An algorithm for finding all preprojective components of the Auslander-Reiten quiver (Q2781222)

Let \(A\) be a finite dimensional algebra over an algebraically closed field and let \(kQ/I\) be the presentation of \(A\) as a quotient of the path algebra \(kQ\) of a quiver \(Q\) by an admissible ideal \(I\). The article is devoted to an algorithm for finding all preprojective components of the Auslander-Reiten quiver \(\Gamma_A\) of the category of finitely generated right \(A\)-modules. A component of \(\Gamma_A\) is preprojective if it contains no oriented cycle and every of its vertices has only finitely many predecessors. First the problem is reduced to the algebra determined by an initial subquiver \(Q^{init}\) of \(Q\) having no oriented cycles. A subquiver \(Q'\) of \(Q\) is initial if every arrow of \(Q\) with the sink in \(Q'\) has also the source in \(Q'\). Assume now that \(Q\) has no oriented cycles. The algorithm is based on an inductive procedure to construct an initial subquiver \(Q^*\) of \(Q\) such that the algebra \(A^*=kQ^*/(I\cap kQ^*)\) has the following properties: (1) every indecomposable projective \(A^*\)-module belongs to a preprojective component of \(\Gamma_{A^*}\), (2) every preprojective component of \(\Gamma_A\) is a preprojective component of \(\Gamma_{A^*}\), (3) a preprojective component of \(\Gamma_{A^*}\) is a preprojective component of \(\Gamma_A\) if and only if it does not contain any indecomposable direct summand of the radical of the projective indecomposable \(A\)-module corresponding to a vertex of \(Q\) being a direct neighbor of \(Q^*\).