Superficial saturation (Q1082387)

For a Cohen-Macaulay semi-local ring A of dimension \(d=1\) a stable ideal I of definition has the equivalent properties \((i)\quad \lambda (A/I^ n)\) \((\lambda =length)\) is a polynomial for all \(n>0\), \((ii)\quad xI=I^ 2\) for some \(x\in I\) [see \textit{J. Lipman}, Am. J. Math. 93, 649-685 (1971; Zbl 0228.13008)]. The present note is largely a contribution to the study of stable ideals for general dimension \(d\) [see \textit{K. Kubota}, Tokyo J. Math. 8, 449-454 (1985; Zbl 0594.13022)]. The paper begins with connections between the integers when first \(I^ n:x=I^{n- 1}\) for a superficial element x (independent of x), when first \(\lambda (A/I^ n)\) can be evaluated from its Hilbert-Samuel polynomial and when first \((x_ 1,...,x_ d)I^ n=I^{n+1}\) for a superficial sequence \(x_ 1,...,x_ d\). A new proof of Kubota's result (loc. cit.) is given that \(\lambda (A/I^{n+1})\) is of the form \(e_ 0\left( \begin{matrix} n+d\\ d\end{matrix} \right)-e_ 1\left( \begin{matrix} n+d-1\\ d-1\end{matrix} \right)\) for all \(n>0\) if and only if, \((x_ 1,...,x_ d)I=I^ 2\). The proof uses the sequence \(\{I^{(n)}\}_{n>0}\) with \(I^{(n)}=U_ kI^{n+k}:x^ k\) for x superficial, called the superficial saturation of I, which is independent of x. The connection with stability is by the fact that \(I^{(n)}=I^ n\) for all \(n\geq 0\), if and only if, I has a stable superficial element x, i.e. \(I^ n:x=I^{n-1}\) for all \(n>0\). Further when I has a stable superficial sequence the normalized coefficients in the Hilbert-Samuel polynomial for \(\lambda (R/I^{n+1})\) are all non-negative. In the case \(d=2\), for each superficial sequence, \((x,y)I^{(n+2)}=I^{(n+3)}\) for \(n\geq 0\).