Towards the Nerves of Steel Conjecture

TL;DR

This paper explores the 'nerves of steel' conjecture in local $ ext{otimes}$-triangulated categories, examining conditions under which certain tensor-nilpotency properties hold and establishing equivalences with stronger forms of the conjecture.

Contribution

It demonstrates the failure of the property in non-rigid categories, identifies classes where it holds, and proves the conjecture's equivalence to a stronger property.

Findings

Property can fail in non-rigid categories

Identifies classes where the property holds

Proves equivalence of the conjecture to a stronger property

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.