On the number of terms in the middle of almost split sequences over cycle-finite artin algebras

Piotr Malicki; José Peña; Andrzej Skowroński

Open Mathematics (2014)

  • Volume: 12, Issue: 1, page 39-45
  • ISSN: 2391-5455

Abstract

top
We prove that the number of terms in the middle of an almost split sequence in the module category of a cycle-finite artin algebra is bounded by 5.

How to cite

top

Piotr Malicki, José Peña, and Andrzej Skowroński. "On the number of terms in the middle of almost split sequences over cycle-finite artin algebras." Open Mathematics 12.1 (2014): 39-45. <http://eudml.org/doc/269744>.

@article{PiotrMalicki2014,
abstract = {We prove that the number of terms in the middle of an almost split sequence in the module category of a cycle-finite artin algebra is bounded by 5.},
author = {Piotr Malicki, José Peña, Andrzej Skowroński},
journal = {Open Mathematics},
keywords = {Auslander-Reiten quiver; Almost split sequence; Cycle-finite algebra; Auslander-Reiten quivers; almost split sequences; cycle-finite algebras; Artin algebras},
language = {eng},
number = {1},
pages = {39-45},
title = {On the number of terms in the middle of almost split sequences over cycle-finite artin algebras},
url = {http://eudml.org/doc/269744},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Piotr Malicki
AU - José Peña
AU - Andrzej Skowroński
TI - On the number of terms in the middle of almost split sequences over cycle-finite artin algebras
JO - Open Mathematics
PY - 2014
VL - 12
IS - 1
SP - 39
EP - 45
AB - We prove that the number of terms in the middle of an almost split sequence in the module category of a cycle-finite artin algebra is bounded by 5.
LA - eng
KW - Auslander-Reiten quiver; Almost split sequence; Cycle-finite algebra; Auslander-Reiten quivers; almost split sequences; cycle-finite algebras; Artin algebras
UR - http://eudml.org/doc/269744
ER -

