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Bicompact torsion classes and conjectures on brick infinite algebras

Sota Asai

Abstract

A torsion class $\mathcal{T}$ of the module category $\operatorname{\mathsf{mod}} A$ of a finite dimensional algebra $A$ over a field $K$ is said to be compact if there exists a module $M \in \operatorname{\mathsf{mod}} A$ such that $\mathcal{T}$ is the smallest torsion class containing $M$. If a torsion class satisfies this and the dual condition, then we call it a bicompact torsion class. We conjecture that bicompact torsion classes are precisely functorially finite torsion classes, and prove it for hereditary algebras and also for semistable torsion classes. This gives that Demonet Conjecture implies Enomoto Conjecture, both of which are important conjectures on brick infiniteness.

Bicompact torsion classes and conjectures on brick infinite algebras

Abstract

A torsion class of the module category of a finite dimensional algebra over a field is said to be compact if there exists a module such that is the smallest torsion class containing . If a torsion class satisfies this and the dual condition, then we call it a bicompact torsion class. We conjecture that bicompact torsion classes are precisely functorially finite torsion classes, and prove it for hereditary algebras and also for semistable torsion classes. This gives that Demonet Conjecture implies Enomoto Conjecture, both of which are important conjectures on brick infiniteness.

Paper Structure

This paper contains 5 sections, 27 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Bicompactness of semistable torsion classes
  4. Numerically disjoint torsion classes
  5. Conjectures on brick infiniteness

Key Result

Theorem 1.3

Assume that $K$ is algebraically closed, and that $A$ is hereditary. Then Conjecture Conj_bicompact holds; that is, if $\mathcal{T} \in \mathop{\mathrm{\mathsf{tors}}}\nolimits A$ is bicompact, then $\mathcal{T}$ is functorially finite. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (53)

  • Definition 1.1: Definition \ref{['Def_compact']}
  • Conjecture 1.2
  • Theorem 1.3: Theorem \ref{['Thm_hered']}
  • Theorem 1.4: Theorems \ref{['Thm_semistable_bicompact']} and \ref{['Thm_ovT_lattice_bicompact']}
  • Theorem 1.5: Theorem \ref{['Thm_bicom_num_dis']}
  • Conjecture 1.6: Enomoto Conjecture, Conjecture \ref{['Conj_Enomoto']}
  • Conjecture 1.7: Demonet Conjecture, Conjecture \ref{['Conj_Demonet']}
  • Theorem 1.8: Theorem \ref{['Thm_Demonet_infin']}
  • Definition 2.1
  • Example 2.2
  • ...and 43 more