A less restrictive Briançon-Skoda theorem with coefficients
Let \((R, \mathfrak{m})\) be a \(d\)-dimensional commutative Noetherian local ring of prime characteristic \(p\). Assume that \(R\) is F-rational and let \(I\) be an ideal of \(R\) of height \(g\) and analytic spread \(l\) with \(g < l\). Also, let \(I_{[l-1]}\) be the intersection of all the primary components of \(I\) of height at most \(l-1\) that appear in a primary decomposition of \(I\). For an ideal \(K\), the integral closure of \(K\) is denoted by \(\overline{K}\).NEWLINENEWLINEThe paper contains extensions of two Briançon-Skoda type theorems by \textit{I. M. Aberbach} and \textit{C. Huneke} from [Mich. Math. J. 49, No. 1, 3--11 (2001; Zbl 1016.13003)] and [Lect. Notes Pure Appl. Math. 220, 1--12 (2001; Zbl 0991.13003)]. The main contributions of the paper are as follows. Let \(J\) be a reduction of \(I\). The first main result states that if \(R\) is Cohen-Macaulay then \(\overline{I^{l+w}} \subseteq J^{w+1}I_{[l-1]}\) for all \(w \geq 0\). For the second main result assume that \(R\) is Gorenstein, \(R/I\) has depth at least \(d-l+1\) and \(I = I_{[l-1]}\). Then, for any reduction of \(J\) of \(I\), we have \(\overline{I^{l-1}} \subseteq J\).
scientific article; zbMATH DE number 6022748