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Algebras with cycle-finite Galois coverings


Authors: José A. de la Peña and Andrzej Skowroński
Journal: Trans. Amer. Math. Soc. 363 (2011), 4309-4336
MSC (2010): Primary 16G60, 16G70
DOI: https://doi.org/10.1090/S0002-9947-2011-05256-6
Published electronically: March 1, 2011
MathSciNet review: 2792989
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Abstract: We prove that the finite dimensional algebras over an algebraically closed field which admit cycle-finite Galois coverings with torsion-free Galois groups are of tame representation type, and derive some consequences.


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  • [ASS] I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras 1: Techniques of Representation Theory. London Mathematical Society Student Texts, Vol. 65, Cambridge University Press, Cambridge, 2006. MR 2197389 (2006j:16020)
  • [AS1] I. Assem and A. Skowroński, Iterated tilted algebras of type $ \widetilde{\mathbb{A}}_n$, Math. Z. 195 (1987), 269-290. MR 892057 (88m:16033)
  • [AS2] I. Assem and A. Skowroński, On some classes of simply connected algebras, Proc. London Math. Soc. 56 (1988), 417-450. MR 931509 (89f:16023a)
  • [AS3] I. Assem and A. Skowroński, Algebras with cycle-finite derived categories, Math. Ann. 280 (1988), 441-463. MR 936322 (89f:16023b)
  • [AS4] I. Assem and A. Skowroński, Minimal representation-infinite coil algebras, Manuscripta Math. 67 (1990), 305-331. MR 1046991 (91h:16025)
  • [AS5] I. Assem and A. Skowroński, On tame repetitive algebras, Fund. Math. 142 (1993), 59-84. MR 1207471 (94a:16022)
  • [BB] R. Bautista and S. Brenner, On the number of terms in the middle of an almost split sequence, In: Representations of Algebras, Lecture Notes in Math., Vol. 903, Springer-Verlag, Berlin-Heidelberg, 1981, pp. 1-8. MR 654699 (83f:16034)
  • [BGRS] R. Bautista, P. Gabriel, A. V. Roiter and L. Salmerón, Representation-finite algebras and multiplicative bases, Invent. Math. 81 (1985), 217-285. MR 799266 (87g:16031)
  • [Bo1] K. Bongartz, Treue einfach zusammenhängende Algebren I, Comment. Math. Helv. 57 (1982), 282-330. MR 684118 (84a:16051)
  • [Bo2] K. Bongartz, A criterion for finite representation type, Math. Ann. 269 (1984), 1-12. MR 756773 (86k:16023)
  • [Bo3] K. Bongartz, Critical simply connected algebras, Manuscripta Math. 46 (1984), 117-136. MR 735517 (85j:16026)
  • [Bo4] K. Bongartz, Indecomposables are standard, Comment. Math. Helv. 60 (1985), 400-410. MR 814147 (87d:16039)
  • [BoG] K. Bongartz and P. Gabriel, Covering spaces in representation theory, Invent. Math. 65 (1982), 337-378. MR 643558 (84i:16030)
  • [BrG] O. Bretscher and P. Gabriel, The standard form of a representation-finite algebra, Bull. Soc. Math. France 111 (1983), 21-40. MR 710374 (85g:16014)
  • [Bru] T. Brüstle, Tame tree algebras, J. Reine Angew. Math. 567 (2004), 51-98. MR 2038305 (2004m:16014)
  • [BPS] T. Brüstle, J. A. de la Peña and A. Skowroński, Tame algebras and Tits quadratic forms, Advances Math. 226 (2011), 887-951.
  • [CB1] W. Crawley-Boevey, On tame algebras and bocses, Proc. London Math. Soc. 56 (1988), 451-483. MR 931510 (89c:16028)
  • [CB2] W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc. 63 (1991), 241-265. MR 1114510 (92m:16019)
  • [CB3] W. Crawley-Boevey, Tameness of biserial algebras, Arch. Math. (Basel) 65 (1995), 399-407. MR 1354686 (96i:16021)
  • [Do1] P. Dowbor, On the category of modules of second kind for Galois coverings, Fund. Math. 149 (1996), 31-54. MR 1372356 (97f:16028)
  • [Do2] P. Dowbor, Galois covering reduction to stabilizers, Bull. Polish Acad. Sci. Sér. Sci. Math. 44 (1996), 341-352. MR 1419407 (98a:16015)
  • [Do3] P. Dowbor, Properties of $ G$-atoms and full Galois covering reduction to stabilizers, Colloq. Math. 83 (2000), 231-265. MR 1758318 (2001b:16014)
