Estimates on the depth of the associated graded ring
Let \((R,m)\) be a Cohen-Macaulay local ring of dimension \(d\), \(I\) an \(m\)-primary ideal of \(R\), and \(\text{gr}(R)=\bigoplus_{n\geq 0}I^n/I^{n+1}\), the associated graded ring of \(I\). Let \(J\) be a minimal reduction of \(I\). A natural question formulated by \textit{A. Guerrieri} [Proc. Am. Math. Soc. 123, No. 1, 11-20 (1995; Zbl 0830.13001)] is: ``Is it true if \(\sum_{k\geq 1}\lambda(J\cap I^{k+1}/JI^k)=t\) (where \(\lambda\) means length), then \(\text{depth gr}_I(R)\geq d-t\)?''. The question is known to be true if \(t=0\) [\textit{P. Valabrega} and \textit{G. Valla}, Nagoya Math. J. 72, 93-101 (1978; Zbl 0362.13007)], \(t=1\) [\textit{A. Guerrieri}, J. Algebra 167, No. 3, 745-757 (1994; Zbl 0810.13009)], and \(t=2\) [\textit{H. Wang}, ``Hilbert coefficients and the associated graded ring'', Proc. Am. Math. Soc. 128, No. 4, 963-973 (2000)]. The main result of the paper under review considers the case \(t=3\): Suppose \(d\geq 4\) and \(J\subseteq I\) is generated by a maximal superficial sequence such that \(\lambda(J\cap I^2/JI)=3\) and \(J\cap I^n=JI^{n-3}\) for all \(n\geq 3\). If \(R/I\) is Gorenstein, then \(\text{depth gr}_I(R)\geq d-3\). The proof uses a formula given by \textit{S. Huckaba} and \textit{T. Marley} [J. Lond. Math. Soc., II. Ser. 56, No. 1, 64-76 (1997; Zbl 0910.13008)] to compare the Hilbert coefficients that arise from three different filtrations.
scientific article; zbMATH DE number 1271333
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