Abstract
The classification of the Auslander-Reiten quivers of the local orders of finite lattice type is completed. For this purpose, it is shown—using the results of [7]—that to the list of the known Auslander-Reiten quivers of the local Bäckström orders of finite lattice type [11], [14] and of the local Gorenstein orders of finite type and their minimal overorders [18] one has to add two remaining types of valued translation quivers to obtain a complete list of all Auslander-Reiten quivers of the local orders of finite lattice type.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auslander, M., Existence theorems for almost split sequences, in: Ring Theory II, Proceedings of the Second Oklahoma Conference (Norman 1975), pp. 1–44. New York-Basel, Marcel Dekker 1977
Auslander, M. and Reiten, I., Representation theory of Artin algebras IV, Comm. Algebra 5 (1977), 443–518
Bongartz, K. and Gabriel, P., Covering spaces in representation theory, Invent. Math. 65 (1982), 331–378
Dade, E.C., Some indecomposable group representations, Ann. of Math. (2) 77 (1963), 406–412
Drozd, Ju.A., Kirichenko, V.V. and Roiter, A.V., On hereditary and Bass orders, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 1415–1436=Math. USSR Izv. 1 (1967), 1357–1376
Drozd, Ju.A. and Kirichenko, V.V., On quasi-Bass orders, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 328–370=Math. USSR Izv. 6 (1972), 323–366
Drozd, Ju.A. and Kirichenko, V.V., Primary orders with a finite number of indecomposable representations, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 715–736=Math. USSR Izv. 7 (1973), 711–732
Drozd, Ju.A. and Roiter, A.V., Commutative rings with a finite number of indecomposable integral representations, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 783–798=Math. USSR Izv. 1 (1967), 757–772
Happel, D., Preiser, U. and Ringel, C.M., Vinberg's characterization of Dynkin diagrams using subadditive functions with application to DTr-periodic modules, in: Representation Theory II, Lecture Notes in Mathematics No. 832, pp. 280–294, Springer-Verlag, Berlin, 1980
Igusa, K. and Todorov, G., A characterization of finite Auslander-Reiten quivers, J. Algebra 89 (1984), 105–147
Ringel, C.M. and Roggenkamp, K.W., Diagrammatic methods in the representation theory of orders, J. Algebra 60 (1979), 11–42
Roggenkamp, K.W., The construction of almost split sequences for integral group rings and orders, Comm. Algebra 5 (1977), 1363–1373
Roggenkamp, K.W., The lattice type of orders II, in: Integral Representations and Applications, Lecture Notes in Mathematics No. 882, pp. 430–477, Springer-Verlag, Berlin, 1981
Roggenkamp, K.W., Auslander-Reiten species of Bäckström orders, J. Algebra 85 (1983), 449–476.
Wiedemann, A., The existence of covering functors for orders and consequences for stable components with oriented cycles, J. Algebra 93 (1985), 292–309
Wiedemann, A., An integral version of the Igusa-Todorov algorithm, J. London Math. Soc. (2) 32 (1985), 75–87
Wiedemann, A., Classification of the Auslander-Reiten quivers of local Gorenstein orders and a characterization of the simple curve singularities, J. Pure Appl. Algebra 41 (1986), 305–329
Wiedemann, A., Die Auslander-Reiten Köcher der gitterendlichen Gorensteinordnungen, Habilitationsschrift, Stuttgart 1986, Bayreuther Math. Schr. 23 (1987), 1–134
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wiedemann, A. Auslander-Reiten quivers of local, orders of finite lattice type. Manuscripta Math 59, 187–207 (1987). https://doi.org/10.1007/BF01158046
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01158046