  • [Do4] P. Dowbor, Stabilizer conjecture for representation-tame Galois coverings of algebras, J. Algebra 239 (2001), 119-149. MR 1827877 (2002e:16024)
  • [DS1] P. Dowbor and A. Skowroński, On Galois coverings of tame algebras, Arch. Math. (Basel) 44 (1985), 522-529. MR 797444 (87a:16035)
  • [DS2] P. Dowbor and A. Skowroński, On the representation type of locally bounded categories, Tsukuba J. Math. 10 (1986), 63-77. MR 846416 (88a:16057)
  • [DS3] P. Dowbor and A. Skowroński, Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987), 311-337. MR 896100 (88m:16020)
  • [Dre] P. Dräxler, Aufrichtige gerichtete Ausnahmealgebren, Bayreuth. Math. Schr. 29 (1989). MR 1007156 (90j:16045)
  • [Dro] Y. A. Drozd, Tame and wild matrix problems, In: Representation Theory II, Lecture Notes in Mathematics, Vol. 832, Springer-Verlag, Berlin-Heidelberg, 1980, pp. 242-258. MR 607157 (83b:16024)
  • [ES] K. Erdmann and A. Skowroński, On Auslander-Reiten components of blocks and self-injective biserial algebras, Trans. Amer. Math. Soc. 330 (1992), 165-189. MR 1144759 (93b:16022)
  • [Ga] P. Gabriel, The universal cover of a representation-finite algebra, In: Representations of Algebras, Lecture Notes in Math., Vol. 903, Springer-Verlag, Berlin-Heidelberg, 1981, pp. 68-105. MR 654725 (83f:16036)
  • [Ge] C. Geiss, On degenerations of tame and wild algebras, Arch. Math. (Basel) 64 (1995), 11-16. MR 1305654 (95k:16014)
  • [GP] C. Geiss and J. A. de la Peña, An interesting family of algebras, Arch. Math. (Basel) 60 (1993), 25-35. MR 1193090 (94a:16023)
  • [HV] D. Happel and D. Vossieck, Minimal algebras of infinite representation type with preprojective component, Manuscripta Math. 42 (1983), 221-243. MR 701205 (84m:16022)
  • [Ke] O. Kerner, Tilting wild algebras, J. London Math. Soc. 39 (1989), 29-47. MR 989917 (90d:16025)
  • [LS] H. Lenzing and A. Skowroński, Quasi-tilted algebras of canonical type, Colloq. Math. 71 (1996), 161-181. MR 1414820 (97j:16019)
  • [Li] S. Liu, Almost split sequences for nonregular modules, Fund. Math. 143 (1993), 183-190. MR 1240634 (94g:16018)
  • [MPS] P. Malicki, J. A. de la Peña and A. Skowroński, Cycle-finite module categories, Preprint 2009.
  • [MS] P. Malicki and A. Skowroński, Algebras with separating almost cyclic coherent Auslander-Reiten components, J. Algebra 291 (2005), 208-237. MR 2158519 (2006e:16034)
  • [NS] R. Nörenberg and A. Skowroński, Tame minimal non-polynomial growth simply connected algebras, Colloq. Math. 73 (1997), 301-330. MR 1446121 (98f:16012)
  • [MP] R. Martinez-Villa and J. A. de la Peña, The universal cover of a quiver with relations, J. Pure Appl. Algebra 30 (1983), 277-292. MR 724038 (85f:16035)
  • [PX] L. G. Peng and J. Xiao, On the number of $ D{Tr}$-orbits containing directing modules, Proc. Amer. Math. Soc. 118 (1993), 753-756. MR 1135078 (93i:16020)
  • [Pe1] J. A. de la Peña, Tame algebras with sincere directing modules, J. Algebra 161 (1993), 171-185. MR 1245849 (95b:16014)
  • [Pe2] J. A. de la Peña, The families of two-parametric tame algebras with sincere directing modules, In: Representations of Algebras, CMS Conf. Proc., Vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 361-392. MR 1265297 (95f:16013)
  • [PTa] J. A. de la Peña and M. Takane, On the number of terms in the middle of almost split sequences over tame algebras, Trans. Amer. Math. Soc. 351 (1999), 3857-3868. MR 1467463 (99m:16030)
  • [PTo] J. A. de la Peña and B. Tomé, Iterated tubular algebras, J. Pure Appl. Algebra 64 (1990), 303-314. MR 1061305 (91h:16028)
  • [Po] Z. Pogorzały, Regularly biserial algebras, In: Topics in Algebra, Banach Center Publ., Vol. 26, Part 1, PWN Warsaw, 1990, pp. 371-384. MR 1171245 (93m:16010)
  • [PoS] Z. Pogorzały and A. Skowroński, Selfinjective biserial standard algebras, J. Algebra 138 (1991), 491-504. MR 1102821 (92f:16012)
