Asymptotic closure of filtrations on rings with Krull dimension 1

Abstract

In the present paper, the concept of asymptotic closure \(\overline {(f,M)}\) of a filtration f relative to a module M is introduced and investigated. The methods used by the authors in previous notes on integral and pruferian closure operations of a filtration have proved to be efficient here, despite the complexity of the asymptotic closure operation comparatively to the integral and pruferian closure operation. Our main result gives a complete description of the asymptotic closure \(\bar f\) of a filtration f on a Dedekind ring A, in terms of the prime ideals of which \(\sqrt f\) is the product, when f belongs to some class containing noetherian filtrations.