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When is the Rees algebra Cohen-Macaulay? (English) Zbl 0696.13015
The authors discuss the Cohen-Macaulay property for the Rees algebra \(R=R({\mathfrak a})=\oplus^{\infty}_{i=0}{\mathfrak a}^ i \) of an ideal \({\mathfrak a}\) in a noetherian local ring (A,\({\mathfrak m})\). The main result is a criterion based on the homogeneous components of the local cohomology of the associated graded ring \(G=G({\mathfrak a})=\oplus^{\infty}_{i=0}{\mathfrak a}^ i/{\mathfrak a}^{i+1} \) with respect to its maximal homogeneous ideal \({\mathfrak M}:\) Suppose that \({\mathfrak a}\) is not contained in the nilradical of A; then R(\({\mathfrak a})\) is Cohen-Macaulay if and only if \([H^ i_{{\mathfrak M}}(G)]_ n=0\) for \(n\neq -1, i=0,...,d-1\) and \(n\geq 0, i=d\) \((d=\dim (A))\). Many special results previously obtained by Goto, Ikeda, Schenzel, Shimoda and others can be derived from this result. As a technical tool the authors introduce the notion of a generalized Cohen-Macaulay ring with respect to an ideal. The proof of the main criterion is then based on a careful discussion of the relationship between the local cohomology modules of A, R, and G induced by standard exact sequences.
The criterion is applied to special cases: \({\mathfrak m}\)-primary ideals \({\mathfrak a}\), the case in which the analytic spread of \({\mathfrak a}\) equals the height of \({\mathfrak a}\), and ideals with small reduction number. We illustrate the results obtained by the following example: Let \(\dim (A)=2\), \({\mathfrak a}\) be \({\mathfrak m}\)-primary and \(x_ 1, x_ 2\) a system of parameters generating a minimal reduction of \({\mathfrak a}\); then R is Cohen-Macaulay if and only if the following conditions are satisfied: \(depth(A)>0\), the natural homomorphism from the Koszul cohomology of \((x_ 1,x_ 2)\) to the local cohomology of A is surjective, \({\mathfrak a}^ 2=(x_ 1,x_ 2){\mathfrak a}\), \((x^ 2_ 1,x^ 2_ 2)\cap {\mathfrak a}^ 3=(x^ 2_ 1,x^ 2_ 2){\mathfrak a}\), and \(((x_ 1):x_ 2)\cap {\mathfrak a}=(x_ 1).\)
Finally the (natural) case in which A is Cohen-Macaulay itself is treated. The condition of the criterion reduces to \([H^ d_{{\mathfrak M}}(G)]_ n=0\) for \(n\geq 0\), and similarly several other theorems have much simpler formulations than in general circumstances.
Reviewer: W.Bruns

MSC:
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13A15 Ideals and multiplicative ideal theory in commutative rings
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