# Representation-finite triangular algebras form an open scheme

Open Mathematics (2003)

- Volume: 1, Issue: 1, page 97-107
- ISSN: 2391-5455

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topStanisław Kasjan. "Representation-finite triangular algebras form an open scheme." Open Mathematics 1.1 (2003): 97-107. <http://eudml.org/doc/268752>.

@article{StanisławKasjan2003,

abstract = {Let V be a valuation ring in an algebraically closed field K with the residue field R. Assume that A is a V-order such that the R-algebra Ā obtained from A by reduction modulo the radical of V is triangular and representation-finite. Then the K-algebra KA ≅ A ⊗V is again triangular and representation-finite. It follows by the van den Dries’s test that triangular representation-finite algebras form an open scheme.},

author = {Stanisław Kasjan},

journal = {Open Mathematics},

keywords = {16G60; 16G30; 03C60},

language = {eng},

number = {1},

pages = {97-107},

title = {Representation-finite triangular algebras form an open scheme},

url = {http://eudml.org/doc/268752},

volume = {1},

year = {2003},

}

TY - JOUR

AU - Stanisław Kasjan

TI - Representation-finite triangular algebras form an open scheme

JO - Open Mathematics

PY - 2003

VL - 1

IS - 1

SP - 97

EP - 107

AB - Let V be a valuation ring in an algebraically closed field K with the residue field R. Assume that A is a V-order such that the R-algebra Ā obtained from A by reduction modulo the radical of V is triangular and representation-finite. Then the K-algebra KA ≅ A ⊗V is again triangular and representation-finite. It follows by the van den Dries’s test that triangular representation-finite algebras form an open scheme.

LA - eng

KW - 16G60; 16G30; 03C60

UR - http://eudml.org/doc/268752

ER -

## References

top- [1] I. Assem, D. Simson and A. Skowroński, “Elements of Representation Theory of Associative Algebras”, Vol I: Techniques of Representation Theory, London Math. Soc. Student Texts, Cambridge University Press, Cambridge, to appear. Zbl1092.16001
- [2] M. Auslander, I. Reiten and S. Smalø, “Representation theory of Artin algebras”, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, 1995. Zbl0834.16001
- [3] I. Assem and A. Skowroński, On some classes of simply connected algebras, Proc. London Math. Soc., 56 (1988), 417–450. Zbl0617.16018
- [4] S. Balcerzyk and T. Józefiak, “Pierścienie przemienne”, Warszawa: Państwowe Wydawnictwo Naukowe (1985), English translation of the Chapters I–IV: “Commutative Noetherian and Krull rings”. PWN-Polish Scientific Publishers, Warsaw. Chichester: Ellis Harwood Limited; New York etc.: Halsted Press. (1989).
- [5] K. Bongartz, Zykellose Algebren sind nicht zügellos. Representation theory II, Proc. 2nd Int. Conf., Ottawa 1979, Lect. Notes Math. 832, (1980) 97–102.
- [6] K. Bongartz, A criterion for finite representation type, Math. Ann. 269 (1984), 1–12. http://dx.doi.org/10.1007/BF01455993 Zbl0552.16012
- [7] K. Bongartz and P. Gabriel, Covering spaces in representation theory, Invent. Math. 65 (1982), 331–378. http://dx.doi.org/10.1007/BF01396624 Zbl0482.16026
- [8] O. Bretcher and P. Gabriel, The standard form of a representation-finite algebra, Bull. Soc. Math. France 111 (1983), 21–40.
- [9] Ch. W. Curtis and I. Reiner, “Methods of Representation Theory”, Vol. I, Wiley Classics Library Edition, New York, 1990.
- [10] H. Ebbinghaus and J. Flum, “Finite Model Theory”, Perspectives in Mathematical Logic. Berlin: Springer-Verlag 1995. Zbl0841.03014
- [11] A. Ehrenfeucht, An application of games to the completness problem for formalized theories, Fund. Math. 49, 1961, 129–141. Zbl0096.24303
- [12] P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71–102. http://dx.doi.org/10.1007/BF01298413
- [13] P. Gariel, Finite representation type is open. in “Representations of algebras”, Lecture Notes in Math. 488, Springer-Verlag, Berlin, Heidelberg and New-York (1975) 132–155.
- [14] P. Gabriel, The universal cover of a representation-finite algebra, in: Lecture Notes in Math. 903, Springer-Verlag, Berlin, Heidelberg and New-York (1981), 68–105.
- [15] Ch. Jensen and H. Lenzing, Homological dimension and representation type of algebras under base field extension, Manuscripta Math., 39, 1–13 (1982). http://dx.doi.org/10.1007/BF01312441 Zbl0498.16023
- [16] Ch. Jensen and H. Lenzing, “Model Theoretic Algebra: with particular emphasis on fields, rings, modules”, Algebra, Logic and Applications, 2. Gordon and Breach Science Publishers, New York, 1989. Zbl0728.03026
- [17] S. Kasjan, On the problem of axiomatization of tame representation type, Fundamenta Mathematicae 171 (2002), 53–67 http://dx.doi.org/10.4064/fm171-1-3 Zbl0997.16008
- [18] S. Kasjan, Representation-directed algebras form an open scheme, Colloq. Math. 93 (2002), 237–250. Zbl1023.16014
- [19] H. Kraft, Geometric methods in representation theory, in: Representations of algebras, 3rd int. Conf., Puebla/Mexico 1980, Lect. Notes Math. 944, 180–258 (1982).
- [20] R. Martinez-Villa and J.A. de la Peña, The universal cover of a quiver with relations, J. Pure. Appl. Algebra 30 (1983), 277–292. http://dx.doi.org/10.1016/0022-4049(83)90062-2 Zbl0522.16028
- [21] B. Poonen, Maximally complete fields, Enseign. Math. 39 (1993), 87–106. Zbl0807.12006
- [22] A. Schrijver, “Theory of Linear And Integer Programming”, Wiley-Interscience Series in Discrete Mathematics. A Wiley-Interscience Publication. Chichester: John Wiley & Sons Ltd. 1986.
- [23] L. van den Dries, Some applications of a model theoretic fact to (semi-) algebraic geometry, Nederl. Akad. Indag. Math., 44 (1982), 397–401. Zbl0538.14017
- [24] H. Weyl, The elementary theory of convex polyhedra. in: Contrib. Theory of Games, Ann. Math. Studies 24, (1950) 3–18. Zbl0041.25401

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