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Borel-Moore motivic homology and weight structure on mixed motives

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Fangzhou Jin
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Abstract

By defining and studying functorial properties of the Borel-Moore motivic homology, we identify the heart of Bondarko-H\'ebert's weight structure on Beilinson motives with Corti-Hanamura's category of Chow motives over a base, therefore answering a question of Bondarko.
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Math. Z. (2016) 283:1149–1183
DOI 10.1007/s00209-016-1636-7
Mathematische Zeitschrift
Borel–Moore motivic homology and weight structure
on mixed motives
Jin Fangzhou1
Received: 5 June 2015 / Accepted: 16 December 2015 / Published online: 24 February 2016
© Springer-Verlag Berlin Heidelberg 2016
Abstract By defining and studying functorial properties of the Borel–Moore motivic homol-
ogy, we identify the heart of Bondarko–Hébert’s weight structure on Beilinson motives with
Corti–Hanamura’s category of Chow motives over a base, therefore answering a question of
Bondarko.
Contents
1 Introduction ...............................................1149
2 Borel–Moore motives and their functorialities .............................1153
2.1 The six functors formalism .....................................1153
2.2 Borel–Moore motives and basic functorialities ...........................1154
2.3 Purity isomorphism, Gysin morphism and refined Gysin morphism ................1157
2.4 Composition laws ..........................................1165
3 Borel–Moore homology and Chow groups ...............................1167
3.1 The niveau spectral sequence ....................................1167
3.2 Operations on Chow groups .....................................1172
3.3 Heart of Chow weight structure ...................................1177
4 Lemmas .................................................1178
4.1 On Lemma 3.6 ............................................1178
4.2 Complements on purity isomorphisms ...............................1180
References ..................................................1183
1 Introduction
Bondarko introduced in [6] the notion of weight structure on triangulated categories as a
counterpart of t-structure [3], with the aim of applying it to the theory of mixed motives.
A weight structure defines a collection of non-positive and non-negative objects, such that
BJin Fangzhou
fangzhou.jin@ens-lyon.fr
1Lyon, France
123
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... g Zq ) is the base change of f Z along the map π q : W q Ñ ZˆB G S. We must show π 0 map DMpW q q pf Zq ! 1, g Zq ! 1pn´m´cqrr´2csq " 0 but via [Fan16][Lem. 2.37] we have the equivalence ...
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... That is F 1 is natural with respect to composition of cycles. The following is a version of [Fan16][Prop. 2.39] for stacks. ...
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