tCG torsion pairs

We investigate conditions when the [Formula: see text]-structure of Happel–Reiten–Smalø associated to a torsion pair is a compactly generated [Formula: see text]-structure. The concept of a [Formula: see text]CG torsion pair is introduced and for any ring [Formula: see text], we prove that [Formula: see text] is a [Formula: see text]CG torsion pair in [Formula: see text] if, and only if, there exists, [Formula: see text] a set of finitely presented [Formula: see text]-modules in [Formula: see text], such that [Formula: see text]. We also show that every [Formula: see text]CG torsion pair is of finite type, and show that the reciprocal is not true. Finally, we give a precise description of the [Formula: see text]CG torsion pairs over Noetherian rings and von Neumman regular rings.

On Auslander’s formula and cohereditary torsion pairs

For a small abelian category [Formula: see text], Auslander’s formula allows us to express [Formula: see text] as a quotient of the category [Formula: see text] of coherent functors on [Formula: see text]. We consider an abelian category with the added structure of a cohereditary torsion pair [Formula: see text]. We prove versions of Auslander’s formula for the torsion-free class [Formula: see text] of [Formula: see text], for the derived torsion-free class [Formula: see text] of the triangulated category [Formula: see text] as well as the induced torsion-free class in the ind-category [Formula: see text] of [Formula: see text]. Further, for a given regular cardinal [Formula: see text], we also consider the category [Formula: see text] of [Formula: see text]-presentable objects in the functor category [Formula: see text]. Then, under certain conditions, we show that the torsion-free class [Formula: see text] can be recovered as a subquotient of [Formula: see text].

FILTRATIONS IN ABELIAN CATEGORIES WITH A TILTING OBJECT OF HOMOLOGICAL DIMENSION TWO

We consider filtrations of objects in an abelian category [Formula: see text] induced by a tilting object T of homological dimension at most two. We define three extension closed subcategories [Formula: see text] and [Formula: see text] with [Formula: see text] for j > i, such that each object in [Formula: see text] has a unique filtration with factors in these categories. In dimension one, this filtration coincides with the classical two-step filtration induced by the torsion pair. We also give a refined filtration, using the derived equivalence between the derived categories of [Formula: see text] and the module category of [Formula: see text].