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 ${\{\overline{I^m}\}}_{m\ge 1}$ and the symbolic integral filtration ${\{I^{⟨m⟩}\}}_{m\ge 1}$ are equivalent whenever ${\overline{Q}}^{\ast }(I)$ 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 ${\{I^{⟨m⟩}\}}_{m\ge 1}$ 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.