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On the global dimension three endomorphism algebras of the minimal generator-cogenerator

Edson Ribeiro Alvares, Clezio Aparecido Braga, Sonia Trepode, Heily Wagner

TL;DR

The paper introduces representation-hereditary algebras as those with $\mathrm{gl.dim}\,\mathrm{End}_A(A \oplus DA)=3$, embedding them in the Auslander framework for representation dimension. It proves these algebras are torsionless-finite and provides exact approximation-theoretic characterizations of the property, linking the kernels and cokernels of right/left $\mathrm{add}(A \oplus DA)$-approximations to projective/injective objects. It then develops sufficient conditions for representation-hereditary-ness, analyzes the case $\mathrm{Hom}_A(DA,A)=0$ (relating to quasi-tilted algebras and providing a dichotomy with tilting behavior), and specializes to tame quasi-tilted and tilted algebras, offering concrete criteria and structural consequences. Overall, the work connects representation-dimension concepts with tilting theory to classify and illuminate the homological anatomy of representation-hereditary algebras and their tilted/quasi-tilted relatives.

Abstract

The main goal of this paper is to study the class of algebras for which the global dimension of the endomorphism ring of the generator-cogenerator, given by the sum of the projective and injective modules, is equal to three. We will refer to these algebras as representation-hereditary algebras. We show that these algebras are torsionless-finite, as defined by Ringel. These algebras do not necessarily have finite global dimension; however, when there is no non-zero morphism from an injective to a projective module, they have global dimension less than or equal to two, with some additional homological properties. By utilizing the general framework provided by the study of the representation dimension of an algebra, we present further homological consequences. In the case where these algebras are tame quasi-tilted algebras, we prove that they belong to certain classes of tilted algebras. Although not all tilted algebras are representation-hereditary, we provide sufficient conditions for them to be representation-hereditary.

On the global dimension three endomorphism algebras of the minimal generator-cogenerator

TL;DR

The paper introduces representation-hereditary algebras as those with , embedding them in the Auslander framework for representation dimension. It proves these algebras are torsionless-finite and provides exact approximation-theoretic characterizations of the property, linking the kernels and cokernels of right/left -approximations to projective/injective objects. It then develops sufficient conditions for representation-hereditary-ness, analyzes the case (relating to quasi-tilted algebras and providing a dichotomy with tilting behavior), and specializes to tame quasi-tilted and tilted algebras, offering concrete criteria and structural consequences. Overall, the work connects representation-dimension concepts with tilting theory to classify and illuminate the homological anatomy of representation-hereditary algebras and their tilted/quasi-tilted relatives.

Abstract

The main goal of this paper is to study the class of algebras for which the global dimension of the endomorphism ring of the generator-cogenerator, given by the sum of the projective and injective modules, is equal to three. We will refer to these algebras as representation-hereditary algebras. We show that these algebras are torsionless-finite, as defined by Ringel. These algebras do not necessarily have finite global dimension; however, when there is no non-zero morphism from an injective to a projective module, they have global dimension less than or equal to two, with some additional homological properties. By utilizing the general framework provided by the study of the representation dimension of an algebra, we present further homological consequences. In the case where these algebras are tame quasi-tilted algebras, we prove that they belong to certain classes of tilted algebras. Although not all tilted algebras are representation-hereditary, we provide sufficient conditions for them to be representation-hereditary.
Paper Structure (11 sections, 23 theorems, 12 equations)

This paper contains 11 sections, 23 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Conventions and notation
  3. On Tilting Theory
  4. The generator-cogenerator
  5. On representation-hereditary algebras: definiton and properties
  6. Representation-hereditary algebras
  7. Torsionless-finite algebras
  8. Some sufficient conditions
  9. Algebras with no morphism from injective to projective
  10. On representation-hereditary quasi-tilted algebras
  11. On representation-hereditary tilted algebras

Key Result

Theorem 1

For an artin algebra $A$ such that $\mathrm{mod} A \ne \mathrm{add} (A \oplus DA)$ the following are equivalent:

Theorems & Definitions (25)

  • Theorem : Theorem \ref{['kernel_projective']}
  • Theorem : Theorem \ref{['torsionless']}
  • Theorem : Theorem \ref{['suficiente']}
  • Theorem : Theorem \ref{['quasitilted']}
  • Theorem : Theorem \ref{['concealed']}
  • Theorem : Theorem \ref{['theo_tilted']}
  • Theorem
  • Definition
  • Theorem
  • Theorem
  • ...and 15 more