  The classical "generalized principal ideal theorems" of Macaulay,
Eagon-Northcott, and others give sharp bounds on the heights of determinantal
ideals in arbitrary rings. But in regular local rings (or graded polynomial
rings) these are far from sharp, and various questions about vector bundles, as
well as other questions in commutative algebra, amount to asking what the real
bounds are. We give partial answers, and more generally we prove new height
bounds in local rings of given embedding codimension. Our theorems extend and
sharpen results of Faltings and Bruns.
