A Briançon-Skoda-type result for a non-reduced analytic space (Q1668306)

The Briançon-Skoda theorem [\textit{H. Skoda} and \textit{J. Briancon}, C. R. Acad. Sci., Paris, Sér. A 278, 949--951 (1974; Zbl 0307.32007)] states that for any ideal \(\mathfrak{a}\subset \mathcal{O} _{\mathbb{C}^{n},0}\) generated by \(m\) germs, we have the inclusion \( \overline{{\mathfrak{a}}^{\min (m,n)+r-1}}\subset {\mathfrak{a}}^{r}\) for \(r\geq 1,\) where \(\overline{I}\) denotes the integral closure of \(I\). This theorem was generalized by \textit{C. Huneke} [Invent. Math. 107, No. 1, 203--223 (1992; Zbl 0756.13001)] to Noetherian reduced local rings \(S\) in the form: there is an integer \(N\) such that \(\overline{{\mathfrak{a}} ^{N+r-1}}\subset {\mathfrak{a}}^{r}\) for all ideals \({\mathfrak{a}}\subset S\) and \(r\geq 1.\) The author generalizes this theorem to non-reduced rings in a very specific case -- \(S:=\mathcal{O}_{ \mathbb{C}^{n},0}/\mathcal{J}\) is the local ring of an arbitrary (not-necessarily reduced) pure dimensional analytic subset of \(\mathbb{C}^{n}\) at \(0\) defined by an ideal \(\mathcal{J}\subset \mathcal{O}_{\mathbb{C}^{n},0}\). Sufficient conditions for the elements \(\phi \in S\) to belong to \({\mathfrak{a}}^{r}\) are expressed in terms of a defining set of Noetherian operators for the ideal \( \mathcal{J}.\)