Balmer's Nerves of Steel Conjecture
Let be a rigid -category. Its comparison map
Balmer's Nerves of Steel Conjecture. The map is bijective. This conjecture seeks to reconcile the homological and triangular spectra of a rigid tensor-triangulated category. The source explains that the comparison map is known to be surjective and that the conjecture holds in the classical stable homotopy example, but does not provide a general resolution.
References
Primary source
Tobias Barthel, Logan Hyslop and Maxime Ramzi, “Geometric Points in Tensor Triangular Geometry”, arXiv:2603.25664 (2026).
Additional references
2 papers in this index state this conjecture (2024–2026). The statement above is taken from the most recent of them; the others are arXiv:2404.04457.
Progress summary
A March preprint claims the conjecture is false, and a September preprint reports another counterexample, but neither claim has independent verification.
The conjecture asserts that the homological and triangular spectra always agree for rigid tensor-triangulated categories, meaning that the comparison map is bijective. Surjectivity is known, and bijectivity holds in classical stable homotopy theory.
Known results
- The comparison map is surjective in general (Balmer; date not stated).
- Bijectivity holds in the classical stable homotopy example (Balmer; date not stated).
- The conjecture is equivalent to an exact-nilpotence condition for every local tensor-triangulated category (Barthel, Hyslop, Ramzi, 2024).
- Exact nilpotence can fail in certain non-rigid categories, which does not settle the rigid case (Barthel, Hyslop, Ramzi, 2024).
March–September 2026 counterexamples
In March 2026, Barthel, Hyslop, and Ramzi’s preprint Geometric Points in Tensor Triangular Geometry claimed a counterexample using free constructions in higher Zariski geometry. A September 2026 preprint, Another counterexample to the Nerves of Steel Conjecture, reports a further counterexample. These claims would disprove bijectivity, but the retrieved evidence is unrefereed and does not independently verify the constructions.
Current status (as of September 2026): the conjecture has claimed counterexamples, but their validity is unverified; the rigid case therefore remains formally unsettled.
Solutions 0
No solutions have been posted yet.