Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura

Abstract

We describe the Voevodsky's category $D{M}_{gm}^{eff}$ of motives in terms of Suslin complexes of smooth projective varieties. This shows that Voeovodsky's $D{M}_{gm}$ is anti-equivalent to Hanamura's one. We give a description of any triangulated subcategory of $D{M}_{gm}^{eff}$ (including the category of effective mixed Tate motives). We descibe 'truncation' functors $t_N$ for $N>0$. $t=t_0$ generalizes the weight complex of Soule and Gillet; its target is $K^b(Cho{w}_{eff})$; it calculates $K_0(D{M}_{gm}^{eff})$, and checks whether a motive is a mixed Tate one. $t_N$ give a weight filtration and a 'motivic descent spectral sequence' for a large class of realizations, including the 'standard' ones and motivic cohomology. This gives a new filtration for the motivic cohomology of a motif. For 'standard realizations' for $l,s\ge 0$ we have a nice description of $W_{l+s}{H}^i/W_{l-1}{H}^i(X)$ in terms of $t_s(X)$. We define the 'length of a motif' that (modulo standard conjectures) coincides with the 'total' length of the weight filtration of singular cohomology. Over a finite field $t_{0}Q$ is (modulo Beilinson-Parshin conjecture) an equivalence.

Several linguistic corrections made; section 2.3 was corrected also

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