  • [RS1] I. Reiten and A. Skowroński, Characterizations of algebras with small homological dimensions, Advances Math. 179 (2003), 122-154. MR 2004730 (2004k:16029)
  • [RS2] I. Reiten and A. Skowroński, Generalized double tilted algebras, J. Math. Soc. Japan 56 (2004), 269-288. MR 2027626 (2005b:16039)
  • [Ri] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Mathematics, Vol. 1099, Springer-Verlag, Berlin-Heidelberg, 1984. MR 774589 (87f:16027)
  • [SS1] D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras 2: Tubes and Concealed Algebras of Euclidean Type. London Mathematical Society Student Texts, Vol. 71, Cambridge University Press, Cambridge, 2007. MR 2360503 (2009f:16001)
  • [SS2] D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras 3: Representation-Infinite Tilted Algebras. London Mathematical Society Student Texts, Vol. 72, Cambridge University Press, Cambridge, 2007.
  • [Sk1] A. Skowroński, Selfinjective algebras of polynomial growth, Math. Ann. 285 (1989), 177-199. MR 1016089 (90k:16024)
  • [Sk2] A. Skowroński, Algebras of polynomial growth, In: Topics in Algebra, Banach Center Publ., Vol. 26, Part 1, PWN Warsaw, 1990, pp. 535-568. MR 1171252 (93k:16026)
  • [Sk3] A. Skowroński, Simply connected algebras and Hochschild cohomologies, In: Representations of Algebras, CMS Conf. Proc., Vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 431-437. MR 1265301
  • [Sk4] A. Skowroński, Regular Auslander-Reiten components containing directing modules, Proc. Amer. Math. Soc. 120 (1994), 19-26. MR 1156473 (94b:16021)
  • [Sk5] A. Skowroński, Cycles in module categories, In: Finite Dimensional Algebras and Related Topics, NATO Adv. Sci. Int. Ser. C, Math. Phys. Sci., Vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 309-345. MR 1308994 (96a:16010)
  • [Sk6] A. Skowroński, Cycle-finite algebras, J. Pure Appl. Algebra 103 (1995), 105-116. MR 1354071 (96i:16023)
  • [Sk7] A. Skowroński, Module categories over tame algebras, In: Representation Theory and Related Topics, Can. Math. Soc. Conf. Proc., Vol. 19, Amer. Math. Soc., 1996, pp. 281-313. MR 1388567 (97e:16033)
  • [Sk8] A. Skowroński, Simply connected algebras of polynomial growth, Compositio Math. 109 (1997), 99-133. MR 1473607 (98k:16019)
  • [Sk9] A. Skowroński, Tame algebras with strongly simply connected Galois coverings, Colloq. Math. 72 (1997), 335-351. MR 1426706 (97k:16016)
  • [Sk10] A. Skowroński, Tame quasi-tilted algebras, J. Algebra 203 (1998), 470-490. MR 1622799 (99b:16019)
  • [Sk11] A. Skowroński, Selfinjective algebras: Finite and tame type, In: Trends in Representation Theory of Algebras and Related Topics, Contemporary Mathematics, Vol. 406, Amer. Math. Soc., Providence, RI, 2006, pp. 169-238. MR 2258046 (2007f:16045)
  • [U] L. Unger, The concealed algebras of the minimal wild, hereditary algebras, Bayreuther Math. Schriften 31 (1990), 145-154. MR 1056151 (91i:16033)

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Additional Information

José A. de la Peña
Affiliation: Instituto de Matemáticas, UNAM, Ciudad Universitaria, 04510 México, D.F. México
Email: jap@matem.unam.mx

Andrzej Skowroński
Affiliation: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Email: skowron@mat.uni.torun.pl

DOI: https://doi.org/10.1090/S0002-9947-2011-05256-6
Keywords: Tame algebras, Galois coverings, cycles of modules
Received by editor(s): June 25, 2009
Received by editor(s) in revised form: November 19, 2009
Published electronically: March 1, 2011
Additional Notes: Both authors acknowledge support from the Consejo Nacional de Ciencia y Technologia of Mexico.
The second author has also been supported by the research grant no. N N201 269135 of the Polish Ministry of Science and Higher Education.
Article copyright: © Copyright 2011 American Mathematical Society

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