On the Heart of a faithful torsion theory

In [R. Colpi, K.R. Fuller, Tilting objects in abelian categories and quasitilted rings, Trans. Amer. Math. Soc., in press] tilting objects in an arbitrary abelian category H are introduced and are shown to yield a version of the classical tilting theorem between H and the category of modules over their endomorphism rings. Moreover, it is shown that given any faithful torsion theory (X,Y) in Mod-R, for a ring R, the corresponding Heart H(X,Y) is an abelian category admitting a tilting object which yields a tilting theorem between the Heart and Mod-R. In this paper we first prove that H(X,Y) is a prototype for any abelian category H admitting a tilting object which tilts to (X,Y) in Mod-R. Then we study AB-type properties of the Heart and commutations with direct limits. This allows us to show, for instance, that any abelian category H with a tilting object is AB4, and to find necessary and sufficient conditions which guarantee that H is a Grothendieck or even a module category. As particular situations, we examine two main cases: when (X,Y) is hereditary cotilting, proving that H(X,Y) is Grothendieck and when (X,Y) is tilting, proving that H(X,Y) is a module category.