Filter regular sequence under small perturbations (Q2200765)

Let \((R,\mathfrak{m})\) denote a Noetherian local ring. A sequence of elements \(f_1,\ldots,f_r\) in \(R\) is called filter-regular if \((f_1,\ldots,f_{i-1} )R :_R f_i/(f_1,\ldots,f_{i-1} )R\) is of finite length for \(i = 1,\ldots,r\). The authors study ``small pertubations'' of filter-regular sequences and answer a question by \textit{V. Srinivas} and \textit{V. Trivedi} [J. Algebra 186, No. 1, 1--19 (1996; Zbl 0870.13008)]. To be more precise: Suppose that \(f_1,\ldots,f_r\) is a filter-regular sequence in \(R\) and \(J\) is an ideal such that \((f_1,\ldots,f_r) + J\) is \(\mathfrak{m}\)-primary. Then there is a positive integer \(N\) such that for every \(\epsilon_1,\ldots,\epsilon_r \in \mathfrak{m}^N\) the form rings \(\operatorname{gr}_J(R/(f_1,\ldots,f_r)) \) and \(\operatorname{gr}_J(R/(f_1+\epsilon_1,\ldots,f_r+\epsilon_r))\) are naturally isomorphic. This implies the equality of the Hilbert functions \(H(J,R/(f_1,\ldots,f_r))(n)\) and \(H(J,R/(f_1+\epsilon_1,\ldots,f_r+\epsilon_r))(n)\). In fact, there is an explicit bound for such an \(N\). Moreover, it is shown that the dimension of the non-Cohen-Macaulay locus does not increase under small pertubations.