Algebras with cycle-finite Galois coverings

de la Peña, José; Skowroński, Andrzej

doi:10.1090/S0002-9947-2011-05256-6

Algebras with cycle-finite Galois coverings
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by José A. de la Peña and Andrzej Skowroński PDF

Trans. Amer. Math. Soc. 363 (2011), 4309-4336 Request permission

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.

References

Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006. Techniques of representation theory. MR 2197389, DOI 10.1017/CBO9780511614309
Ibrahim Assem and Andrzej Skowroński, Iterated tilted algebras of type $\tilde \textbf {A}_n$, Math. Z. 195 (1987), no. 2, 269–290. MR 892057, DOI 10.1007/BF01166463
Ibrahim Assem and Andrzej Skowroński, On some classes of simply connected algebras, Proc. London Math. Soc. (3) 56 (1988), no. 3, 417–450. MR 931509, DOI 10.1112/plms/s3-56.3.417
Ibrahim Assem and Andrzej Skowroński, Algebras with cycle-finite derived categories, Math. Ann. 280 (1988), no. 3, 441–463. MR 936322, DOI 10.1007/BF01456336
Ibrahim Assem and Andrzej Skowroński, Minimal representation-infinite coil algebras, Manuscripta Math. 67 (1990), no. 3, 305–331. MR 1046991, DOI 10.1007/BF02568435
Ibrahim Assem and Andrzej Skowroński, On tame repetitive algebras, Fund. Math. 142 (1993), no. 1, 59–84. MR 1207471
Raymundo Bautista and Sheila Brenner, On the number of terms in the middle of an almost split sequence, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin-New York, 1981, pp. 1–8. MR 654699
R. Bautista, P. Gabriel, A. V. Roĭter, and L. Salmerón, Representation-finite algebras and multiplicative bases, Invent. Math. 81 (1985), no. 2, 217–285. MR 799266, DOI 10.1007/BF01389052
Klaus Bongartz, Treue einfach zusammenhängende Algebren. I, Comment. Math. Helv. 57 (1982), no. 2, 282–330 (German). MR 684118, DOI 10.1007/BF02565862
Klaus Bongartz, A criterion for finite representation type, Math. Ann. 269 (1984), no. 1, 1–12. MR 756773, DOI 10.1007/BF01455993
Klaus Bongartz, Critical simply connected algebras, Manuscripta Math. 46 (1984), no. 1-3, 117–136. MR 735517, DOI 10.1007/BF01185198
Klaus Bongartz, Indecomposables are standard, Comment. Math. Helv. 60 (1985), no. 3, 400–410. MR 814147, DOI 10.1007/BF02567423
K. Bongartz and P. Gabriel, Covering spaces in representation-theory, Invent. Math. 65 (1981/82), no. 3, 331–378. MR 643558, DOI 10.1007/BF01396624
O. Bretscher and P. Gabriel, The standard form of a representation-finite algebra, Bull. Soc. Math. France 111 (1983), no. 1, 21–40 (English, with French summary). MR 710374
Thomas Brüstle, Tame tree algebras, J. Reine Angew. Math. 567 (2004), 51–98. MR 2038305, DOI 10.1515/crll.2004.013
T. Brüstle, J. A. de la Peña and A. Skowroński, Tame algebras and Tits quadratic forms, Advances Math. 226 (2011), 887–951.
W. W. Crawley-Boevey, On tame algebras and bocses, Proc. London Math. Soc. (3) 56 (1988), no. 3, 451–483. MR 931510, DOI 10.1112/plms/s3-56.3.451
William Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc. (3) 63 (1991), no. 2, 241–265. MR 1114510, DOI 10.1112/plms/s3-63.2.241
William Crawley-Boevey, Tameness of biserial algebras, Arch. Math. (Basel) 65 (1995), no. 5, 399–407. MR 1354686, DOI 10.1007/BF01198070
Piotr Dowbor, On the category of modules of second kind for Galois coverings, Fund. Math. 149 (1996), no. 1, 31–54. MR 1372356, DOI 10.4064/fm-149-1-31-54
Piotr Dowbor, Galois covering reduction to stabilizers, Bull. Polish Acad. Sci. Math. 44 (1996), no. 3, 341–352. MR 1419407
