%0 Journal Article
%T Weight structures in localizations (revisited) and the weight lifting property
%A Vladimir Sosnilo
%J Mathematics
%D 2015
%I arXiv
%X The goal of the current paper is to study in detail the behaviour of weight structures under localizations. Let $\underline{D}$ be a triangulated subcategory of a triangulated category $\underline{C}$ endowed with a weight structure $w$; assume that any object of $\underline{D}$ admits a weak weight decomposition inside $\underline{D}$ (i.e., $\underline{D}$ is compatible with $w$ in a certain sense). Then we prove that the localized category $\underline{C}/\underline{D}$ possesses a weight structure $w_{\underline{C}/\underline{D}}$ such that the localization functor $\underline{C} \stackrel{L}\to \underline{C}/\underline{D}$ is weight-exact (i.e., "respects weights"). Suppose moreover that $\underline{D}$ is generated by $B = \operatorname{Cone}(S)$, where $S \subset \operatorname{Mor}(\underline{Hw})$ is a class of morphisms satisfying the Ore condition. In this case we give a certain explicit description of those $X\in \operatorname{Obj} \underline{C}$ such that $L(X)$ has non-positive weights in $\underline{C}/\underline{D}$. This yields a new simplified description of $w_{\underline{C}/\underline{D}}$. We apply this result to the setting of Tate motives and prove some new properties of weight $0$ motivic cohomology of Tate motives. This result also will be applied to the study of the so-called Chow-weight homology. Furthermore, we ask whether there exists a description as above for objects having non-positive weights in $\underline{C}/\underline{D}$ for $\underline{D}$ being an arbitrary subcategory of $\underline{C}$ admitting weak weight decompositions. Under certain assumptions we find a necessary and sufficient condition in terms of $K_0(\underline{Hw})$ for having such a description (and we give an example where such a description is not possible).
%U http://arxiv.org/abs/1510.03403v2