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Reduction number bounds on analytic deviation 2 ideals and Cohen- Macaulayness of associated graded rings. (English) Zbl 0831.13002
Let \(I\) be an ideal of analytic deviation 2 in a Cohen-Macaulay local ring \(R\). The authors are interested in situations when the associated graded ring \({\mathcal G} (I) = \bigoplus_{n \geq 0} I^n/I^{n + 1}\) of \(I\) is Cohen-Macaulay. In theorem 2.10, they prove that \({\mathcal G} (I)\) is Cohen-Macaulay if the reduction exponent \(r(I) \leq 3\) and \(\text{depth} (R/I^j) \geq \dim (R/I) - j + 1\) for \(j = 1,2,3\) (provided that the reduction number of certain localizations of \(I\) are 0 or 1); these conditions were originally stated by Z. Tang [Commun. Algebra 22, No. 12, 4855-4898 (1994; Zbl 0803.13003)] in the case \(R\) is a Gorenstein ring. Next, following ideas of a joint paper by the authors and C. Huneke [J. Pure Appl. Algebra 102, No. 1, 1-15 (1995)] who considered ideals of analytic deviation 1, the authors give bounds for the reduction number of \(I\), provided \({\mathcal G} (I)\) and \(R/I\) are Cohen-Macaulay rings and \(I\) is minimially generated by height\((I)\)+3 elements (theorem 3.4). Finally, theorem 4.3 provides an analytic deviation 2 version of a result of I. M. Aberbach and C. Huneke [Math. Ann. 297, No. 2, 343- 369 (1993; Zbl 0788.13001)], characterizing the Cohen-Macaulay property of \({\mathcal G} (I)\) and the Rees ring \(\bigoplus_{n \geq 0}I^n\) of a 5- generated perfect height 2 ideal in a four-dimensional Cohen-Macaulay ring by properties of a presentation matrix of \(I\), by a small reduction number \((r(I) \leq 3)\), and by a small relation type number \((N(I) \leq 4)\).
Reviewer: J.Ribbe (Köln)

MSC:
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13A15 Ideals and multiplicative ideal theory in commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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