Bounds for the minimum numbers of generators of generalized Cohen- Macaulay ideals
An ideal I of a local ring (R,\({\mathfrak m})\) is called a generalized Cohen- Macaulay ideal if \(S=R/I\) is a generalized Cohen-Macaulay ring, i.e. \(\ell (H^ i_{{\mathfrak m}}(S))<\infty\) for \(i=0,...,\dim S-1.\) The class of such ideals is rather large, e.g. it includes the defining prime ideals of isolated singularities. Inspired of an article of \textit{J. Sally} on Cohen-Macaulay ideals [Pac. J. Math. 63, 517-520 (1976; Zbl 0314.13009)], this paper will give bounds for the minimum number of generators v(I) of a generalized Cohen-Macaulay ideal I in terms of the embedding dimensions, the dimensions, the multiplicities of R, S, and \({\mathcal L}(H^ i_{{\mathfrak m}}(S))\). There are some interesting applications. For example, if I is the defining prime ideal of an arithmetically Buchsbaum curve C in \(P^ 3_ k\), i.e. \(H^ 1_ m(S)\) is a vector space over the field k, where S is the local ring of the vertex of the affine cone over C, then \(v(I)\leq 3e/2+1,\) where e denotes the degree of C.
scientific article; zbMATH DE number 3871524