Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)

In this paper we introduce a new notion of weight structure (w) for a triangulated category (C) under bar; this notion is an important natural counterpart of the notion of t-structure. It allows extending several results of the preceding paper [Bon09] to a large class of triangulated categories and functors. The heart of w is an additive category (Hw) under bar subset of (C) under bar. We prove that a weight structure yields Postnikov towers for any X is an element of Obj (C) under bar (whose ' factors' X-i is an element of Obj (Hw) under bar). For any (co) homological functor H : (C) under bar -> A (A is abelian) such a tower yields a weight spectral sequence T : H (X-i [j]) double right arrow H (X[i +j]); T is canonical and functorial in X starting from E-2. T specializes to the usual (Deligne) weight spectral sequences for ' classical' realizations of Voevodsky's motives DMgmeff (if we consider w = w(Chow) with (Hw) under bar = Chow(eff)) and to Atiyah-Hirzebruch spectral sequences in topology. We prove that there often exists an exact conservative weight complex functor (C) under bar -> K ((Hw) under bar). This is a generalization of the functor t : DM (eff)(gm) -> K-b (C how(eff)) constructed in [Bon09] (which is an extension of the weight complex of Gillet and Soul,). We prove that K0. C / S K0. Hw/ under certain restrictions. We also introduce the concept of adjacent structures: a t-structure is adjacent to w if their negative parts coincide. This is the case for the Postnikov t-structure for the stable homotopy category SH (of topological spectra) and a certain weight structure for it that corresponds to the cellular filtration. We also define a new (Chow) t-structure t(Chow) for DM_(eff) superset of DMgmeff which is adjacent to the Chow weight structure. We have (Ht) under bar Chow congruent to AddFun (Chow(eff) (op) , Ab): t(Chow) is related to unramified cohomolgy. Functors left adjoint to those that are t-exact with respect to some t-strures are weight-exact with respect to the corresponding adjacent weight structures, and vice versa. Adjacent structures identify two spectral sequences converging to (C) under bar (X, Y) : the one that comes from weight truncations of X with the one coming from t-truncations of Y (for X, Y is an element of Obj (C) under bar). Moreover, the philosophy of adjacent structures allows expressing torsion motivic cohomology of certain motives in terms of the etale cohomology of their 'submotives'. This is an extension of the calculation of E-2 of coniveau spectral sequences (by Bloch and Ogus).