  Prompted by \cite{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, and inspired by \cite{Hrb18}, we
study the local coherence of the hearts associated with Thomason filtrations of
the prime spectrum of a commutative ring, achieving a useful recursive
characterisation in case of finite length filtrations. Low length cases involve
hereditary torsion classes of finite type of the ring, and even their
Happel-Reiten-Smal\o\ hearts; in these cases, the relevant characterisations
are given by few module-theoretic conditions.
