On the weight lifting property for localizations of triangulated 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 $w^{\prime }$ on the Verdier quotient ${\underline{C}}^{\prime }=\underline{C}/\underline{D}$ such that the localization functor $\underline{C}\to {\underline{C}}^{\prime }$ is weight-exact (i.e., "respects weights"). The goal of this paper is to find conditions ensuring that for any object of ${\underline{C}}^{\prime }$ 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 ${\underline{Hw}}^{\prime }$ 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.

arXiv commentv3: 25 pages, A full revision has been made, an author added