On the Composition of Four Irreducible Morphisms in the Fifth Power of the Radical

摘要

Given A an artin algebra we study when the non-zero composition of four irreducible morphisms between indecomposable A-modules belongs to the fifth power of the radical of its module category. In particular, when A is a finite dimensional algebra over an algebraically closed field we prove that if the composition of four irreducible morphisms between indecomposable A-modules belongs to the fifth power of the radical of their module category then any composition of four irreducible morphisms between the same indecomposable A-modules so is.