Piotr Dowbor, Properties of $G$-atoms and full Galois covering reduction to stabilizers, Colloq. Math. 83 (2000), no. 2, 231–265. MR 1758318, DOI 10.4064/cm-83-2-231-265
Piotr Dowbor, Stabilizer conjecture for representation-tame Galois coverings of algebras, J. Algebra 239 (2001), no. 1, 112–149. MR 1827877, DOI 10.1006/jabr.2000.8635
Piotr Dowbor and Andrzej Skowroński, On Galois coverings of tame algebras, Arch. Math. (Basel) 44 (1985), no. 6, 522–529. MR 797444, DOI 10.1007/BF01193992
Piotr Dowbor and Andrzej Skowroński, On the representation type of locally bounded categories, Tsukuba J. Math. 10 (1986), no. 1, 63–72. MR 846416, DOI 10.21099/tkbjm/1496160389
Piotr Dowbor and Andrzej Skowroński, Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987), no. 2, 311–337. MR 896100, DOI 10.1007/BF02564450
P. Dräxler, Aufrichtige gerichtete Ausnahmealgebren, Bayreuth. Math. Schr. 29 (1989), 191 (German). MR 1007156
Ju. A. Drozd, Tame and wild matrix problems, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin-New York, 1980, pp. 242–258. MR 607157
Karin Erdmann and Andrzej Skowroński, On Auslander-Reiten components of blocks and self-injective biserial algebras, Trans. Amer. Math. Soc. 330 (1992), no. 1, 165–189. MR 1144759, DOI 10.1090/S0002-9947-1992-1144759-7
P. Gabriel, The universal cover of a representation-finite algebra, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin-New York, 1981, pp. 68–105. MR 654725
Christof Geiss, On degenerations of tame and wild algebras, Arch. Math. (Basel) 64 (1995), no. 1, 11–16. MR 1305654, DOI 10.1007/BF01193544
Christof Geiss and José Antonio de la Peña, An interesting family of algebras, Arch. Math. (Basel) 60 (1993), no. 1, 25–35. MR 1193090, DOI 10.1007/BF01194235
Dieter Happel and Dieter Vossieck, Minimal algebras of infinite representation type with preprojective component, Manuscripta Math. 42 (1983), no. 2-3, 221–243. MR 701205, DOI 10.1007/BF01169585
Otto Kerner, Tilting wild algebras, J. London Math. Soc. (2) 39 (1989), no. 1, 29–47. MR 989917, DOI 10.1112/jlms/s2-39.1.29
Helmut Lenzing and Andrzej Skowroński, Quasi-tilted algebras of canonical type, Colloq. Math. 71 (1996), no. 2, 161–181. MR 1414820, DOI 10.4064/cm-71-2-161-181
Shi Ping Liu, Almost split sequences for nonregular modules, Fund. Math. 143 (1993), no. 2, 183–190. MR 1240634, DOI 10.4064/fm-143-2-183-190
P. Malicki, J. A. de la Peña and A. Skowroński, Cycle-finite module categories, Preprint 2009.
Piotr Malicki and Andrzej Skowroński, Algebras with separating almost cyclic coherent Auslander-Reiten components, J. Algebra 291 (2005), no. 1, 208–237. MR 2158519, DOI 10.1016/j.jalgebra.2005.03.021
Rainer Nörenberg and Andrzej Skowroński, Tame minimal non-polynomial growth simply connected algebras, Colloq. Math. 73 (1997), no. 2, 301–330. MR 1446121, DOI 10.4064/cm-73-2-301-330
R. Martínez-Villa and J. A. de la Peña, The universal cover of a quiver with relations, J. Pure Appl. Algebra 30 (1983), no. 3, 277–292. MR 724038, DOI 10.1016/0022-4049(83)90062-2
Lian Gang Peng and Jie Xiao, On the number of $D\textrm {Tr}$-orbits containing directing modules, Proc. Amer. Math. Soc. 118 (1993), no. 3, 753–756. MR 1135078, DOI 10.1090/S0002-9939-1993-1135078-X
J. A. de la Peña, Tame algebras with sincere directing modules, J. Algebra 161 (1993), no. 1, 171–185. MR 1245849, DOI 10.1006/jabr.1993.1211
