Open Access
October, 2016 On hearts which are module categories
Carlos E. PARRA, Manuel SAORÍN
J. Math. Soc. Japan 68(4): 1421-1460 (October, 2016). DOI: 10.2969/jmsj/06841421


Given a torsion pair $\boldsymbol{t}=(\mathcal{T},\mathcal{F})$ in a module category $R\text{-}\mathrm{Mod}$ we give necessary and sufficient conditions for the associated Happel–Reiten–Smalø t-structure in $\mathcal{D}(R)$ to have a heart $\mathcal{H}_{\boldsymbol{t}}$ 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 $\boldsymbol{t}$ for which $\mathcal{H}_{\boldsymbol{t}}$ is a module category in the following cases: i) when $\boldsymbol{t}$ is the left constituent of a TTF triple, showing that $\boldsymbol{t}$ need not be HKM; ii) when $\boldsymbol{t}$ is faithful; iii) when $\boldsymbol{t}$ 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.


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Carlos E. PARRA. Manuel SAORÍN. "On hearts which are module categories." J. Math. Soc. Japan 68 (4) 1421 - 1460, October, 2016.


Published: October, 2016
First available in Project Euclid: 24 October 2016

zbMATH: 06669084
MathSciNet: MR3564438
Digital Object Identifier: 10.2969/jmsj/06841421

Primary: 16Exx
Secondary: 16B50 , 18Gxx

Keywords: derived category , Happel–Reiten–Smalø t-structure , heart of a t-structure , module category , tilting module , torsion pair , TTF triple

Rights: Copyright © 2016 Mathematical Society of Japan

Vol.68 • No. 4 • October, 2016
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