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<p>Let <i>k</i> be an algebraically closed field. Let <i>R</i> be a local commutative finite dimensional <i>k</i>-algebra and let <i>Q</i> be a quiver with no loops or oriented cycles. We show that mutation of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-exceptional sequences over <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\Lambda = R\otimes _k kQ \cong RQ\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo>=</mo> <mi>R</mi> <msub> <mo>⊗</mo> <mi>k</mi> </msub> <mi>k</mi> <mi>Q</mi> <mo>≅</mo> <mi>R</mi> <mi>Q</mi> </mrow> </math></EquationSource> </InlineEquation> in the sense of Buan, Hanson, and Marsh coincides with the classical mutation of exceptional sequences defined by Crawley-Boevey and Ringel. In particular, the braid group acts transitively on the set of complete <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-exceptional sequences in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\textrm{mod}{\Lambda }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>mod</mtext> <mi mathvariant="normal">Λ</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Mutation of \(\tau \)-exceptional Sequences for Acyclic Quivers Over Local Algebras

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月6日

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  • Iacopo Nonis

摘要

Let k be an algebraically closed field. Let R be a local commutative finite dimensional k-algebra and let Q be a quiver with no loops or oriented cycles. We show that mutation of \(\tau \) τ -exceptional sequences over \(\Lambda = R\otimes _k kQ \cong RQ\) Λ = R ⊗ k k Q ≅ R Q in the sense of Buan, Hanson, and Marsh coincides with the classical mutation of exceptional sequences defined by Crawley-Boevey and Ringel. In particular, the braid group acts transitively on the set of complete \(\tau \) τ -exceptional sequences in \(\textrm{mod}{\Lambda }\) mod Λ .