Integral closures of ideals generated by regular sequences
In a Noetherian ring A, let q be an ideal generated by an A-regular sequence \(x_ 1,...,x_ d\). Let R be the Rees ring \(\sum_{n\in Z}q^ nt^ n\quad and\) let \(R'=\sum (q^ n)^*t^ n\quad where\) \((\quad)^*\) denotes integral closure. Theorem: \(q^ n\cap (q^{n+1})^*=q^ nq^*\) for all \(n\geq 1\). This follows from the vanishing of the cohomology modules \(H^ 2_ J(R')_ n\) when \(n\leq 0\), \(J=(t^{-1},x_ 1t,...,x_ rt)R\) and \(2\leq r\leq d.\) Various applications are given. In Hiroshima Math. J. 17, 373-375 (1987; Zbl 0642.13006), the author deduced the theorem for q an ideal of the principal class and A locally quasi-unmixed.
scientific article; zbMATH DE number 4065130
scientific article