Skip to main content
SpringerLink
Log in

Approximations and Mittag-Leffler conditions the applications

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

A classic result by Bass says that the class of all projective modules is covering if and only if it is closed under direct limits. Enochs extended the if-part by showing that every class of modules C, which is precovering and closed under direct limits, is covering, and asked whether the converse is true. We employ the tools developed in [18] and give a positive answer when C = A, or C is the class of all locally A≤ω-free modules, where A is any class of modules fitting in a cotorsion pair (A, B) such that B is closed under direct limits. This setting includes all cotorsion pairs and classes of locally free modules arising in (infinite-dimensional) tilting theory. We also consider two particular applications: to pure-semisimple rings, and Artin algebras of infinite representation type.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Angeleri Hügel, Covers and envelopes via endoproperties of modules, Proceedings of the London Mathematical Society 86 (2003), 649–665.

    Article  MathSciNet  MATH  Google Scholar 

  2. L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, Journal of Algebra 307 (2007), 361–376.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Angeleri Hügel and D. Herbera, Mittag-Leffler conditions on modules, Indiana university Mathematics Journal 57 (2008), 2459–2517.

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Angeleri Hügel and D. Herbera, Auslander–Reiten-components over puresemisimple hereditary rings, Journal of Algebra 331 (2011), 285–303.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. Angeleri Hügel, D. Herbera and J. Trlifaj, Tilting modules and Gorenstein rings, Forum Mathematicum 18 (2006), 211–229.

    MathSciNet  MATH  Google Scholar 

  6. L. Angeleri Hügel, D. Herbera and J. Trlifaj, Baer modules and Mittag-Leffler modules over tame hereditary algebras, Mathematische Zeitschrift 265 (2010), 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Angeleri Hügel, O. Kerner and J. Trlifaj, Large tilting modules and representation type, Manuscripta Mathematica 132 (2010), 483–499.

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Angeleri Hügel and M. Saorín, Modules with perfect decompositions, Mathematica Scandinavica 98 (2006), 19–43.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Angeleri Hügel, J. Šaroch and J. Trlifaj, On the telescope conjecture for module categories, Journal of Pure and Applied Algebra 212 (2008), 297–310.

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Baer, Homological properties of wild hereditary Artin algebras, in Representation Theory. I (Ottawa, Ont. 1984), Lecture Notes in Mathematics, Vol. 1177, Springer, Berlin, 1986, pp. 1–12.

    MathSciNet  MATH  Google Scholar 

  11. N. V. D˜ung and J. L. García, Indecomposable modules over pure semisimple hereditary rings, Journal of Algebra 371 (2012), 577–595.

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Göbel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, de Gruyter Expositions in Mathematics, Vol. 41, Walter de Gruyter, Berlin, 2012.

  13. A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents, Institut de Hautes Études Scientifiques. Publications Mathématiques 11 (1961).

    MATH  Google Scholar 

  14. D. Herbera, Definable classes and Mittag-Leffler conditions, in Ring Theory and its Applications, Contemporary Mathematics, Vol. 609, American Mathematical Society, Providence, RI, 2014, pp. 137–166.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Herbera and J. Trlifaj, Almost free modules and Mittag-Leffler conditions, Advances in Mathematics 229 (2012) 3436–3467.

    Article  MathSciNet  MATH  Google Scholar 

  16. O. Kerner and M. Takane, Mono orbits, epi orbits, and elementary vertices of representation infinite quivers, Communications in Algebra 25 (1997), 51–77.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Prest, Purity, Spectra and Localisation, Encyclopedia of Mathematics and its Applications, Vol. 121, Cambridge University Press, Cambridge, 2009.

  18. J. Šaroch, Approximations and Mittag-Leffler conditions. The tools, Israel Journal of mathematics 226 (2018), 737–756.

    Article  Google Scholar 

  19. J. Šaroch and J. Št’ovíček, The countable Telescope Conjecture for module categories, Advances in Mathematics 219 (2008), 1002–1036.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Slávik and J. Trlifaj, Approximations and locally free modules, Bulletin of the London Mathematical Society 46 (2014), 76–90.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Xu, Flat Covers of Modules, Lecture Notes in Mathematics, Vol. 1634, Springer, Berlin, 1996.

  22. W. Zimmermann, Existenz von AR-Folgen, Archiv der Mathematik 40 (1983), 40–49.

    Article  MATH  Google Scholar 

  23. W. Zimmermann, (Σ-)algebraic compactness of rings, Journal of Pure and Applied Algebra 23 (1982), 319–328.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica - Settore di Matematica, Università degli Studi di Verona, Strada le Grazie 15 - Ca’ Vignal, 37134, Verona, Italy

    Lidia Angeleri Hügel

  2. Faculty of Mathematics and Physics, Department of Algebra, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Jan Śaroch & Jan Trlifaj

Authors
  1. Lidia Angeleri Hügel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jan Śaroch
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan Trlifaj
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lidia Angeleri Hügel.

Additional information

The research of Angeleri Hügel has been supported by DGI MICIIN MTM2011-28992-C02-01, by Generalitat de Catalunya through Project 2009 SGR 1389, and by Fondazione Cariparo, Progetto di Eccellenza ASATA.

The research of Šaroch and Trlifaj supported by grant GAČR 14-15479S. Received May 16, 2017 and in revised form July 12, 2017

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Angeleri Hügel, L., Śaroch, J. & Trlifaj, J. Approximations and Mittag-Leffler conditions the applications. Isr. J. Math. 226, 757–780 (2018). https://doi.org/10.1007/s11856-018-1711-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-018-1711-3

Search

Navigation