References

top
  1. [1] Assem I., Simson D., Skowroński A., Elements of the Representation Theory of Associative Algebras, 1, London Math. Soc. Stud. Texts, 65, Cambridge University Press, Cambridge, 2006 http://dx.doi.org/10.1017/CBO9780511614309 Zbl1092.16001
  2. [2] Assem I., Skowroński A., Algebras with cycle-finite derived categories, Math. Ann., 1988, 280(3), 441–463 http://dx.doi.org/10.1007/BF01456336 Zbl0617.16017
  3. [3] Assem I., Skowroński A., Minimal representation-infinite coil algebras, Manuscripta Math., 1990, 67(3), 305–331 http://dx.doi.org/10.1007/BF02568435 Zbl0696.16023
  4. [4] Auslander M., Representation theory of artin algebras II, Comm. Algebra, 1974, 1(4), 269–310 http://dx.doi.org/10.1080/00927877409412807 Zbl0285.16029
  5. [5] Auslander M., Reiten I., Representation theory of artin algebras III. Almost split sequences, Comm. Algebra, 1975, 3(3), 239–294 http://dx.doi.org/10.1080/00927877508822046 Zbl0331.16027
  6. [6] Auslander M., Reiten I., Uniserial functors, In: Representation Theory, 2, Ottawa, August 13–25, 1979, Lecture Notes in Math., 832, Springer, Berlin, 1980, 1–47 
  7. [7] Auslander M., Reiten I., Smalø S.O., Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., 36, Cambridge University Press, Cambridge, 1995 http://dx.doi.org/10.1017/CBO9780511623608 Zbl0834.16001
  8. [8] Bautista R., Brenner S., On the number of terms in the middle of an almost split sequence, In: Representations of Algebras, Puebla, August 4–8, 1980, Lecture Notes in Math., 903, Springer, Berlin, 1981, 1–8 Zbl0483.16030
  9. [9] Brenner S., Butler M.C.R., Generalizations of the Bernstein-Gel’fand-Ponomarev reflection functors, In: Representation Theory, II, Ottawa, August 13–25, 1979, Lecture Notes in Math., 832, Springer, Berlin, 1980, 103–169 
  10. [10] Brenner S., Butler M.C.R., Wild subquivers of the Auslander-Reiten quiver of a tame algebra, In: Trends in the Representation Theory of Finite-Dimensional Algebras, Seattle, July 20–24, 1997, Contemp. Math., 229, American Mathematical Society, Providence, 1998, 29–48 http://dx.doi.org/10.1090/conm/229/03309 
  11. [11] Butler M.C.R., Ringel C.M., Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra, 1987, 15(1–2), 145–179 http://dx.doi.org/10.1080/00927878708823416 Zbl0612.16013
  12. [12] Coelho F.U., Marcos E.N., Merklen H.A., Skowroński A., Module categories with infinite radical square zero are of finite type, Comm. Algebra, 1994, 22(11), 4511–4517 http://dx.doi.org/10.1080/00927879408825084 Zbl0812.16019
  13. [13] Crawley-Boevey W., Tame algebras and generic modules, Proc. London Math. Soc., 1991, 63(2), 241–265 http://dx.doi.org/10.1112/plms/s3-63.2.241 Zbl0741.16005
  14. [14] Crawley-Boevey W., Modules of finite length over their endomorphism rings, In: Representations of Algebras and Related Topics, Tsukuba, 1990, London Math. Soc. Lecture Note Series, 168, Cambridge University Press, Cambridge, 1992, 127–184 http://dx.doi.org/10.1017/CBO9780511661853.005 Zbl0805.16028
  15. [15] Dlab V., Ringel C.M., Indecomposable Representations of Graphs and Algebras, Mem. Amer. Math. Soc., 173, American Mathematical Society, Providence, 1976 Zbl0332.16015
  16. [16] Dlab V., Ringel C.M., The representations of tame hereditary algebras, In: Representation Theory of Algebras, Philadelphia, May 24–28, 1976, Lecture Notes in Pure Appl. Math., 37, Marcel Dekker, New York, 1978, 329–353 
  17. [17] Dowbor P., Skowroński A., Galois coverings of representation-infinite algebras, Comment. Math. Helv., 1987, 62(2), 311–337 http://dx.doi.org/10.1007/BF02564450 Zbl0628.16019
  18. [18] Happel D., Ringel C.M., Tilted algebras, Trans. Amer. Math. Soc., 1982, 274(2), 399–443 http://dx.doi.org/10.1090/S0002-9947-1982-0675063-2 Zbl0503.16024
  19. [19] Jaworska A., Malicki P., Skowroński A., Tilted algebras and short chains of modules, Math. Z., 2013, 273(1–2), 19–27 http://dx.doi.org/10.1007/s00209-012-0993-0 Zbl1331.16008
  20. [20] Kerner O., Tilting wild algebras, J. London Math. Soc., 1989, 39(1), 29–47 http://dx.doi.org/10.1112/jlms/s2-39.1.29 Zbl0675.16013
  21. [21] Kerner O., Stable components of wild tilted algebras, J. Algebra, 1992, 142(1), 37–57 http://dx.doi.org/10.1016/0021-8693(91)90215-T 
  22. [22] Kerner O., Skowroński A., On module categories with nilpotent infinite radical, Compositio Math., 1991, 77(3), 313–333 Zbl0717.16012
