Almost split morphisms, preprojective algebras and multiplication maps of maximal rank

With a grading previously introduced by the second-named author, the multiplication maps in the preprojective algebra satisfy a maximal rank property that is similar to the maximal rank property proven by Hochster and Laksov for the multiplication maps in the commutative polynomial ring. The result follows from a more general theorem about the maximal rank property of a minimal almost split morphism, which also yields a quadratic inequality for the dimensions of indecomposable modules involved.