On the structure of Gorenstein ideals of deviation two
In this article, the authors prove: Theorem 3.1. Let \((R, {\mathfrak m})\) be a regular local ring with infinite residue field, let \(I\subset {\mathfrak m}^2\) be a reduced Gorenstein ideal of deviation two such that \(\dim R/I= 2\), \(I\) is a complete intersection locally on the punctured spectrum and \(T^2 (R/ I)= 0\). If \(I^k\) is integrally closed for some \(k\geq 3\), then \(g= \text{grade } I\) is odd, and \(I\) is a Huneke-Ulrich ideal, i.e., there exist a \(g+1\) by 1 matrix \(X\) and an alternating \(g+1\) by \(g+1\) matrix \(Y\) both of which have entries in \({\mathfrak m}\) such that \(I= I_1 (Y\cdot X)+ Pf (Y)\). Corollary 3.2. Let \((R, {\mathfrak m})\) be a regular local ring with infinite residue field, let \(I \subset {\mathfrak m}^2\) be a reduced Gorenstein ideal of deviation two such that \(\dim R/I= 2\), \(I\) is a complete intersection locally on the punctured spectrum and \(I\) is licci. If \(I^k\) is integrally closed for some \(k\geq 3\), then \(g= \text{grade } I\) is odd, and \(I\) is a Huneke-Ulrich ideal.
scientific article; zbMATH DE number 872702
scientific article