Let A be the path algebra of a quiver of Dynkin type <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {A}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">A</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. The module category <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$mod \,A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mi>o</mi> <mi>d</mi> <mspace width="0.166667em" /> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> has a combinatorial model as the category of diagonals in a polygon S with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> vertices. The recently introduced notion of almost rigid modules is a weakening of the classical notion of rigid modules. The importance of this new notion stems from the fact that maximal almost rigid A-modules are in bijection with the triangulations of the polygon S. In this article, we give a different realization of maximal almost rigid modules. We introduce a non-standard exact structure <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {E}_{\diamond }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">E</mi> <mo>⋄</mo> </msub> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$mod \,A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mi>o</mi> <mi>d</mi> <mspace width="0.166667em" /> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> such that the maximal almost rigid A-modules in the usual exact structure are exactly the maximal rigid A-modules in the new exact structure. A maximal rigid module in this setting is the same as a tilting module. Thus the tilting theory relative to the exact structure <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {E}_{\diamond }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">E</mi> <mo>⋄</mo> </msub> </math></EquationSource> </InlineEquation> translates into a theory of maximal almost rigid modules in the usual exact structure. As an application, we show that with the exact structure <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {E}_{\diamond }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">E</mi> <mo>⋄</mo> </msub> </math></EquationSource> </InlineEquation>, the module category becomes a 0-Auslander category in the sense of Gorsky, Nakaoka and Palu. We also discuss generalizations to quivers of type <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation> and gentle algebras.

An Exact Structure Approach to Almost Rigid Modules Over Quivers of Type $\mathbb {A}$

期刊论文
发表日期：

2026年7月20日

查看全文

Thomas Brüstle,
Eric J. Hanson,
Sunny Roy,
Ralf Schiffler

摘要

Let A be the path algebra of a quiver of Dynkin type $\mathbb {A}_n$ $A_{n}$ . The module category $mod \,A$ $m o d A$ has a combinatorial model as the category of diagonals in a polygon S with $n+1$ $n + 1$ vertices. The recently introduced notion of almost rigid modules is a weakening of the classical notion of rigid modules. The importance of this new notion stems from the fact that maximal almost rigid A-modules are in bijection with the triangulations of the polygon S. In this article, we give a different realization of maximal almost rigid modules. We introduce a non-standard exact structure $\mathcal {E}_{\diamond }$ $E_{⋄}$ on $mod \,A$ $m o d A$ such that the maximal almost rigid A-modules in the usual exact structure are exactly the maximal rigid A-modules in the new exact structure. A maximal rigid module in this setting is the same as a tilting module. Thus the tilting theory relative to the exact structure $\mathcal {E}_{\diamond }$ $E_{⋄}$ translates into a theory of maximal almost rigid modules in the usual exact structure. As an application, we show that with the exact structure $\mathcal {E}_{\diamond }$ $E_{⋄}$ , the module category becomes a 0-Auslander category in the sense of Gorsky, Nakaoka and Palu. We also discuss generalizations to quivers of type $\mathbb {D}$ $D$ and gentle algebras.

An Exact Structure Approach to Almost Rigid Modules Over Quivers of Type \(\mathbb {A}\)

摘要