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PreprintarXiv licencearXiv

Another counterexample to the Nerves of Steel Conjecture

  • Kevin Coulembier
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Abstract

In [B2] Balmer introduced the homological spectrum of a rigid tensor-triangulated category, and had the ``nerves of steel'' not to conjecture that it was in bijection with the prime spectrum, despite an ``avalanche of examples''. Naturally, the statement came to be known as Balmer's Nerves of Steel Conjecture. More recently, in [NVY] the conjecture was stated explicitly and extended to a more general non-symmetric context. In this paper we provide a counterexample to the original suggestion in [B2], so also to the conjecture from [NVY]. A first counterexample was recently constructed in [BHR].

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Cite

BibTeXRIS
BibTeX
@misc{coulembier2026another,
  title = {{Another counterexample to the Nerves of Steel Conjecture}},
  author = {Kevin Coulembier},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03489},
  eprint = {2609.03489},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03489},
}
RIS
TY  - UNPB
TI  - Another counterexample to the Nerves of Steel Conjecture
AU  - Kevin Coulembier
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03489
UR  - https://arxiv.org/abs/2609.03489
AB  - In [B2] Balmer introduced the homological spectrum of a rigid tensor-triangulated category, and had the ``nerves of steel'' not to conjecture that it was in bijection with the prime spectrum, despite an ``avalanche of examples''. Naturally, the statement came to be known as Balmer's Nerves of Steel Conjecture. More recently, in [NVY] the conjecture was stated explicitly and extended to a more general non-symmetric context. In this paper we provide a counterexample to the original suggestion in [B2], so also to the conjecture from [NVY]. A first counterexample was recently constructed in [BHR].
ER  -