Order Ideals and a Generalized Krull Height Theorem
  Let N be a finitely generated module over a Noetherian local ring (R,m). We give criteria for the height of the order ideal N^*(x) of an element x \in N to be bounded by the rank of N. The Generalized Principal Ideal Theorem of Bruns, Eisenbud and Evans says that this inequality always holds if x \in mN, but if x is a minimal generator the question is more subtle; for example, when R is graded and N represents a vector bundle on Proj R, the question is connected with the vanishing of the top chern class. We show that the inequality does hold if the hypothesis x \in mN becomes true after first extending scalars to some local domain and then factoring out torsion. We also give conditions under which it holds in terms of residual intersections and integral closures of modules.   We derive information about order ideals that leads to bounds on the heights of trace ideals of modules--even in circumstances where we do not have the expected bounds for the heights of the order ideals. 
David Eisenbud, Craig Huneke, and Bernd Ulrich