Uniform equivalence of symbolic and adic topologies
Let \((R,\mathfrak{m})\) denote a local ring. For two ideals \(I, J \subset R\) denote by \(I : \langle J\rangle = \bigcup_{m \geq 1} I : J^m.\) Very often it is of some interest whether the family of ideals \(\{I^n :\langle J\rangle\}_{n \geq 1}\) is equivalent to the \(I\)-adic topology, i.e., for each \(n \geq 1\) there is an \(m(n) \geq 1\) such that \(I^{m(n)}:\langle J\rangle \subseteq I^n.\) The situation when \(m(n) = n +k\) for a certain integer \(k \geq 1\) was characterized by the reviewer [Math. Nachr. 129, 123--148 (1986; Zbl 0606.13001)]. He posed the problem whether the equivalence of \(\{I^n :\langle J\rangle\}_{n \geq 1}\) to the \(I\)-adic topology implies that one might choose \(m(n) = nk\) for a certain integer \(k \geq 1,\) now called linear equivalence. This problem was solved by \textit{I. Swanson} [Math. Z. 234, No. 4, 755--775 (2000; Zbl 1010.13015)]. For symbolic powers of a prime ideal \(\mathfrak{p} \subset R\) this implies that \(\mathfrak{p}^{(nk)} \subseteq \mathfrak{p}^n\) for all \(n \geq 1.\) While a priori \(k\) depends on the ideal \(\mathfrak{p},\) in their paper \textit{M. Hochster} and \textit{C. Huneke} [Invent. Math. 147, No. 2, 349--369 (2002; Zbl 1061.13005)] have shown that \(\mathfrak{p}^{(nd)} \subseteq \mathfrak{p}^n\) for all \(n \geq 1\) and \(\dim R = d\) for a prime ideal \(\mathfrak{p}\) of a regular local ring \(R.\) In the paper under review the authors study when there is a uniform \(k \geq 1\) such that \(I^{nk}:\langle J\rangle \subseteq I^n\) for all \(n \geq 1\) and all ideals \(I,\) provided \(\{I^n :\langle J\rangle\}_{n \geq1}\) is equivalent to the \(I\)-adic topology. As the main result of the paper the authors prove the following Theorem. Let \(R\) be an equicharacteristic, local domain such that \(R\) is an isolated singularity. Assume that \(R\) is either essentially of finite type over a field of characteristic zero or \(R\) has positive characteristic and is \(F\)-finite. Then there exists an integer \(k \geq 1\) with the property that for all ideal \(I \subset R\) such that the symbolic topology of \(I\) is equivalent to the \(I\)-adic topology, \(I^{(nk)} \subseteq I^n\) for all \(n \geq 1.\) The main ingredients of the proof of their result is threefold: (1) The relation between the jacobian ideal and symbolic powers as established by Hochster and Huneke [loc. cit.]. (2) The uniform Artin-Rees Theorem as it was shown by the first author [Invent. Math. 107, No. 1, 203--223 (1992; Zbl 0756.13001)]. (3) A uniform Chevalley Theorem proved in the paper: Let \(R\) be an analytically unramified local ring. Then there is an integer \(k \geq 1\) with the property: For all ideals \(I \subset R\) and all multiplicatively closed subsets \(S\) in \(R\) such that the topology \(\{ I^n R_S \cap R\}_{n \geq 1}\) is finer then the \(\mathfrak{m}\)-adic topology it follows that \(I^{nk}R_S \cap R \subset \mathfrak{m}^n\) for all \(n \geq 1.\)
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