in: arXiv.org 2022-10
• Main Authors: Arpin, Sarah, Bozlee, Sebastian, Herr, Leo, Smith, Hanson
• Format: Article
• Language: English
• Subjects: Topology
• ISSN: 2331-8422

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 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 \(\theta\).