The height of ideals and regular sequences
The following theorem is proved. Let \(R\) be a Noetherian ring that is a homomorphic image of a regular ring and let \(M\) be a finitely generated \(R\)-module. Then there exists a finite set \(\Lambda_M\) of prime ideals of \(R\) such that for every ideal \(I\) of \(R\), \[ \text{depth}_I(M)= \min_{p\in\Lambda_M} \left\{\text{ht}\left({I+ p\over p}\right)+\text{depth}(M_p)\right\}. \] As a corollary, the authors obtain the following result first proved by \textit{D. Eisenbud} [Nagoya Math. J. 54, 61-67 (1974; Zbl 0299.13010)]: Let \((R,m)\) be a Noetherian local ring and \(M\) a finitely generated \(R\)-module. Then the set \(S_r= \{(x_1,\dots, x_r)\in m^{\oplus r}\mid x_1,\dots, x_r\) is an \(M\)-sequence\} is an open subset of \(m^{\oplus r}\) with respect to the \(m\)-adic topology, i.e., given an \(M\)-sequence \(x_1,\dots, x_r\) there exists \(N>0\) such that \(x_1+ g_1,\dots, x_r+ g_r\) is an \(M\)-sequence for any \(g_i\in m^N\).
scientific article; zbMATH DE number 1038302