Motivic obstruction to rationality of a general cubic hypersurface in $\mathbb P^5$

We introduce integrally essentially indecomposable motives and prove that, if the integral motive of a smooth projective surface over a field of sufficiently big transcendence degree is integrally essentially indecomposable, then a very general cubic fourfold in $\mathbb P^5$ is not rational. We also prove a lifting theorem saying that, given a smooth projective family of surfaces over a Henselian DVR, if the motive of the special fibre is integrally essentially indecomposable, then so is the motive of the generic fibre. This suggests a possible reduction of the cubic fourfold conjecture to certain arithmetic phenomena in positive characteristic.