The Scheme of Monogenic Generators II: Local Monogenicity and Twists

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 $\theta$.