Chow motives without projectivity

TL;DR

This paper explores the construction of a motivic version of interior cohomology for smooth, non-projective schemes within the category of geometrical motives, leveraging weight structures introduced by Bondarko.

Contribution

It introduces a method to define interior cohomology motivically in $ ext{DgM}$, extending previous work to non-projective schemes under specific weight conditions.

Findings

Constructs a motivic interior cohomology for non-projective schemes.

Utilizes Bondarko's weight structures to achieve this construction.

Prepares groundwork for applications to Shimura varieties.

Abstract

In paper 0704.4003, Bondarko recently defined the notion of weight structure, and proved that the category $\DgM$ of geometrical motives over a perfect field k, as defined and studied by Voevodsky, Suslin and Friedlander, is canonically equipped with such a structure. Building on this result, and under a condition on the weights avoided by the boundary motive, we describe a method to construct intrinsically in $\DgM$ a motivic version of interior cohomology of smooth, but possibly non-projective schemes. In a sequel to this work, this method will be applied to Shimura varieties.