A conjecture on the composition of localizations on a stratified tensor triangulated category
Nicola Bellumat
TL;DR
This work extends the study of compositions of Bousfield localizations from a finite linear ordering to general Balmer-spectra stratified by Balmer-Favi support. The authors formulate a conjecture: two iterated localizations ${\mathbb{L}}_{\mathbb{A}}$ and ${\mathbb{L}}_{\mathbb{B}}$ are canonically isomorphic whenever their thread sets coincide, tying the outcome to combinatorics of chains in the Balmer spectrum. They develop axiomatic reductions, fracture-cube techniques, and chain-based holim decompositions to analyze ${\mathbb{L}}_{\mathbb{A}}$, proving the conjecture in cases with a model and in Balmer spectra that are finite or of low Krull dimension (dim 0, dim 1, and certain dim 2 scenarios). These results generalize chromatic-type fracture arguments and illuminate how local-to-global gluing data in tensor triangulated categories is governed by spectral combinatorics. The findings underscore a deep link between the topology of the Balmer spectrum and the algebraic behavior of iterated localizations, with potential applications in chromatic-type contexts beyond the classical finite-linear setting.
Abstract
We study the composition of Bousfield localizations on a tensor triangulated category stratified via the Balmer-Favi support and with noetherian Balmer spectrum. Our aim is to provide reductions via purely axiomatic arguments, allowing us general applications to concrete categories examined in mathematical practice. We propose a conjecture which states that the behaviour of the composition of the localizations depends on the chains of inclusions of the Balmer primes indexing said localizations. We prove this conjecture in the case of finite or low dimensional Balmer spectra.