an:06669084
Zbl 1397.16026
Parra, Carlos E.; Saor??n, Manuel
On hearts which are module categories
EN
J. Math. Soc. Japan 68, No. 4, 1421-1460 (2016).
00361713
2016
j
16S90 16E35 18E30
derived category; Happel-Reiten-Smal?? \(\mathrm{t}\)-structure; heart of a \(\mathrm{t}\)-structure; module category; torsion pair; TTF triple; tilting module
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-Smal?? \(\mathrm{t}\)-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 \textit{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.
Zbl 1006.18011