Triangulated categories of relative 1-motives

TL;DR

This paper constructs a candidate for the motivic t-structure on the triangulated category of relative 1-motives over a scheme, relating it to Deligne 1-motives and studying its properties.

Contribution

It introduces a non-degenerate t-structure on the category of relative 1-motives and explores its relation to Deligne 1-motives and the motivic Picard functor.

Findings

The t-structure is non-degenerate.

Deligne 1-motives embed fully faithfully into the heart when S is regular.

The motivic Picard functor preserves compact objects.

Abstract

We construct and study a candidate for the standard motivic t-structure on the triangulated category of relative cohomological 1-motives with rational coefficients over a noetherian finite dimensional scheme S. This t-structure is defined as a generated t-structure, and we show it is non-degenerate. We relate its heart MM^1(S) with Deligne 1-motives over S; in particular, when S is regular, the category of Deligne 1-motives embeds in MM^1(S) fully faithfully. We also study the inclusion of DA^1(S) into the larger category DA^{coh}(S) of relative cohomological motives on S, and prove that its right adjoint, the motivic Picard functor, preserves compact objects.