On the stable module category of a self-injective algebra
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- by Karin Erdmann and Otto Kerner
- Trans. Amer. Math. Soc. 352 (2000), 2389-2405
- DOI: https://doi.org/10.1090/S0002-9947-00-02232-7
- Published electronically: February 14, 2000
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Abstract:
Let $\Lambda$ be a finite-dimensional self-injective algebra. We study the dimensions of spaces of stable homomorphisms between indecomposable $\Lambda$-modules which belong to Auslander-Reiten components of the form $\mathbf {Z}A_\infty$ or $\mathbf {Z}A_\infty /\langle \tau ^k\rangle$. The results are applied to representations of finite groups over fields of prime characteristic, especially blocks of wild representation type.References
- Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1997. Corrected reprint of the 1995 original. MR 1476671
- Dagmar Baer, Homological properties of wild hereditary Artin algebras, Representation theory, I (Ottawa, Ont., 1984) Lecture Notes in Math., vol. 1177, Springer, Berlin, 1986, pp. 1–12. MR 842456, DOI 10.1007/BFb0075255
- David Benson, Modular representation theory: new trends and methods, Lecture Notes in Mathematics, vol. 1081, Springer-Verlag, Berlin, 1984. MR 765858
- D. J. Benson, Representations and cohomology. II, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1991. Cohomology of groups and modules. MR 1156302
- 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
- Jon F. Carlson, Varieties for modules, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 37–44. MR 933347, DOI 10.1090/pspum/047.1/933347
- 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
- J.F. Carlson, R. Rouquier, Self-equivalences of stable categories of modules over modular group algebras, in preparation.
- Everett C. Dade, Endo-permutation modules over $p$-groups. I, Ann. of Math. (2) 107 (1978), no. 3, 459–494. MR 496843, DOI 10.2307/1971125
- K. Erdmann, O. Kerner, A. Skowroński, Self-injective algebras of wild tilted type; preprint (Bielefeld 97-080)
- 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
- Karin Erdmann, Blocks of tame representation type and related algebras, Lecture Notes in Mathematics, vol. 1428, Springer-Verlag, Berlin, 1990. MR 1064107, DOI 10.1007/BFb0084003
- Karin Erdmann, On Auslander-Reiten components for group algebras, J. Pure Appl. Algebra 104 (1995), no. 2, 149–160. MR 1360172, DOI 10.1016/0022-4049(94)00133-7
- M. Hoshino , Modules without self-extensions and Nakayama’s conjecture, Arch. Math. 43 (1984), 493-500
- 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
- Otto Kerner, Stable components of wild tilted algebras, J. Algebra 142 (1991), no. 1, 37–57. MR 1125203, DOI 10.1016/0021-8693(91)90215-T
- Maria Alicia Aviñó Díaz and Raymundo Bautista Ramos, On Artin algebras over Artinian commutative local rings, Arch. Math. (Basel) 66 (1996), no. 5, 366–371. MR 1383900, DOI 10.1007/BF01781554
- Otto Kerner and Frank Lukas, Regular modules over wild hereditary algebras, Representations of finite-dimensional algebras (Tsukuba, 1990) CMS Conf. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 1991, pp. 191–208. MR 1143852
- Zygmunt Pogorzały, On the stable Grothendieck groups, Proceedings of the Sixth International Conference on Representations of Algebras (Ottawa, ON, 1992) Carleton-Ottawa Math. Lecture Note Ser., vol. 14, Carleton Univ., Ottawa, ON, 1992, pp. 11. MR 1206956
- Claus Michael Ringel, Finite dimensional hereditary algebras of wild representation type, Math. Z. 161 (1978), no. 3, 235–255. MR 501169, DOI 10.1007/BF01214506
- 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
- Peter J. Webb, The Auslander-Reiten quiver of a finite group, Math. Z. 179 (1982), no. 1, 97–121. MR 643050, DOI 10.1007/BF01173918
Bibliographic Information
- Karin Erdmann
- Affiliation: Mathematical Institute, University of Oxford, 24-29 St. Giles, Oxford OX1 3LB, United Kingdom
- MR Author ID: 63835
- ORCID: 0000-0002-6288-0547
- Email: erdmann@maths.ox.ac.uk
- Otto Kerner
- Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany
- MR Author ID: 194039
- Email: kerner@cs.uni-duesseldorf.de
- Received by editor(s): June 4, 1996
- Received by editor(s) in revised form: October 2, 1997
- Published electronically: February 14, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 2389-2405
- MSC (2000): Primary 18G25; Secondary 16G70, 20C20
- DOI: https://doi.org/10.1090/S0002-9947-00-02232-7
- MathSciNet review: 1487612