Skip to main content
SpringerLink
Log in

On strongly flat and weakly cotorsion modules

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

The aim of this paper is to describe the classes of strongly flat and weakly cotorsion modules with respect to a multiplicative subset or a finite collection of multiplicative subsets in a commutative ring. The strongly flat modules are characterized by a set of conditions, while the weakly cotorsion modules are produced by a generation procedure. Passing to the collection of all countable multiplicative subsets, we define quite flat and almost cotorsion modules, and show that, over a Noetherian ring with countable spectrum, all flat modules are quite flat and all almost cotorsion modules are cotorsion.

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. Angeleri Hügel, L., Herbera, D., Trlifaj, J.: Divisible modules and localizations. J. Algebra 294(2), 519–551 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bass, H.: Finitistic dimension and a homological generalization of semi-primary rings. Trans. Am. Math. Soc. 95, 466–488 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bazzoni, S., Positselski, L.: \(S\)-almost perfect commutative rings. arXiv:1801.04820 [math.AC]

  4. Bazzoni, S., Salce, L.: Strongly flat covers. J. Lond. Math. Soc. 66(2), 276–294 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bazzoni, S., Salce, L.: On strongly flat modules over integral domains. Rocky Mt. J. Math. 34(2), 417–439 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bican, L., El Bashir, R., Enochs, E.: All modules have flat covers. Bull. Lond. Math. Soc. 33(4), 385–390 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Colbert, C.H.: Cardinality Restrictions on Noetherian Spectra. Ph. D. Dissertation, University of Texas at Austin (2017)

  8. Eklof, P.C., Trlifaj, J.: How to make Ext vanish. Bull. Lond. Math. Soc. 33(1), 41–51 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Enochs, E.: Flat covers and flat cotorsion modules. Proc. Am. Math. Soc. 92(2), 179–184 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fuchs, L., Salce, L.: \(S\)-divisible modules over domains. Forum Math. 4(4), 383–394 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fuchs, L., Salce, L.: Modules over non-Noetherian domains. Mathematical Surveys and Monographs, vol. 84. American Math. Society, Providence (2001)

    MATH  Google Scholar 

  12. Fuchs, L., Salce, L.: Almost perfect commutative rings. J. Pure Appl. Algebra 222(12), 4223–4238 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Göbel, R., Trlifaj, J.: Approximations and endomorphism algebras of modules. Second Revised and Extended Edition. De Gruyter Expositions in Mathematics, vol. 41. De Gruyter, Berlin (2012)

    Book  Google Scholar 

  14. Govorov, V.E.: On flat modules (Russian). Sibir. Mat. Zh. 6, 300–304 (1965)

    MATH  Google Scholar 

  15. Gruson, L.: Dimension homologique des modules plats sur un anneau commutatif noetherien. In: Convegno di Algebra Commutativa, INDAM, Rome, 1971, Symposia Mathematica, vol. XI, pp. 243–254. Academic Press, London (1973)

  16. Hamsher, R.M.: On the structure of a one-dimensional quotient field. J. Algebra 19(3), 416–425 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  17. Harrison, D.K.: Infinite abelian groups and homological methods. Ann. Math. 69(2), 366–391 (1959). (Correction: Ann. Math. 71(1), 197 (1960))

    Article  MathSciNet  MATH  Google Scholar 

  18. Lazard, D.: Autour de la platitude. Bull. Soc. Math. France 97, 81–128 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  19. McAdam, S.: Intersections of height \(2\) primes. J. Algebra 49(2), 315–321 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  20. Matlis, E.: Cotorsion modules. Mem. Am. Math. Soc., vol. 49 (1964)

  21. Positselski, L.: Weakly curved A\(_{\infty }\)-algebras over a topological local ring. arXiv:1202.2697 [math.CT]

  22. Positselski, L.: Contraherent cosheaves. arXiv:1209.2995 [math.CT]

  23. Positselski, L.: Contraadjusted modules, contramodules, and reduced cotorsion modules. Moscow Math. J. 17(3), 385–455 (2017). arXiv:1605.03934 [math.CT]

  24. Positselski, L.: Triangulated Matlis equivalence. J. Algebra Appl. 17(4), article ID 1850067 (2018). arXiv:1605.08018 [math.CT]

  25. Positselski, L.: Abelian right perpendicular subcategories in module categories. arXiv:1705.04960 [math.CT]

  26. Positselski, L., Rosický, J.: Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories. J. Algebra 483, 83–128, (2017). arXiv:1512.08119 [math.CT]

  27. Positselski, L., Slávik, A.: Flat morphisms of finite presentation are very flat. arXiv:1708.00846 [math.AC]

  28. Raynaud, M., Gruson, L.: Critères de platitude et de projectivité: techniques de “platification” d’un module. Invent. Math. 13(1–2), 1–89 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  29. Salce, L.: Cotorsion theories for abelian groups. In: Conference on abelian groups and their relationship to the theory of modules, INDAM, Rome, 1977, Symposia Mathematica, vol. XXIII, pp. 11–32. Academic Press, London (1979)

  30. Šaroch, J., Št’ovíček, J.: The countable telescope conjecture for module categories. Adv. Math.219(3), 1002–1036 (2008). arXiv:0801.3936 [math.RA]

  31. Trlifaj, J.: Cotorsion theories induced by tilting and cotilting modules. In: Abelian groups, rings and modules (Perth, 2000), Contemp. Math. vol. 273, pp. 285–300. AMS, Providence (2001)

  32. Trlifaj, J.: Covers, envelopes, and cotorsion theories. Lecture Notes for the Workshop “Homological Methods in Module Theory”, Cortona, September 2000, p. 39 (2000). http://matematika.cuni.cz/dl/trlifaj/NALG077cortona.pdf

  33. Yekutieli, A.: Flatness and completion revisited. Algebras Represent. Theory 21(4), 717–736 (2018). arXiv:1606.01832 [math.AC]

Download references

Acknowledgements

The first author is grateful to Silvana Bazzoni for very helpful discussions and communications. This paper grew out of the first author’s visits to Padova and Prague, and he wishes to thank Silvana Bazzoni and Jan Trlifaj for the kind invitations. We also wish to thank Jan Trlifaj for very helpful discussions. The first author’s research is supported by the Israel Science Foundation Grant # 446/15 and by the Grant Agency of the Czech Republic under the Grant P201/12/G028. The second author’s research is supported by the Grant Agency of the Czech Republic under the Grant 17-23112S and by the SVV project under the Grant SVV-2017-260456.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Natural Sciences, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Leonid Positselski

  2. Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Moscow, 119048, Russia

    Leonid Positselski

  3. Sector of Algebra and Number Theory, Institute for Information Transmission Problems, Moscow, 127051, Russia

    Leonid Positselski

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

    Leonid Positselski & Alexander Slávik

Authors
  1. Leonid Positselski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Slávik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leonid Positselski.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Positselski, L., Slávik, A. On strongly flat and weakly cotorsion modules. Math. Z. 291, 831–875 (2019). https://doi.org/10.1007/s00209-018-2116-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-018-2116-z

Search

Navigation