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

Abstract

Abstract In this paper we introduce a new notion of weight structure (w) for a triangulated category C ; 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 ⊂ C . We prove that a weight structure yields Postnikov towers for any X ∈ Obj C (whose 'factors' X i ∈ Obj Hw ). For any (co)homological functor H : C → A ( A is abelian) such a tower yields a weight spectral sequence T : H(X i [j] ) ⇒ 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 DM eff gm (if we consider w = w Chow with Hw = Chow eff ) and to Atiyah-Hirzebruch spectral sequences in topology. We prove that there often exists an exact conservative weight complex functor C → K ( Hw ). This is a generalization of the functor t : DM eff gm → K b (Chow eff ) constructed in [Bon09] (which is an extension of the weight complex of Gillet and Soulé). We prove that K 0 ( C ) ≅ K 0 ( 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 tChow for DM eff _ ⊃ DM eff gm which is adjacent to the Chow weight structure. We have Ht Chow ≅ AddFun( Chow effop ,Ab ); tChow is related to unramified cohomology. Functors left adjoint to those that are t -exact with respect to some t -structures are weight-exact with respect to the corresponding adjacent weight structures, and vice versa. Adjacent structures identify two spectral sequences converging to C (X,Y): the one that comes from weight truncations of X with the one coming from t -truncations of Y (for X,Y ∈ Obj C ). Moreover, the philosophy of adjacent structures allows expressing torsion motivic cohomology of certain motives in terms of the étale cohomology of their 'submotives'. This is an extension of the calculation of E 2 of coniveau spectral sequences (by Bloch and Ogus).