Equivalence of symbolic and adic topologies

Abstract

Let [Formula: see text] denote an ideal of a Noetherian ring [Formula: see text] and [Formula: see text] a nonzero finitely generated [Formula: see text]-module. It is shown that if the [Formula: see text]-symbolic topology is equivalent to the [Formula: see text]-adic topology on [Formula: see text], for all [Formula: see text], then the [Formula: see text]-symbolic topology on [Formula: see text] is equivalent to the [Formula: see text]-adic topology on [Formula: see text]. Moreover, we show that if [Formula: see text] consists of a single prime ideal, for all [Formula: see text], then the [Formula: see text]-adic and the [Formula: see text]-symbolic topologies on [Formula: see text] are equivalent. Finally, it is shown that if for every one-dimensional prime ideal [Formula: see text] in [Formula: see text], the [Formula: see text]-adic and the [Formula: see text]-symbolic topologies are equivalent on [Formula: see text], then [Formula: see text] is unmixed and [Formula: see text] has only one element.