Over a commutative noetherian ring R , the prime spectrum controls, via the assignment of support, the structure of both \mathsf{Mod}(R) and \mathsf{D}(R) . We show that, just like in \mathsf{Mod}(R) , the assignment of support classifies hereditary torsion pairs in the heart of any nondegenerate compactly generated t -structure of \mathsf{D}(R) . Moreover, we investigate whether these t -structures induce derived equivalences, obtaining a new source of Grothendieck categories which are derived equivalent to \mathsf{Mod}(R) .