Algebras and Representation Theory

, Volume 22, Issue 5, pp 1183–1207 | Cite as

Weakly Stable Torsion Classes

  • Rishi VyasEmail author


Weakly stable torsion classes were introduced by the author and Yekutieli to provide a torsion theoretic characterisation of the notion of weak proregularity from commutative algebra. In this paper we investigate weakly stable torsion classes, with a focus on aspects related to localisation and completion. We characterise when torsion classes arising from left denominator sets and idempotent ideals are weakly stable. We show that every weakly stable torsion class T can be associated with a dg ring AT; in well behaved situations there is a homological epimorphism AAT. We end by studying torsion and completion with respect to a single regular and normal element.


Noncommutative ring theory Torsion theories Derived categories Derived functors 

Mathematics Subject Classification (2010)

(Primary) 16S90 (Secondary) 16E35 18E30 18G10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author would like to thank Amnon Yekutieli for his assistance and many suggestions regarding the material in this paper, and the anonymous referee for their comments.


  1. 1.
    Alonso, L., Jeremias, A., Lipman, J.: Local homology and cohomology on schemes. Ann. Sci. Éc. Norm. Supér. 30, 1–39 (1997). Correction, available onlineMathSciNetCrossRefGoogle Scholar
  2. 2.
    Auslander, M., Platzeck, M.I., Todorov, G.: Homological theory of idempotent ideals. Trans. Amer. Math. Soc. 332(2), 667–692 (1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Avramov, L.L., Iyengar, S.B., Lipman, J., Nayak, S.: Reduction of derived Hochschild functors over commutative algebras and schemes. Adv. Math. 223(2), 735–772 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Goodearl, K.R., Jordan, D.A.: Localizations of essential extensions. Proc. Edinb. Math. Soc. 31, 243–247 (1988)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Goodearl, K.R., Warfield Jr., R.B.: An Introduction to Noncommutative Noetherian Rings, 2nd edn., vol. 61. London Mathematical Society Student Texts, Cambridge University Press, Cambridge (2004)Google Scholar
  6. 6.
    Greenlees, J.P.C., May, J.P.: Derived functors of I-adic completion and local homology. J. Algebra 149, 438–453 (1992)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Grothendieck, A., Raynaud, M.: Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA2), Advanced Studies in Pure Mathematics 2, North-Holland Publishing Company. Updated version by Yves Laszlo available at (1968)
  8. 8.
    Hartshorne, R.: Residues and Duality, Lecture Notes in Mathematics, vol. 20. Springer, Berlin (1966)Google Scholar
  9. 9.
    Hartshorne, R.: Local Cohomology: A Seminar Given by A. Grothendieck, Lecture Notes in Mathematics, vol. 41. Springer, Berlin (1967)CrossRefGoogle Scholar
  10. 10.
    Kashiwara, M., Schapira, P.: Categories and Sheaves, Grundlehren der mathematischen Wissenschaften, vol. 332. Springer, Berlin (2006)Google Scholar
  11. 11.
    Krause, H.: Localization Theory for Triangulated Categories, Triangulated Categories, London Mathematical Society Lecture Note Series, vol. 375, pp. 161–253. Cambridge University Press (2010)Google Scholar
  12. 12.
    Matlis, E.: The higher properties of R-sequences. J. Algebra 50, 77–112 (1978)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Porta, M., Shaul, L., Yekutieli, A.: On the homology of completion and torsion. Algebr. Represent. Theory 17, 31–67 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Porta, M., Shaul, L., Yekutieli, A.: Completion by derived double centralizer. Algebr. Represent. Theory 17, 481–494 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Positselski, L.: Dedualizing complexes and MGM duality. J. Pure Appl. Algebra 220(12), 3866–3909 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nicolás, P., Saorín, M.: Parametrizing recollement data for triangulated categories. J. Algebra 322, 1220–1250 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Schenzel, P.: Proregular sequences, local cohomology, and completion. Math. Scand. 92, 161–180 (2003)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Stenström, B.: Rings of quotients, Grundlehren der mathematischen Wissenschaften, vol. 217. Springer, Berlin (1975)Google Scholar
  19. 19.
    Vyas, R., Yekutieli, A.: Weak proregularity, weak stability, and the noncommutative MGM equivalence. In press, J. Algebra. 61 pages. MathSciNetCrossRefGoogle Scholar
  20. 20.
    Vyas, R., Yekutieli, A.: Dualizing complexes in the noncommutative arithmetic context, in preparationGoogle Scholar
  21. 21.
    Weibel, C.A.: An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  22. 22.
    Yekutieli, A.: Derived categories of bimodules, in preparationGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsBen Gurion UniversityBe’er ShevaIsrael
  2. 2.DelhiIndia

Personalised recommendations