Semi-prime operation and superficial sequence

Abstract

Let [Formula: see text] be a Cohen–Macaulay local ring with infinite residue field [Formula: see text] and [Formula: see text] be an [Formula: see text]-primary ideal of [Formula: see text]. Let [Formula: see text] be a semi-prime operation in the set of ideals of [Formula: see text] and [Formula: see text] be a [Formula: see text]-good filtration on [Formula: see text]. Let [Formula: see text] be an integer and [Formula: see text] for all large integers [Formula: see text]. In this paper, we give some relationships between [Formula: see text] and [Formula: see text]-superficial sequence. Thus, we show that if [Formula: see text] is a maximal [Formula: see text]-superficial sequence on [Formula: see text] and [Formula: see text], then it exists a positive integer [Formula: see text] such that [Formula: see text] for all [Formula: see text].