Abstract
The well-known tame theorem tells that for a given tame bocs and a positive integer n there exist finitely many minimal bocses, such that any representation of the original bocs of dimension at most n is isomorphic to the image of a representation of some minimal bocses under a certain reduction functor. In the present paper we will give an alternative statement of the tame theorem in terms of matrix problem, by constructing a unified minimal matrix problem whose indecomposable matrices cover all the canonical forms of the indecomposable representations of dimension at most n for each non-negative integer n.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Drozd Y A. On tame and wild matrix problems. Matrix Problems. Institute of Mathematics, Academy of Sciences, Ukranian SSR. Kive: 1977, 104–114
Crawley-Boevey W W. On tame algebras and bocses. Proc London Math Soc, 56(3): 451–483 (1988)
Bautista R, Drozd Yu A, Zeng X Y, et al. On Hom-spaces of tame algebras. Central European J Math, 5(2): 215–263 (2007)
Crawley-Boevey W W. Matrix problems and Drozd’s theorem. Topics Algebra, 26: 199–222 (1990)
Sergeichuk V V. Canonical matrices for basic matrix problems. Linear Algebra Appl, 317(1/3): 53–102 (2000)
Gabriel P, Roiter A V. Representation of finite-dimensional algebras. Encyclopaedia of Math Sci, Vol 73, Algebra VIII. New York: Springer-Verlag, 1992
Han Y. Controlled wild algebras. Proc London Math Soc, 83(3): 279–298 (2001)
Jacobson N. Basic Algebra. San Francisco: WH Freemann and Company, 1974
Crawley-Boevey W W. Tame algebras and generic modules. Proc London Math Soc, 63(3): 241–165 (1991)
Krause H. Generic modules over artin algebras. Proc London Math Soc, 76(3): 276–306 (1998)
Ovsienko S A. Generic representations of free bocses, Preprint 93-010, Univ Bielefeld, 1993
Bautista R, Zhang Y B. Representatons of a k-algebra over the rational functions over k. J Algebra, 267: 342–358 (2003)
Ringel C M. Tame Algebras and Integral Quadratic Forms. LNM 1099. New York: Springer, 1984
Auslander M, Reiten I. Representation theory of artin algebras III. Comm Algebra, 3(3): 239–294 (1975)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by National Natural Science Foundation of China (Grant Nos. 10731070, 10501010)
Rights and permissions
About this article
Cite this article
Zhang, Y., Xu, Y. Unified tame theorem. Sci. China Ser. A-Math. 52, 2036–2068 (2009). https://doi.org/10.1007/s11425-008-0171-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-008-0171-3