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On Degenerations of Modules With Nondirecting Indecomposable Summands

Published online by Cambridge University Press:  20 November 2018

A. Skowroński  and
G. Zwara
A. Skowroński
Affiliation:
Faculty of Mathematics and Informatics Nicholas Copernicus University Chopina 12/18, 87-100 Toruń, POLAND email: skowron@mat.uni.torun.plgzwara@mat.uni.torun.pl
G. Zwara
Affiliation:
Faculty of Mathematics and Informatics Nicholas Copernicus University Chopina 12/18, 87-100 Toruń, POLAND email: skowron@mat.uni.torun.plgzwara@mat.uni.torun.pl
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Abstract

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Let A be a finite dimensional associative K-algebra with an identity over an algebraically closed field K, d a natural number, and modA(d) the affine variety of d-dimensional A-modules. The general linear group Gld(K) acts on modA(d) by conjugation, and the orbits correspond to the isomorphism classes of d-dimensional modules. For M and N in modA(d), N is called a degeneration of M, if TV belongs to the closure of the orbit of M. This defines a partial order ≤deg on modA(d). There has been a work [1], [10], [11], [21] connecting ≤deg with other partial orders ≤ext and ≤deg on modA(d) defined in terms of extensions and homomorphisms. In particular, it is known that these partial orders coincide in the case A is representation-finite and its Auslander-Reiten quiver is directed. We study degenerations of modules from the additive categories given by connected components of the Auslander-Reiten quiver of A having oriented cycles. We show that the partial orders ≤ext, ≤deg and < coincide on modules from the additive categories of quasi-tubes [24], and describe minimal degenerations of such modules. Moreover, we show that M ≤degN does not imply M ≤ext N for some indecomposable modules M and N lying in coils in the sense of [4].

Keywords

14L3016G1016G70
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 5 , 01 October 1996 , pp. 1091 - 1120
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Abeasis, S. and del, A. Fra, Degenerations for the representations of a quiver of type 𝔸 m, J., Algebra 93(1985), 376–412.Google Scholar
2. Assem, I. and Skowroński, A., Minimal representation-infinite coil algebras, Manuscripta, Math. 67(1990), 305–331.Google Scholar
3. Assem, I., Indecomposable modules over multicoil algebras, Math., Scand. 71(1992), 31–61.Google Scholar
4. Assem, I., Multicoil algebras, Representations of Algebras, CMS Conference, Proceedings 14(1993), 29–68.Google Scholar
5. Assem, I., Skowroński, A. and Tome, B., Coil enlargements of algebras, Tsukuba J., Math. 19(1995), 453- 479.Google Scholar
6. Auslander, M., Representation theory of finite dimensional algebras, Contemp., Math. 13(1982), 27–39.Google Scholar
7. Auslander, M. and Reiten, I., Modules determined by their composition factors, Illinois Math., J. 29(1985), 280–301.Google Scholar
8. Bongartz, K., On a result of Bautista and Smah, Comm., Algebra 11(1983), 2123–2124.Google Scholar
9. Bongartz, K., A generalization of a theorem of M. Auslander, Bull. London Math., Soc. 21(1989), 255–256.Google Scholar
10. Bongartz, K., On degenerations and extensions of finite dimensional modules, Adv. Math., to appear.Google Scholar
11. Bongartz, K., Minimal singularities for representations ofDynkin quivers, Commentari Math., Helv. 69(1994), 575–611.Google Scholar
12. Bongartz, K. and Gabriel, P., Covering spaces in representation theory, Invent., Math. 65(1982), 331–378.Google Scholar
13. Crawley-Boevey, W.W., On tame algebras and bocses, Proc. London Math., Soc. 62(1988), 451–483.Google Scholar
14. D'Este, G. and Ringel, C.M., Coherent tubes, J., Algebra 87(1984), 150–201.Google Scholar
15. Happel, D., Preiser, U. and Ringel, C.M., Vingbergs characterization ofDynkin diagrams using subadditive functions with application to DTr-periodic modules, Representation Theory II, Springer Lecture Notes in, Math. 832(1980), 280–294.Google Scholar
16. Kraft, H., Geometric methods in representation theory, Representations of Algebras, Springer Lecture Notes in, Math. 944(1982), 180–258.Google Scholar
17. Liu, S., Degrees of irreducible maps and the shapes of Auslander-Reiten quivers, J. London Math., Soc. 45(1992), 32–54.Google Scholar
18. Liu, S., Infinite radicals in standard Auslander-Reiten component, J., Algebra 166(1994), 245–254.Google Scholar
19. Peng, L.G. and Xiao, J., On the number of DTr-orbits containing directing modules, Proc. Amer. Math., Soc. 118(1993), 753–756.Google Scholar
20. Reiten, I., Skowroński, A. and Smalo, S.O., Short chains and short cycles of modules, Proc. Amer. Math., Soc. 117(1993), 343–354.Google Scholar
21. Riedtmann, C., Degenerations for representations of quivers with relations, Ann. Sci. Ecole Normal, Sup. 4(1986), 275–301.Google Scholar
22. Ringel, C.M., Tame algebras and integral quadratic forms, Springer Lecture Notes in, Math. 1099(1984).Google Scholar
23. Simson, D., Linear representations of partially ordered sets and vector space categories, Algebra, Logic and Appl. Series 4, Gordon and Breach Science Publ., Amsterdam, 1992.Google Scholar
24. Skowronski, A., Algebras of polynomial growth. In: Topics in Algebra, Banach Center Publications 26, Part I, PWN, Warszawa, 1990. 535–568.Google Scholar
25. Skowronski, A., Regular Auslander-Reiten components containing directing modules, Proc. Amer. Math., Soc. 120(1994), 19–26.Google Scholar
26. Skowronski, A., Cycles in module categories. In: Finite Dimenional Algebras and Related Topics, NATO ASI Series, Series C 424, Kluwer Acad. Publ., Dordrecht, 1994. 309–345.Google Scholar
27. Skowronski, A., Generalized standard Auslander-Reiten components, J. Math. Soc., Japan 46(1994), 517–543.Google Scholar
28. Skowronski, A., Criteria for polynomial growth of algebras, Bull. Polish Acad. Sci.,, Mathematics 42(1994), 173–183.Google Scholar
29. Skowronski, A., Tame algebras with simply connected Galois coverings, Toruń, 1995. preprint.Google Scholar