  We study the behavior of direct limits in the heart of a t-structure. We
prove that, for any compactly generated t-structure in a triangulated category
with exact coproducts, countable direct limits are exact in its heart. Then,
for a given Grothendieck category G and a torsion pair t = (T ;F) in G, we show
that if the heart of the associated t-structure in the derived category D(G) is
AB5, then F is closed under taking direct limits. The reverse implication is
true, even implying that the heart is a Grothendieck category, for a wide class
of torsion pairs which include the hereditary ones, those for which T is a
cogenerating class and those for which F is a generating class. The results
allow to extend well-known results by Buan-Krause, Bazzoni and Colpi-Gregorio
to the general context of Grothendieck categories and to improve some results
of (co)tlting theory of modules.
