Journal of the Mathematical Society of Japan

On hearts which are module categories

Carlos E. PARRA and Manuel SAORÍN

Full-text: Open access

Abstract

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.

Article information

Source
J. Math. Soc. Japan, Volume 68, Number 4 (2016), 1421-1460.

Dates
First available in Project Euclid: 24 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1477327220

Digital Object Identifier
doi:10.2969/jmsj/06841421

Mathematical Reviews number (MathSciNet)
MR3564438

Zentralblatt MATH identifier
06669084

Subjects
Primary: 16Exx: Homological methods {For commutative rings, see 13Dxx; for general categories, see 18Gxx}
Secondary: 18Gxx: Homological algebra [See also 13Dxx, 16Exx, 20Jxx, 55Nxx, 55Uxx, 57Txx] 16B50: Category-theoretic methods and results (except as in 16D90) [See also 18-XX]

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

Citation

PARRA, Carlos E.; SAORÍN, Manuel. On hearts which are module categories. J. Math. Soc. Japan 68 (2016), no. 4, 1421--1460. doi:10.2969/jmsj/06841421. https://projecteuclid.org/euclid.jmsj/1477327220


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