The Gorenstein property of the associated graded rings of powers of an ideal
This paper gives a second proof of the following theorem: Let \((R,m)\) be a local Noetherian ring with \(\dim(R)=d\geq 1\) and with infinite residue field. Let \(I\) be an \(m\)-primary ideal of \(R\) such that the associated graded ring \(G(I)\) is Gorenstein. Then for any integer \(r\geq 1\), \(G(I^ r)\) is Gorenstein if and only if \(\delta(I)\equiv d-1\pmod r\), where \(\delta(I)\) is the smallest integer \(i\) such that \(I^{i+1}=JI^ i\) for some parameter ideal \(J\) contained in \(I\). The author had given an earlier proof using canonical modules in J. Algebra 155, No. 2, 397-414 (1993). The proof given in this paper uses Hilbert functions and the fact that under certain conditions \(G(I)\) is Gorenstein if and only if it is Cohen-Macaulay and the Hilbert function is symmetric. The paper ends by determining when \(G(m^ r)\) and \(R(m^ r)\) are Gorenstein for \(R\) a local Gorenstein ring with emb\((R)=e(R)+\dim R-3\).
scientific article; zbMATH DE number 64644