Prime divisors, asymptotic R-sequences and unmixed local rings
Let (R,M) be a local ring and \(R^*\) be its completion. The author introduces the notion of asymptotic R-sequence as follows: Elements \(x_ 1,...x_ d\) form an asymptotic R-sequence iff \(x_{i+1}\not\in \cup \{P| P\in \tilde A^*(x_ 1,...,x_ i)\},\quad 1\leq i\leq d-1,\) where: \(\tilde A^*(I)=\cap_{j}\{A^*(J)| \quad (\exists n) I^ nR\leq JR\leq \overline{I^ nR}\}\) with: \(A^*(J)=Ass R/J^ n\), \(n\gg 0\), \(\overline{I^ nR} =\) integral closure of \(I^ n\). The main results of this paper are the following: (i) The length of any maximal asymptotic R-sequence characterizes the minimal depth among prime divisors of zero in R. - (ii) R is unmixed iff each system of parameters forms an asymptotic R-sequence. - (iii) R is unmixed iff \(S=\sum_{n}| \cup_{k}(I^ n:M^ k))t^ n\) is a finite module over the Rees ring of I, for all ideals I with analytic spread less than dim R.
scientific article; zbMATH DE number 3968735