### Tensor Structure on Smooth Motives, by Anandam Banerjee

Grothendieck first defined the notion of a "motif" as a way of finding a
universal cohomology theory for algebraic varieties. 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. The construction of the triangulated category of motives has been
extended by Cisinski-D\'{e}glise to a triangulated category of motives over a
base-scheme S. 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 rational
coefficients. In my thesis, I 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. This is done by defining a pseudo-tensor
structure on the DG category of motives constructed by Levine, which induces a
tensor structure on its homotopy category.

Anandam Banerjee <banerjee.a@neu.edu>