Towards the Nerves of Steel Conjecture
Logan Hyslop
TL;DR
The paper investigates Balmer's nerves of steel conjecture by recasting it as an exact-nilpotence condition in local tt-categories. It shows that non-rigid (non-tt) settings can violate this property through universal free-construction frameworks, while a broad class of rigid tt-categories generated by the unit satisfy exact-nilpotence under natural Noetherian and rational hypotheses. The work further strengthens the conjecture by proving an equivalent formulation with a uniform nilpotence bound $n$ that works across all local tt-categories, using ultraproduct techniques. Collectively, the results clarify the role of rigidity, establish concrete positive cases, and provide a pathway toward a uniform, bound-based understanding of nilpotence in tensor triangular geometry.
Abstract
Given a local $\otimes$-triangulated category, and a fiber sequence $y\to 1 \to x$, one may ask if there is always a nonzero object $z$ such that either $z\otimes f$ or $z\otimes g$ is $\otimes$-nilpotent. The claim that this property holds for all local $\otimes$-triangulated categories is equivalent to Balmer's "nerves of steel conjecture" from arXiv:2001.00284. In the present paper, we will see how this property can fail if the category we start with is not rigid, discuss a large class of categories where the property holds, and ultimately prove that the nerves of steel conjecture is equivalent to a stronger form of this property.