Serre functors and complete torsion pairs

Given a torsion pair (T,F) in an abelian category A, there is a t-structure (UT,VT) determined by T on the derived category Dᵇ(A). The existence of derived equivalence between heart B of the t-structure and A which naturally extends the embedding B→ Dᵇ(A) is determined by the completeness of the torsion pair [6]. When A is the module category of a finite-dimensional hereditary algebra and UT is closed under Serre functor, then there exists a triangle equivalence Dᵇ(B)→ Dᵇ(A) [21]. In this case, we give a straightforward proof of the fact torsion pair (T,F) is complete if and only if UT is closed under the Serre functor.