Tensor Structure on Smooth Motives

TL;DR

This paper develops a tensor structure on the homotopy category of motives over certain base schemes, extending previous constructions by Levine to a broader setting with semi-local schemes over characteristic zero fields.

Contribution

It introduces a pseudo-tensor structure on Levine's DG category of motives, enabling tensor operations on its homotopy category over semi-local, essentially smooth schemes.

Findings

Tensor structure established over semi-local schemes.

Extends Levine's DG category to broader base schemes.

Enables new tensor operations in motivic homotopy theory.

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-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 $\mathbb{Q}$-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.