  Any Thomason filtration of a commutative ring yields (at least) two
t-structures in the derived category of the ring, one of which is compactly
generated [Hrb20,HHZ21]. We study the hearts of these two t-structures and
prove that they coincide in case of a weakly bounded below filtration. Prompted
by [SS20], in which it is proved that the heart of a compactly generated
t-structure in a triangulated category with coproduct is a locally finitely
presented Grothendieck category, we study the local coherence of the hearts
associated with a weakly bounded below Thomason filtration, achieving a useful
recursive characterisation in case of a finite length filtration. Low length
cases involve hereditary torsion classes of finite type of the ring, and even
their Happel-Reiten-Smalo hearts; in these cases, the relevant
characterisations are given by few module-theoretic conditions.
