On the derived category of 1-motives, I

TL;DR

This paper studies the derived category of 1-motives over a perfect field, embedding it into Voevodsky's motives, and introduces a functor \\LAlb that approximates a left adjoint, enabling explicit computations and motivic proofs of classical theorems.

Contribution

It provides a fully faithful embedding of the derived category of 1-motives into Voevodsky's motives and introduces the \\LAlb functor, advancing the understanding of 1-motives in the motivic framework.

Findings

Embedding into Voevodsky's motives is fully faithful.

The \\LAlb functor approximates a left adjoint to the embedding.

Motivic proofs of Roitman type theorems in characteristic 0.

Abstract

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).