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

Part of: Higher algebraic $K$-theory Homological methods (Co)homology theory Singularities

Published online by Cambridge University Press:  16 October 2008

M. V. Bondarko
M. V. Bondarko
Affiliation:
Saint-Petersburg State University, Faculty of Higher Algebra and Number Theory, Bibliotechnaya Pl. 2, 198904, Saint Petersburg, Russia (mbond77@mail.ru)
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Abstract

We describe explicitly the Voevodsky's triangulated category of motives (and give a ‘differential graded enhancement’ of it). This enables us to able to verify that DMgm ℚ is (anti)isomorphic to Hanamura's (k).

We obtain a description of all subcategories (including those of Tate motives) and of all localizations of . We construct a conservative weight complex functor ; t gives an isomorphism . A motif is mixed Tate whenever its weight complex is. Over finite fields the Beilinson–Parshin conjecture holds if and only if tℚ is an equivalence.

For a realization D of we construct a spectral sequence S (the spectral sequence of motivic descent) converging to the cohomology of an arbitrary motif X. S is ‘motivically functorial’; it gives a canonical functorial weight filtration on the cohomology of D(X). For the ‘standard’ realizations this filtration coincides with the usual one (up to a shift of indices). For the motivic cohomology this weight filtration is non-trivial and appears to be quite new.

We define the (rational) length of a motif M; modulo certain ‘standard’ conjectures this length coincides with the maximal length of the weight filtration of the singular cohomology of M.

Keywords

motivesalgebraic cyclesrealizations and cohomologyweight filtrations and spectral sequences

MSC classification

Secondary: 14F42: Motivic cohomology; motivic homotopy theory 16E45: Differential graded algebras and applications 14F20: Étale and other Grothendieck topologies and (co)homologies 19D55: $K$-theory and homology; cyclic homology and cohomology 32S20: Global theory of singularities; cohomological properties
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 1 , January 2009 , pp. 39 - 97
Copyright
Copyright © Cambridge University Press 2008

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