On hearts which are module categories

Abstract

Given a torsion pair t = $(\mathcal{T},\mathcal{F})$ in a module category R-Mod we give necessary and sufficient conditions for the associated Happel–Reiten–Smalø t-structure in $\mathcal{D}$(R) to have a heart $\mathcal{H}$_t which is a module category. We also study when such a pair is given by a 2-term complex of projective modules in the way described by Hoshino–Kato–Miyachi ([HKM]). Among other consequences, we completely identify the hereditary torsion pairs t for which $\mathcal{H}$_t is a module category in the following cases: i) when t is the left constituent of a TTF triple, showing that t need not be HKM; ii) when t is faithful; iii) when t is arbitrary and the ring R is either commutative, semi-hereditary, local, perfect or Artinian. We also give a systematic way of constructing non-tilting torsion pairs for which the heart is a module category generated by a stalk complex at zero.