The plus closure in mixed characteristic
Let \(I\) be an arbitrary ideal in an integrally closed domain \(R\). The plus closure of \(I\) is defined to be the ideal \(I^+ = IR^+\cap R\), where \(R^+\) denotes the integral closure of \(R\) in an algebraic closure of its quotient field. \textit{M. Hochster} and \textit{C. Huneke} [Adv. Math. 113, No. 1, 45-117 (1995; Zbl 0834.13013)] have used this notion to prove a generalized Briançon-Skoda theorem which says that if \(R\) is a noetherian ring of characteristic \(p\) and if \(I\) is generated by \(n\) elements, then \(\overline{I^{m+n}}\subseteq (I^{m+1})^+\) for any integer \(m\geq 0\), where \(\overline{J}\) denotes the integral closure of the ideal \(J\). The aim of this paper is to develop a mixed characteristic analogue of this theorem. If \(p \in \sqrt{I}\), the author obtains the desired result in a number of cases which contains the equicharacteristic case as a corollary.
scientific article; zbMATH DE number 1051583