J. A. de la Peña, The families of two-parametric tame algebras with sincere directing modules, Representations of algebras (Ottawa, ON, 1992) CMS Conf. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 361–392. MR 1265297
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), no. 9, 3857–3868. MR 1467463, DOI 10.1090/S0002-9947-99-02137-6
J. A. de la Peña and B. Tomé, Iterated tubular algebras, J. Pure Appl. Algebra 64 (1990), no. 3, 303–314. MR 1061305, DOI 10.1016/0022-4049(90)90064-O
Zygmunt Pogorzały, Regularly biserial algebras, Topics in algebra, Part 1 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 371–384. MR 1171245
Zygmunt Pogorzały and Andrzej Skowroński, Self-injective biserial standard algebras, J. Algebra 138 (1991), no. 2, 491–504. MR 1102821, DOI 10.1016/0021-8693(91)90183-9
Idun Reiten and Andrzej Skowroński, Characterizations of algebras with small homological dimensions, Adv. Math. 179 (2003), no. 1, 122–154. MR 2004730, DOI 10.1016/S0001-8708(02)00029-4
Idun Reiten and Andrzej Skowroński, Generalized double tilted algebras, J. Math. Soc. Japan 56 (2004), no. 1, 269–288. MR 2027626, DOI 10.2969/jmsj/1191418706
Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589, DOI 10.1007/BFb0072870
Daniel Simson and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 2, London Mathematical Society Student Texts, vol. 71, Cambridge University Press, Cambridge, 2007. Tubes and concealed algebras of Euclidean type. MR 2360503
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.
Andrzej Skowroński, Selfinjective algebras of polynomial growth, Math. Ann. 285 (1989), no. 2, 177–199. MR 1016089, DOI 10.1007/BF01443513
Andrzej Skowroński, Algebras of polynomial growth, Topics in algebra, Part 1 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 535–568. MR 1171252
Andrzej Skowroński, Simply connected algebras and Hochschild cohomologies [ MR1206961 (94e:16016)], Representations of algebras (Ottawa, ON, 1992) CMS Conf. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 431–447. MR 1265301
Andrzej Skowroński, Regular Auslander-Reiten components containing directing modules, Proc. Amer. Math. Soc. 120 (1994), no. 1, 19–26. MR 1156473, DOI 10.1090/S0002-9939-1994-1156473-X
Andrzej Skowroński, Cycles in module categories, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 309–345. MR 1308994
Andrzej Skowroński, Cycle-finite algebras, J. Pure Appl. Algebra 103 (1995), no. 1, 105–116. MR 1354071, DOI 10.1016/0022-4049(94)00094-Y
Andrzej Skowroński, Module categories over tame algebras, Representation theory of algebras and related topics (Mexico City, 1994) CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 281–313. MR 1388567
Andrzej Skowroński, Simply connected algebras of polynomial growth, Compositio Math. 109 (1997), no. 1, 99–133. MR 1473607, DOI 10.1023/A:1000245728528
Andrzej Skowroński, Tame algebras with strongly simply connected Galois coverings, Colloq. Math. 72 (1997), no. 2, 335–351. MR 1426706, DOI 10.4064/cm-72-2-335-351
Andrzej Skowroński, Tame quasi-tilted algebras, J. Algebra 203 (1998), no. 2, 470–490. MR 1622799, DOI 10.1006/jabr.1997.7328
Andrzej Skowroński, Selfinjective algebras: finite and tame type, Trends in representation theory of algebras and related topics, Contemp. Math., vol. 406, Amer. Math. Soc., Providence, RI, 2006, pp. 169–238. MR 2258046, DOI 10.1090/conm/406/07658
Luise Unger, The concealed algebras of the minimal wild, hereditary algebras, Bayreuth. Math. Schr. 31 (1990), 145–154. MR 1056151