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Ratliff-Rush ideal and reduction numbers. (English) Zbl 1391.13019
Let \((R,\mathfrak{m})\) be a Cohen-Macaulay local ring of positive dimension \(d\) and infinite residue field. Let \(I\) be an \(\mathfrak{m}\)-primary ideal and \(J\) a minimal reduction of \(I\). The author shows that \(\widetilde{r_{J}(I)} \leq r_{J}(I)\). This answer to a question that made by M. E. Rossi and I. Swanson [Contemp. Math. 331, 313–328 (2003; Zbl 1089.13501), Question 4.6]. The author shows that every minimal reduction of \(I\) can be generated by a tame supercial sequence of \(I\) and uses inductive arguments for to show the most result.

13C14 Cohen-Macaulay modules
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Full Text: DOI arXiv
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