Bounds for the number of generators of generalized Cohen-Macaulay ideals
Let \((S,M)\) be a Cohen-Macaulay local ring and \(I\) an ideal. The author deals with the problem of finding the number of generators for such ideals. Numerically this number will depend on properties of \(R = S/I\). For example one may try to find bounds depending on \(e(R)\), the multiplicity of \(R\). Of course this is still hopeless. Earlier papers and books dealt with this problem with more restrictions on \(I\), like Cohen-Macaulay property for \(R\). Here the author deals with the problem when \(I\) is a generalized Cohen-Macaulay ideal. That is to say, the local cohomology modules \(H^i_M (R)\) are of finite length for all \(i < \dim R\). \textit{Ngô Viêt Trung} and others proved bounds for the number of generators for \(I\) as above which depend on the length of these local cohomology modules. Here the author proves estimates for these numbers which depend on a much cruder number \(l\), where one only needs that \(M^l\) is an \(R\)-standard ideal and such an \(l\) exists if \(R\) is a generalized Cohen-Macaulay ring.
scientific article; zbMATH DE number 826752