On the weight lifting property for localizations of triangulated categories

TL;DR

This paper investigates conditions under which weight structures in triangulated categories can be lifted through localizations, especially when certain Ore conditions are satisfied, with applications to motives and stable homotopy theory.

Contribution

It establishes criteria for weight lifting properties in localized triangulated categories based on Ore conditions, extending the understanding of weight structures in these contexts.

Findings

Weight lifting properties hold under Ore conditions.

Objects in the heart of the localized category lift to the original category.

Applications to Tate motives and stable homotopy categories.

Abstract

As we proved earlier, for a triangulated category $\underline{C}$ endowed with a weight structure $w$ and a triangulated subcategory $\underline{D}$ of $\underline{C}$ (strongly) generated by cones of a set of morphisms $S$ in the heart $\underline{Hw}$ of $w$ there exists a weight structure $w'$ on the Verdier quotient $\underline{C}'=\underline{C}/\underline{D}$ such that the localization functor $\underline{C} \to \underline{C}'$ is weight-exact (i.e., "respects weights"). The goal of this paper is to find conditions ensuring that for any object of $\underline{C}'$ of non-negative (resp. non-positive) weights there exists its preimage in $\underline{C}$ satisfying the same condition; we call a certain stronger version of the latter assumption the left (resp., right) weight lifting property. We prove that these weight lifting properties are fulfilled whenever the set $S$ satisfies the corresponding (left or right) Ore conditions. Moreover, if $\underline{D}$ is generated by objects of $\underline{Hw}$ then any object of $\underline{Hw}'$ lifts to $\underline{Hw}$. We apply these results to obtain some new results on Tate motives and finite spectra (in the stable homotopy category). Our results are also applied to the study of the so-called Chow-weight homology in another paper.