  23. [23] Lenzing H., Skowroński A., Quasi-tilted algebras of canonical type, Colloq. Math., 1996, 71(2), 161–181 Zbl0870.16007
  24. [24] Liu S.-P., Almost split sequences for nonregular modules, Fund. Math., 1993, 143(2), 183–190 Zbl0801.16009
  25. [25] Liu S.-P., Semi-stable components of an Auslander-Reiten quiver, J. London Math. Soc., 1993, 47(3), 405–416 http://dx.doi.org/10.1112/jlms/s2-47.3.405 Zbl0818.16015
  26. [26] Liu S.-P., Tilted algebras and generalized standard Auslander-Reiten components, Arch. Math. (Basel), 1993, 61(1), 12–19 http://dx.doi.org/10.1007/BF01258050 Zbl0809.16015
  27. [27] Malicki P., Skowroński A., Algebras with separating almost cyclic coherent Auslander-Reiten components, J. Algebra, 2005, 291(1), 208–237 http://dx.doi.org/10.1016/j.jalgebra.2005.03.021 Zbl1121.16017
  28. [28] de la Peña J.A., Skowroński A., Algebras with cycle-finite Galois coverings, Trans. Amer. Math. Soc., 2011, 363(8), 4309–4336 http://dx.doi.org/10.1090/S0002-9947-2011-05256-6 Zbl1253.16018
  29. [29] de la Peña J.A., Takane M., On the number of terms in the middle of almost split sequences over tame algebras, Trans. Amer. Math. Soc., 1999, 351(9), 3857–3868 http://dx.doi.org/10.1090/S0002-9947-99-02137-6 Zbl0944.16019
  30. [30] de la Peña J.A., Tomé B., Iterated tubular algebras, J. Pure Appl. Algebra, 1990, 64(3), 303–314 http://dx.doi.org/10.1016/0022-4049(90)90064-O Zbl0704.16006
  31. [31] Peng L.G., Xiao J., On the number of DTr-orbits containing directing modules, Proc. Amer. Math. Soc., 1993, 118(3), 753–756 http://dx.doi.org/10.1090/S0002-9939-1993-1135078-X Zbl0787.16014
  32. [32] Pogorzały Z., Skowroński A., On algebras whose indecomposable modules are multiplicity-free, Proc. London Math. Soc., 1983, 47(3), 463–479 http://dx.doi.org/10.1112/plms/s3-47.3.463 Zbl0559.16016
  33. [33] Reiten I., Skowroński A., Characterizations of algebras with small homological dimensions, Adv. Math., 2003, 179(1), 122–154 http://dx.doi.org/10.1016/S0001-8708(02)00029-4 Zbl1051.16011
  34. [34] Reiten I., Skowroński A., Generalized double tilted algebras, J. Math. Soc. Japan, 2004, 56(1), 269–288 http://dx.doi.org/10.2969/jmsj/1191418706 Zbl1071.16011
  35. [35] Reiten I., Skowroński A., Smalø S.O., Short chains and regular components, Proc. Amer. Math. Soc., 1993, 117(3), 601–612 http://dx.doi.org/10.1090/S0002-9939-1993-1124149-X Zbl0782.16008
  36. [36] Ringel C.M., Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., 1099, Springer, Berlin, 1984 
  37. [37] Skowroński A., Selfinjective algebras of polynomial growth, Math. Ann., 1989, 285(2), 177–199 http://dx.doi.org/10.1007/BF01443513 Zbl0653.16021
  38. [38] Skowroński A., Generalized standard Auslander-Reiten components without oriented cycles, Osaka J. Math., 1993, 30(3), 515–527 Zbl0818.16017
  39. [39] Skowroński A., Regular Auslander-Reiten components containing directing modules, Proc. Amer. Math. Soc., 1994, 120(1), 19–26 Zbl0831.16014
  40. [40] Skowroński A., Cycles in module categories, In: Finite-Dimensional Algebras and Related Topics, Ottawa, August 10–18, 1992, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 424, Kluwer, Dordrecht, 1994, 309–345 Zbl0819.16013
  41. [41] Skowroński A., Cycle-finite algebras, J. Pure Appl. Algebra, 1995, 103(1), 105–116 http://dx.doi.org/10.1016/0022-4049(94)00094-Y 
  42. [42] Skowroński A., Simply connected algebras of polynomial growth, Compositio Math., 1997, 109(1), 99–133 http://dx.doi.org/10.1023/A:1000245728528 Zbl0889.16004
  43. [43] Skowroński A., Tame algebras with strongly simply connected Galois coverings, Colloq. Math., 1997, 72(2), 335–351 Zbl0876.16007
  44. [44] Skowroński A., Tame quasi-tilted algebras, J. Algebra, 1998, 203(2), 470–490 http://dx.doi.org/10.1006/jabr.1997.7328 
  45. [45] Skowroński A., Selfinjective algebras: finite and tame type, In: Trends in Representation Theory of Algebras and Related Topics, Querétaro, August 11–14, 2004, Contemp. Math., 406, American Mathematical Society, Providence, 2006, 169–238 Zbl1129.16013
  46. [46] Skowroński A., Waschbüsch J., Representation-finite biserial algebras, J. Reine Angew. Math., 1983, 345, 172–181 Zbl0511.16021
  47. [47] Wald B., Waschbüsch J., Tame biserial algebras, J. Algebra, 1985, 95(2), 480–500 http://dx.doi.org/10.1016/0021-8693(85)90119-X 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.