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Geometric interactions between bricks and $τ$-rigidity

Kaveh Mousavand, Charles Paquette

TL;DR

This work develops a unified algebro-geometric framework connecting bricks, $τ$-rigid modules, and their geometric counterparts across module varieties. It establishes a two-out-of-three principle linking the algebraic sets $ ext{ind}(A)$, $ ext{brick}(A)$, and $iτ ext{-rigid}(A)$, and characterizes locally representation-directed algebras via these equalities. The authors extend the perspective to the geometry of irreducible components, introducing the $τ$-tilting fan, $τ$-regular components, and the $τ$-convergence property, which together yield new tools for constructing vectors outside the $τ$-tilting fan and for approaching long-standing conjectures (2nd bBT, Demonets conjecture) in tame and minimal brick-infinite settings. They show that, under suitable hypotheses, minimal brick-infinite algebras exhibit $E$-infinity and thus satisfy Demonets conjecture, with a coherent story across both algebraic and geometric viewpoints, including a robust analysis of the torsion-class structure induced by components. Overall, the paper provides new structural insights, explicit geometric constructions, and cross-cutting results that advance understanding of brick-infinity, rigidity, and their connection to $E$-finiteness in representation theory of algebras.

Abstract

For finite-dimensional algebras over algebraically closed fields, we consider two fundamental classes of modules and their geometric counterparts: bricks and $τ$-rigid modules, as well as brick components and $τ$-regular components. We then apply our results in the study of some open conjectures. First, we investigate the situation where every brick is $τ$-rigid. We prove that this occurs exactly when the algebra is locally representation-directed; a family of algebras introduced by Dräxler in the 1990s, which are always representation-finite. Then, we adopt a geometric perspective and analyze the brick and $τ$-regular components of module varieties. In this greater generality, we establish new properties of such components. Inspired by some recent conjectures, we apply our results to the study of minimal brick-infinite algebras. Along the way, we construct some limits of rigid $g$-vectors, under a condition that we call the $τ$-convergence property. This construction is novel and, in certain cases, yields an integral $g$-vector lying outside the $τ$-tilting fan (a.k.a. $g$-vector fan). We show how our results provide new tools to the study of some open conjectures and particularly illustrate that for $E$-tame algebras.

Geometric interactions between bricks and $τ$-rigidity

TL;DR

This work develops a unified algebro-geometric framework connecting bricks, -rigid modules, and their geometric counterparts across module varieties. It establishes a two-out-of-three principle linking the algebraic sets , , and , and characterizes locally representation-directed algebras via these equalities. The authors extend the perspective to the geometry of irreducible components, introducing the -tilting fan, -regular components, and the -convergence property, which together yield new tools for constructing vectors outside the -tilting fan and for approaching long-standing conjectures (2nd bBT, Demonets conjecture) in tame and minimal brick-infinite settings. They show that, under suitable hypotheses, minimal brick-infinite algebras exhibit -infinity and thus satisfy Demonets conjecture, with a coherent story across both algebraic and geometric viewpoints, including a robust analysis of the torsion-class structure induced by components. Overall, the paper provides new structural insights, explicit geometric constructions, and cross-cutting results that advance understanding of brick-infinity, rigidity, and their connection to -finiteness in representation theory of algebras.

Abstract

For finite-dimensional algebras over algebraically closed fields, we consider two fundamental classes of modules and their geometric counterparts: bricks and -rigid modules, as well as brick components and -regular components. We then apply our results in the study of some open conjectures. First, we investigate the situation where every brick is -rigid. We prove that this occurs exactly when the algebra is locally representation-directed; a family of algebras introduced by Dräxler in the 1990s, which are always representation-finite. Then, we adopt a geometric perspective and analyze the brick and -regular components of module varieties. In this greater generality, we establish new properties of such components. Inspired by some recent conjectures, we apply our results to the study of minimal brick-infinite algebras. Along the way, we construct some limits of rigid -vectors, under a condition that we call the -convergence property. This construction is novel and, in certain cases, yields an integral -vector lying outside the -tilting fan (a.k.a. -vector fan). We show how our results provide new tools to the study of some open conjectures and particularly illustrate that for -tame algebras.
Paper Structure (17 sections, 38 theorems, 18 equations)

This paper contains 17 sections, 38 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries, setting and notations
  3. Irreducible components of module varieties
  4. The $\tau$-tilting fan
  5. Wall-and-chamber structure, canonical decomposition
  6. Bricks and $\tau$-rigid modules
  7. Locally representation-directed algebras
  8. $\tau$-regular components
  9. Minimal brick-infinite algebras
  10. Cartan matrix as quadratic form
  11. The $\tau$-convergence property
  12. Constructing rational vectors outside $\mathcal{F}$
  13. Faithful $\tau$-regular components
  14. Geometry of bricks and $\tau$-rigidity
  15. Torsion classes of modules varieties and $E$-tameness
  16. ...and 2 more sections

Key Result

Theorem 1.1

For an algebra $A$, if two of the three sets $\mathop{\mathrm{ind}}\limits(A)$, $\mathop{\mathrm{brick}}\limits(A)$, and $\textit{i}\operatorname{\tau-rigid}(A)$ are equal, then all the three sets coincide. Consequently, the following are equivalent:

Theorems & Definitions (76)

  • Theorem 1.1: Theorem \ref{['Thm:new characterization of locally rep-directed']} and Corollary \ref{['Cor: locally rep-directed']}
  • Conjecture 1.1: 2nd bBT Conj.
  • Conjecture 1.2: Demonet's Conj.
  • Theorem 1.2
  • Theorem 1.3: Theorem \ref{['thm:rational_g_vector']}
  • Corollary 1.4: Remark \ref{["2nd bBT and Demonet's conj for tame algebras"]} and Prop. \ref{['Prop: 2nd bBT and tau-convergence for tame algebras']}
  • Proposition 1.5
  • Theorem 1.6: Corollary \ref{['Cor: E-tame and bricks']} and Prop. \ref{['prop: E-tame almost all brick tau-rigid']}
  • Theorem 2.1: AIR
  • Remark 2.2
  • ...and 66 more