The HRS tilting process and Grothendieck hearts of t-structures

In this paper we revisit the problem of determining when heart a t-structure is Grothendieck category, with special attention to case Happel-Reiten-Smalo (HSR) in derived category associated torsion pair latter. We HRS tilting process deriving from it lot information on t-structures which have projective generator or an injective cogenerator, and obtain several bijections between classes pairs $(\mathcal{A},\mathbf{t})$ consisting abelian it. use these re-prove, by different methods, recent result Tilting Theory fact that if $\mathbf{t}=(\mathcal{T},\mathcal{F})$ $\mathcal{G}$, then itself if, only $\mathbf{t}$ finite type. survey last results after its solution.