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Maximal $ n$-orthogonal modules for selfinjective algebras


Authors: Karin Erdmann and Thorsten Holm
Journal: Proc. Amer. Math. Soc. 136 (2008), 3069-3078
MSC (2000): Primary 16G10, 16D50, 16E10, 16G70
DOI: https://doi.org/10.1090/S0002-9939-08-09297-6
Published electronically: April 29, 2008
MathSciNet review: 2407069
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Abstract: Let $ A$ be a finite-dimensional selfinjective algebra. We show that, for any $ n\ge 1$, maximal $ n$-orthogonal $ A$-modules (in the sense of Iyama) rarely exist. More precisely, we prove that if $ A$ admits a maximal $ n$-orthogonal module, then all $ A$-modules are of complexity at most 1.


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  • 1. M. Auslander, Representation dimension of Artin algebras. Queen Mary College, Mathematics Notes, University of London, 1971. Also in: I. Reiten, S. Smalø, Ø. Solberg (eds.), Selected works of Maurice Auslander, Part I, Amer. Math. Soc., Providence, RI, 1999, 505-574. MR 1674397 (2000j:01119a)
  • 2. M. Auslander, I. Reiten, Representation theory of Artin algebras III. Comm. Algebra 3 (1975), 239-294. MR 0379599 (52:504)
  • 3. M. Auslander, I. Reiten, S. Smalø, Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, 1995. MR 1314422 (96c:16015)
  • 4. J. Białkowski, A. Skowroński, Selfinjective algebras of tubular type. Colloq. Math. 94 (2002), no. 2, 175-194. MR 1967373 (2004a:16019)
  • 5. J. Białkowski, A. Skowroński, On tame weakly symmetric algebras having only periodic modules. Arch. Math. (Basel) 81 (2003), no. 2, 142-154. MR 2009556 (2004k:16039)
  • 6. K. Bongartz, Tilted algebras. Representations of algebras (Puebla, 1980), pp. 26-38, Lecture Notes in Math., 903, Springer, Berlin-New York, 1981. MR 654701 (83g:16053)
  • 7. S. Brenner, M.C.R. Butler, A.D. King, Periodic algebras which are almost Koszul. Algebr. Represent. Theory 5 (2002), 331-367. MR 1930968 (2003i:16011)
  • 8. J. Białkowski, K. Erdmann, A. Skowroński, Deformed preprojective algebras of generalized Dynkin type. Trans. Amer. Math. Soc. 359 (2007), 2625-2650. MR 2286048
  • 9. A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics. Adv. Math. 204 (2006), no. 2, 572-618. MR 2249625 (2007f:16033)
  • 10. P. Caldero, F. Chapoton, R. Schiffler, Quivers with relations arising from clusters ($ A_n$ case). Trans. Amer. Math. Soc. 358 (2006), 1347-1364. MR 2187656 (2007a:16025)
  • 11. K. Erdmann, Blocks of tame representation type and related algebras. Lecture Notes in Mathematics, 1428, Springer-Verlag, Berlin, 1990. MR 1064107 (91c:20016)
  • 12. K. Erdmann, M. Holloway, N. Snashall, Øyvind Solberg, R. Taillefer, Support varieties for selfinjective algebras. K-Theory 33 (2004), 67-87. MR 2199789 (2007f:16014)
  • 13. K. Erdmann, O. Kerner, On the stable module category of a self-injective algebra. Trans. Amer. Math. Soc. 352 (2000), no. 5, 2389-2405. MR 1487612 (2000j:16027)
  • 14. K. Erdmann, N. Snashall, Hochschild cohomology of preprojective algebras I. J. Algebra 205 (1998), 413-434. MR 1632808 (99e:16013)
  • 15. K. Erdmann, N. Snashall, Preprojective algebras of Dynkin type, periodicity and the second Hochschild cohomology. In: I. Reiten, S. Smalø, Ø, Solberg (eds.), Algebras and Modules II, CMS Conf. Proc. 24, American Math. Soc., Providence, RI, 1998, pp. 183-193. MR 1648626 (99h:16016)
  • 16. S. Fomin, A. Zelevinsky, Cluster algebras I. Foundations. J. Amer. Math. Soc. 15 (2002), no. 2, 497-529. MR 1887642 (2003f:16050)
  • 17. C. Geiß, B. Leclerc, J. Schröer, Rigid modules over preprojective algebras. Invent. Math. 165 (2006), 589-632. MR 2242628 (2007g:16023)
  • 18. K. Igusa, G. Todorov, On the finitistic global dimension conjecture for Artin algebras. Representations of algebras and related topics, 201-204, Fields Inst. Commun., 45, Amer. Math. Soc., Providence, RI, 2005. MR 2146250 (2006f:16012)
  • 19. O. Iyama, Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories. Adv. Math. 210 (2007), no. 1, 22-50. MR 2298819
  • 20. O. Iyama, Auslander correspondence. Adv. Math. 210 (2007), no. 1, 51-82. MR 2298820
  • 21. O. Iyama, Maximal orthogonal subcategories of triangulated categories satisfying Serre duality. Mathematisches Forschungsinstitut Oberwolfach, Report, no. 6 (2005), 353-355.
  • 22. O. Iyama, Y. Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules. Preprint, math/0607736. Invent. Math., to appear.
  • 23. B. Keller, I. Reiten, Cluster-tilted algebras are Gorenstein and stably Calabi-Yau. Adv. Math. 211 (2007), 123-151. MR 2313531
  • 24. S. Liu, R. Schulz, The existence of bounded infinite $ DTr$-orbits. Proc. Amer. Math. Soc. 122 (1994), 1003-1005. MR 1223516 (95b:16007)

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Additional Information

Karin Erdmann
Affiliation: Mathematical Institute, 24-29 St. Giles, Oxford OX1 3LB, United Kingdom
Email: erdmann@maths.ox.ac.uk

Thorsten Holm
Affiliation: Institut für Algebra und Geometrie, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, 39016 Magdeburg, Germany – and – Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Address at time of publication: Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Welfengarten 1, 30167 Hannover, Germany
Email: thorsten.holm@mathematik.uni-magdeburg.de, holm@math.uni-hannover.de

DOI: https://doi.org/10.1090/S0002-9939-08-09297-6
Keywords: Selfinjective algebras, maximal $n$-orthogonal modules.
Received by editor(s): August 8, 2006
Received by editor(s) in revised form: July 20, 2007
Published electronically: April 29, 2008
Additional Notes: We gratefully acknowledge the support of the Mathematisches Forschungsinstitut Oberwolfach through a Research in Pairs (RiP) project, and also the support through a London Mathematical Society Scheme 4 grant.
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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