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<p>In this article, we give a family of examples of algebras, showing that for every <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(m \ge 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, there is an algebra displaying a path of <i>n</i> irreducible morphisms between indecomposable modules whose composite lies in the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((n+m+3)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-th power of the radical, but not in the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((n+m+4)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-th power. Such an algebra may be also supposed to be string and representation-finite.</p>

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Leaps in the depth of compositions of irreducible morphisms

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月19日

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  • Viktor Chust,
  • Flávio U. Coelho

摘要

In this article, we give a family of examples of algebras, showing that for every \(n \ge 2\) n ≥ 2 and \(m \ge 0\) m ≥ 0 , there is an algebra displaying a path of n irreducible morphisms between indecomposable modules whose composite lies in the \((n+m+3)\) ( n + m + 3 ) -th power of the radical, but not in the \((n+m+4)\) ( n + m + 4 ) -th power. Such an algebra may be also supposed to be string and representation-finite.