Skip to main content
SpringerLink
Log in

τ-Categories III: Auslander Orders and Auslander–Reiten Quivers

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

Abstract

We study the relationship between representation theoretic properties and homological properties of orders. We show that there is a close relationship among Auslander orders, τ-categories and Auslander regular rings. As an application, we give a combinatorial characterization of finite Auslander–Reiten quivers of orders.

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. Auslander, M.: Representation theory of Artin algebras. I, II, Comm. Algebra 1 (1974), 177–268; ibid. 1 (1974), 269–310.

    Article  MATH  MathSciNet  Google Scholar 

  2. Anderson, F. W. and Fuller, K. R.: Rings and Categories of Modules, Grad. Texts in Math. 13, Springer, New York, 1992.

    MATH  Google Scholar 

  3. Auslander, M. and Reiten, I.: k-Gorenstein algebras and syzygy modules, J. Pure Appl. Algebra 92(1) (1994), 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  4. Auslander, M. and Roggenkamp, K. W.: A characterization of orders of finite lattice type, Invent. Math. 17 (1972), 79–84.

    Article  MathSciNet  MATH  Google Scholar 

  5. Cartan, H. and Eilenberg, S.: Homological Algebra, Princeton University Press, Princeton, NJ, 1956.

    MATH  Google Scholar 

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

    Google Scholar 

  7. Fujita, H.: Minimal injective resolution of an order over a local Dedekind domain, Comm. Algebra 26(2) (1998), 447–451.

    MATH  MathSciNet  Google Scholar 

  8. Fossum, R. M., Griffith, P. and Reiten, I.: Trivial Extensions of Abelian Categories. Homological Algebra of Trivial Extensions of Abelian Categories with Applications to Ring Theory, Lecture Notes in Math. 456, Springer, Berlin, 1975.

    MATH  Google Scholar 

  9. Goto, S. and Nishida, K.: Minimal injective resolutions of Cohen–Macaulay isolated singularities, Arch. Math. (Basel) 73(4) (1999), 249–255.

    MathSciNet  MATH  Google Scholar 

  10. Iyama, O.: Some categories of lattices associated to a central idempotent, J. Math. Kyoto Univ. 38(3) (1998), 487–501.

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  12. Iyama, O.: τ-categories II: Nakayama pairs and rejective subcategories, Algebr. Represent. Theory 8 (2005), 449–477.

    MATH  MathSciNet  Google Scholar 

  13. Igusa, K. and Todorov, G.: A characterization of finite Auslander–Reiten quivers, J. Algebra 89(1) (1984), 148–177.

    MathSciNet  MATH  Google Scholar 

  14. Matsumura, H.: Commutative Ring Theory (translated from the Japanese by M. Reid), 2nd edn, Cambridge Stud. Adv. Math. 8, Cambridge University Press, Cambridge, 1989.

    MATH  Google Scholar 

  15. McConnell, J. C. and Robson, J. C.: Noncommutative Noetherian Rings (with the cooperation of L. W. Small), Pure Appl. Math., Wiley, Chichester, 1987.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Osamu Iyama

Authors
  1. Osamu Iyama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Osamu Iyama.

Additional information

Mathematics Subject Classifications (2000)

Primary: 16G30; secondary: 16E65, 16G70, 18E05.

Current address: Department of Mathematics, University of Hyogo, Himeji, 671-2201, Japan. e-mail: iyama@sci.u-hyogo.ac.jp

Rights and permissions

Reprints and permissions

About this article

Cite this article

Iyama, O. τ-Categories III: Auslander Orders and Auslander–Reiten Quivers. Algebr Represent Theor 8, 601–619 (2005). https://doi.org/10.1007/s10468-005-0970-y

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-005-0970-y

Keywords

Search

Navigation