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Reduction numbers, Briançon-Skoda theorems and the depth of Rees rings. (English) Zbl 0854.13003
During the last two decades there was – for several reasons – a strong interest in the blow-up rings, i.e. the Rees algebra $$R[ It]=\bigoplus_{n\geq 0} I^n t^n$$ and the form ring $$G(I)=\bigoplus_{n\geq 0} I^n/I^{n+1}$$. Here $$I$$ denotes an ideal of a local ring $$(R, {\mathfrak m})$$. One of the main questions has been a characterization of their Cohen-Macaulay property. In a certain sense the authors present a complete solution for $$R$$ a Cohen-Macaulay ring.
In order to describe their results let $$J$$ denote a minimal reduction of $$I$$, i.e. an ideal minimal among those satisfying $$I^{n+1}=JI^n$$ for a certain integer $$n\in \mathbb{N}$$. The smallest possible integer $$n\in \mathbb{N}$$ with this property is called the reduction number $$r(I)$$ of $$I$$. The analytic spread $$l(I)$$ of $$I$$ is defined as the dimension of the fibre ring $$G(I)/{\mathfrak m} G(I)$$.
Suppose that $$R$$ is a Cohen-Macaulay ring. Then the following two results are proved:
(1) For $$\text{height }I \geq 1$$ the Rees algebra $$R[ It]$$ is a Cohen-Macaulay ring if and only if $$G(I)$$ is a Cohen-Macaulay ring and $$r(I_{\mathfrak p})\leq \text{height } {\mathfrak p}-1$$ for every prime ideal $${\mathfrak p} \supseteq I$$ with $$l (I_{\mathfrak p})=\text{height } {\mathfrak p}$$.
(2) For $$\text{height }I\geq 2$$ the Rees algebra $$R[ It]$$ is a Gorenstein ring if and only if $$G(I)$$ is a Gorenstein ring, $$r(I_{\mathfrak p})=\text{height } {\mathfrak p}-2$$ for every prime ideal $${germ p}\supseteq I$$ with height $${\mathfrak p}/I=0$$, and $$r(I_{\mathfrak p})\leq \text{height } {\mathfrak p}-2$$ for every prime ideal $${\mathfrak p} \supseteq I$$ with $$l(I_{\mathfrak p})=\text{height } {\mathfrak p}\leq l(I)$$.
This generalizes several results previously known in the literature. In particular for an $${\mathfrak m}$$-primary ideal $$I$$ it turns out that $$R[ It]$$ is a Cohen-Macaulay ring if and only if $$G(I)$$ is a Cohen-Macaulay ring and $$r(I)< \dim R$$, see S. Goto and Y. Shimoda [in Commutative algebra: analytical methods, Conf. Fairfax 1979, Lect. Notes Pure Appl. Math. 68, 213-231 (1982; Zbl 0482.13011)], which was one of the first results in this direction and a motivation for the characterization of the Cohen-Macaulayness of the blow-up rings.
The method of the authors’ proofs are based on filter-regular sequences, minimal reduction, and Castelnuovo-Mumford regularity. There are several applications of the authors’ fundamental characterization concerning Serre’s condition $$(S_k)$$ for $$R[ It]$$ and $$G(I)$$, reduction numbers of ideals in regular local rings, and the Briançon-Skoda theorem for equicharacteristic local rings [see also the first two authors in Proc. Am. Math. Soc. 124, No. 3, 707-713 (1996; see the preceding review)]. – For related results about the Cohen-Macaulayness of Rees and form rings on pseudo-rational local rings [see also J. Lipman in Math. Res. Lett. 1, No. 2, 149-157 (1994; Zbl 0844.13006)].
Reviewer: P.Schenzel (Halle)

##### MSC:
 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13C40 Linkage, complete intersections and determinantal ideals
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