On the derived category of 1-motives, I

By L. Barbieri-Viale, B. Kahn. 2007

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We consider the category of Deligne 1-motives over a perfect field k of exponential characteristic p and its derived category for a suitable exact structure after inverting p. As a first result, we provide a fully faithful embedding into an etale version of Voevodsky's triangulated category of geometric motives. Our second main result is that this full embedding"almost"has a left adjoint, that we call \LAlb. Applied to the motive of a variety we thus get a bounded complex of 1-motives, that we compute fully for smooth varieties and partly for singular varieties. As an application we give motivic proofs of Roitman type theorems (in characteristic 0).

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/347baec248e30091a39f5af7f913086a7060e589