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Rees rings and associated graded rings of ideals having higher analytic deviation. (English) Zbl 0803.13003
Given an ideal $$I$$ of positive height in a local Cohen-Macaulay ring $$R$$, the following questions have recently been the focus of intensive research:
(1) Which conditions on $$I$$ and the graded ring $$G(I): = \oplus_{n \geq 0} I^ n/I^{n + 1}$$ guarantee that the Rees ring $$R(I) = R[It]$$ is a Cohen-Macaulay ring?
(2) Under which assumptions is $$G(I)$$ a Cohen-Macaulay ring?
(3) When are $$G(I)$$ and $$R(I)$$ Gorenstein rings?
For all three questions, the paper under review offers new and rather general answers. With respect to the first question, the author proves that for ideals $$I$$ of positive analytic deviation $$\text{ad} (I)$$ with the property that $$I_ P$$ is a complete intersection of height $$\text{ht}(I)$$ for all primes $$P \supseteq I$$ such that $$\text{ht} (P/I) \leq \text{ad} (I) - 1$$, the Rees ring $$R(I)$$ is Cohen-Macaulay if and only if $$G(I)$$ is Cohen-Macaulay and $$I^{\ell (I)} = JI^{\ell (I) - 1}$$. Here $$\ell (I)$$ denotes the analytic spread of $$I$$, and $$J$$ is a minimal reduction of $$I$$. – This result generalizes earlier work of S. Goto and S. Huckaba [Am. J. Math. 116, No. 4, 905-919 (1994; see the preceding review)] in case $$\text{ad} (I) = 1$$, and by S. Goto and Y. Nakamura [“Cohen-Macaulay Rees algebras of ideals having analytic deviation two” (preprint)] in case $$\text{ad}(I) =2$$. The crucial point of the argument is a system of parameters for $$G(I)$$ which is constructed in theorem 2.4 and which has a particular simple form.
Also with respect to question 2 the author is able to generalize earlier results byS. Huckaba and C. Huneke [J. Math. 114, 367-403 (1992; Zbl 0758.13001)] in case $$\text{ad} (I) = 1$$, and by S. Goto and Y. Nakamura [loc. cit.] in case $$\text{ad} (I) = 2$$. For instance, in case $$\text{ad} (I) \geq 3$$, he shows that under suitable assumptions on $$I$$, the graded ring $$G(I)$$ is Cohen-Macaulay if and only if depth$$(R/I^{\text{ad} (I)}) \geq \dim (R) - \ell (I)$$. Further results in this direction in the case $$\text{ad} (I) = 3$$ are contained in a recent paper by S. Goto [“Cohen-Macaulayness in graded rings associated to ideals”, in: Bruns, Herzog, Hochster, Vetter (eds.), Proc. Conf. Commutative Algebra, Vechta 1994, Vechtaer Universitätsschriften 13)].
As for question 3, the author’s main result says that, again under suitable assumptions on $$R$$ and $$I$$, for ideals $$I$$ of analytic deviation $$\text{ad} (I) \geq 3$$, the ring $$G(I)$$ is Gorenstein if and only if $$r_ J(I) \leq 1$$ with a minimal reduction $$J$$ of $$I$$. This is similar to the results of S. Goto and Y. Nakamura [loc. cit.] in case $$\text{ad} (I) = 2$$.
The complexity of the involved hypotheses and necessary lengthy calculations make the paper appear very technical, although the presentation is clear and concise.

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