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Length of triangulated categories

Yuki Hirano, Martin Kalck, Genki Ouchi

TL;DR

The paper introduces composition series for triangulated categories as maximal chains of thick subcategories and defines invariants such as the length, length spectrum, and Jordan–Dedekind (JD) index. It proves the JD property for derived categories of path algebras of acyclic quivers and, more broadly, for many hereditary settings, showing that all composition series arise from full exceptional sequences. It then provides geometric counterexamples to the JD property, notably on smooth toric surfaces with negative curves and in certain toric and singular contexts, while also detailing how length spectra can contain multiple values. The work develops a framework connecting bouquet sphere–like objects, categorical resolutions, and singularity categories to explain when JD fails or holds, and ends with open questions about JD in Fano and other geometric situations, highlighting rich interactions between lattice invariants of thick subcategories and geometric features.

Abstract

We introduce the notion of composition series of triangulated categories, which generalizes full exceptional sequences. The lengths of composition series yield invariants for triangulated categories. We study composition series of derived categories for some classes of projective varieties and finite-dimensional algebras. We prove that certain negative rational curves on rational surfaces cause composition series of different lengths in the derived categories of the surfaces. On the other hand, we show that for derived categories of finite-dimensional hereditary algebras, for nontrivial admissible subcategories of ${\rm D}^{\rm b}(\mathbb{P}^2)$ and for derived categories of some singular varieties, all composition series have the same length.

Length of triangulated categories

TL;DR

The paper introduces composition series for triangulated categories as maximal chains of thick subcategories and defines invariants such as the length, length spectrum, and Jordan–Dedekind (JD) index. It proves the JD property for derived categories of path algebras of acyclic quivers and, more broadly, for many hereditary settings, showing that all composition series arise from full exceptional sequences. It then provides geometric counterexamples to the JD property, notably on smooth toric surfaces with negative curves and in certain toric and singular contexts, while also detailing how length spectra can contain multiple values. The work develops a framework connecting bouquet sphere–like objects, categorical resolutions, and singularity categories to explain when JD fails or holds, and ends with open questions about JD in Fano and other geometric situations, highlighting rich interactions between lattice invariants of thick subcategories and geometric features.

Abstract

We introduce the notion of composition series of triangulated categories, which generalizes full exceptional sequences. The lengths of composition series yield invariants for triangulated categories. We study composition series of derived categories for some classes of projective varieties and finite-dimensional algebras. We prove that certain negative rational curves on rational surfaces cause composition series of different lengths in the derived categories of the surfaces. On the other hand, we show that for derived categories of finite-dimensional hereditary algebras, for nontrivial admissible subcategories of and for derived categories of some singular varieties, all composition series have the same length.
Paper Structure (20 sections, 65 theorems, 84 equations)

This paper contains 20 sections, 65 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Composition series of triangulated categories
  4. Main results
  5. Notations and conventions
  6. Acknowledgements
  7. Length of triangulated categories
  8. Semi-simple triangulated categories
  9. Prime and maximal thick subcategories
  10. Composition series, length and JD property
  11. Krah's phantom subcategory
  12. Length spectra
  13. Triangulated categories with the JD property
  14. Hereditary algebras
  15. Application of results on hereditary algebras
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $X$ be a smooth projective toric surface containing a rational curve $C$ with $C^2< -1$. Then the derived category $\operatorname{D^b}(X)$ contains an admissible subcategory $\EuScript{A}$ such that $\ell(\EuScript{A})<\infty$, $\ell(\operatorname{D^b}(X)/\EuScript{A})<\infty$ and $\EuScript{A}$

Theorems & Definitions (137)

  • Theorem 1.1: Corollary \ref{['cor:toric surface']}
  • Theorem 1.2: Theorem \ref{['thm:toric']}
  • Theorem 1.4: Corollary \ref{['cor:appendix']}
  • Theorem 1.5: Theorem \ref{['thm:JD wild']}
  • Corollary 1.6
  • Definition 2.1
  • Theorem 2.2: kel2
  • proof
  • Proposition 2.3: takahashi2
  • Proposition 2.4
  • ...and 127 more