Skip to main content
Log in

Vanishing of Ext modules over algebras

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

Recently, Kimura, Otake, and Takahashi proved a theorem about the vanishing of Ext of finitely generated modules over Cohen–Macaulay rings. The aim of this paper is to obtain extensions of their result over algebras.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Araya, T.: The Auslander–Reiten conjecture for Gorenstein rings. Proc. Amer. Math. Soc. 137(6), 1941–1944 (2009)

  2. Auslander, M., Functors and morphisms determined by objects. In: Proc. Conf. on Representation Theory, pp. 1-244 (Philadelphia 1967). Dekker, New York (1978)

  3. Auslander, M., Bridger, M.: Stable module theory. Mem. Amer. Math. Soc. 94 (1969)

  4. Auslander, M., Reiten, I.: On a generalized version of the Nakayama conjecture. Proc. Amer. Math. Soc. 52, 69–74 (1975)

  5. Bahlekeh, A., Mahin Fallah, A., Salarian, Sh.: On the Auslander–Reiten conjecture for algebras. J. Algebra 427, 252-263 (2015)

  6. Brown, K.S.: Cohomology of Groups. Graduate Texts in Mathematics, 87. Springer, New York-Berlin (1982)

  7. Christensen, L.W., Holm, H.: Algebras that satisfy Auslander’s condition on vanishing of cohomology. Math. Z. 265, 21–40 (2010)

    Article  MathSciNet  Google Scholar 

  8. Celikbas, O., Takahashi, R.: Auslander–Reiten conjecture and Auslander–Reiten duality. J. Algebra 382, 100–114 (2013)

    Article  MathSciNet  Google Scholar 

  9. Curtis, C.W., Reiner, I.: Methods of Representation Theory. Vol. I. With Applications to Finite Groups and Orders. Reprint of the 1981 Original.Wiley, New York (1990)

  10. Enochs, E., Jenda, O.M.G.: Relative Homological Algebra. Walter de Gruyter, Berlin New York (2000)

    Book  Google Scholar 

  11. Happel, D.: Homological Conjectures in Representation Theory of Finite-Dimensional Algebras. Sherbrook Lecture Notes Series (1991)

  12. Hoshino, M.: Modules without self-extensions and Nakayama’s conjecture. Arch. Math. (Basel) 43, 494–500 (1984)

  13. Huneke, C., Leuschke, G.J.: On a conjecture of Auslander and Reiten. J. Algebra 275, 781–790 (2004)

    Article  MathSciNet  Google Scholar 

  14. Kimura, K., Otake, Y., Takahashi, R.: Maximal Cohen–Macaulay tensor products and vanishing of Ext modules. Bull. Lond. Math. Soc. 54(6), 2456–2468 (2022)

    Article  MathSciNet  Google Scholar 

  15. Lam, T.Y.: Exercises in Classical Ring Theory. Problem Books in Mathematics, 2nd edn. Springer, Berlin (2003)

    Google Scholar 

  16. Mahin Fallah, A.: Auslander-Reiten conjucture of the tilting theorem. J. Algebra Appl. 23(1), Paper No. 2450013, 8 pp. (2024)

  17. Mahin Fallah, A.: Generalization of Auslander–Reiten duality for algebras. Comm. Algebra 49(10), 4427-4433 (2021)

  18. Mahin Fallah, A.: On the vanishing of self extensions over algebras. Arch. Math. (Basel) 115, 619-625 (2020)

  19. Matsumura, H.: Commutative Ring Theory. Translated from the Japanese by M. Reid. Second Edition. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge (1989)

  20. Nakayama, T.: On algebras with complete homology. Abh. Math. Sem. Univ. Hamburg 22, 300–307 (1958)

    Article  MathSciNet  Google Scholar 

  21. Scheja, G., Storch, U.: Quasi-Frobenius–Algebren und lokal vollständige Durchschnitte. Manuscripta Math. 19(1), 75–104 (1976)

    Article  MathSciNet  Google Scholar 

  22. Tachikawa, H.: Quasi-Frobenius Rings and Generalizations. QF–3 and QF–1 Rings. Notes by Claus Michael Ringel. Lecture Notes in Mathematics, vol. 351. Springer, Berlin-New York (1973)

  23. Zhang, X.: A note on Gorenstein projective conjecture II. Nanjing Daxue Xuebao Shuxue Bannian Kan 29(2), 155–162 (2012)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ali Mahin Fallah.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was also in part supported by a Grant from IPM (No. 1401130014).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mahin Fallah, A. Vanishing of Ext modules over algebras. Arch. Math. (2024). https://doi.org/10.1007/s00013-024-01981-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00013-024-01981-3

Keywords

Mathematics Subject Classification

Navigation