On the asymptotic cograde of an ideal (Q790885)

Let I be an ideal in a local ring (R,M). The elements \(x_ 1,...,x_ n\) of M are an asymptotic sequence over I if for each i, \(x_ i\) is a regular element modulo the integral closure of \((I,x_ 1,...,x_{i- 1})^ m\) for all \(m>0\). The length of a maximal such sequence is the asymptotic cograde of I, s(I). Rees has shown that \(s(I)\leq altitude R- a(I),\) with a(I) the analytic spread of I. This paper proves several more inequalities. For example, \(s(I)\leq depth P,\) where P is any prime divisor of the integral closure of \(I^ m\), for any \(m>0\).