#### Advisor(s)

Marc Levine (1952-)

#### Date of Award

2010

#### Date Accepted

4-2010

#### Degree Grantor

Northeastern University

#### Degree Level

Ph.D.

#### Degree Name

Doctor of Philosophy

#### Department or Academic Unit

College of Arts and Sciences. Department of Mathematics.

#### Keywords

DG categories, tensor structure

#### Subject Categories

Motives (Mathematics)

#### Disciplines

Mathematics

#### Abstract

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-Deglise 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 coefficients. In my thesis, I show how to provide a tensor structure on the homotopy category mentioned above, when S is semilocal 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.

#### Document Type

Dissertation

#### Rights Holder

Anandam Banerjee

#### Permanent URL

#### Recommended Citation

Banerjee, Anandam, "Tensor structure on smooth motives" (2010). *Mathematics Dissertations.* Paper 10. http://hdl.handle.net/2047/d20000154

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