Degenerations for modules over representation-finite algebras. (English) Zbl 0927.16008
Summary: 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
\[
\dim_K\operatorname{Hom}_A(M,X)\leq\dim_K\operatorname{Hom}_A(N,X)
\]
hold for all \(A\)-modules \(X\). We prove also that if \(\text{Ext}_A^1(X,X)=0\) for any indecomposable \(A\)-module \(X\), then any degeneration of \(A\)-modules is given by a chain of short exact sequences.
MSC:
16G10 | Representations of associative Artinian rings |
16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
14L30 | Group actions on varieties or schemes (quotients) |
16P10 | Finite rings and finite-dimensional associative algebras |