Let \((R,{\mathfrak m})\) be a Cohen-Macaulay local ring, \(I\) an ideal of \(R\) such that \(\dim(R/I)=0\). The main background of this paper is coming from the following question: What information about \(I\) and its associated graded ring \(G(I)\) can be obtained from the Hilbert function \(H_I(n)=\lambda(R/I^n)\) and the Hilbert polynomial \(P_I(n)=\sum^d_{i=0}(-1)^ie_i(I){n+d-i-1\choose d-i}\) of \(I\)?
One of the main results is corollary 4.8 which shows that if \(J\) is any minimal reduction of \(I\) and \(r\geq 0\), then \(e_1(I)=\sum^r_{n=1}\lambda((I^n,J)/J)\), with equality if and only if \(G(I)\) is Cohen-Macaulay and \(JI^r=I^{r+1}\). – The case \(r=0\) extends previous results of D. G. Northcott [J. Lond. Math. Soc., II. Ser. 35, 209-214 (1960; Zbl 0118.04502)], C. Huneke [Mich. Math. J. 34, 293-318 (1987; Zbl 0628.13012)] and A. Ooishi [Hiroshima Math. J. 17, 361-372 (1987; Zbl 0639.13016)]. Then corollary 4.10 gives the following criterion:
The Rees ring \(R(I)\) is Cohen-Macaulay if and only if \(e_1(I)=\sum^{d-1}_{n=1}\lambda((I^n,J)/J)\).
The results are shown to hold even in a more general situation, namely for Hilbert functions of some general filtrations called Hilbert filtrations, including the filtrations given by \(I^n\) (the integral closure of \(I^n)\) and \(\widetilde{I^n}\) (the Ratliff-Rush closure of \(I^n)\).