Néron models of 1-motives and duality

Abstract

In this paper, we propose a definition of Néron models of arbitrary Deligne 1-motives over Dedekind schemes, extending Néron models of semi-abelian varieties. The key property of our Néron models is that they satisfy a generalization of Grothendieck's duality conjecture in SGA 7 when the residue fields of the base scheme at closed points are perfect. The assumption on the residue fields is unnecessary for the class of 1-motives with semistable reduction everywhere. In general, this duality holds after inverting the residual characteristics. The definition of Néron models involves careful treatment of ramification of lattice parts and its interaction with semi-abelian parts. This work is a complement to Grothendieck's philosophy on Néron models of motives of arbitrary weights.