On the Weight Lifting Property for Localizations of Triangulated Categories

Abstract

As we proved earlier, for any 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 morphism 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 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.