On the equivalence of the integral symbolic and adic topologies

Abstract

Let R be a commutative Noetherian ring and I an ideal of R. The purpose of this paper is to show that the topologies defined by the integral filtration {Im¯}m≥1 and the symbolic integral filtration {I⟨m⟩}m≥1 are equivalent whenever Q¯∗(I) consists all of the minimal prime ideals of I. As an application of this result, by using the Jacobian theorem of Lipman and Sathaye we deduce that the symbolic integral topology {I⟨m⟩}m≥1 is equivalent to the I-adic topology whenever R is a regular ring. Also, applying these results we provide extensions of classical results of Hartshorne and Zariski on the equivalence of symbolic and adic topologies.