Tensor functor from Smooth Motives to motives over a base
  Recently, Levine constructed a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme $S$ generated by the motives of smooth projective $S$-schemes, assuming that $S$ is itself smooth over a perfect field. In his construction, the tensor structure required $\mathbb{Q}$-coefficients. The author has previously shown how to provide a tensor structure on the homotopy category mentioned above, when $S$ is semi-local and essentially smooth over a field of characteristic zero, extending Levine's tensor structure with $\mathbb{Q}$-coefficients. In this article, it is shown that, under these conditions, the fully faithful functor $\rho_S$ that Levine constructed from his category of smooth motives to the category $DM_S$ of motives over a base (defined by Cisinski-D\'{e}glise) is a tensor functor. 
Anandam Banerjee
math.AG math.KT