Are there conditions which guarantee that the heart of a triangulated category is Grothendieck? Is the compatibility between the t-structure with filtered colimits enough?



When is the heart of a triangulated category Grothendieck?
This topic has been studied by Parra & Saorín. For details, see their


Direct limits in the heart of a t-structure: the case of a torsion pair. 
*J. Pure Appl. Algebra*, **219** (2015), no. 9, 4117–4143. 


The case of the derived category of a commutative Noetherian ring is treated in


Hearts of t-structures in the derived category of a commutative Noetherian ring. *Trans. Amer. Math. Soc.*, **369** (2017), no. 11, 7789–7827. 


