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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