  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 $\mathsf{Mod}(R)$.
