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Cohen-Macaulay Rees algebras of ideals having analytic deviation two. (English) Zbl 0812.13003
Let $$I$$ be an ideal in a Gorenstein local ring $$A$$ with infinite residue class field. We assume that (i) $$A/I$$ is a Cohen-Macaulay ring, (ii) $$s = \text{ht}_ AI > 0$$ and $$IA_ P$$ is generated by an $$A_ P$$-regular sequence of length $$s$$ for all $$P \in \text{Spec} A$$ such that $$P \supseteq I$$ and $$\dim A_ P \leq s + 1$$, and (iii) the analytic spread of $$I$$ is equal to $$s + 2$$.
We investigate the Cohen-Macaulay property of the Rees algebra $$R(I) = \oplus_{n \geq 0} I^ n$$ of $$I$$ under these assumptions. Let $$J$$ be a minimal reduction of $$I$$. In this paper, it is shown that $$R(I)$$ is Cohen-Macaulay if and only if the associated graded ring $$G(I) = \oplus_{n \geq 0} I^ n/I^{n + 1}$$ is Cohen-Macaulay and the equality $$I^{s+2} = JI^{s+1}$$ holds, which can be regarded as an analytic deviation 2 version of the result by the first author and S. Huckaba [Am. J. Math. 116, No. 4, 905-919 (1994; Zbl 0803.13002)]. And then, it is shown that $$R(I)$$ is Cohen-Macaulay if and only if depth $$A/I^ 2 \geq \dim A/I - 2$$ under the assumption that $$I^ 3 = JI^ 2$$.
Reviewer: S.Goto (Tokyo)

##### MSC:
 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13C14 Cohen-Macaulay modules
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