On low Gorenstein colength
Let \(A = R/I\) be a quotient of the ring \(R = kx_1,\dots,x_n\) of formal power series, for some \(n \geq 1\), where \(k\) is a field. The ring \(A\) is \textit{Teter} if \(A\) has the form \(G / \text{soc}(G)\), where \(G\) is Gorenstein. The \textit{Gorenstein colength} \(\text{gcl}(A)\) is defined as the minimum of \(\ell (G) - \ell(A)\), where \(G\) is an Artin Gorenstein ring such that \(A \cong G/H\) for some ideal \(H \subset G\). We know that \(A\) is Teter if and only if \(\text{gcl}(A) = 1\). The goal of this paper is to extend and improve the characterization of Artin rings \(A = R/I\) of Gorenstein colength two due to H. Ananthnarayan, and to extend the characterization of Teter rings given by the first author and M. Silva-Takatuji to Artin rings of Gorenstein colength two. Both improvements are in terms of their Macaulay inverse systems \(I^\perp\). In particular, the authors give an explicit generator of \(I^\perp\). They also give several explicit examples and families of Gorenstein colength two rings in terms of their analytic type.
scientific article; zbMATH DE number 6942903
scientific article