Local reduction numbers and Cohen-Macaulayness of associated graded rings.

*(English)*Zbl 0918.13001From the introduction: Given an ideal \(I\) in a Noetherian ring \(R\), the associated graded ring \(G(I)= R/I\oplus I/I^2\oplus\cdots\) and the Rees (or blow-up) algebra \(R[It]= R\oplus It\oplus I^2t^2\oplus\cdots\) (\(t\) an indeterminate) contain information on the analytic properties of \(I\). In particular it is useful to know when one or both are Cohen-Macaulay (CM from now on). We will restrict our attention to the case that \(R\) is a CM local ring. In this case \(R[It]\) is CM if and only if \(G(I)\) is CM and certain local reduction number bounds hold for \(I\). Thus, interest has shifted slightly to determining when \(G(I)\) is CM, especially for ideals of positive analytic deviation.

Let \((R,{\mathfrak m})\) be local. Recall that a reduction of \(I\) is an ideal \(J\subseteq I\) such that \(I^{r+1}= JI^r\) for some \(r\). The smallest such \(r\) is the reduction number of \(I\) with respect to \(J\), denoted \(r_J(I)\). The reduction \(J\) is minimal if it properly contains no other reduction of \(I\). The reduction number of \(I\) is \(r(I)= \min\{r_J(I)\mid J\) is a minimal reduction of \(I\}\). The analytic spread of \(I\) is \(l(I)= \dim G(I)/ {\mathfrak m}G(I)\). If \(R/{\mathfrak m}\) is finite then every minimal reduction of \(I\) is minimally generated by \(l(I)\) elements. The analytic deviation of \(I\) is \(\text{ad}(I)= l(I)- \text{ht}(I)\). – The main result of this paper is:

Theorem 3.1. Let \((R,{\mathfrak m})\) be a Cohen-Macaulay ring having an infinite residue field. Let \(I\) be an ideal of height \(s\) such that

(1) \(\text{depth } R/I^k\geq \dim R/I-k+1\) for \(1\leq k\leq \text{ad}(I)+1\),

(2) \(I\) has locally bounded reduction numbers, and

(3) \(r(I)\leq \text{ad}(I)+1\).

Then \(G(I)\) is Cohen-Macaulay.

Let \((R,{\mathfrak m})\) be local. Recall that a reduction of \(I\) is an ideal \(J\subseteq I\) such that \(I^{r+1}= JI^r\) for some \(r\). The smallest such \(r\) is the reduction number of \(I\) with respect to \(J\), denoted \(r_J(I)\). The reduction \(J\) is minimal if it properly contains no other reduction of \(I\). The reduction number of \(I\) is \(r(I)= \min\{r_J(I)\mid J\) is a minimal reduction of \(I\}\). The analytic spread of \(I\) is \(l(I)= \dim G(I)/ {\mathfrak m}G(I)\). If \(R/{\mathfrak m}\) is finite then every minimal reduction of \(I\) is minimally generated by \(l(I)\) elements. The analytic deviation of \(I\) is \(\text{ad}(I)= l(I)- \text{ht}(I)\). – The main result of this paper is:

Theorem 3.1. Let \((R,{\mathfrak m})\) be a Cohen-Macaulay ring having an infinite residue field. Let \(I\) be an ideal of height \(s\) such that

(1) \(\text{depth } R/I^k\geq \dim R/I-k+1\) for \(1\leq k\leq \text{ad}(I)+1\),

(2) \(I\) has locally bounded reduction numbers, and

(3) \(r(I)\leq \text{ad}(I)+1\).

Then \(G(I)\) is Cohen-Macaulay.

##### 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.) |