Skip to main content
Log in

The Ziegler Spectrum and Ringel’s Quilt of the A-infinity Plane Curve Singularity

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

Abstract

We describe the Cohen–Macaulay part of the Ziegler spectrum and calculate Ringel’s quilt of the category of finitely generated Cohen–Macaulay modules over the A-infinity plane curve singularity.

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. Auslander, M.: Functors and morphisms determined by objects. In: Representation Theory of Algebras, Lecture Notes Pure Appl. Math., vol. 37, pp 1–112 (1978)

    Google Scholar 

  2. Auslander, M.: A survey on existence theorems for almost split sequences. In: Representations of Algebras, London Math. Soc. Lecture Notes, vol. 116, pp. 81–89. Cambridge University Press (1986)

  3. Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bruns, W., Herzog, J.: Cohen–Macaulay Rings, revised edition, Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press (2005)

  5. Buchweitz, R.O., Greuel, G.M., Schreyer, F.O.: Cohen–Macaulay modules over hypersurface singularities II. Invent. Math. 88, 165–182 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burban, I., Gnedin, W.: Cohen–Macaulay modules over some non-reduced curve singularities. arXiv:1301.3305v1

  7. Eisenbud, D.: Commutative Algebra with a View Towards Algebraic Geometry, Graduate Texts in Mathematics, vol. 150. Springer (1994)

  8. Gelfand, I.M., Ponomarev, V.A.: Indecomposable representations of the Lorenz group. Russ. Math. Surv. 23, 1–58 (1968)

    Article  Google Scholar 

  9. Herzog, I.: The Auslander–Reiten translate. In: Contemporary Mathematics, vol. 130, pp 153–165 (1992)

    Google Scholar 

  10. Herzog, I., Puninskaya, V.: The model theory of divisible modules over a domain. Fund. Applied Math. 2(2), 563–594 (1996)

    MATH  Google Scholar 

  11. Herzog, I., Rothmaler, P.: Pure-projective approximations. Math. Proc. Camb. Philos. Soc. 146, 83–94 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jensen, C.U., Lenzing, H.: Model-theoretic Algebra with Particular Emphasis on Fields, Rings, Modules, Algebra, Logic and Applications, vol. 2. Gordon and Breach (1989)

  13. Klingler, L., Levy, L.S.: Representation type of commutative rings I: Local wildness. Pac. J. Math. 200, 345–386 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Leuschke, G.J., Wiegand, R.: Cohen–Macalay Representations, vol. 181. AMS Mathematicas Surveys and Monographs (2012)

  15. Prest, M.: Ziegler spectra of tame hereditary algebras. J. Algebra 207, 146–164 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Prest, M.: Purity, Spectra and Localization, Encyclopedia of Mathematics and its Applications, vol. 121. Cambridge Univesity Press (2009)

  17. Prest, M., Puninski, G.: One-directed indecomposable pure injective modules over string algebras. Colloq. Math. 101, 89–112 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Puninski, G.: Serial rings. Kluwer (2001)

  19. Ringel, C.M.: The indecomposable representation of the 2-dihedral groups. Math. Ann. 214, 19–34 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ringel, C.M.: Tame algebas (On algorithm of solving vector space problems. II). In: Proceedings Ottawa Conference in Representation Theory, Lecture Notes in Math, vol. 831, pp 137–287 (1980)

    Google Scholar 

  21. Ringel, C. M.: The Ziegler spectrum of a tame hereditary algebra. Colloq. Math. 76, 105–115 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ringel, C.M.: Infinite length modules. Some Examples as Introduction. In: Krause, H., Ringel, C.M. (eds.) Infinite Length Modules, pp 1–73. Birkhäuser (2000)

  23. Ringel, C.M.: The minimal representation-infinite algebras which are special biserial. In: Representations of Algebras and Related Topics, EMS Ser. Congr. Rep., Eur. Math Soc., Zürich, pp 501–560 (2011)

  24. Schreyer, F.O.: Finite and countable CM-representation type. In: Singularities, Representation of Algebras, and Vector Bundles, Lecture Notes in Math, vol. 1273, pp 9–34. Springer (1987)

  25. Schröer, J.: On the infinite radical of the module category. Proc. Lond. Math. Soc. 81, 651–674 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yoshino, Y.: Cohen–Macaulay Modules over Cohen–Macaulay Rings, London Math. Soc. Lecture Note Series, vol. 146. Cambridge University Press (1990)

  27. Ziegler, M.: Model theory of modules. Ann. Pure Appl. Logic 26, 149–213 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gena Puninski.

Additional information

Presented by Peter Littelmann.

The article is published posthumously.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Puninski, G. The Ziegler Spectrum and Ringel’s Quilt of the A-infinity Plane Curve Singularity. Algebr Represent Theor 21, 419–446 (2018). https://doi.org/10.1007/s10468-017-9720-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-017-9720-1

Keywords

Mathematics Subject Classification (2010)

Navigation