The homological spectrum for monoidal triangulated categories

TL;DR

This paper introduces a new homological prime spectrum for monoidal triangulated categories, extending Balmer's framework, and proves a conjecture relating this spectrum to the Balmer spectrum under certain conditions.

Contribution

It defines the homological spectrum via maximal elements in fibers of an extended comparison map and proves the Nerves of Steel Conjecture in stratified and uniform cases.

Findings

Constructed a surjective continuous homological comparison map.

Proved the Nerves of Steel Conjecture under stratification and uniformity.

Extended validity of the conjecture to categories with group actions.

Abstract

The authors develop a notion of homological prime spectrum for an arbitrary monoidal triangulated category, ${\mathbf C}$. Unlike the symmetric case due to Balmer, the homological primes of ${\mathbf C}$ are not defined as the maximal Serre ideals of the small module category ${\sf mod}\text{-}{{\mathbf C}}$, or via a noncommutative ring theory inspired version of this construction. Instead, the authors work with an extended comparison map from the Serre prime spectrum $\operatorname{Spc} ({\sf mod}\text{-}{{\mathbf C}})$ to the Balmer spectrum $\operatorname{Spc}{\mathbf C}$, and select the maximal elements of each fiber to define the homological spectrum $\operatorname{Spc}^{\text{h}}{\mathbf C}$. A surjective continuous homological comparison map $\operatorname{Spc}^{\text{h}}{\mathbf C} \to \operatorname{Spc}{\mathbf C}$ is constructed and used to formulate an extended Nerves of Steel Conjecture stating that this map is a homeomorphism. The conjecture is proved to hold under stratification and uniformity conditions on ${\mathbf C}$. The proof is based on a general theorem giving an explicit description of the Balmer--Favi support of the pure-injectives associated to all Serre primes of ${\sf mod}\text{-}{{\mathbf C}}$. It is shown that the validity of the conjecture carries over from ${\mathbf C}$ to a semidirect product ${\mathbf C} \rtimes G$ with an arbitrary group $G$, and when $G$ is infinite, this provides examples of monoidal triangulated categories satisfying the conjecture and whose thick ideals are not centrally generated. Important cases in which the conjecture is verified include the stable module categories of the coordinate rings of all finite group schemes and the Benson--Witherspoon Hopf algebras.