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Cohen-Macaulayness in graded rings associated to ideals. (English) Zbl 0878.13003
Let $$(A,m)$$ be a Noetherian local ring with Krull dimension $$d$$ and let $$I$$ be an ideal of $$A$$ with $$\text{ht} (I)=s$$. Let $$R(I)= \sum_{n\geq 0}I^nt^n$$ be the Rees algebra and $$G(I)= R(I)/IR(I)$$ be the associated graded ring on $$I$$. The purpose of this paper is to find practical conditions under which $$R(I)$$ or $$G(I)$$ are Cohen-Macaulay or Gorenstein. Let $$\lambda(I)=l$$ be the analytic spread of $$I$$ and $$\text{ad} (I)=l-s$$ be the analytic deviation. For a reduction $$J$$ of $$I$$ let $$r_J(I)= \min\{n\geq 0\mid I^{n+1}= JI^n\}$$. Recall that a minimal reduction $$J$$ of $$I$$ is said to be special if $$I$$ contains a system of generators $$a_1,\dots, a_l$$ for $$J$$ which satisfies the equality $$I_Q= (a_1,\dots,a_h)A_Q$$ for all primes $$Q\supseteq I$$ with $$\text{ht }Q= h\leq l-1$$. The main result of the paper (with notation as above) are as follows:
Theorem (1.1). Let $$A$$ be Cohen-Macaulay. Assume that $$I$$ contains a special reduction $$J$$ with $$r_J(I)\leq \text{ad}(I)$$ and that $$I$$ satisfies the following inequalities: $\text{depth} (A/I^n)_Q\geq \min\{\text{ad} (I)-n,\text{ht } Q-s-n\}$ and $$\text{depth } A/I^n\geq d-s-n+1$$ for all prime ideals $$Q\supseteq I$$ and for all integers $$n$$ with $$1\leq n\leq\text{ad}(I)$$. Then $$G(I)$$ is Cohen-Macaulay of $$a(G(I))=-s$$ (here $$a(G(I))$$ denotes the $$a$$-invariant of $$G(I)$$) and $$G(I)$$ is Gorenstein if $$A$$ is Gorenstein. Hence $$R(I)$$ is Cohen-Macaulay if $$s\geq 1$$. Suppose $$s\geq 2$$, then $$R(I)$$ is Gorenstein if and only if $$A$$ is Gorenstein and $$s=2$$.
Theorem (1.3). Let $$A$$ be Gorenstein and $$\text{ad}(I)\geq1$$. Suppose that $$A/I$$ is Cohen-Macaulay and $$I$$ contains a special reduction $$J$$. Then:
(1) $$r_J(I)\leq \text{ad}(I)-1$$ if $$G(I)$$ is Gorenstein.
(2) Assume the following inequalities: $$\text{depth} (A/I^n)_Q\geq \min\{\text{ad}(I)-1-n$$, $$\text{ht } Q-s-n\}$$ and $$\text{depth } A/I^n\geq d-s-n$$ for all prime ideals $$Q\supseteq I$$ and for all integers $$n$$ with $$1\leq n\leq \text{ad} (I)-1$$. Then $$G(I)$$ is Gorenstein if and only if $$r_J(I)\leq \text{ad} (I)-1$$. Thus if $$I$$ satisfies the inequalities in (2), $$R(I)$$ is Gorenstein if $$s=2$$ and $$r_J(I)\leq \text{ad} (I)-1$$.
Theorem (1.5). Let $$A$$ be Cohen-Macaulay, $$\text{ad} (I)=2$$, and assume that $$A/I$$ is Cohen-Macaulay and that $$I$$ contains a special reduction $$J$$ with $$r_J(I)\leq 2$$. Then $$G(I)$$ is Cohen-Macaulay if and only if $$\text{depth } A/I^n\geq d-s-2$$. Hence $$R(I)$$ is Cohen-Macaulay if $$s\geq 1$$ and $$\text{depth } A/I^2\geq d-s-2$$.
Theorem (1.6). Let $$A$$ be Gorenstein, $$\text{ad}(I)=3$$, and assume that $$A/I$$ is Cohen-Macaulay and that $$I$$ contains a special reduction $$J$$ with $$r_J(I)\leq 2$$. Then $$G(I)$$ is Gorenstein if and only if $$\text{depth } A/I^n\geq d-s-3$$.
Theorem (1.7). Let $$A$$ be Gorenstein, $$\text{ad} (I)=3$$, and assume that $$A/I$$ is Cohen-Macaulay and that $$I$$ contains a special reduction $$J$$ with $$r_J(I)\leq 1$$. Then $$G(I)$$ is Gorenstein if and only if $$\text{depth }A/I^n\geq d-s-2$$.

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