Homology, Homotopy and Applications

Weight structures and 'weights' on the hearts of $t$-structures

Mikhail V. Bondarko

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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 $w_{Chow}$ 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 $w_{Chow}$) 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 $D^b_{\bar{H}_p}$ (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 $K$.

Article information

Homology Homotopy Appl., Volume 14, Number 1 (2012), 239-261.

First available in Project Euclid: 12 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 4C15 18E30: Derived categories, triangulated categories 19E15: Algebraic cycles and motivic cohomology [See also 14C25, 14C35, 14F42] 14C30: Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture 14K05: Algebraic theory

Weight structure motive triangulated category $t$-structure


Bondarko, Mikhail V. Weight structures and 'weights' on the hearts of $t$-structures. Homology Homotopy Appl. 14 (2012), no. 1, 239--261. https://projecteuclid.org/euclid.hha/1355321073

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