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

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

2007-04-30

arxiv.org/abs/0704.4003v8

Journal of K-theory, Volume 6, Issue 03, pp. 387-504, 2010

Mathematics

K-Theory and Homology

Several minor corrections made

Scientific paper

This paper is dedicated to triangulated categories endowed with weight structures (a new notion). This axiomatizes the properties of stupid truncations of complexes in $K(B)$. We also construct weight structures for Voevodsky's 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 $H:C\to A$ 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' (Deligne's) 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 Voevodsky's $DM^{eff}_-$ (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 with unramified cohomology. We also express torsion motivic cohomology of certain motives in terms of certain etale cohomology groups; this extends well-known coniveau spectral sequence calculations (originating from Bloch and Ogus).

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