On the transitivity of degeneration of modules

  • Ryo TakahashiEmail author


This paper investigates the transitivity of the relation defined by degeneration of finitely generated modules over an associative algebra. It is proved in this paper that if L degenerates to M and M degenerates to N, then \(L^{\oplus e}\) degenerates to \(N^{\oplus e}\) for some (but explicitly given) integer \(e>0\).

Mathematics Subject Classification

16D70 14D06 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abeasis, S., Del Fra, A.: Degenerations for the representations of a quiver of type \(\mathscr {A}_m\). J. Algebra 93(2), 376–412 (1985)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aehle, R., Riedtmann, Ch., Zwara, G.: Complexity of degenerations of modules. Comment. Math. Helv. 76(4), 781–803 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Eisenbud, D.: Commutative Algebra, With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)zbMATHGoogle Scholar
  4. 4.
    Hiramatsu, N., Yoshino, Y.: Examples of degenerations of Cohen–Macaulay modules. Proc. Am. Math. Soc. 141(7), 2275–2288 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Miyata, T.: Note on direct summands of modules. J. Math. Kyoto Univ. 7, 65–69 (1967)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Riedtmann, C.: Degenerations for representations of quivers with relations. Ann. Sci. École Norm. Sup. (4) 19(2), 275–301 (1986)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Yoshino, Y.: On degenerations of modules. J. Algebra 278(1), 217–226 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Yoshino, Y.: Stable degenerations of Cohen–Macaulay modules. J. Algebra 332, 500–521 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zwara, G.: A degeneration-like order for modules. Arch. Math. (Basel) 71(6), 437–444 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Zwara, G.: Degenerations of finite-dimensional modules are given by extensions. Compos. Math. 121(2), 205–218 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan
  2. 2.Department of MathematicsUniversity of KansasLawrenceUSA

Personalised recommendations