Cohen-Macaulayness in graded rings associated to ideals.

*(English)*Zbl 0878.13003Let \((A,m)\) 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 \(\text{depth }A/I^n\geq d-s-2\).

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 \(\text{depth }A/I^n\geq d-s-2\).

Reviewer: D.D.Anderson (Iowa City)

##### 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.) |