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On hearts which are module categories. (English) Zbl 1397.16026
Summary: Given a torsion pair $$\mathbf{t}=(\mathcal{T,F})$$ in a module category $$R$$-Mod we give necessary and sufficient conditions for the associated Happel-Reiten-Smalø $$\mathrm{t}$$-structure in $$\mathcal{D}(R)$$ to have a heart $$\mathcal{H}_{\mathbf{t}}$$ 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 M. Hoshino et al. [J. Pure Appl. Algebra 167, No. 1, 15–35 (2002; Zbl 1006.18011)] (HKM). Among other consequences, we completely identify the hereditary torsion pairs $$\mathbf{t}$$ for which $$\mathcal{H}_{\mathbf{t}}$$ is a module category in the following cases: i) when $$\mathbf{t}$$ is the left constituent of a TTF triple, showing that $$\mathbf{t}$$ need not be HKM; ii) when $$\mathbf{t}$$ is faithful; iii) when $$\mathbf{t}$$ 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.

MSC:
 16S90 Torsion theories; radicals on module categories (associative algebraic aspects) 16E35 Derived categories and associative algebras 18E30 Derived categories, triangulated categories (MSC2010)
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