Analytic spread of nitrations, asymptotic nature and some stability properties

Abstract

Many concepts of analytic spread for filtrations exist in the current literature. As far as we know, none of them is explicitly of asymptotic nature. Here we give a definition of analytic spread for filtrations which is shown to be of asymptotic nature. On the other hand, it is known that, in a noetherian local ring, the analytic spread of ideals is invariant under the integral closure operation. In this paper we investigate too the behaviour of the analytic spread of a filtration f under some closure operations, the analytic spread of f= (In ) being indifferently one of the following numbers: Where J is an ideal of the Rees rings of the filtration f,X an indeterminate and u=X-1.Here we give also in a noetherian local ring (A, m) a class of filtrations f= (In ), containing I-good filtrations, such that the analytic spread of fcoincides with that of In for all n ≥1, in answer to a question raised by J.S.Okon in 1984 which was shown to be false for general noetherian filtrations. We give equally an upper bound of the analytic spread of a strongly noetherian filtration In in terms of the asymptotic value of depth which is the analogue of a result of Brodmann.