---
title: "Another counterexample to the Nerves of Steel Conjecture"
authors:
  - "Kevin Coulembier"
year: 2026
venue: "arXiv:math.CT"
arxiv: "2609.03489"
doi: null
source: "https://arxiv.org/abs/2609.03489"
topics:
  - "curiosity"
  - "maths-foundations"
seed_rank: 1614
seed_batch: "curiosity-2026-09-07"
reviewed: "2026-09-07"
pool: "maths-foundations"
relevance_score: 10
lineage: tensor-triangular-geometry
cites:
  - title: "Another counterexample to the Nerves of Steel Conjecture"
    url: "https://arxiv.org/abs/2609.03489"
    year: 2026
    arxiv: "2609.03489"
    doi: null
see:
  - "1584-generalized-telescope-conjecture"
---

# Another counterexample to the Nerves of Steel Conjecture

## One-sentence takeaway

Balmer's "Nerves of Steel" hope — that the homological spectrum always matches the Balmer spectrum — fails again: another rigid tt-category where they diverge.

## Why it's lovely

Why you might love this: Balmer famously had the "nerves of steel" not to conjecture that the homological spectrum equals the prime spectrum despite an avalanche of examples — and the community named the statement after that restraint. Coulembier supplies another counterexample. Pure spectator sport for tensor-triangular geometry, and a reminder that spectra of ideals can disagree in more than one way.

## Key ideas

- Homological spectrum vs Balmer spectrum for rigid tensor-triangulated categories.
- The Nerves of Steel statement: hoped bijection between those spectra.
- New counterexample showing the spectra can still diverge.
- Fits the post-Telescope / Balmer shelf (1584) where spectrum comparisons are the plot.

## Caveats

Counterexample paper — does not give a classification of when the spectra agree. Read beside positive stratification results (e.g. Artin-motive stratification 1615).

## Implications for Broadside

Keeps the Balmer / tt-geometry shelf honest: spectrum comparisons are theorems, not slogans. Pairs with Telescope/Balmer work already in the library (1584).

## Links

- arXiv abs: https://arxiv.org/abs/2609.03489
- PDF: https://arxiv.org/pdf/2609.03489.pdf