This paper is dedicated to triangulated categories endowed with weight structures (a new notion; D. Pauksztello has independently introduced them as co-t-structures). This axiomatizes the properties of stupid truncations of complexes in $K(B)$. We al
so construct weight structures for Voevodskys categories of motives and for various categories of spectra. A weight structure $w$ defines Postnikov towers of objects; these towers are canonical and functorial up to morphisms that are zero on cohomology. For $Hw$ being the heart of $w$ (in $DM_{gm}$ we have $Hw=Chow$) we define a canonical conservative weakly exact functor $t$ from our $C$ to a certain weak category of complexes $K_w(Hw)$. For any (co)homological functor $H:Cto A$ for an abelian $A$ we construct a weight spectral sequence $T:H(X^i[j])implies H(X[i+j])$ where $(X^i)=t(X)$; it is canonical and functorial starting from $E_2$. This spectral sequences specializes to the usual (Delignes) weight spectral sequences for classical realizations of motives and to Atiyah-Hirzebruch spectral sequences for spectra. Under certain restrictions, we prove that $K_0(C)cong K_0(Hw)$ and $K_0(End C)cong K_0(End Hw)$. The definition of a weight structure is almost dual to those of a t-structure; yet several properties differ. One can often construct a certain $t$-structure which is adjacent to $w$ and vice versa. This is the case for the Voevodskys $DM^{eff}_-$ (one obtains certain new Chow weight and t-structures for it; the heart of the latter is dual to $Chow^{eff}$) and for the stable homotopy category. The Chow t-structure is closely related to unramified cohomology.
We construct a triangulated analogue of coniveau spectral sequences: the motif of a variety over a countable field is decomposed (in the sense of Postnikov towers) into the twisted (co)motives of its points; this is generalized to arbitrary Voevodsky
s motives. To this end we construct a Gersten weight structure for a certain triangulated category of comotives: the latter is defined to contain comotives for all projective limits of smooth varieties; the definition of a weight structure was introduced in a preceding paper. The corresponding weight spectral sequences are essentially coniveau one; they are $DM^{eff}_{gm}$-functorial (starting from $E_2$) and can be computed in terms of the homotopy $t$-structure for the category $DM^-_{eff}$ (similarly to the case of smooth varieties). This extends to motives the seminal coniveau spectral sequence computations of Bloch and Ogus. We also obtain that the cohomology of a smooth semi-local scheme is a direct summand of the cohomology of its generic fibre; cohomology of function fields contain twisted cohomology of their residue fields (for all geometric valuations). We also develop further the general theory of weight structures for triangulated categories (independently from the motivic part of the paper). Besides, we develop a certain theory of nice pairings of triangulated categories; this subject seems to be new.
The paper is suspended. The reason: as was noted by prof. H. Esnault, Theorem 2.1.1 of the previous version (as well as the related Theorem 6.1.1 of http://arxiv.org/PS_cache/math/pdf/9908/9908037v2.pdf of D. Arapura and P. Sastry) is wrong unless on
e assumes H to be a generic hyperplane section. Hence the proofs of all results starting from 2.3 contain gaps. The author hopes to correct this (somehow) in a future version. At least, most of the results follow from certain standard motivic conjectures (see part 1 of Remark 3.2.4 in the previous version). If the author would not find a way to prove Theorems 2.3.1 and 2.3.2 (without 2.1.1), then in the next version of the preprint the results of section 4 will be deduced from certain conjectures; certainly this is not a very exiting result.
