The Scheme of Monogenic Generators I: Representability

2021/08/16 by Sarah Arpin, Arpin, Sarah, Sebastian Bozlee +5 · 1 citation
Mathematics · #13E15 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #FOS: Mathematics #Number Theory (math.NT) #Primary 14D20 #Rings, Modules, and Algebras #Secondary 11R04

paper · pdf · doi:10.48550/arxiv.2108.07185

openalex publication_date 2021/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This is the first in a series of two papers that study monogenicity of number rings from a moduli-theoretic perspective. Given an extension of algebras B/A, when is B generated by a single element θ∈ B over A? In this paper, we show there is a scheme M_B/A parameterizing the choice of a generator θ∈ B, a "moduli space" of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples.

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