Geodesic
Algebras stably equivalent to selfinjective algebras whose Auslander-Reiten quivers consist only of generalized standard components
Glasgow mathematical journal, Tome 40 (1998) no. 1, pp. 1-19

Voir la notice de l'article provenant de la source Cambridge University Press

Throughout the paper K denotes a fixed algebraically closed field. All algebras considered are finite-dimensional associative K-algebras with a unit element. Moreover, they are assumed to be basic and connected. For an algebra A we denote by mod(A) the category of all finitely generated right A-modules, and mod(A) denotes the stable category of mod(A), i.e. mod(A)/℘ where ℘ is the two-sided ideal in mod(A) of all morphisms that factorize through projective A-modules. Two algebras A and B are said to be stably equivalent if the stable categories mod(A) and mod(B) are equivalent. The study of stable equivalences of algebras has its sources in modular representation theory of finite groups. It is of importance in this theory whether two stably equivalent algebras have the same number of pairwise non-isomorphic nonprojective simple modules. Another motivation for studying stable equivalences appears in the following context. If E is a K-algebra of finite global dimension then its derived category Db(E) is equivalent to the stable category mod(Ê) of the repetitive category Ê of E [15]. Thus the problem of a classification of derived equivalent algebras leads in many cases to a classification of stably equivalent selfinjective algebras. 
Pogorzały, Zygmunt. Algebras stably equivalent to selfinjective algebras whose Auslander-Reiten quivers consist only of generalized standard components. Glasgow mathematical journal, Tome 40 (1998) no. 1, pp. 1-19. doi: 10.1017/S0017089500032316
@article{10_1017_S0017089500032316,
     author = {Pogorza{\l}y, Zygmunt},
     title = {Algebras stably equivalent to selfinjective algebras whose {Auslander-Reiten} quivers consist only of generalized standard components},
     journal = {Glasgow mathematical journal},
     pages = {1--19},
     year = {1998},
     volume = {40},
     number = {1},
     doi = {10.1017/S0017089500032316},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500032316/}
}
TY  - JOUR
AU  - Pogorzały, Zygmunt
TI  - Algebras stably equivalent to selfinjective algebras whose Auslander-Reiten quivers consist only of generalized standard components
JO  - Glasgow mathematical journal
PY  - 1998
SP  - 1
EP  - 19
VL  - 40
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500032316/
DO  - 10.1017/S0017089500032316
ID  - 10_1017_S0017089500032316
ER  - 
%0 Journal Article
%A Pogorzały, Zygmunt
%T Algebras stably equivalent to selfinjective algebras whose Auslander-Reiten quivers consist only of generalized standard components
%J Glasgow mathematical journal
%D 1998
%P 1-19
%V 40
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500032316/
%R 10.1017/S0017089500032316
%F 10_1017_S0017089500032316

[1] 1.Assem, I., Nehring, J. and Schewe, W., Fundamental domains and duplicated algebras, Canadian Math. Soc. Conference Proceedings 11 (1991), 25–51. Google Scholar

[2] 2.Assem, I. and Skowroński, A., On some class of simply connected algebras, Proc. London Math. Soc. 56 (1988), 417–450. Google Scholar | DOI

[3] 3.Assem, I. and Skowroriski, A., Algebras with cycle-finite derived categories, Math. Ann. 280 (1988), 441–463. Google Scholar

[4] 4.Auslander, M. and Reiten, I., Representation theory of artin algebras III, Comm. Algebra 3 (1975), 239–294. Google Scholar

[5] 5.Auslander, M. and Reiten, I., Representation theory of artin algebras IV, Comm. Algebra 5 (1977), 443–518. Google Scholar

[6] 6.Auslander, M. and Reiten, I., Representation theory of artin algebras VI, Comm. Algebra 6 (1978), 257–300. Google Scholar | DOI

[7] 7.Bongartz, K., Tilted algebras in: Representations of Algebras, Lecture Notes in Math. 903 (Springer, Berlin, 1981), 26–38. Google Scholar

[8] 8.Bongartz, K. and Gabriel, P., Covering spaces in representation theory, Invent. Math. 65 (1981/1982), 331–378. Google Scholar

[9] 9.Bretscher, O., Laser, C. and Riedtmann, C., Selfinjective and simply connected algebras, Manuscr. Math. 36 (1981/1982), 253–307. Google Scholar

