A pre-triangulated category which is not triangulated

Xiao-Wu Chen, Jian Liu, Xue-Song Lu, Chencheng Zhang

arXiv:2608.09777v1 Primary: math.CT Secondary: math.AG, math.RT

Abstract

In this article, we construct an explicit pre-triangulated category which is not a triangulated category. Its underlying additive category is the category of finitely generated projective modules of the type-$A_5$ preprojective algebra over $\mathbb F_2$, and the suspension is induced by the graph-reflection automorphism.

Disclosure

“trary to Theorem 3.14. Hence TR4 fails. Acknowledgements. The mathematical search leading to the construction was carried out with sub- stantial assistance from Eureka, an autonomous multi-agent mathematical-reasoning system. We also used GPT-5.6 Sol to check and revise this manuscript. We would like to thank the JIUCHONG team at the University of Science and Technology of China for providing us with access to Eureka. We are grateful to Jie Li and Tianyang Sun for their assistance”
PDF page 13

Classification: Substantial mathematical content or result generation
Multiplier: 10
Verified: 2026-08-23T05:08:01.184691Z

Structural counts

Pages 14 pdf

Theorems 3 source

Lemmas 5 source

Propositions 5 source

Corollaries 1 source

Definitions 6 source

Displayed equations 60 source

Bibliography entries 15 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file 2026CLLZ.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.