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Local forms for the double $A_n$ quiver

Hao Zhang

TL;DR

This work develops a noncommutative framework for the double $A_n$ quiver with loops, introducing intrinsic Type $A$ potentials and proving a monomialization that yields a canonical monomial form. It then establishes a precise correspondence between Jacobi algebras of these potentials and contraction algebras of crepant resolutions of $cA_n$ singularities, thereby realizing Brown–Wemyss conjectures in the Type $A$ setting. The results provide a unified bridge between noncommutative deformation theory and the geometry of $cA_n$ singularities, enabling explicit classifications and a complete geometric realization for Type $A$ crepant resolutions, including concrete derived-equivalence data. In low-dimensional cases, notably $n\le3$, the paper offers complete classifications of Type $A$ potentials (with and without loops) and their derived-equivalence classes, with connections to quaternion-type tame algebras. The approach yields both intrinsic characterizations and a robust geometric interpretation, enhancing tools for the Minimal Model Program through noncommutative invariants.

Abstract

This paper studies the noncommutative singularity theory of the double $A_n$ quiver $Q_n$ (with a single loop at each vertex), with applications to algebraic geometry and representation theory. We give various intrinsic definitions of a Type A potential on $Q_n$, then via coordinate changes we (1) prove a monomialization result that expresses these potentials in a particularly nice form, (2) prove that Type A potentials precisely correspond to crepant resolutions of cAn singularities, (3) solve the Realisation Conjecture of Brown-Wemyss in this setting. For $n \leq 3$, we furthermore give a full classification of Type A potentials (without loops) up to isomorphism, and those with finite-dimensional Jacobi algebras up to derived equivalence. There are various algebraic corollaries, including to certain tame algebras of quaternion type due to Erdmann, where we describe all basic algebras in the derived equivalence class.

Local forms for the double $A_n$ quiver

TL;DR

This work develops a noncommutative framework for the double quiver with loops, introducing intrinsic Type potentials and proving a monomialization that yields a canonical monomial form. It then establishes a precise correspondence between Jacobi algebras of these potentials and contraction algebras of crepant resolutions of singularities, thereby realizing Brown–Wemyss conjectures in the Type setting. The results provide a unified bridge between noncommutative deformation theory and the geometry of singularities, enabling explicit classifications and a complete geometric realization for Type crepant resolutions, including concrete derived-equivalence data. In low-dimensional cases, notably , the paper offers complete classifications of Type potentials (with and without loops) and their derived-equivalence classes, with connections to quaternion-type tame algebras. The approach yields both intrinsic characterizations and a robust geometric interpretation, enhancing tools for the Minimal Model Program through noncommutative invariants.

Abstract

This paper studies the noncommutative singularity theory of the double quiver (with a single loop at each vertex), with applications to algebraic geometry and representation theory. We give various intrinsic definitions of a Type A potential on , then via coordinate changes we (1) prove a monomialization result that expresses these potentials in a particularly nice form, (2) prove that Type A potentials precisely correspond to crepant resolutions of cAn singularities, (3) solve the Realisation Conjecture of Brown-Wemyss in this setting. For , we furthermore give a full classification of Type A potentials (without loops) up to isomorphism, and those with finite-dimensional Jacobi algebras up to derived equivalence. There are various algebraic corollaries, including to certain tame algebras of quaternion type due to Erdmann, where we describe all basic algebras in the derived equivalence class.

Paper Structure

This paper contains 18 sections, 68 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main Results
  4. Special cases: $A_3$
  5. Algebraic Preliminaries
  6. Geometric Preliminaries
  7. NCCRs and Contraction Algebras
  8. $cA_n$ singularities
  9. Monomialization
  10. Monomialization
  11. Transform monomialized Type A potentials on $Q_{n, I}$ to $Q_n$
  12. Geometric Realisation
  13. Geometric Realisation
  14. Type $A_{n,I}$ crepant resolutions and potentials
  15. Special cases: A3
  16. ...and 3 more sections

Key Result

Theorem 1.1

For any Type $A$ potential $f$ on $Q_n$ where $n \geq 1$, there exists a crepant resolution $\uppi \colon \EuScript{X} \rightarrow \mathop{\mathrm{Spec}}\EuScript{R}$ where $\EuScript{R}$ is $cA_n$, such that $\EuScript{J}\mathrm{ac}(f) \cong \Lambda_{\mathrm{con}}(\uppi)$.

Theorems & Definitions (150)

  • Theorem 1.1: \ref{['511']}
  • Corollary 1.2: \ref{['cor: TypeA_intrin']}
  • Corollary 1.3: \ref{['516']}
  • Theorem 1.4: \ref{['cor']}
  • Theorem 1.5: \ref{['610']}
  • Theorem 1.6: \ref{['611']}
  • Corollary 1.7: \ref{['631']}
  • Definition 2.1
  • Example 2.2
  • Definition 2.4
  • ...and 140 more