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.
Diaz, Steven P. and Kleiner, Mark, "Almost Split Morphisms, Preprojective Algebras and Multiplication Maps of Maximal Rank" (2005). Mathematics - Faculty Scholarship. 75.
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