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Local reduction numbers and Cohen-Macaulayness of associated graded rings. (English) Zbl 0918.13001
From 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.

##### 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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