Triangulated categories of relative 1-motives

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.