Degenerations for representations of extended Dynkin quivers

Abstract

Let A be the path algebra of a quiver of extended Dynkin type $\tilde {\Bbb {A}}_n, \tilde {\Bbb {D}}_n, \tilde {\Bbb {E}}_6, \tilde {\Bbb {E}}_7$ or $\tilde {\Bbb {E}}_8$ . We show that a finite dimensional A-module M degenerates to another A-module N if and only if there are short exact sequences $0 \to U_i \to M_i \to V_i \to 0$ of A-modules such that $M = M_1$ , $M_{i+1} = U_i \oplus V_i$ for $1 \leq i \leq s$ and $N = M_{s+1}$ are true for some natural number s.