Abstract
We prove that the class of finite dimensional selfinjective algebras over a field which admit Galois coverings by the repetitive algebras of the quasitilted algebras, with Galois groups generated by compositions of the Nakayama automorphisms with strictly positive automorphisms, is invariant under stable and derived equivalences.
Similar content being viewed by others
References
Asashiba, H.: The derived equivalence classification of representation-finite selfinjective algebras. J. Algebra 214, 182–221 (1999)
Asashiba, H.: Derived and stable equivalence classification of twisted multifold extensions of piecewise hereditary algebras of tree type. J. Algebra 249, 345–376 (2002)
Assem, I., Nehring, J., Skowroński, A.: Domestic trivial extensions of simply connected algebras. Tsukuba J. Math. 13, 31–72 (1989)
Auslander, M., Reiten, I., Smalø, S.O.: Representation theory of artin algebras. Cambridge Studies in Advanced Math. 36, Cambridge Univ. Press, 1995
Assem, I., Skowroński, A.: Multicoil algebras. In: Representations of algebras, CMS Conf. Proc. 14, CMS/AMS 1993, pp. 29–68
Assem, I., Skowroński, A., Tomé, B.: Coil enlargements of algebras. Tsukuba J. Math. 19, 453–478 (1995)
Białkowski, J., Erdmann, K., Skowroński, A.: Deformed preprojective algebras of generalized Dynkin type. Trans. Amer. Math. Soc., in press
Białkowski, J., Holm, T., Skowroński, A.: Derived equivalences for tame weakly symmetric algebras having only periodic modules. J. Algebra 269, 652–668 (2003)
Białkowski, J., Skowroński, A.: Selfinjective algebras of tubular type. Colloq. Math. 34, 175–194 (2002)
Białkowski, J., Skowroński, A.: On tame weakly symmetric algebras having only periodic modules. Arch. Math. 81, 142–152 (2003)
Białkowski, J., Skowroński, A.: Socle deformations of selfinjective algebras of tubular type. J. Math. Soc. Japan 56, 687–716 (2004)
Bocian, R., Holm, T., Skowroński, A.: Derived equivalence classification of weakly symmetric algebras of Euclidean type. J. Pure Appl. Algebra 191, 43–74 (2004)
Bocian, R., Holm, T., Skowroński, A.: Derived equivalence classification of one-parametric selfinjective algebras. J. Pure Appl. Algebra, in press
Bocian, R., Skowroński, A.: Symmetric special biserial algebras of Euclidean type. Colloq. Math. 96, 121–148 (2003)
Bocian, R., Skowroński, A.: Weakly symmetric algebras of Euclidean type. J. reine angew. Math. 580, 157–199 (2005)
Bocian, R., Skowroński, A.: One-parametric selfinjective algebras. J. Math. Soc. Japan 57, 491–512 (2005)
Bretscher, O., Läser, C., Riedtmann, C.: Selfinjective algebras and simply connected algebras. Manuscr. Math. 36, 253–307 (1982)
Bongartz, K., Gabriel, P.: Covering spaces in representation theory. Invent. Math. 65, 331–378 (1981/82)
Dowbor, P., Skowroński, A.: Galois coverings of representation-infinite algebras. Comment. Math. Helv. 62, 311–337 (1987)
Erdmann, K., Kerner, O., Skowroński, A.: Self-injective algebras of wild tilted type. J. Pure Appl. Algebra 149, 127–176 (2000)
Erdmann, K., Skowroński, A.: The stable Calabi-Yau dimension of tame symmetric algebras. J. Math. Soc. Japan, in press
Farnsteiner, R., Skowroński, A.: Classification of restricted Lie algebras with tame principal block. J. reine angew. Math. 546, 1–45 (2002)
Farnsteiner, R., Skowroński, A.: The tame infinitesimal groups of odd characteristic. Advances. Math., in press
Farnsteiner, R., Skowroński, A.: Galois actions and blocks of tame infinitesimal group schemes. Trans. Amer. Math. Soc., in press
Gabriel, P.: The universal cover of a representation-finite algebra. In: Representations of algebras, Lecture Notes in Math. 903, Springer-Verlag, 1981, pp. 68–105
Geigle, W., Lenzing, H.: A class of weighted projective curves arising in representation theory of finite dimensional algebras. In: Singularities, representations of algebras and vector bundles, Lecture Notes in Math. 1273, Springer-Verlag, 1987, pp. 265–297
Happel, D.: On the derived category of a finite-dimensional algebra. Comment. Math. Helv. 62, 339–389 (1987)
Happel, D.: Triangulated categories in the representation theory of finite dimensional algebras. London Math. Soc. Lecture Notes Ser. 119 (1988)
Happel, D.: A characterization of hereditary categories with tilting object. Invent. Math. 144, 381–398 (2001)
Happel, D., Reiten, I.: Hereditary abelian categories with tilting object over arbitrary basic fields. J. Algebra 256, 414–432 (2002)
Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and quasitilted algebras. Mem. Amer. Math. Soc. 575 (1996)