[10] 10.Dowbor, P. and Skowroński, A., On Galois coverings of tame algebras, Arch. Math. 44 (1985), 522–529. Google Scholar | DOI

[11] 11.Dowbor, P. and Skowroński, A., Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987), 311–337. Google Scholar | DOI

[12] 12.Drozd, Y. A., Tame and wild matrix problems, in Representations and quadratic forms, (Kiev, Akad. Nauk USSR, 1979), 39–74. Google Scholar

[13] 13.d'Este, G. and Ringel, C. M., Coherent tubes, J. Algebra 87 (1984), 150–201. Google Scholar

[14] 14.Gabriel, P., The universal cover of a representation-finite algebra, in Representations of Algebras, Lecture Notes in Math. 903 (Springer, Berlin, 1981), 68–105. Google Scholar | DOI

[15] 15.Happel, D., On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), 339–389. Google Scholar | DOI

[16] 16.Happel, D. and Ringel, C. M., The derived category of a tubular algebra, Lecture Notes in Math. 1177 (Springer, Berlin, 1986), 156–180. Google Scholar

[17] 17.Happel, D. and Ringel, C. M., Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 339–443. Google Scholar

[18] 18.Hughes, D. and Waschbüsch, J., Trivial extensions of tilted algebras, Proc. London Math. Soc. 46 (1983), 347–364. Google Scholar | DOI

[19] 19.Nehring, J. and Skowroński, A., Polynomial growth trivial extensions of simply connected algebras, Fund. Math. 132 (1989), 117–134. Google Scholar

[20] 20.Peng, L. and Xiao, J., Invariability of trivial extensions of tilted algebras under stable equivalence, J. London Math. Soc. (2) 52 (1995), 61–72. Google Scholar

[21] 21.Pogorzaly, Z., On the stable Grothendieck groups, Canadian Mathematical Society, Conference Proceedings, 14 (1993), 393–406. Google Scholar

[22] 22.Pogorzaly, Z., Algebras stably equivalent to the trivial extensions of hereditary and tubular algebras, preprint, Toruń (1994). Google Scholar

[23] 23.Pogorzaly, Z. and Skorwoński, A., Symmetric algebras stably equivalent to the trivial extensions of tubular algebras, J. Algebra, 181 (1996), 95–111. Google Scholar | DOI

[24] 24.Riedtmann, C., Algebren, , Darstellungköcher, , Überlagerungen and zurück, Comment. Math. Helv. 55 (1980), 199–224. Google Scholar

[25] 25.Riedtmann, C., Representation-finite selfinjective algebras of type , Lecture Notes in Math. 832(Springer, Berlin, 1980), 449–520. Google Scholar

[26] 26.Riedtmann, C., Representation-finite selfinjective algebras of class ,, Compos. Math. 49 (1983), 231–282. Google Scholar

[27] 27.Ringel, C. M., Representation theory of finite-dimensional algebras, in Representations of algebras, Proc. Durham Symposium 1985, London Math. Soc. Lecture Notes Series 116, (Cambridge U.P.) 7–79. Google Scholar

[28] 28.Ringel, C. M., Tame algebras and integral quadratic forms, Lecture Notes in Math. 1099 (Springer, Berlin, 1984). Google Scholar

[29] 29.Skowroński, A., Algebras of polynomial growth, in Banach Center Publications, 26 (PWN, Warszawa, 1990), 535–568. Google Scholar

[30] 30.Skowroński, A., Generalized standard Auslander-Reiten components, J. Math. Soc. Japan 46 (1994), 517–543. Google Scholar | DOI

[31] 31.Skowroński, A., Group algebras of polynomial growth, Manuscr. Math. 59 (1987), 499–516. Google Scholar | DOI

[32] 32.Skowroński, A., Selfinjective algebras of polynomial growth, Math. Ann. 285 (1989), 177–199. Google Scholar | DOI

[33] 33.Skowroński, A., Cycle-finite algebras, J. Pure Appl. Algebra, 103 (1995), 105–116. Google Scholar

[34] 34.Skowroński, A., Module categories over tame algebras, Canadian Math. Soc. Conference Proceedings 19 (1996), 281–313. Google Scholar

[35] 35.Skowroński, A. and Yamagata, K., Galois coverings of selfinjective algebras by repetitive algebras, preprint, Tsukuba 1996. Google Scholar

Cité par Sources :