### The boundary motive definition and basic properties, by J. Wildeshaus

[This preprint has been updated on March 22, 2005.]

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 <wildesh@math.univ-paris13.fr>