Journal of the Mathematical Society of Japan

On hearts which are module categories

Carlos E. PARRA and Manuel SAORÍN

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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


Export citation

References

  • P. Ara, Extensions of exchange rings, J. Algebra, 197 (1997), 409–423.
  • I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, Vol.,1: Techniques of Representation Theory, London Math. Soc. Student Texts, 65, Cambridge Univ. Press, 2006.
  • A. Beilinson, J. Bernstein and P. Deligne, Faisceaux Pervers, Analysis and topology on singular spaces, I, Luminy 1981, Astérisque, 100, Soc. Math. France, Paris, 1982, 5–171.
  • R. Colpi, Tilting in Grothendieck categories, Forum Math., 11 (1999), 735–759.
  • R. Colpi, G. D'Este and A. Tonolo, Quasi-tilting modules and counter equivalences, J. Algebra, 191 (1997), 461–494.
  • R. Colpi, G. D'Este and A. Tonolo, Corrigendum: Quasi-tilting modules and counter equivalences, J. Algebra, 206 (1998), 370.
  • R. Colpi and E. Gregorio, The heart of cotilting theory pair is a Grothendieck category, preprint.
  • R. Colpi, E. Gregorio and F. Mantese, On the heart of a faithful torsion theory, J. Algebra, 307 (2007), 841–863.
  • R. Colpi, F. Mantese and A. Tonolo, When the heart of a faithful torsion pair is a module category, J. Pure and Appl. Algebra, 215 (2011), 2923–2936.
  • R. Colpi and C. Menini, On the structure of $\star$-modules, J. Algebra, 158 (1993), 400–419.
  • R. Colpi and J. Trlifaj, Tilting modules and tilting torsion pairs, J. Algebra, 178 (1995), 614–634.
  • K. R. Fuller, Algebras from diagrams, J. Pure and Appl. Algebra, 48 (1987), 23–37.
  • J. L. Gómez Pardo and J. L. García Hernández, On endomorphism rings of quasi-projective modules, Math. Zeithschr., 196 (1987), 87–108.
  • D. Happel, I. Reiten and S. O. Smal\o, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc., 120, 1996.
  • M. Hoshino, Y. Kato and J.-I. Miyachi, On t-structures and torsion theories induced by compact objects, J. Pure and Appl. Algebra, 167 (2002), 15–35.
  • F. Kasch, Modules and Rings, Academic Press Inc., London, New York, Paris, 1982.
  • D. Lazard, Autour de la platitude, Bull. Soc. Math. France, 97 (1969), 81–128.
  • F. Mantese and A. Tonolo, On the heart associated with a torsion pair, Topology Appl., 159 (2012), 2483–2489.
  • Y. Miyashita, Tilting modules of finite projective dimension, Math. Zeitschr., 193 (1986), 113–146.
  • A. Neeman, Triangulated categories, Ann. of Math. Stud., Princeton University Press, 148, 2001.
  • P. Nicolás and M. Saorín, Classification of split torsion torsionfree triples in modules categories, J. Pure Appl. Algebra, 208 (2007), 979–988.
  • B. Pareigis, Categories and functors, Academic Press, 1970.
  • C. E. Parra and M. Saorín, Direct limits in the heart of a t-structure: the case of a torsion pair, J. Pure Appl. Algebra, 219 (2015), 4117–4143.
  • C. E. Parra and M. Saorín, Addendum to “Direct limits in the heart of a t-structure: the case of a torsion pair” [J. Pure Appl. Algebra, 219(9) (2015), 4117–4143], J. Pure Appl. Algebra, 220 (2016), 2467–2469.
  • N. Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monogr., 3, Academic Press, 1973.
  • J. Rickard, Morita theory for Derived Categories, J. London Math. Soc., 39 (1989), 436–456.
  • J. Rickard, Derived equivalences as derived functors, J. London Math. Soc., 43 (1991), 37–48.
  • B. Stenström, Rings of quotients, Grundlehren der math, Wissensch., 217, Springer-Verlag, 1975.
  • J. L. Verdier, Des catégories dérivées des catégories abéliennes, Asterisque, 239, 1996.