On the Gorenstein property of the associated graded ring of a power of an ideal
Let \(I\) be an ideal of positive height \(h\) in a local ring \(A\) such that the form ring \(gr_ A (I) = \oplus_{n \geq 0} I^ n/I^{n+1}\) is Cohen-Macaulay. The author proves (theorem 2.4) that, for \(r \geq 1\), the form ring \(gr_ A (I^ r)\) is Gorenstein if and only if \(gr_ A (I)\) is Gorenstein and \(a(gr_ A (I)) \equiv -1\) mod \(r\), where \(a(gr_ A(I))\) denotes the \(a\)-invariant of the graded ring \(gr_ A(I)\). That was known before only when \(h\geq 2\) [cf. \textit{M. Herrmann}, \textit{J. Ribbe} and \textit{P. Schenzel}, Math. Z. 213, No. 2, 301-309 (1993; Zbl 0797.13001)]; the author provides a proof which works for all \(h \geq 1\).
scientific article; zbMATH DE number 492803