On the derived category of 1-motives

TL;DR

This paper advances the understanding of 1-motives by providing a motivic proof of a significant conjecture, developing an abstract theory of realizations with weight filtrations, and expanding the derived category framework.

Contribution

It offers a motivic proof of Deligne's conjecture on 1-motives with rational coefficients and develops an abstract theory of realizations with weight filtrations.

Findings

Proves a substantial part of Deligne's conjecture using motivic methods.

Develops an abstract theory of realizations with weight filtrations.

Expands the derived category of 1-motives with new theoretical insights.

Abstract

This is the final version of the 2007 preprint titled "On the derived category of 1-motives, I". It has been substantially expanded to contain a motivic proof of (two thirds of) Deligne's conjecture on 1-motives with rational coefficients, hence the new title. Compared to the 2007 preprint, the additions mainly concern an abstract theory of realisations with weight filtrations; Deligne's conjecture is tackled though them by an adjunction game.