On the asymptotic cograde of an ideal

Abstract

Let I be an ideal in a local ring ( R, M) and let s( I) be the asymptotic cograde of I (= the maximum length of an asymptotic sequence over I). Rees has shown that s( I) ⩽ altitude R − l( I), where l( I) is the analytic spread of I. We prove several more bounds on s( I): (i) s( I) ⩽ min{little depth P; P is an asymptotic prime divisor of I; (ii) s(I) ⩽ grade ∗ M I (= the maximum length of an asymptotic sequence contained in M I ) ; (iii) s(I) ⩽ grade ∗ M- grade ∗ I ; (iv) grade ∗ M − l(I) ⩽ s(I) ; and. (v) if R is quasi-unmixed, then grade R I n ⩽ s(I) for all large n. Also, it is shown that the asymptotic prime divisors of ( I, b 1,…, b s ) R contain those of I, when b 1,…, b s are an asymptotic sequence over I, and that the residue classes modulo I of such a sequence are an asymptotic sequence in R I . Finally, several sufficient conditions for l( I) to be equal to l( I p ) for some asymptotic prime divisor P of I are given.