Chennai Mathematical Institute


2.00 p.m.
Tensor Structure on Smooth Motives (Part II)

Anandam Banerjee



Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric motives over a perfect field, which has many of the properties expected of the derived category of the conjectural abelian category of motives. Recently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky's category of effective geometric motives, assuming resolution of singularities. Soon after, Levine extended this idea to construct 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 both constructions, the tensor structure requires $\mathbb{Q}$-coefficients. In the first talk, I will go describe the category of smooth motives defined by Levine. In the second, I will talk about a pseudo-tensor structure on the DG category of motives and show 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.