Towards the nerves of steel conjecture

Abstract

Given a local ⊗-triangulated category, and a fiber sequence $y\stackrel{g}{\to }1\stackrel{f}{\to }x$, one may ask if there is always a nonzero object z such that either $z\otimes f$ or $z\otimes g$ is ⊗-nilpotent. The claim that this property holds for all local ⊗-triangulated categories is equivalent to Balmer's “nerves of steel conjecture” [7, Remark 5.15]. 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.