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PreprintCC BY 4.0arXiv

Relative Suspension and Higher Delooping for Support τ-Tilting Modules

  • Mingfei Xu
  • Xiaojin Zhang
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Abstract

We construct relative suspension for the exact category generated by a support τ-tilting module and characterize higher delooping by the units of the resulting adjunction, which are the generalizations of Gélinas's results in Adv. Math.(2022). We prove that comparison with ordinary higher delooping requires a shift of one in the index. More precisely, for every k,d≥1, there is a module whose ordinary k-delooping level is zero and whose relative level is d. We also prove that tensor induction preserves the relative constructions and higher delooping levels on induced modules. As an application, we study cyclic Nakayama algebras with arbitrary local coefficients and automorphism twists. Their left and right little and big finitistic dimensions and ordinary and derived delooping levels all equal the finitistic dimension of the underlying Nakayama algebra.

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Cite

BibTeXRIS
BibTeX
@misc{xu2026relative,
  title = {{Relative Suspension and Higher Delooping for Support $τ$-Tilting Modules}},
  author = {Mingfei Xu and Xiaojin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36444},
  eprint = {2609.36444},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36444},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Relative Suspension and Higher Delooping for Support τ-Tilting Modules
AU  - Mingfei Xu
AU  - Xiaojin Zhang
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.36444
UR  - https://arxiv.org/abs/2609.36444
AB  - We construct relative suspension for the exact category generated by a support τ-tilting module and characterize higher delooping by the units of the resulting adjunction, which are the generalizations of Gélinas's results in Adv. Math.(2022). We prove that comparison with ordinary higher delooping requires a shift of one in the index. More precisely, for every k,d≥1, there is a module whose ordinary k-delooping level is zero and whose relative level is d. We also prove that tensor induction preserves the relative constructions and higher delooping levels on induced modules. As an application, we study cyclic Nakayama algebras with arbitrary local coefficients and automorphism twists. Their left and right little and big finitistic dimensions and ordinary and derived delooping levels all equal the finitistic dimension of the underlying Nakayama algebra.
ER  -