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PreprintCC BY 4.0arXiv

A pre-(n+2)-angulated category which is not (n+2)-angulated

  • Jian He
  • Jing He
  • Panyue Zhou
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Abstract

We construct an explicit pre-$9$-angulated category which is not $9$-angulated, thereby giving a genuinely higher counterexample to the implication from pre-(n+2)-angulated to (n+2)-angulated. The underlying additive category is the category of finitely generated projective right modules over the preprojective algebra Π(A5) over 𝔽2. The construction is obtained by taking an odd power of the twisted complete comparison used by Chen-Liu-Lu-Zhang in their pre-triangulated counterexample and by showing that the resulting pre-$9$-angulation fails the higher mapping-cone axiom.

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Cite

BibTeXRIS
BibTeX
@misc{he2026pren2angulated,
  title = {{A pre-$(n+2)$-angulated category which is not $(n+2)$-angulated}},
  author = {Jian He and Jing He and Panyue Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29221},
  eprint = {2608.29221},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29221},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A pre-(n+2)-angulated category which is not (n+2)-angulated
AU  - Jian He
AU  - Jing He
AU  - Panyue Zhou
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.29221
UR  - https://arxiv.org/abs/2608.29221
AB  - We construct an explicit pre-$9$-angulated category which is not $9$-angulated, thereby giving a genuinely higher counterexample to the implication from pre-(n+2)-angulated to (n+2)-angulated. The underlying additive category is the category of finitely generated projective right modules over the preprojective algebra Π(A_5) over 𝔽_2. The construction is obtained by taking an odd power of the twisted complete comparison used by Chen-Liu-Lu-Zhang in their pre-triangulated counterexample and by showing that the resulting pre-$9$-angulation fails the higher mapping-cone axiom.
ER  -