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Mutation of τ-exceptional pairs and sequences

Aslak B. Buan, Eric J. Hanson, Bethany R. Marsh

Abstract

We introduce a notion of mutation for $τ$-exceptional sequences of modules over arbitrary finite dimensional algebras. For hereditary algebras, we show that this coincides with the classical mutation of exceptional sequences. For rank two algebras, we show that mutation of $τ$-exceptional sequences is transitive if and only if mutation of support $τ$-tilting modules in the sense of Adachi-Iyama-Reiten is transitive.

Mutation of τ-exceptional pairs and sequences

Abstract

We introduce a notion of mutation for -exceptional sequences of modules over arbitrary finite dimensional algebras. For hereditary algebras, we show that this coincides with the classical mutation of exceptional sequences. For rank two algebras, we show that mutation of -exceptional sequences is transitive if and only if mutation of support -tilting modules in the sense of Adachi-Iyama-Reiten is transitive.
Paper Structure (21 sections, 68 theorems, 97 equations)

This paper contains 21 sections, 68 theorems, 97 equations.

Table of Contents

  1. Tau-rigid objects and torsion pairs
  2. Torsion pairs
  3. Projective and injective objects
  4. Functorially finite torsion pairs and $\tau$-rigid modules
  5. Support $\tau$-rigid objects
  6. Wide subcategories
  7. $\tau$-perpendicular wide subcategories
  8. Left and right finite wide subcategories
  9. Projectives in $\tau$-perpendicular subcategories
  10. Sincere $\tau$-perpendicular subcategories
  11. Tau-exceptional pairs and sequences
  12. Definitions and background
  13. Sincere $\tau$-perpendicular categories
  14. Irregular $\tau$-exceptional pairs
  15. Mutation of tau-exceptional pairs
  16. ...and 6 more sections

Key Result

Theorem 1

The maps $\varphi$ and $\psi$ are mutually inverse bijections.

Theorems & Definitions (134)

  • Theorem 1: Theorem \ref{['thm:mutation_pairs']}
  • Theorem 2: Theorem \ref{['prop:one_mutation']}
  • Theorem 3: Theoerm \ref{['prop:mutation_complete']}
  • Corollary 4: Corollary \ref{['cor:mutation_complete']}
  • Theorem 5: Theorem \ref{['thm:hereditary_case']}
  • Theorem 6: Corollary \ref{['cor:connected']}
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Definition 1.5
  • ...and 124 more