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A note on the moduli spaces of free algebras of rank 2

Sophie Marques

TL;DR

The paper addresses classifying free algebras of rank $2$ over a base ring via moduli spaces, formulating the problem functorially and showing that, in key families, the moduli functors admit explicit presentations as presheaf quotients of affine schemes by group-scheme actions. It centers the classification on a single quadratic polynomial $x^2 + a x + b$, with the discriminant $a^2 - 4b$ governing separability and isomorphism classes, and develops concrete parameter spaces together with Hopf-algebra coactions to compute inertia and automorphism structures. The main contributions include explicit affine-model descriptions of isomorphisms and automorphisms, a detailed description of moduli spaces as quotients under natural group actions, and a framework that yields one-parameter parametrizations in several cases, linking to prior work by Voight. Together, these results provide computable moduli descriptions for rank-$2$ algebras and set the stage for further stacky or categorical analyses of these moduli problems.

Abstract

In this paper, we present a formulation of the moduli problem for rank-2 algebras over general base rings in functorial terms, providing presentations as presheaf quotients of affine schemes by group scheme actions.

A note on the moduli spaces of free algebras of rank 2

TL;DR

The paper addresses classifying free algebras of rank over a base ring via moduli spaces, formulating the problem functorially and showing that, in key families, the moduli functors admit explicit presentations as presheaf quotients of affine schemes by group-scheme actions. It centers the classification on a single quadratic polynomial , with the discriminant governing separability and isomorphism classes, and develops concrete parameter spaces together with Hopf-algebra coactions to compute inertia and automorphism structures. The main contributions include explicit affine-model descriptions of isomorphisms and automorphisms, a detailed description of moduli spaces as quotients under natural group actions, and a framework that yields one-parameter parametrizations in several cases, linking to prior work by Voight. Together, these results provide computable moduli descriptions for rank- algebras and set the stage for further stacky or categorical analyses of these moduli problems.

Abstract

In this paper, we present a formulation of the moduli problem for rank-2 algebras over general base rings in functorial terms, providing presentations as presheaf quotients of affine schemes by group scheme actions.

Paper Structure

This paper contains 6 sections, 14 theorems, 25 equations.

Table of Contents

  1. Notations and preliminaries
  2. Characterizing certain types of free algebras of rank $2$
  3. The structure of the functor of isomorphisms
  4. The structure of the moduli spaces
  5. The structure of the moduli spaces at a given algebra
  6. The structure of the moduli spaces as a functor

Key Result

Lemma 1.4

Let $p$ be a prime number. The following statements are equivalent: Moreover, if $k(\mathfrak{p})$ is the residue field at a prime ideal $\mathfrak{p}$ of $R$ of characteristic $p$, then $p \in \mathfrak{p}$.

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • proof
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • ...and 22 more