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Tilting theory on continuous quivers (I) maximal rigid and tilting subcategories

  • Haowei Liu
  • Zonghui Liu
  • Shijie Zhu
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Abstract

Let rep(𝕀) be the category of finitely generated persistence modules on the interval 𝕀⊆ℝ. We study the maximal rigid subcategories of rep(𝕀). Maximal rigid interval consistent subcategories are classified via tilting modules over the corresponding discrete type A quiver. We compare several related notions, (weak) tilting and maximal rigidity, in both the context of hereditary abelian categories with enough projective objects and rep(𝕀) and prove that every maximal rigid interval consistent subcategory is weak tilting and every maximal rigid consistent below subcategory is tilting. We also establish a continuous analogue of Ingalls-Thomas's bijection by showing a bijection between integrally supported maximal rigid consistent below subcategories and integrally generated torsion classes.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026tilting,
  title = {{Tilting theory on continuous quivers (I) maximal rigid and tilting subcategories}},
  author = {Haowei Liu and Zonghui Liu and Shijie Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31689},
  eprint = {2610.31689},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31689},
}
RIS
TY  - UNPB
TI  - Tilting theory on continuous quivers (I) maximal rigid and tilting subcategories
AU  - Haowei Liu
AU  - Zonghui Liu
AU  - Shijie Zhu
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.31689
UR  - https://arxiv.org/abs/2610.31689
AB  - Let rep(𝕀) be the category of finitely generated persistence modules on the interval 𝕀⊆ℝ. We study the maximal rigid subcategories of rep(𝕀). Maximal rigid interval consistent subcategories are classified via tilting modules over the corresponding discrete type A quiver. We compare several related notions, (weak) tilting and maximal rigidity, in both the context of hereditary abelian categories with enough projective objects and rep(𝕀) and prove that every maximal rigid interval consistent subcategory is weak tilting and every maximal rigid consistent below subcategory is tilting. We also establish a continuous analogue of Ingalls-Thomas's bijection by showing a bijection between integrally supported maximal rigid consistent below subcategories and integrally generated torsion classes.
ER  -