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.