On the number of tame concealed convex subcategories in cycle-finite algebras

Malicki, Piotr

doi:10.1090/tran/8238

On the number of tame concealed convex subcategories in cycle-finite algebras
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Abstract:

We prove that for every cycle-finite algebra $A$ the number of pairwise different tame concealed convex subcategories of the convex hull of the support of an indecomposable $A$-module is bounded by $3$.

References

Ibrahim Assem and Flávio U. Coelho, Two-sided gluings of tilted algebras, J. Algebra 269 (2003), no. 2, 456–479. MR 2015287, DOI 10.1016/S0021-8693(03)00436-8
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, Indecomposable modules over multicoil algebras, Math. Scand. 71 (1992), no. 1, 31–61. MR 1216102, DOI 10.7146/math.scand.a-12409
Ibrahim Assem and Andrzej Skowroński, Multicoil algebras [ MR1206930 (93k:16020)], Representations of algebras (Ottawa, ON, 1992) CMS Conf. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 29–68. MR 1265275
Ibrahim Assem, Andrzej Skowroński, and Bertha Tomé, Coil enlargements of algebras, Tsukuba J. Math. 19 (1995), no. 2, 453–479. MR 1366647, DOI 10.21099/tkbjm/1496162881
Raymundo Bautista, On algebras of strongly unbounded representation type, Comment. Math. Helv. 60 (1985), no. 3, 392–399. MR 814146, DOI 10.1007/BF02567422
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, Critical simply connected algebras, Manuscripta Math. 46 (1984), no. 1-3, 117–136. MR 735517, DOI 10.1007/BF01185198
Klaus Bongartz, A criterion for finite representation type, Math. Ann. 269 (1984), no. 1, 1–12. MR 756773, DOI 10.1007/BF01455993
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, DOI 10.24033/bsmf.1975
Flávio Ulhoa Coelho and Marcelo Américo Lanzilotta, Algebras with small homological dimensions, Manuscripta Math. 100 (1999), no. 1, 1–11. MR 1714397, DOI 10.1007/s002290050191
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
Gabriella D’Este and Claus Michael Ringel, Coherent tubes, J. Algebra 87 (1984), no. 1, 150–201. MR 736774, DOI 10.1016/0021-8693(84)90165-0
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
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
Mitsuo Hoshino, Modules without self-extensions and Nakayama’s conjecture, Arch. Math. (Basel) 43 (1984), no. 6, 493–500. MR 775735, DOI 10.1007/BF01190950
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, Degrees of irreducible maps and the shapes of Auslander-Reiten quivers, J. London Math. Soc. (2) 45 (1992), no. 1, 32–54. MR 1157550, DOI 10.1112/jlms/s2-45.1.32
Piotr Malicki, Generalized coil enlargements of algebras, Colloq. Math. 76 (1998), no. 1, 57–83. MR 1611356, DOI 10.4064/cm-76-1-57-83
Piotr Malicki, José A. de la Peña, and Andrzej Skowroński, Cycle-finite module categories, Algebras, quivers and representations, Abel Symp., vol. 8, Springer, Heidelberg, 2013, pp. 209–252. MR 3183887, DOI 10.1007/978-3-642-39485-0_{1}0
Piotr Malicki, José Antonio de la Peña, and Andrzej Skowroński, Finite cycles of indecomposable modules, J. Pure Appl. Algebra 219 (2015), no. 5, 1761–1799. MR 3299706, DOI 10.1016/j.jpaa.2014.07.011
Piotr Malicki, José Antonio de la Peña, and Andrzej Skowroński, Existence of cycle-finite algebras of infinite representation type without directing projective or injective modules, Colloq. Math. 148 (2017), no. 2, 165–190. MR 3660285, DOI 10.4064/cm7190-2-2017
Piotr Malicki and Andrzej Skowroński, Almost cyclic coherent components of an Auslander-Reiten quiver, J. Algebra 229 (2000), no. 2, 695–749. MR 1769296, DOI 10.1006/jabr.2000.8314
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
Piotr Malicki and Andrzej Skowroński, On the additive categories of generalized standard almost cyclic coherent Auslander-Reiten components, J. Algebra 316 (2007), no. 1, 133–146. MR 2354856, DOI 10.1016/j.jalgebra.2007.07.011
Piotr Malicki and Andrzej Skowroński, Algebras with separating Auslander-Reiten components, Representations of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 251–353. MR 2931900, DOI 10.4171/101-1/7
Piotr Malicki and Andrzej Skowroński, The structure and homological properties of generalized standard Auslander-Reiten components, J. Algebra 518 (2019), 1–39. MR 3872856, DOI 10.1016/j.jalgebra.2018.10.015
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
José A. de la Peña and Andrzej Skowroński, Algebras with cycle-finite Galois coverings, Trans. Amer. Math. Soc. 363 (2011), no. 8, 4309–4336. MR 2792989, DOI 10.1090/S0002-9947-2011-05256-6
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
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
Daniel Simson and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 3, London Mathematical Society Student Texts, vol. 72, Cambridge University Press, Cambridge, 2007. Representation-infinite tilted algebras. MR 2382332
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, Generalized standard Auslander-Reiten components, J. Math. Soc. Japan 46 (1994), no. 3, 517–543. MR 1276836, DOI 10.2969/jmsj/04630517
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, 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, Simply connected algebras of polynomial growth, Compositio Math. 109 (1997), no. 1, 99–133. MR 1473607, DOI 10.1023/A:1000245728528
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
Bertha Tomé, Iterated coil enlargements of algebras, Fund. Math. 146 (1995), no. 3, 251–266. MR 1323732
Ying Bo Zhang, The structure of stable components, Canad. J. Math. 43 (1991), no. 3, 652–672. MR 1118014, DOI 10.4153/CJM-1991-038-1