The Chow Motive of a Locally Trivial Fibration

Abstract

Guillet and Soul\'e have shown that, for a fibration $\pi: Y \to X$ with fibre $Z$, locally trivial in the Zariski topology, we have a decomposition \[ [Y] = [X] \cdot [Z], \] where $[\cdot]$ denotes a class in the Grothendieck group $K_{0}(\mathcal{M}_{Rat}(k))$ associated to the category of (pure effective) Chow motives $\mathcal{M}_{Rat}(k)$ for a field $k$. By assuming some additional properties for the fibre $Z$, we construct an explicit isomorphism $h(Y) \cong h(X) \otimes h(Z)$ in the category $\mathcal{M}_{Rat}(k)$, and we use it to prove, for this type of fibrations, some conjectures disscussed by Murre.