On Hilbert functions and cohomology
The paper starts with the definition and some basic properties of the Ratliff-Rush closure of a good filtration \({\mathcal F}= \{I_n\}_{n\geq 0}\) in a Noetherian ring \(A\). Suppose that \((A,m)\) is \(d\)-dimensional local, with \({\mathcal F}\) a Hilbert filtration (so the ideal \(I_1\) is \(m\)-primary). Complexes closely related to those of \textit{D. Kirby} and \textit{H. A. Mehran} [J. Lond Math. Soc., II. Ser. 24, 459-466 (1981; Zbl 0442.13010)] and \textit{S. Huckaba} and \textit{T. Marley} [``Hilbert coefficients and the depths of associated graded rings'', J. Lond. Math. Soc., II. Ser. (to appear)] are used to compute the local cohomology of the Rees algebra \({\mathcal R}\) of \({\mathcal F}\). In particular, for \(n\geq 0\), \([H_{{\mathcal R}_+}^1 ({\mathcal R}]_n\cong \bigcup_{k\geq 1} (I_{n+k}: I_1^k)/I_n\). \textit{B. Johnston} and \textit{J. Verma} [Proc. Am. Math. Soc. 123, No. 1, 1-10 (1995; Zbl 0830.13013)] gave a formula, in terms of local cohomology of \({\mathcal R}\), for the difference between the Hilbert-Samuel function and polynomial for an \(I\)-adic \({\mathcal F}\). Their formula is extended here to the extended Rees algebra for a general Hilbert filtration, and (amongst other results) a question of theirs about the \(\text{depth} (\text{gr}_A{\mathcal F})\geq d-1\) case is answered.
scientific article; zbMATH DE number 1026099
scientific article