We study the behavior of direct limits in the heart of a t-structure. We prove that, for any compactly generated t-structure in a triangulated category with coproducts, countable direct limits are exact in its heart. Then, for a given Grothendieck category G and a torsion pair t = (T , F) in G, we show that the heart Ht of the associated t-structure in the derived category D(G) is AB5 if, and only if, it is a Grothendieck category. If this is the case, then F is closed under taking direct limits. The reverse implication is true for a wide class of torsion pairs which include the hereditary ones, those for which T is a cogenerating class and those for which F is a generating class. The results allow to extend results by Buan-Krause and Colpi-Gregorio to the general context of Grothendieck categories and to improve some results of (co)tilting theory of modules.

Let R be a commutative Noetherian ring and let D(R) be its (unbounded) derived category. We show that all compactly generated t-structures in D(R) associated to a left bounded filtration by supports of Spec(R) have a heart which is a Grothendieck category. Moreover, we identify all compactly generated t-structures in D(R) whose heart is a module category. As geometric consequences for a compactly generated t-structure (U, U ⊥ [1]) in the derived category D(X) of an affine Noetherian scheme X, we get the following: 1) If the sequence (U[−n]∩D ≤0 (X)) n∈N is stationary, then the heart H is a Grothendieck category; 2) If H is a module category, then H is always equivalent to Qcoh(Y), for some affine subscheme Y ⊆ X; 3) If X is connected, then: a) when ⋂ k∈Z U[k] = 0, the heart H is a module category if, and only if, the given t-structure is a translation of the canonical t-structure in D(X); b) when X is irreducible, the heart H is a module category if, and only if, there are an affine subscheme Y ⊆ X and an integer m such that U consists of the complexes U ∈ D(X) such that the support of H j (U ) is in X ∖ Y, for all j > m. ) and from the Fundación 'Séneca' of Murcia (04555/GERM/06), with a part of FEDER funds. The authors thank these institutions for their help. We also thank Leovigildo Alonso and Ana Jeremías for their remarks and for calling our attention on Ana's thesis. Finally, we thank the referee for the careful reading of the manuscript and his/her suggestions and comments.1. If H is a module category, then there exists an affine subscheme Y of X such that H ≅ Qcoh(Y).2. When X is connected, the following statements hold true:. Then H is a module category if, and only if, there exists an integer m such that (U, U ⊥ [1]) = (D ≤m (X), D ≥m (X)).(b) Suppose that X is irreducible. Then H is a module category if, and only if, there exist an integer m and a nonempty affine subscheme Y ⊆ X such that U consists of the complexes U ∈ D(X) such that Supp(H i (U )) ⊆ X ∖ Y, for all i > m. In this case H is equivalent to Qcoh(Y).The organization of the paper goes as follows. The reader is referred to Section 2 for the pertinent definitions. In Section 2 we introduce and recall the concepts and results which are most relevant for the latter sections of the paper. In particular, we recall the bijection (see [AJS, Theorem 3.1]) between compactly generated t-structures of D(R) and filtrations by supports of Spec(R). In Section 3 we give some properties of the heart H of a compactly generated t-structure in the derived category D(R) of a commutative Noetherian ring, showing in particular that H always has a generator (Proposition 3.10) and that the AB5 condition on

We study the notions of n-hereditary rings and its connection to the classes of finitely n-presented modules, FPn-injective modules, FPn-flat modules and n-coherent rings. We give characterizations of n-hereditary rings in terms of quotients of injective modules and submodules of flat modules, and a characterization of n-coherent using an injective cogenerator of the category of modules. We show two torsion pairs with respect to the FPn-injective modules and the FPn-flat modules over n-hereditary rings. We also provide an example of a Bézout ring which is 2-hereditary, but not 1-hereditary, such that the torsion pairs over this ring are not trivial.

Given a torsion pair t = (T , F ) in a Grothendieck category G, we study when the heart H t of the associated Happel-Reiten-Smalø t-structure in the derived category D(G) is a locally finitely presented or a locally coherent Grothendieck category. Since H t is Grothendieck precisely when t is of finite type (i.e., F is closed under direct limits), we first study the latter torsion pairs showing that, as in modules, they are precisely the quasi-cotilting ones, that in turn coincide with the cosilting ones.We then prove that, for G chosen in a wide class of locally finitely presented Grothendieck categories that includes the locally coherent ones, the module categories and several categories of quasi-coherent sheaves over schemes, the heart H t is locally finitely presented if, and only if, t is generated by finitely presented objects. For the same class of Grothendieck categories, it is then proved that if F is a generating class in G, in which case it is known that t is given by a (1-)cotilting object Q, the heart H t is locally coherent if, and only if, it is generated by finitely presented objects and there is a set X ⊂ F ∩ fp(G) that is a set of generators of G and satisfies the following two conditions:(1) Ext 1 G (X, −) vanishes on direct limits of objects in Prod(Q), for all X ∈ X ; (2) each epimorphism p :has a kernel which is a direct summand of (1 : t)(N ), for some N ∈ fp(G). A consequence of this is that, when G = Mod-A is the module category over small pre-additive category A (e.g., over an associative unital ring) and F is generating in Mod-A, the heart H t is locally coherent if, and only if, t is generated by finitely presented modules and, for each M ∈ mod-A := fp(Mod-A), the module (1 : t)(M ) admits a projective resolution with finitely generated terms.A further consequence is that if R is a right coherent ring and t = (T , F ) is a torsion pair such that the torsion ideal t(R) is finitely generated on the right, then H t is a locally coherent Grothendieck category if, and only if, t is generated by finitely presented modules and it is given by a cosilting module which is an elementary cogenerator in Mod-R.

We investigate conditions for when the t-structure of Happel-Reiten-Smalø associated to a torsion pair is a compactly generated t-structure. The concept of a tCG torsion pair is introduced and for any ring R, we prove that t = (T , F ) is a tCG torsion pair in R-Mod if, and only if, there exists, {T λ } a set of finitely presented R-modules in T , such that F = Ker(Hom R (T λ , ?)). We also show that every tCG torsion pair is of finite type, and show that the reciprocal is not true. Finally, we give a precise description of the tCG torsion pairs over Noetherian rings and von Neumman regular rings.