Weight structures and “weights” on the hearts of $t$-structures

We define and study transversal weight and $t$-structures (for triangulated categories); if a weight structure is transversal to a $t$-one, then it defines certain “weights” for its heart. Our results axiomatize and describe in detail the relations between the Chow weight structure ${w_{\textit{Chow}}}$ for Voevodsky’s motives (introduced in a preceding paper), the (conjectural) motivic $t$-structure, and the conjectural weight filtration for them. This picture becomes non-conjectural when restricted to the derived categories of Deligne’s $1$-motives (over a smooth base) and of Artin-Tate motives over number fields. In particular, we prove that the “weights” for Voevodsky’s motives (that are given by ${w_{\textit{Chow}}}$) are compatible with those for $1$-motives (that were “classically” defined using a quite distinct method); this result is new. Weight structures transversal to the canonical $t$-structures also exist for the Beilinson’s $D^b_{{\tilde{H}}_p}$ (the derived category of graded polarizable mixed Hodge complexes) and for the derived category of (Saito’s) mixed Hodge modules.

We also study weight filtrations for the heart of $t$ and (the degeneration of) weight spectral sequences. The corresponding relation between $t$ and $w$ is strictly weaker than transversality; yet it is easier to check, and we still obtain a certain filtration for (objects of) the heart of $t$ that is strictly respected by morphisms.

In a succeeding paper we apply the results obtained in order to reduce the existence of Beilinson’s mixed motivic sheaves (over a base scheme $S$) and “weights” for them to (certain) standard motivic conjectures over a universal domain $K$.

Keywords

weight structure, motive, triangulated category, $t$-structure

2010 Mathematics Subject Classification

14C30, 14K05, 18E30, 19E15

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Published 13 July 2012