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Ringel’s conjecture for domestic string algebras

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Abstract

We classify indecomposable pure injective modules over domestic string algebras, verifying Ringel’s conjecture on the structure of such modules.

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References

  1. Baratella, S., Prest, M.: Pure-injective modules over the dihedral algebras. Commun. Algebra 25(1), 11–31 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Burke, K., Prest, M.: The Ziegler and Zariski spectra of some domestic string algebras. Algebras Represent. Theory 5, 211–234 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Butler, M.C.R., Ringel, C.M.: Auslander–Reiten sequences with few middle terms and applications to string algebras. Commun. Algebra 15, 145–179 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Crawley-Boevey, W.: Infinite-dimensional modules in the representation theory of finite-dimensional algebras. In: Reiten, I., Smalø, S., Solberg, Ø. (eds.) Algebras and Modules I, Canadian Mathematics Society, Conference Proceedings, vol. 23, pp. 29–54. American Mathematical Society (1998)

  5. Harland, R.J.: Pure injective modules over tubular algebras and string algebras. Ph.D. Thesis, University of Manchester (2011)

  6. Krause, H.: Maps between tree and band modules. J. Algebra 137, 186–194 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Puninski, G.: Super decomposable pure injective modules exist over some string algebras. Proc. Am. Math. Soc. 132, 1891–1898 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Puninski, G.: Band combinatorics of domestic string algebras. Colloq. Math. 108, 285–296 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Puninski, G.: Krull–Gabriel dimension and Cantor–Bendixson rank of 1-domestic string algebras. Colloq. Math. 127, 185–210 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Puninski, G.: Pure injective indecomposable modules over 1-domestic string algebras. Algebras Represent. Theory 17, 643–673 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ringel, C.M.: Some algebraically compact modules. I. In: Facchini, A., Menini, C. (eds.) Abelian Groups and Modules, pp. 419–439. Kluwer, Alphen aan den Rijn (1995)

    Chapter  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Ringel, C.M.: Infinite length modules. Some examples as introduction. In: Krause, H., Ringel, C.M. (eds.) Infinite Length Modules, pp. 1–73. Birkhäuser, Basel (2000)

    Chapter  Google Scholar 

  17. Schröer, J.: Hammocks for string algebras. Ph.D. Thesis, Bielefeld, SFB 343 E97–010 (1997)

  18. Ziegler, M.: Model theory of modules. Ann. Pure Appl. Logic 26(2), 149–213 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Belarusian State University, Praspekt Nezalezhnosti 4, 220030, Minsk, Belarus

    Gena Puninski

  2. School of Mathematics, Alan Turing Building, University of Manchester, Manchester, M13 9PL, UK

    Mike Prest

Authors
  1. Gena Puninski
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  2. Mike Prest
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Corresponding author

Correspondence to Mike Prest.

Additional information

The first author is grateful for EPSRC support on Grant EP/K022490/1 for his one month visit to Manchester University during which this paper was written. He also is indebted to the University for an encouraging scientific environment. Both authors are greatly indebted to a referee for many useful suggestions.

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Puninski, G., Prest, M. Ringel’s conjecture for domestic string algebras. Math. Z. 282, 61–77 (2016). https://doi.org/10.1007/s00209-015-1532-6

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  • DOI: https://doi.org/10.1007/s00209-015-1532-6

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