On the Gorenstein property of form rings
This paper deals with the Gorenstein property of the Rees algebra \(R(I) = \bigoplus_{n \geq 0} I^ nt^ n\) and the form ring \(G(I) = \bigoplus_{n \geq 0} I^ n/I^{n+1}\) of an ideal \(I\) in a local Gorenstein ring \((A,{\mathfrak m})\). Here are the main results of the paper for an ideal \(I\) of height \(\geq 2\): \(\bullet\) The following two conditions are equivalent: (1) \(R(I^ r)\) is Gorenstein and \(R(I)\) is Cohen-Macaulay. (2) \(G(I)\) is Gorenstein with \(a(G(I)) = - (r+1)\), where \(a=a(G(I))\) is the \(a\)-invariant of \(G(I)\). \(\bullet\) The following three conditions are equivalent: (1) \(G(I)\) is a Gorenstein ring. (2) \(G(I^ r)\) is a Gorenstein ring for all \(r| a+1\). (3) \(G(I^ r)\) is a Gorenstein ring for some \(r \geq 1\). Examples are provided to delimit the scope of the presented results.
scientific article; zbMATH DE number 549773
scientific article