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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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