Symbolic powers of ideals and their topology over a module

Abstract

Let $I$ denote an ideal of a Noetherian ring $R$ and $N$ a non-zero finitely generated $R$-module. In the present paper, some necessary and sufficient conditions are given to determine when the $I$-adic topology on $N$ is equivalent to the $I$-symbolic topology on $N$. Among other things, we shall give a complete solution to the question raised by R. Hartshorne in [{\it Affine duality and cofiniteness}, Invent. Math. {\bf9}(1970), 145-164], for a prime ideal $\frak p$ of dimension one in a local Noetherian ring $R$, by showing that the $\frak{p}$-adic topology on $N$ is equivalent to the $\frak{p}$-symbolic topology on $N$ if and only if for all $z\in \Ass_{R^*}N^*$ there exists $\frak{q}\in \Supp(N^*)$ such that $z\subseteq \frak{q}$ and $\frak{q}\cap R=\frak{p}.$ Also, it is shown that if for every ${\mathfrak{p}}\in \Supp(N)$ with $\dim R/\mathfrak{p}=1$, the $\mathfrak{p}$-adic and the $\mathfrak{p}$-symbolic topologies are equivalent on $N$, then $N$ is unmixed and $\Ass_{R} N$ has only one element. Finally, we show that if $\Ass_{R_{\mathfrak{p}}^*}{N^*_{\mathfrak{p}}}$ consists of a single prime ideal, for all ${\mathfrak{p}}\in {A^*}(I,N)$, then the $I$-adic and the $I$-symbolic topologies on $N$ are equivalent. \end{abstract}