[image omitted]-Compatible torsion theories. The sheaves Pic(Σ) and B(Σ)

When [image omitted]  are Grothendieck categories and [image omitted]  are functors such that F is exact and F ⊣ G the extension of torsion theories in [image omitted]  and the restriction of torsion theories in C are denned. In the case that the category [image omitted]  is considered, if σ≤τ are [image omitted] -compatible idempotent kernel functors or with support it is possible to construct an exact sequence of sheaves of Brauer and Picard relative groups.