Complexe de Poids, Dualit\'e et Motifs de Beilinson

TL;DR

This paper generalizes the weight complex construction for Beilinson motives, introduces a dual motivic Euler characteristic, and proves that duality exchanges weights similarly to t-structures.

Contribution

It extends Gillet and Soulé's weight complex to Beilinson motives and defines a dual motivic Euler characteristic, showing duality exchanges weights as in t-structures.

Findings

Generalization of weight complex for Beilinson motives

Definition of dual motivic Euler characteristic

Duality exchanges weights in motivic context

Abstract

In the article [GS96], Gillet and Soul\'e define a weight complex on the category of Voevodsky motives over a field of characteristic 0. In [Bon07], Bondarko generalizes this construction for any f-category with a bounded weight structure, as is the case for Beilinson motives (following Cisinski-D\'eglise ; [CD09]). The first purpose of this note is to generalize [GS96, thm. 2] in the world of Beilinson motives. This done, we will naturally be led to define the motivic Euler characteristic dual to that considered by Bondarko in [Bon10]. This fact will motivate the second line of this note : proving that the duality operation exchanges the weight as is the case for t-structure ([BBD, 5.1.14.(iii)]). ----- Dans l'article [GS96], Gillet et Soul\'e d\'efinissent un complexe de poids sur la cat\'egorie des motifs de Voevodsky d\'efinie sur un corps de caract\'eristique 0. Dans [Bon07], Bondarko g\'en\'eralise cette construction pour toute f-cat\'egorie munie d'une structure de poids born\'ee, comme c'est le cas pour les motifs de Beilinson (suivant Cisinski-D\'eglise ; [CD09]). Le premier but de cette note est de g\'en\'eraliser [GS96, thm. 2] dans le monde des motifs de Beilinson. Ceci fait, on sera naturellement amen\'e \`a d\'efinir la caract\'eristique d'Euler motivique duale \`a celle consid\'er\'ee par Bondarko dans [Bon10]. Ce fait motivera le second axe de cette note : prouver que l'op\'eration de dualit\'e \'echange les poids comme c'est le cas pour les t-structures ([BBD, 5.1.14.(iii)]).