Smooth motives

TL;DR

This paper constructs a differential graded (DG) category for motives over a base scheme, extending Bondarko's ideas, to better understand the structure of motives generated by smooth projective schemes over smooth bases.

Contribution

It introduces a DG category model for motives over a base scheme, providing a new framework for studying motives in algebraic geometry.

Findings

DG category equivalent to motives generated by smooth projective schemes

Framework applicable when base scheme is smooth over a perfect field

Advances the understanding of motives in the context of DG categories

Abstract

Following ideas of Bondarko, we 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.