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

  • M.V. Bondarko (a1)

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 HwC. We prove that a weight structure yields Postnikov towers for any XObjC (whose 'factors' XiObjHw). For any (co)homological functor H : CA (A is abelian) such a tower yields a weight spectral sequenceT : H(Xi [j]) ⇒ H(X[i + j]); T is canonical and functorial in X starting from E2. T specializes to the usual (Deligne) weight spectral sequences for 'classical' realizations of Voevodsky's motives DMeffgm (if we consider w = wChow with Hw = Choweff ) and to Atiyah-Hirzebruch spectral sequences in topology.

We prove that there often exists an exact conservative weight complex functor CK(Hw). This is a generalization of the functor t : DMeffgmKb(Choweff) constructed in [Bon09] (which is an extension of the weight complex of Gillet and Soulé). We prove that K0(C) ≅ 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 tChow for DMeff_DMeffgm which is adjacent to the Chow weight structure. We have HtChow ≅ AddFun(Choweffop,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,YObjC). 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 E2 of coniveau spectral sequences (by Bloch and Ogus).

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