Skip to main content
SpringerLink
Log in

Dual F-Signature of Cohen-Macaulay Modules Over Rational Double Points

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

The dual F-signature is a numerical invariant defined via the Frobenius morphism in positive characteristic. It is known that the dual F-signature characterizes some singularities. However, the value of the dual F-signature is not known except in only a few cases. In this paper, we determine the dual F-signature of Cohen-Macaulay modules over two-dimensional rational double points. The method for determining the dual F-signature is also valid for determining the Hilbert-Kunz multiplicity. We discuss it in Appendix.

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. Aberbach, I., Leuschke, G.: The F-signature and strongly F-regularity. Math. Res. Lett. 10, 51–56 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Auslander, M.: Rational singularities and almost split sequences. Trans. Amer. Math. Soc. 293(2), 511–531 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  3. Auslander, M., Reiten, I.: McKay quivers and extended Dynkin diagrams. Trans. Amer. Math. Soc. 293(1), 293–301 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brieskorn, E.: Rationale singularitäten komplexer flächen. Invent. Math. 4, 336–358 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gabriel, P.: Auslander-Reiten sequences and representation-finite algebras, Lecture Notes in Math. 831, Representation theory I (Proceedings, Ottawa, Carleton Univ., 1979), pp 1–71. Springer (1980)

  6. Hara, N., Sawada, T.: Splitting of Frobenius sandwiches. RIMS Kôkyûroku Bessatsu B24, 121–141 (2011)

    MATH  MathSciNet  Google Scholar 

  7. Hashimoto, M., Nakajima, Y.: Generalized F-signature of invariant subrings. arXiv:1311.5963

  8. Huneke, C.: Hilbert-Kunz Multiplicity and the F-signature, Commutative algebra. Expository papers dedicated to David Eisenbud on the occasion of his 65th birthday, Springer-Verlag, New York, 485–525

  9. Huneke, C., Leuschke, G.: Two theorems about maximal Cohen-Macaulay modules. Math. Ann. 324(2), 391–404 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Huneke, C., Yao, Y.: Unmixed local rings with minimal Hilbert-Kunz multiplicity are regular. Proc. Amer. Math. Soc. 130, 661–665 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kunz, E.: Characterizations of regular local rings for characteristic p. Amer. J. Math. 41, 772–784 (1969)

    Article  MathSciNet  Google Scholar 

  12. Kunz, E.: On Noetherian ring of characteristic p. Amer. J. Math. 88, 999–1013 (1976)

    Article  MathSciNet  Google Scholar 

  13. Iyama, O.: τ-categories I: Ladders. Algebr. Represent. Theory 8(3), 297–321 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Iyama, O., Wemyss, M.: The classification of special Cohen Macaulay modules. Math. Z 265(1), 41–83 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Leuschke, G., Wiegand, R.: Cohen-Macaulay Representations, vol. 181 of Mathematical Surveys and Monographs. American Mathematical Society (2012)

  16. Monsky, P.: The Hilbert-Kunz function. Math. Ann. 263, 43–49 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  17. Nakajima, Y.: Dual F-signature of special Cohen-Macaulay modules over cyclic quotient surface singularities. arXiv:1311.5967

  18. Riemenschneider, O.: Die Invarianten der endlichen Untergruppen von GL(2, ℂ). Math. Z 153, 37–50 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sannai, A.: Dual F-signature, to appear in Int. Math. Res. Not. IMRN. arXiv:2381.1301

  20. Smith, K.E., Van den Bergh, M.: Simplicity of rings of differential operators in prime characteristic. Proc. Lond. Math. Soc. (3) 75(1), 32–62 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Tucker, K.: F-signature exists. Invent. Math. 190(3), 743–765 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Watanabe, K.: Certain invariant subrings are Gorenstein. I. Osaka J. Math. 11, 1–8 (1974)

    MATH  MathSciNet  Google Scholar 

  23. Watanabe, K., Yoshida, K.: Hilbert-Kunz multiplicity and an inequality between multiplicity and colength. J. Algebra 230, 295–317 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Watanabe, K., Yoshida, K.: Minimal relative Hilbert-Kunz multiplicity. Illinois J. Math. 48(1), 273–294 (2004)

    MATH  MathSciNet  Google Scholar 

  25. Wemyss, M.: The GL(2, ℂ) McKay Correspondence. Math. Ann. 350(3), 631–659 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wunram, J.: Reflexive modules on quotient surface singularities. Math. Ann. 279(4), 583–598 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  27. Yao, Y.: Modules with Finite F-Representation Type. J. Lond. Math. Soc. (2) 72(2), 53–72 (2005)

    Article  MATH  Google Scholar 

  28. Yao, Y.: Observations on the F-signature of local rings of characteristic p. J. Algebra 299(1), 198–218 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. Yoshino, Y.: Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, vol. 146. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School Of Mathematics, Nagoya University, Chikusa-Ku, Nagoya, 464-8602, Japan

    Yusuke Nakajima

Authors
  1. Yusuke Nakajima
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yusuke Nakajima.

Additional information

Presented by Claus Michael Ringel.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nakajima, Y. Dual F-Signature of Cohen-Macaulay Modules Over Rational Double Points. Algebr Represent Theor 18, 1211–1245 (2015). https://doi.org/10.1007/s10468-015-9538-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-015-9538-7

Keywords

Mathematics Subject Classifiation (2010)

Search

Navigation