On certain d-sequence on Rees algebra (Q1100516)

Let (A,m) be a local Noetherian ring, such that for all \(i<Dim(A)\), the i-th local cohomology module \(H^ i_ m(A)\) has finite length. (Call A a ``generalized Cohen-Macaulay ring''.) Let \(q=(a_ 1,...,a_ r)\) be an ideal of A, with \(a_ 1,...,a_ r\) a subsystem of parameters for m. The first theorem of this paper says \(a_ 1,a_ 2-a_ 1X,...,a_ r- a_{r-1}X,a_ rX\) form a d-sequence in the Rees ring \(R(q)=R[a_ 1X,...,a_ rX]\), and R(q) modulo the ideal generated by that sequence has dimension \(= Dim(A)-r\). The second theorem says that if \(0\leq n<r\), then \(a_ 1,a_ 2-a_ 1X,...,a_ n-a_{n-1}X,a_{n+1}X,...,a_ rX\) (with \(a_ 0=a_{-1}=0)\) forms a d-sequence in R(q), and moding out by the ideal generated by that sequence gives a ring of dimension \(= Dim(A)- n\). Recall that in a commutative ring, the sequence \(x_ 1,...,x_ r\) is a d-sequence if for all \(0\leq i<k\leq r\), \((x_ 1,...,x_ i):x_{i+1}x_ k=(x_ 1,...,x_ i):x_ k\).