On The Motive of G-bundles

Details On The Motive of G-bundles On The Motive of G-bundles

2011-12-18

arxiv.org/abs/1112.4110v2

Mathematics

Algebraic Geometry

15 pages

Scientific paper

Let $G$ be a reductive algebraic group over a perfect field $k$ and $\mathcal{G}$ a $G$-bundle over a scheme $X/k$. The main aim of this article is to study the motive associated with $\mathcal{G}$, inside the Veovodsky Motivic categories. We consider the case that $\charakt k=0$ (resp. $\charakt k\geq 0$), the motive associated to $X$ is geometrically mixed Tate (resp. geometrically cellular) and $\mathcal{G}$ is locally trivial for the Zariski (resp. \'etale) topology on $X$ and show that the motive of $\mathcal{G}$ is a geometrically mixed Tate motive. Moreover for a general $X$ we construct a filtration on the motive associated to $\mathcal{G}$ in terms of weight polytopes. Along the way we give some applications and examples.

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