  We show that, under particular conditions, if a t-structure in the unbounded
derived category of a locally coherent Grothendieck category restricts to the
bounded derived category of its category of finitely presented objects, then
its heart is itself a locally coherent Grothendieck category. Those particular
conditions are always satisfied when the Grothendieck category is arbitrary and
one considers the t-structure associated to a torsion pair in the category of
finitely presented objects. They are also satisfied when one takes any
compactly generated t-structure in the derived category of a commutative
noetherian ring which restricts to the bounded derived category of finitely
generated modules. As a consequence, any t-structure in this latter bounded
derived category has a heart which is equivalent to the category of finitely
presented objects of some locally coherent Grothendieck category.
