Rees rings and associated graded rings of ideals having higher analytic deviation.

*(English)*Zbl 0803.13003Given 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 UniversitĂ¤tsschriften 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.

(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 UniversitĂ¤tsschriften 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.

Reviewer: M.Kreuzer (Regensburg)