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Classification of modules for infinite-dimensional string algebras


Author: William Crawley-Boevey
Journal: Trans. Amer. Math. Soc. 370 (2018), 3289-3313
MSC (2010): Primary 16D70; Secondary 13C05
DOI: https://doi.org/10.1090/tran/7032
Published electronically: December 20, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: We relax the definition of a string algebra to also include infinite-dimensional algebras such as $ k[x,y]/(xy)$. Using the functorial filtration
method, which goes back to Gelfand and Ponomarev, we show that finitely generated modules and artinian modules (and more generally finitely controlled and pointwise artinian modules) are classified in terms of string and band modules. This subsumes the known classifications of finite-dimensional modules for string algebras and of finitely generated modules for $ k[x,y]/(xy)$. Unlike in the finite-dimensional case, the words parameterizing string modules may be infinite.


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  • [1] Igor Burban and Yuriy Drozd, Derived categories of nodal algebras, J. Algebra 272 (2004), no. 1, 46-94. MR 2029026, https://doi.org/10.1016/j.jalgebra.2003.07.025
  • [2] M. C. R. Butler and Claus Michael Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), no. 1-2, 145-179. MR 876976, https://doi.org/10.1080/00927878708823416
  • [3] William Crawley-Boevey, Infinite-dimensional modules in the representation theory of finite-dimensional algebras, Algebras and modules, I (Trondheim, 1996) CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 1998, pp. 29-54. MR 1648602
  • [4] P. W. Donovan and M.-R. Freislich, The indecomposable modular representations of certain groups with dihedral Sylow subgroup, Math. Ann. 238 (1978), no. 3, 207-216. MR 514428, https://doi.org/10.1007/BF01420248
  • [5] I. M. Gelfand and V. A. Ponomarev, Indecomposable representations of the Lorentz group, Uspehi Mat. Nauk 23 (1968), no. 2 (140), 3-60 (Russian). MR 0229751
  • [6] Henning Krause, A note on infinite string modules [MR1206952 (93k:16027)], Representations of algebras (Ottawa, ON, 1992) CMS Conf. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 309-312. MR 1265293
  • [7] Reinhard C. Laubenbacher and Bernd Sturmfels, A normal form algorithm for modules over $ k[x,y]/\langle xy\rangle$, J. Algebra 184 (1996), no. 3, 1001-1024. MR 1407881, https://doi.org/10.1006/jabr.1996.0295
  • [8] Lawrence S. Levy, Modules over Dedekind-like rings, J. Algebra 93 (1985), no. 1, 1-116. MR 780485, https://doi.org/10.1016/0021-8693(85)90176-0
  • [9] Hideyuki Matsumura, Commutative ring theory, with translated from the Japanese by M. Reid, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. MR 879273
  • [10] L. A. Nazarova and A. V. Roĭter, Finitely generated modules over a dyad of two local Dedekind rings, and finite groups which possess an abelian normal divisor of index $ p$, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 65-89 (Russian). MR 0260859
  • [11] L. A. Nazarova, A. V. Roĭter, V. V. Sergeĭčuk, and V. M. Bondarenko, Application of modules over a dyad to the classification of finite $ p$-groups that have an abelian subgroup of index $ p$ and to the classification of pairs of mutually annihilating operators, Investigations on the theory of representations, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 28 (1972), 69-92 (Russian). MR 0332963
  • [12] Claus Michael Ringel, The indecomposable representations of the dihedral $ 2$-groups, Math. Ann. 214 (1975), 19-34. MR 0364426, https://doi.org/10.1007/BF01428252
  • [13] Claus Michael Ringel, Some algebraically compact modules. I, Abelian groups and modules (Padova, 1994) Math. Appl., vol. 343, Kluwer Acad. Publ., Dordrecht, 1995, pp. 419-439. MR 1378216
  • [14] Andrzej Skowroński and Josef Waschbüsch, Representation-finite biserial algebras, J. Reine Angew. Math. 345 (1983), 172-181. MR 717892
  • [15] Burkhard Wald and Josef Waschbüsch, Tame biserial algebras, J. Algebra 95 (1985), no. 2, 480-500. MR 801283, https://doi.org/10.1016/0021-8693(85)90119-X

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Additional Information

William Crawley-Boevey
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Address at time of publication: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
Email: wcrawley@math.uni-bielefeld.de

DOI: https://doi.org/10.1090/tran/7032
Received by editor(s): March 7, 2016
Received by editor(s) in revised form: July 26, 2016
Published electronically: December 20, 2017
Additional Notes: This material is based upon work supported by the National Science Foundation under grant No. 0932078 000 while the author was in residence at the Mathematical Science Research Institute (MSRI) in Berkeley, California, during the spring semester 2013.
Article copyright: © Copyright 2017 American Mathematical Society

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