The boundary motive: definition and basic properties

Details The boundary motive: definition and basic properties The boundary motive: definition and basic properties

2004-08-22

arxiv.org/abs/math/0408295v3

Compositio Math. 142 (2006), 631-656

Mathematics

Algebraic Geometry

33 pages; accepted for publication in Comp. Math. Following the request of the referee, a section on the case of normal crossi

Scientific paper

10.1112/S0010437X06001989

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.

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