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Tensor triangular geometry of band algebras

  • Peng Xu
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Abstract

In this paper, we study the tensor triangular geometry of band algebras RB, where B is a finite left regular band and R is a commutative Noetherian ring. We show that the Balmer spectrum of Perf(RB) is homeomorphic to Spec R× LB, where LB is the support lattice of B. We also show that there is a bijection between the localizing ideals of D(RB) and the stable subsets of Spec R× LB. As a consequence, the telescope conjecture holds for D(RB). Finally, we construct a band B such that the generalized Nerves of Steel Conjecture fails for the non-rigid tensor triangulated category Perf(RB).

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Cite

BibTeXRIS
BibTeX
@misc{xu2026tensor,
  title = {{Tensor triangular geometry of band algebras}},
  author = {Peng Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11979},
  eprint = {2610.11979},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11979},
}
RIS
TY  - UNPB
TI  - Tensor triangular geometry of band algebras
AU  - Peng Xu
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11979
UR  - https://arxiv.org/abs/2610.11979
AB  - In this paper, we study the tensor triangular geometry of band algebras RB, where B is a finite left regular band and R is a commutative Noetherian ring. We show that the Balmer spectrum of Perf(RB) is homeomorphic to Spec R× L_B, where L_B is the support lattice of B. We also show that there is a bijection between the localizing ideals of D(RB) and the stable subsets of Spec R× L_B. As a consequence, the telescope conjecture holds for D(RB). Finally, we construct a band B such that the generalized Nerves of Steel Conjecture fails for the non-rigid tensor triangulated category Perf(RB).
ER  -