Open Access
2012 Weight structures and 'weights' on the hearts of $t$-structures
Mikhail V. Bondarko
Homology Homotopy Appl. 14(1): 239-261 (2012).


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 $K$.


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Mikhail V. Bondarko. "Weight structures and 'weights' on the hearts of $t$-structures." Homology Homotopy Appl. 14 (1) 239 - 261, 2012.


Published: 2012
First available in Project Euclid: 12 December 2012

zbMATH: 1251.18006
MathSciNet: MR2954675

Primary: 14C30 , 14K05 , 18E30 , 19E15 , 4C15

Keywords: $t$-structure , motive , triangulated category , Weight structure

Rights: Copyright © 2012 International Press of Boston

Vol.14 • No. 1 • 2012
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