Independent elements, integrally closed ideals, and quasi-unmixedness

Abstract

Let I a denote the integral closure of an ideal I in a Noetherian ring A. Then it is shown that sup a ∞ I, the maximum number of elements which are ( I n ) a -independent as n → ∞, exists and is equal to min{height (I(A P) ∗ + z) z : P is a prime divisor of ( I n ) a for all large n and z is a minimal prime ideal in the completion (A p) ∗ of A p} . From this it follows that A is locally quasi-unmixed if and only if sup a ∞ I = height I for all ideals I in A.