On the Heart of a faithful torsion theory

It was shown in [R. Colpi and K. R. Fuller, Trans. Amer. Math. Soc. 359 (2007), no. 2, 741--765 (electronic); MR2255195] that for any ring R and faithful torsion theory (X,Y) on Mod-R there is a cocomplete abelian category H(X,Y), the heart associated to (X,Y), and a tilting object V in H(X,Y) with End(V)≃R and which induces a counterequivalence between the torsion theory, (T,F), generated in H(X,Y) by V and (X,Y). In this paper it is shown first that if A is any abelian category with a tilting object W such that End(W)≃R and which tilts the torsion theory it generates in A to (X,Y), then A must be equivalent to H(X,Y). Next, it is shown that if H is an abelian category containing a tilting object V then H must have arbitrary coproducts, indeed must be AB4, and both functors HV=HomA(V,−) and HV′=Ext1A(V,−) from H to Mod-R must commute with coproducts. Furthermore, HV commutes with direct limits iff H is Grothendieck. Also, if (X,Y) is a faithful torsion theory in a module category then H(X,Y) has an injective cogenerator iff (X,Y) is cogenerated by a cotilting module. Further results include that if (X,Y) is a hereditary cotilting torsion theory then (X,Y) is Grothendieck and that, if (X,Y) is faithful, then H(X,Y) is equivalent to a module category iff it is the heart of the t-structure on Db(Mod-R) generated by a tilting complex (in particular this will be so if X is generated by a tilting module).