On Degenerations and Extensions of Finite Dimensional Modules

Abstract

We derive a cancellation theorem for degenerations of modules that says in particular, that projective or injective common direct summands can always be neglected. Combining the cancellation result with the existence of almost split sequences we characterize the orbit closure of a module living on preprojective components by the fact that the dimension of the homomorphism space to any other module does not decrease. For representation-directed algebras, whence in particular for path algebras of Dynkin quivers, we provide an alternative proof which shows in addition that any minimal degenerationNofMcomes from an exact sequence with middle termMwhose end terms add up toN. By a careful examination, the same is true for degenerations of matrix pencils. Having used so far the existence of certain extensions to obtain degenerations we then turn the tables and use degenerations to produce a lot of interesting short exact sequences. In particular, we show that any non-simple indecomposable over a tame quiver is an extension of an indecomposable and a simple.