Hughes, D., Waschbüsch, J.: Trivial extensions of tilted algebras. Proc. London Math. Soc. 46, 347–364 (1983)
Lenzing, H., Meltzer H.: Tilting sheaves and concealed-canonical algebras. In: Representations of algebras, CMS Conf. Proc. 18, CMS/AMS 1996, pp. 455–473
Lenzing, H., de la Peña, J.A.: Concealed-canonical algebras and separating tubular families. Proc. London Math. Soc. 78, 513–540 (1999)
Lenzing, H., Skowroński, A.: Quasi-tilted algebras of canonical type. Colloq. Math. 71, 161–181 (1996)
Lenzing, H., Skowroński, A.: Selfinjective algebras of wild canonical type. Colloq. Math. 96, 245–275 (2003)
Malicki, P., Skowroński, A.: Almost cyclic coherent components of an Auslander-Reiten quiver. J. Algebra 229, 695–749 (2000)
Malicki, P., Skowroński, A.: Algebras with separating almost cyclic coherent Auslander-Reiten components. J. Algebra 291, 208–237 (2005)
Nehring, J., Skowroński, A.: Polynomial growth trivial extensions of simply connected algebras. Fund. Math. 132, 117–134 (1989)
Peng, L., Xiao, J.: Invariability of trivial extensions of tilted algebras under stable equivalences. J. London Math. Soc. 52, 61–72 (1995)
Pogorzały, Z.: Algebras stably equivalent to selfinjective algebras whose Auslander-Reiten quivers consist only of generalized standard components. Glasgow Math. J. 40, 1–19 (1998)
Pogorzały, Z.: Forms of algebras stably equivalent to self-injective algebras of polynomial growth. J. Algebra 205, 168–206 (1998)
Pogorzały, Z., Skowroński, A.: Symmetric algebras stably equivalent to the trivial extensions of tubular algebras. J. Algebra 181, 95–111 (1996)
Reiten, I., Skowroński, A.: Sincere stable tubes. J. Algebra 232, 64–75 (2000)
Riedtmann, C.: Representation-finite selfinjective algebras of class . In: Representation theory II, Lecture Notes in Math. 832, Springer-Verlag, 1980, pp. 449–520
Riedtmann, C.: Representation-finite selfinjective algebras of class , Compositio Math. 49, 231–282 (1983)
Rickard, J.: Morita theory for derived categories. J. London Math. Soc. 39, 436–456 (1989)
Rickard, J., Derived categories and stable equivalence, J. Pure Appl. Algebra 61, 303–317 (1989)
Ringel, C.M.: Representaions of K-species and bimodules. J. Algebra 41, 269–302 (1976)
Ringel, C.M.: Tame algebras and integral quadratic forms, Lecture Notes in Math. 1099, Springer-Verlag, 1984
Ringel, C.M.: The canonical algebras, with an appendix by W. Crawley-Boevey, In: Topics in Algebra, Banach Center Publications 26, Part 1, PWN Warsaw, 1990, pp. 407–432
Skowroński, A.: Selfinjective algebras of polynomial growth. Math. Ann. 285, 177–199 (1989)
Skowroński, A.: Generalized standard Auslander-Reiten components. J. Math. Soc. Japan 46, 517–543 (1994)
Skowroński, A.: On the composition factors of periodic modules. J. London Math. Soc. 49, 477–492 (1994)
Skowroński, A.: Generalized canonical algebras and standard stable tubes. Colloq. Math. 90, 77–93 (2001)
Skowroński, A.: Selfinjective algebras: finite and tame type. Contemporary Math., in press
Skowroński, A., Yamagata, K.: Socle deformations of self-injective algebras. Proc. London Math. Soc. 72, 545–566 (1996)
Skowroński, A., Yamagata, K.: Stable equivalence of selfinjective algebras of tilted type. Arch. Math. 70, 341–350 (1988)
Skowroński, A., Yamagata, K.: Galois coverings of selfinjective algebras by repetitive algebras. Trans. Amer. Math. Soc. 351, 715–734 (1999)
Skowroński, A., Yamagata, K.: On selfinjective artin algebras having nonperiodic generalized standard Auslander-Reiten components. Colloq. Math. 96, 235–244 (2003)
Skowroński, A., Yamagata, K.: On invariability of selfinjective algebras of tilted type under stable equivalences. Proc. Amer. Math. Soc. 132, 659–667 (2004)
Skowroński, A., Yamagata, K.: Stable equivalence of selfinjective algebras of Dynkin type. Algebras and Representation Theory, in press
Skowroński, A., Yamagata, K.: Positive Galois coverings of selfinjective algebras. Advances Math. 194, 398–436 (2005).
Yamagata, K.: Frobenius algebras, In: Handbook of algebra, Vol. 1, Amsterdam/New York, Elsevier 1996, pp. 841–887
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday
Rights and permissions
About this article
Cite this article
Kerner, O., Skowroński, A. & Yamagata, K. Invariance of selfinjective algebras of quasitilted type under stable equivalences. manuscripta math. 119, 359–381 (2006). https://doi.org/10.1007/s00229-005-0623-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-005-0623-9