On the Hilbert-Samuel function
Let A be a Cohen-Macaulay semi-local ring in which each maximal ideal has height d. Let I be an open ideal with Hilbert-Samuel polynomial \(\sum (- 1)^ ie_ i\binom {n+d-i}{d-i}.\) Starting from results, for \(d=1\), of \textit{J. Lipman} [Am. J. Math. 93, 649-685 (1971; Zbl 0228.13008)] on stable ideals, the author proves the equivalence of: (1) the length \(\ell (A/I^{n+1})=e_ 0\binom{n+d}{d}-e_ 1\binom{n+d-1}{d-1}\) for all \(n\geq 0\); (2) \(\ell (A/I)=e_ 0-e_ 1\) and \(\ell (A/I^ 2)=e_ 0(d+1)-e_ 1d,\) (3) \(\ell (A/I^ 2)=e_ 0+d\ell (A/I),\) and also, under suitable conditions, (4) \(I^ 2=XI\) for some ideal X generated by d elements.
scientific article; zbMATH DE number 3955077