Tensor functor from Smooth Motives to motives over a base
Abstract
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 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 Q-coefficients. In this article, it is shown that, under these conditions, the fully faithful functor S that Levine constructed from his category of smooth motives to the category DMS of motives over a base (defined by Cisinski-D\'eglise) is a tensor functor.