Degenerations for modules over representation-finite algebras
Grzegorz Zwara
Faculty of Mathematics and Informatics, Nicholas Copernicus University, 
Chopina 12/18, 87-100 Torun, Poland 
Proc. Amer. Math. Soc. 127 (1999), 1313-1322. 

Abstract. Let $A$ be a representation-finite algebra. We show that a finite 
dimensional $A$-module $M$ degenerates to another $A$-module $N$ if and only 
if the inequalities $\dim_K\text{Hom}_A(M,X) \le \dim_K\text{Hom}_A(N,X)$
hold for all $A$-modules $X$. We prove also that if 
$\text{\rm Ext}^1_A(X,X) = 0$ for any indecomposable $A$-module $X$, then any
degeneration of $A$-modules is given by a chain of short exact sequences.