×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI arXiv