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Splitting torsion classes in higher homological algebra

  • Erlend D. Børve
  • Peter Jorgensen
  • Mads Hustad Sandøy
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Abstract

Let d geqslant 1 be an integer, ℳ a suitable d-abelian category. There is a notion of higher torsion classes in ℳ, also known as d-torsion classes. We generalise a classic theorem of Hoshino by providing criteria for a d-torsion class 𝒰 to be splitting, notably that 𝒰 is stable under τd-, the inverse d-Auslander--Reiten translation. We apply the criteria to show that under additional assumptions, which are satisfied in Dynkin type A, the elements of the spine of the lattice of d-torsion classes are precisely the splitting d-torsion classes. The spine consists of the elements which belong to a chain of maximal length.

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Cite

BibTeXRIS
BibTeX
@misc{borve2026splitting,
  title = {{Splitting torsion classes in higher homological algebra}},
  author = {Erlend D. Børve and Peter Jorgensen and Mads Hustad Sandøy},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07953},
  eprint = {2609.07953},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07953},
}
RIS
TY  - UNPB
TI  - Splitting torsion classes in higher homological algebra
AU  - Erlend D. Børve
AU  - Peter Jorgensen
AU  - Mads Hustad Sandøy
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07953
UR  - https://arxiv.org/abs/2609.07953
AB  - Let d geqslant 1 be an integer, ℳ a suitable d-abelian category. There is a notion of higher torsion classes in ℳ, also known as d-torsion classes. We generalise a classic theorem of Hoshino by providing criteria for a d-torsion class 𝒰 to be splitting, notably that 𝒰 is stable under τ_d^-, the inverse d-Auslander--Reiten translation. We apply the criteria to show that under additional assumptions, which are satisfied in Dynkin type A, the elements of the spine of the lattice of d-torsion classes are precisely the splitting d-torsion classes. The spine consists of the elements which belong to a chain of maximal length.
ER  -