The boundary motive: definition and basic properties
  We introduce the notion of the boundary motive of a scheme X over a perfect field. By definition, it measures the difference between the motive X and the motive with compact support of X. We develop three tools to compute the boundary motive in terms of the geometry of a compactification of X: co-localization, invariance under abstract blow-up, and analytical invariance. We then prove auto-duality of the boundary motive of a smooth scheme X. As a formal consequence of this, and of co-localization, we obtain a fourth computational tool, namely localization for the boundary motive. 
J. Wildeshaus