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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

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