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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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