Codimension of Jacobian ideals and \((R_ n)\) conditions for complete intersection (Q1080483)

Let \(R=k[[ X_ 1,X_ 2,...,X_ m]]\) where k is a field of characteristic zero, and let \(f_ 1,f_ 2,...,f_ r\) be nonunits of R. In this paper is is shown that the height of \(J_ s(f_ 1,f_ 2,...,f_ r)\) is greater than or equal to the height of \((J_ s(f_ 1,f_ 2,...,f_ r),f_ 1,f_ 2,...,f_ s))-s+1\), where \(1\leq s\leq r\) and \(J_ s(f_ 1,f_ 2,...,f_ r)\) denotes the ideal generated by the \(s\times s\) minors of the Jacobian matrix of \((f_ 1,f_ 2,...,f_ r)\). Several consequences of this are given, the main one being the following theorem: Let R be an excellent regular local ring of characteristic zero and let \(f_ 1,f_ 2,...,f_ r\) be a regular sequence of R. Assume that \(R/(f_ 1,f_ 2,...,f_ r)\) satisfies \((R_ n)\) for some \(n>0\). Then there exists integers \(n_ 2,n_ 3,...,n_ r\) such that \(R/(f_ 1+n_ 2f_ 2,f_ 2+n_ 3f_ 3,...,f_{r-1}+n_ rf_ r)\) satisfies (R\({}_{n+1}\)). This has the following corollary: If R is as above and I is an ideal of R generated by a R-sequence and R/I satisfies \((R_ n)\), \(n\geq 0\), then there exists a minimal generating set \(f_ 1,f_ 2,...,f_ r\) of I with the property that \(A_ i=R/(f_ 1,f_ 2,...,f_{r-i})\) satisfies \((R_{n+i})\) for \(0\leq i\leq r\). In particular if \(A_ 0\) is reduced then \(A_ 1\) is normal. If \(A_ 0\) is an isolated singularity, then so is any \(A_ i\).