Abstract
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.
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