On upper bounds for asymptotic ideal-grade

Abstract

Let [Formula: see text] and [Formula: see text] be ideals in a Noetherian ring [Formula: see text] and let [Formula: see text] be nonunits in [Formula: see text]. Then [Formula: see text] is said to be an asymptotic sequence over [Formula: see text] if [Formula: see text] and if for all [Formula: see text], [Formula: see text] is not in any associated prime of the integral closure [Formula: see text] of [Formula: see text], where [Formula: see text] is very large. Let [Formula: see text] be the maximum number of elements in [Formula: see text] which form an asymptotic sequence over [Formula: see text]. It is characterized when [Formula: see text] is equal to: (i) [Formula: see text], the analytic spread of [Formula: see text], when [Formula: see text] is local; (ii) [Formula: see text], where [Formula: see text] is the maximum number of elements in [Formula: see text] which form an asymptotic sequence over [Formula: see text], and several consequences of these characterizations are given. Finally, if [Formula: see text] is local with maximal ideal [Formula: see text] then we reprove a known upper bound for [Formula: see text].