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Artin-Nagata properties and Cohen-Macaulay associated graded rings. (English) Zbl 0866.13001
This paper is a welcome addition to the existing literature on blow-up algebras. For sometime to come it will contain among the most general results on the subject of the Cohen-Macaulayness of the Rees algebra $${\mathcal R}={\mathcal R}(I)$$ and the associated graded ring $$G=\text{gr}_I(R)$$, where $$I\subset R$$ is an ideal of the local ring $$R$$. There has been quite some intense activity on this subject in recent years – one can have an idea of its size by looking at the very references of the present paper. In the case in which the ring $$R$$ is regular (pseudo-rational singularity would be enough), J. Lipman [Math. Res. Lett. 1, No. 2, 149-157 (1994; Zbl 0844.13006)] proved that if $$G$$ is Cohen-Macaulay then so is $${\mathcal R}$$. This settled completely, in the regular case, one of the main problems in the subject. However, the more general situation where $$R$$ is Cohen-Macaulay – or even Gorenstein for that matter – has as yet a lot to go.
The authors base their main results on assumptions regarding the Cohen-Macaulayness of certain residual intersection loci of the ideal $$I$$. The precise notion entering these assumptions – dubbed by the authors the AN (for Artin-Nagata) property – is perhaps a little too technical to be reproduced here. What they show, in a nonprecise wording, is that the conjunction of the following three properties imply the Cohen-Macaulayness of $$G$$ (and of $${\mathcal R}$$ in codimension at least 2):
(1) the AN property (sufficiently qualified),
(2) a reasonable growth estimate for the local number of generators of $$I$$, and
(3) sufficient “sliding-depth” for the residue rings $$R/I^j$$.
Alas! It is not so easy to recognize when one is in presence of all such properties. This is perhaps felt by the very authors who, therefore, took pains in showing that most of the technical features of the hypotheses fall off under some further natural restrictions. They do this in the form of several interesting corollaries.
The proof of the main theorem (theorem 3.1) is quite involved as expected from a result that contains significant improvement on all the previous ones by many authors, including the present second author himself. The proof itself takes most of section 3. The subsequent sections (4 and 5) are respectively concerned with the Castelnuovo regularity and bounds for the reduction number, and the Gorensteinness of $$G$$ and the form of the canonical module. The results obtained in these sections also generalize to quite some extent the previously known theorems.