Computing the extensions of preinjective and preprojective Kronecker modules

Let I  , I′I′ be preinjective Kronecker modules (i.e. all their indecomposable components are preinjective). We describe the modules M   for which there exists an exact sequence 0→I′→M→I→00→I′→M→I→0 by explicit, easy to check numerical conditions, resulting in an algorithm (linear in the number of indecomposable components) for the decision problem. We also propose a method to generate all extensions of I′I′ by I and we give a different proof for a theorem in [13] providing numerical criteria in terms of Kronecker invariants for the existence of a monomorphism f:I′→If:I′→I. All these results apply dually to preprojective modules as well.