Intersecting the dimension and slice filtrations for motives

In this note we prove that the intersections of the levels of the dimension filtration on Voevodsky’s motivic complexes over a field with the levels of the slice filtration are “as small as possible”. This statement is applied to prove that a conjecture of Ayoub is equivalent to a certain orthogonality assumption.

Keywords

Voevodsky motive, dimension filtration, slice filtration, weight structure, triangulated category, Chow motive

2010 Mathematics Subject Classification

Primary 14C15, 18E30. Secondary 14C25, 18E35, 19E15.

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Research is supported by the Russian Science Foundation grant No. 16-11-10200.

Received 16 June 2017

Received revised 13 November 2017

Published 21 February 2018