The boundary motive: definition and basic properties

Abstract

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 ${\mathop{M_{\rm gm}}\nolimits} (X)$ and the motive with compact support ${\mathop{M_{\rm gm}^{\rm c}}\nolimits}(X)$, as defined and studied by Voevodsky et al. in Cycles, transfers, and motivic homology theories (Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000)). 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\break boundary motive. In a sequel to this work (J. Wildeshaus, On the boundary motive of a Shimura variety, Prépublications du Laboratoire d'Analyse, Géométrie et Applications de l'Université Paris 13, no. 2004-23), these tools will be applied to Shimura varieties.